Add all bibliographic notes and zettels

content/article/_index.md

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+title = "Papers"
+author = ["Thomas Dehaeze"]
+type = "paper"
+draft = false
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+
+Here is the list of papers I took note about.

content/article/abir16_optim_estim_real_time_dynam.md

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+title = "Optimized estimator for real-time dynamic displacement measurement using accelerometers"
+author = ["Dehaeze Thomas"]
+draft = true
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+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Abir et al. 2016</a>)
+
+Author(s)
+: Abir, J., Longo, S., Morantz, P., &amp; Shore, P.
+
+Year
+: 2016
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Abir, Jonathan, Stefano Longo, Paul Morantz, and Paul Shore. 2016. “Optimized Estimator for Real-Time Dynamic Displacement Measurement Using Accelerometers.” <i>Mechatronics</i> 39: 1–11. doi:<a href="https://doi.org/10.1016/j.mechatronics.2016.07.003">10.1016/j.mechatronics.2016.07.003</a>.</div>
+</div>

content/article/alkhatib03_activ_struc_vibrat_contr.md

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+title = "Active structural vibration control: a review"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Alkhatib and Golnaraghi 2003</a>)
+
+Author(s)
+: Alkhatib, R., &amp; Golnaraghi, M. F.
+
+Year
+: 2003
+
+
+## Process of designing an active vibration control system {#process-of-designing-an-active-vibration-control-system}
+
+1.  Analyze the structure to be controled
+2.  Obtain an idealized mathematical model with FEM or experimental modal analysis
+3.  Reduce the model order is necessary
+4.  Analyze the resulting model: dynamics properties, types of disturbances, ...
+5.  Quantify sensors and actuators requirements. Decide on their types and location
+6.  Analyze the impact of the sensors and actuators on the overall dynamic characteristics
+7.  Specify performance criteria and stability tradeoffs
+8.  Device of the type of control algorythm to be employed and design a controller to meet the specifications
+9.  Simulate the resulting controlled system on a computer
+10. If the controller does not meet the requirements, adjust the specifications or modify the type of controller
+11. Choose hardware and software and integrate the components on a pilot plant
+12. Formulate experiments and perform system identification and model updating
+13. Implement controller and carry out system test to evaluate the performance
+
+
+## Feedback control {#feedback-control}
+
+
+### Active damping {#active-damping}
+
+The objective is to reduce the resonance peaks of the closed loop transfer function.
+
+\\[T(s) = \frac{G(s)H(s)}{1+G(s)H(s)}\\]
+
+Then \\(T(s) \approx G(s)\\) except near the resonance peaks where the amplitude is reduced.
+
+This method can be realized without a model of the structure with **guaranteed stability**, granted that the actuators and sensors are **collocated**.
+
+
+### Model based feedback {#model-based-feedback}
+
+Objective: keep a control variable (position, velocity, ...) to a desired value in spite of external disturbances \\(d(s)\\).
+
+We have \\[\frac{y(s)}{d(s)} = \frac{1}{1+G(s)H(s)}\\] so we need large values of \\(G(s)H(s)\\) in the frequency range where the disturbance has considerable effect.
+
+To do so, we need a mathematical model of the system, then the control bandwidth and effectiveness are restricted by the accuracy of the model.
+Unmodeled structural dynamics may destabilize the system.
+
+
+## Feedforward Control {#feedforward-control}
+
+We need a signal that is correlated to the disturbance. Then feedforward can improve performance over simple feedback control.
+
+An adaptive filter manipulates the signal correlated to the disturbance and the output is applied to the system by the actuator.
+The filter coefficients are adapted in such a way that an error signal is minimized.
+The idea is to generate a secondary disturbance, which destructively interferes with the effect of the primary distance at the location of the error sensor.
+However, there is no guarantee that the global response is also reduced at other locations.
+
+The method is considered to be a **local technique**, in contrast to feedback which is global.
+
+Contrary to active damping which can only reduce the vibration near the resonance, **feedforward control can be effective for any frequency**.
+The major restriction to the application of feedforward adaptive filtering is the accessibility of a reference signal correlated to the disturbance.
+
+<a id="table--table:comparison-constrol-strat"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--table:comparison-constrol-strat">Table 1</a>:</span>
+  Comparison of control strategies
+</div>
+
+| Type of control                | Advantages                                  | Disadvantages                                 |
+|--------------------------------|---------------------------------------------|-----------------------------------------------|
+| Active Damping                 | Simple to implement                         | Effective only near resonance                 |
+|                                | Does not required accurate model            |                                               |
+|                                | Guaranteed stability (collocated)           |                                               |
+| Model Based                    | Global method                               | Requires accurate model                       |
+|                                | Attenuate all disturbance within bandwidth  | Required low delay                            |
+|                                |                                             | Limited bandwidth                             |
+|                                |                                             | Spillover                                     |
+| Feedforward Adaptive filtering | No model is necessary                       | Error signal required                         |
+|                                | Robust to change in plant transfer function | Local method: may amplify vibration elsewhere |
+|                                | More effective for narrowband disturbance   | Large amount of real-time computation         |
+
+
+## Controllability and Observability {#controllability-and-observability}
+
+Controllability and Observability are two fundamental qualitave properties of dynamic systems.
+
+A system is said to be **controllable** if every state vector can be transform to a desirate state in finite time by the application of unconstrained control inputs.
+
+A system is said to be **observable** at time \\(t\_0\\) if for a state \\(z(t\_0)\\), there is a finite time \\(t\_1>t\_0\\) such that the knowledge of the input \\(u(t)\\) and output \\(y(t)\\) from \\(t\_0\\) to \\(t\_1\\) are sufficient to determine the state \\(z(t\_0)\\).
+
+
+## Coordinate Coupling Control {#coordinate-coupling-control}
+
+Coordinate coupling control (CCC) is an **energy-basded method**.
+
+The idea is to **transfer the vibrations** from a low or undamped oscilatory system (the plant) to a damped system (the controller).
+
+This can be implemented passively using tuned mass damper. But the key advantage of this technique is that one can replace the physical absorber with a computer model. The coupling terms can then be selected to maximise the energy transfer.
+
+
+## Robust control {#robust-control}
+
+Robust control concentrates on the **tradeoffs between performance and stability** in the presence of uncertainty in the system model as well as the exogenous inputs to which it is subjected.
+
+Uncertainty can be divided into four types:
+
+-   parameter errors
+-   error in model order
+-   neglected disturbances
+-   neglected nonlinearities
+
+The \\(\mathcal{H}\_\infty\\)  controller is developed to address uncertainty by systematic means.
+A general block diagram of the control system is shown [Figure 1](#figure--fig:alkhatib03-hinf-control).
+
+A **frequency shaped filter** \\(W(s)\\) coupled to selected inputs and outputs of the plant is included.
+The outputs of this frequency shaped filter define the error ouputs used to evaluate the system performance and generate the **cost** that will be used in the design process.
+
+<a id="figure--fig:alkhatib03-hinf-control"></a>
+
+{{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="<span class='figure-number'>Figure 1: </span>Block diagram for robust control" >}}
+
+The generalized plan \\(G\\) can be partitionned according to the input-output variables. And we have that the transfer function matrix from \\(d\\) to \\(z\\) is:
+\\[ H\_{z/d} = G\_{z/d} + G\_{z/u} K (I - G\_{y/u} K)^{-1} G\_{y/d} \\]
+This transfer function matrix contains measures of performance and stability robustness.
+
+The objective of \\(\mathcal{H}\_\infty\\) control is to design an admissible control \\(u(s)=K(s)y(s)\\) such that \\(\\| H\_{z/d} \\|\_\infty\\) is minimum.
+
+
+## Optimal Control {#optimal-control}
+
+The control \\(u(t)\\) is designed to minimize a cost function \\(J\\), given the initial conditions \\(z(t\_0)\\) and \\(\dot{z}(t\_0)\\) subject to the constraint that:
+
+\begin{align\*}
+\dot{z} &= Az + Bu\\\\
+y &= Cz
+\end{align\*}
+
+One such cost function appropriate to a vibration control is
+\\[J = 1/2 \int\_{t\_0}^{t\_f} ( z^T A z + u^T R u ) dt\\]
+Where \\(Q\\) and \\(R\\) and positive definite symmetric weighting matrices.
+
+
+## State Observers (Estimators) {#state-observers--estimators}
+
+It is not always possible to determine the entire state variables. There are usualy too many degrees of freedom and only limited measurements.
+
+The state vector \\(z(t)\\) can be estimated independently of the control problem, and the resulting estimate \\(\hat{z}(t)\\) can be used.
+
+
+## Intelligent Structure and Controller {#intelligent-structure-and-controller}
+
+Intelligent structure would have the capability to:
+
+-   recognize the present dynamic state of its own structure and evaluate the functional performance of the structure
+-   identify functional descriptions of external and internal disturbances
+-   detect changes in structural properties and changes in external and internal disturbances
+-   predict possible future changes in structural properties
+-   make intelligent decisions regarding compensations for disturbances and adequately generale actuation forces
+-   learn from past performance to improve future actions
+
+Two main methodologies:
+
+-   artificial neural networks
+-   fuzzy logic
+
+
+## Adaptive Control {#adaptive-control}
+
+Adaptive control is frequently used to control systems whose parameters are unknown, uncertain, or slowly varying.
+
+The design of an adaptive controller involves several steps:
+
+-   selection of a controller structure with adjustable parameters
+-   selection of an adaptation law for adjusting those parameters
+-   selection of a performance index
+-   real-time evaluation of the performance with respect to some desired behavior
+-   real-time plant identification and model updating
+-   real-time adjustment of the controller parameters to bring the performance closer to the desired behavior
+
+It essentially consists of a real-time system identification technique integrated with a control algorithm.
+
+Two different methods
+
+-   **Direct method**: the controller parameters are adjusted directly based on the error between the measured and desired outputs.
+-   **Indirect method**: the computations are divided into two consecutive phases. First, the plant model is first estimated in real time. Second, the controller parameters are modified based on the most recent updated plant parameters.
+
+
+## Active Control Effects on the System {#active-control-effects-on-the-system}
+
+<a id="figure--fig:alkhatib03-1dof-control"></a>
+
+{{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="<span class='figure-number'>Figure 2: </span>1 DoF control of a spring-mass-damping system" >}}
+
+Consider the control system [Figure 2](#figure--fig:alkhatib03-1dof-control), the equation of motion of the system is:
+\\[ m\ddot{x} + c\dot{x} + kx = f\_a + f \\]
+
+The controller force can be expressed as: \\(f\_a = -g\_a \ddot{x} + g\_v \dot{x} + g\_d x\\). The equation of motion becomes:
+\\[ (m+g\_a)\ddot{x} + (c+g\_v)\dot{x} + (k+g\_d)x = f \\]
+
+Depending of the type of signal used, the active control adds/substracts mass, damping and stiffness.
+
+
+## Time Delays {#time-delays}
+
+One of the limits to the performance of active control is the time delay in controllers and actuators. Time delay introduces phase shift, which deteriorates the controller performance or even causes instability in the system.
+
+
+## Optimal Placement of Actuators {#optimal-placement-of-actuators}
+
+The problem of optimizing the locations of the actuators can be more significant than the control law itself.
+
+If the actuator is placed at the wrong location, the system will require a greater force control. In that case, the system is said to have a **low degree of controllability**.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Alkhatib, Rabih, and M. F. Golnaraghi. 2003. “Active Structural Vibration Control: A Review.” <i>The Shock and Vibration Digest</i> 35 (5): 367–83. doi:<a href="https://doi.org/10.1177/05831024030355002">10.1177/05831024030355002</a>.</div>
+</div>

content/article/bibel92_guidel_h.md

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+title = "Guidelines for the selection of weighting functions for h-infinity control"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [H Infinity Control]({{< relref "h_infinity_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Bibel and Malyevac 1992</a>)
+
+Author(s)
+: Bibel, J. E., &amp; Malyevac, D. S.
+
+Year
+: 1992
+
+
+## Properties of feedback control {#properties-of-feedback-control}
+
+<a id="figure--fig:bibel92-control-diag"></a>
+
+{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="<span class='figure-number'>Figure 1: </span>Control System Diagram" >}}
+
+From the [Figure 1](#figure--fig:bibel92-control-diag), we have:
+
+\begin{align\*}
+y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\
+e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\\
+u(s) &= S(s)K(s) r(s) - S(s)K(s) d(s) - S(s)K(s) n(s)
+\end{align\*}
+
+With the following definitions
+
+-   \\(L(s) = G(s)K(s)\\) is the **loop transfer matrix**
+-   \\(S(s) = [I+G(s)K(s)]^{-1}\\) is the **Sensitivity** function matrix
+-   \\(T(s) = [I+G(s)K(s)]^{-1}G(s)K(s)\\) is the **Transmissibility** function matrix
+
+<div class="cbox">
+
+\\[ S(s) + T(s) = 1 \\]
+
+</div>
+
+<div class="cbox">
+
+-   **Command following**: \\(S=0\\) and \\(T=1\\) =&gt; large gains
+-   **Disturbance rejection**: \\(S=0\\) =&gt; large gains
+-   **Sensor noise attenuation**: \\(T\\) small where the noise is concentrated
+-   **Control Sensitivity minimization**: \\(K S\\) small
+-   **Robustness to modeling errors**: \\(T\\) small in the frequency range of the expected model undertainties
+
+</div>
+
+
+## SISO tradeoff {#siso-tradeoff}
+
+We want \\(S\\) small for command following and disturbance rejection.
+We want \\(T\\) small to remain insensitive to sensor noise and modeling errors and to reduce control sensitivity.
+
+However we cannot keep both \\(S\\) and \\(T\\) small as \\(S(s)+T(s)=1\\).
+
+We must determine some **tradeoff** between the sensitivity and the complementary sensitivity functions.
+
+Usually, reference signals and disturbances occur at low frequencies, while noise and modeling errors are concentrated at high frequencies. The tradeoff, in a SISO sense, is to make \\(|S(j\omega)|\\) small as low frequencies and \\(|T(j\omega)|\\) small at high frequencies.
+
+
+## \\(H\_\infty\\) and weighting functions {#h-infty-and-weighting-functions}
+
+<div class="cbox">
+
+\\(\mathcal{H}\_\infty\\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller's performance and robustness characteristics.
+
+</div>
+
+<a id="figure--fig:bibel92-general-plant"></a>
+
+{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="<span class='figure-number'>Figure 2: </span>\\(\mathcal{H}\_\infty\\) control framework" >}}
+
+New design framework ([Figure 2](#figure--fig:bibel92-general-plant)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
+
+-   \\(w\\): exogenous inputs
+-   \\(z\\): regulated performance output
+-   \\(u\\): control inputs
+-   \\(y\\): measured output variables
+
+The plant \\(P\\) has two inputs and two outputs, it can be decomposed into four sub-transfer function matrices:
+\\[P = \begin{bmatrix}P\_{11} & P\_{12} \\\ P\_{21} & P\_{22} \end{bmatrix}\\]
+
+
+## Lower Linear Fractional Transformation {#lower-linear-fractional-transformation}
+
+The transformation from the input \\(w\\) to the output \\(z\\), \\(T\_{zw}\\) is called the **Lower Linear Fractional Transformation** \\(F\_l (P, K)\\).
+
+<div class="cbox">
+
+\\[T\_{zw} = F\_l (P, K) = P\_{11} + P\_{12}K (I-P\_{22})^{-1} P\_{21}\\]
+
+</div>
+
+The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\| T\_{zw} \\|\_\infty\\) over the space of all realizable controllers \\(K(s)\\) that stabilize the closed-loop system.
+
+
+## Weights for inputs/outputs signals {#weights-for-inputs-outputs-signals}
+
+Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs ([Figure 3](#figure--fig:bibel92-hinf-weights)).
+
+<a id="figure--fig:bibel92-hinf-weights"></a>
+
+{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="<span class='figure-number'>Figure 3: </span>Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
+
+The weights on the input and output variables are selected to reflect the spatial and **frequency dependence** of the respective signals and performance specifications.
+
+These inputs and output weighting functions are defined as rational, stable and **minimum-phase transfer function** (no poles or zero in the right half plane).
+
+
+## General Guidelines for Weight Selection: \\(W\_S\\) {#general-guidelines-for-weight-selection-w-s}
+
+\\(W\_S\\) is selected to reflect the desired **performance characteristics**.
+The sensitivity function \\(S\\) should have low gain at low frequency for good tracking performance and high gain at high frequencies to limit overshoot.
+We have to select \\(W\_S\\) such that \\({W\_S}^-1\\) reflects the desired shape of \\(S\\).
+
+<div class="cbox">
+
+-   **Low frequency gain**: set to the inverse of the desired steady state tracking error
+-   **High frequency gain**: set to limit overshoot (\\(0.1\\) to \\(0.5\\) is a good compromise between overshoot and response speed)
+-   **Crossover frequency**: chosen to limit the maximum closed-loop time constant (\\(\omega\_c \approx 1/\tau\\))
+
+</div>
+
+
+## General Guidelines for Weight Selection: \\(W\_T\\) {#general-guidelines-for-weight-selection-w-t}
+
+We want \\(T\\) near unity for good tracking of reference and near zero for noise suppresion.
+
+<div class="cbox">
+
+A high pass weight is usualy used on \\(T\\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:
+
+-   The **crossover frequency** is chosen to **limit the closed-loop bandwidth**
+-   The **high frequency gain** is set high to proide **sensor noise rejection** and high frequency gain attenuation
+
+</div>
+
+When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the magnitude of theise weights at the crossover frequency is less that one to not violate \\(S+T=1\\).
+
+
+## Unmodeled dynamics weighting function {#unmodeled-dynamics-weighting-function}
+
+Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output ([Figure 4](#figure--fig:bibel92-unmodeled-dynamics)).
+
+<a id="figure--fig:bibel92-unmodeled-dynamics"></a>
+
+{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="<span class='figure-number'>Figure 4: </span>Unmodeled dynamics model" >}}
+
+The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown [Figure 5](#figure--fig:bibel92-weight-dynamics).
+
+<a id="figure--fig:bibel92-weight-dynamics"></a>
+
+{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="<span class='figure-number'>Figure 5: </span>Example of unmodeled dynamics weight" >}}
+
+
+## Inputs and Output weighting function {#inputs-and-output-weighting-function}
+
+It is possible to **weight the control input and actuator rate**.
+This is used to **prevent actuator saturation** and **limit amplification of sensor noise signals** on the control input signal.
+
+Typically actuator input weights are constant over frequency and set at the inverse of the saturation limit.
+
+
+## Order of the weighting functions {#order-of-the-weighting-functions}
+
+**The order of the optimal controller is equal to the order of the nominal plant model plus the order of the weights**. The complexity of the controller is increase as the order of the weights increases.
+
+**The order of the weights should be kept reasonably low** to reduce the order of th resulting optimal compensator and avoid potential convergence problems in the DK interactions.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bibel, J. E., and D. S. Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” Naval Surface Warfare Center Dahlgren div va.</div>
+</div>

content/article/bryson93_contr_spacec_aircr.md

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+title = "Control of spacecraft and aircraft"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Bryson 1993</a>)
+
+Author(s)
+: Bryson, A. E.
+
+Year
+: 1993
+
+
+## 9.2.3 Roll-Off Filters {#9-dot-2-dot-3-roll-off-filters}
+
+[Spillover Effect]({{< relref "spillover_effect.md" >}})
+
+> Synthesizing control logic using only one vibration mode means we are consciously **neglecting the higher-order vibration modes**.
+> When doing this, it is a good idea to insert "roll-off" into the control logic, so that the loop-transfer gain decreases rapidly with frequency beyond the control bandwidth.
+> This reduces the possibility of destabilizing the unmodelled higher frequency dynamics ("**spillover**").
+
+
+## 9.5 Robust Compensator Synthesis {#9-dot-5-robust-compensator-synthesis}
+
+> LQG synthesis using feedback of estimated states will produce almost the same good response as LQR [...] for systems with control system bandwidths that are well below the frequency of the first vibration mode.
+> However, it may not be true for systems with higher control system bandwidths, even when one or more vibration modes are included in the control design model.
+
+<!--quoteend-->
+
+> If a rate sensor is co-located with an actuator on a flexible body, and its signal is fed back to the actuator, all vibration modes are stabilized.
+> If a rate sensor is not co-located with an actuator on a flexible body, ans its signal is fed back to the actuator, some vibration modes are stabilized and others are destabilized, depending on the location of the sensor relative to the actuator.
+
+
+## 9.5.2 Low-Authority Control/High-Authority Control [HAC-HAC]({{< relref "hac_hac.md" >}}) {#9-dot-5-dot-2-low-authority-control-high-authority-control-hac-hac--hac-hac-dot-md}
+
+> [Figure 1](#figure--fig:bryson93-hac-lac) shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
+> LAC uses a co-located rate sensor to add damping to all the vibratory modes (but not the rigid-body mode).
+> HAC uses a separated displacement sensor to stabilize the rigid body mode, which slightly decreases the damping of the vibratory modes but not enough to produce instability (called "spillover")
+
+<a id="figure--fig:bryson93-hac-lac"></a>
+
+{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="<span class='figure-number'>Figure 1: </span>HAC-LAC control concept" >}}
+
+> LAC/HAC is usually insensitive to small deviation of the plant dynamics away from the design values, that is, it is **robust** to plant parameter changes.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bryson, Arthur Earl. 1993. <i>Control of Spacecraft and Aircraft</i>. Princeton university press Princeton, New Jersey.</div>
+</div>

content/article/butler11_posit_contr_lithog_equip.md

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++++
+title = "Position control in lithographic equipment"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Multivariable Control]({{< relref "multivariable_control.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Butler 2011</a>)
+
+Author(s)
+: Butler, H.
+
+Year
+: 2011
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Butler, H. 2011. “Position Control in Lithographic Equipment.” <i>IEEE Control Systems</i> 31 (5): 28–47. doi:<a href="https://doi.org/10.1109/mcs.2011.941882">10.1109/mcs.2011.941882</a>.</div>
+</div>

content/article/chen00_ident_decoup_contr_flexur_joint_hexap.md

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+title = "Identification and decoupling control of flexure jointed hexapods"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>)
+
+Author(s)
+: Chen, Y., &amp; McInroy, J.
+
+Year
+: 2000
+
+
+## Abstract {#abstract}
+
+> By exploiting properties of the joint space mass-inertia matrix of flexure jointed hexapods, a new **decoupling method** is proposed.
+> The new decoupling method, through a **static** input-output mapping, transforms the highly coupled 6 inputs 6 outputs dynamics into 6 independent single-input single-output channels.
+> Prior decoupling control algorithms imposed severe constraints on the allowable geometry, workspace and payload.
+> This paper derives a new algorithm which removes these constraints, thus greatly expanding the applications.
+> Based on the new decoupling algorithm, an **identification algorithm** is introduced to identify the **joint space mass-inertia matrix** using payload acceleration and base forces.
+> This algorithm can be used for precision payload calibration, thus improving performance and removing the labor required to design the control for different payloads.
+> The new decoupling algorithm is experimentally compared to earlier techniques.
+> These experimental results indicate that the new approach is practical, and improves performance.
+
+
+## Introduction {#introduction}
+
+Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control.md" >}})) impose two constraints:
+
+-   the payload mass/inertia matrix is diagonal
+-   the geometry of the platform and attachment of the payload must be carefully chosen
+
+This limits the applications significantly.
+
+The algorithm derived herein removes these constraints, thus greatly expanding the potential applications.
+
+
+## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
+
+The derivation of the dynamic model is done in (<a href="#citeproc_bib_item_2">McInroy 1999</a>) ([Notes]({{< relref "mcinroy99_dynam.md" >}})).
+
+<a id="figure--fig:chen00-flexure-hexapod"></a>
+
+{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="<span class='figure-number'>Figure 1: </span>A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}}
+
+In the joint space, the dynamics of a flexure jointed hexapod are written as:
+
+\begin{equation}
+  \vec{f}\_b = \vec{f}\_m - \bm{K}(\vec{l} - \vec{l}\_r) - \bm{B} \dot{\vec{l}}
+\end{equation}
+
+\begin{aligned}
+  & \left( {}^U\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} \right) \ddot{\vec{l}} + \\\\
+  & {}^U\_B\bm{R} \bm{J}^T \bm{B} \dot{\vec{l}} + {}^U\_B\bm{R}\bm{J}^T \bm{K}(\vec{l} - \vec{l}\_r) = \\\\
+  & {}^U\_B\bm{R} \bm{J}^T \vec{f}\_m + \vec{\mathcal{F}}\_e + \vec{\mathcal{F}} + \vec{\mathcal{C}} - \\\\
+  & \left( {}^U\_B\bm{R} \bm{J}^T \bm{M}\_s + {}^U\_P\bm{R} {}^P\bm{M}\_x {}^U\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1} \right) \ddot{\vec{q}}\_s
+\end{aligned}
+
+where:
+
+-   \\(\bm{J}\\) is the \\(6 \times 6\\) hexapod Jacobian relating payload Cartesian movements, expressed in {P}, to strut length changes in the joint space
+-   \\({}^B\_U\bm{R}\\) is the \\(6 \times 6\\) rotation matrix from the base frame {B} to the universal inertial frame of reference {U} (it consists of two identical \\(3 \times 3\\) rotation matrices forming a block diagonal \\(6 \times 6\\) matrix)
+-   \\(\bm{J}\_c\\) and \\(\bm{J}\_B\\) are \\(6 \times 6\\) Jacobian matrices capturing base motion
+-   \\({}^P\bm{M}\_x\\) is the \\(6 \times 6\\) mass-inertia matrix of the payload found with respect to the payload frame {P}
+-   \\(\bm{M}\_s\\) is a diagonal \\(6 \times 6\\) matrix containing the moving mass of each strut
+-   \\(\bm{B}\\) and \\(\bm{K}\\) are \\(6 \times 6\\) diagonal matrices containing the damping of stiffness, respectively, of each strut
+-   \\(\vec{l}\\) is the \\(6 \times 1\\) vector of strut lengths, and \\(\vec{l}\_r\\) is the constant vector of relaxed strut length
+-   \\(\vec{f}\_b\\) is the vector of forces exerted at the bottom of the strut
+-   \\(\vec{f}\_m\\) is the vector of strut motor forces
+-   \\(\ddot{\vec{q}}\_s\\) is a \\(6 \times 1\\) vector of base accelerations along each strut plus some Coriolis terms
+-   \\(\vec{\mathcal{F}}\_e\\) is a vector of payload exogenous generalized forces
+-   \\(\vec{\mathcal{C}}\\) is a vector containing all the Coriolis and centripetal terms except the Coriolis terms in \\(\ddot{\vec{q}}\_s\\)
+-   \\(\vec{\mathcal{G}}\\) is a vector containing all gravity terms
+
+\begin{aligned}
+  \bm{M}\_p & \ddot{\vec{p}}\_s + \bm{B} \dot{\vec{p}}\_s + \bm{K} \vec{p}\_s = \vec{f}\_m + \\\\
+  & \bm{M}\_q \ddot{\vec{q}}\_s + \bm{B} \dot{\vec{q}}\_s + \bm{J}^{-T} {}^U\_B\bm{R}^T \vec{\mathcal{F}}\_e
+\end{aligned}
+
+where
+
+-   \\(\bm{M}\_p = \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} + \bm{M}\_s\\)
+-   \\(\bm{M}\_q = \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} - \bm{J}^{-T} {}^B\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1}\\)
+
+\\(\bm{M}\_p\\) and \\(\bm{M}\_q\\) are joint space mass-inertia matrices.
+
+
+## Decoupling the Dynamics of Flexure Jointed Hexapods {#decoupling-the-dynamics-of-flexure-jointed-hexapods}
+
+
+## Identification of Joint Space Mass-Inertia Matrix {#identification-of-joint-space-mass-inertia-matrix}
+
+
+## Experimental Results {#experimental-results}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)</i>. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>McInroy, J. E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
+</div>

content/article/chen04_decoup_contr_flexur_joint_hexap.md

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+title = "Decoupled control of flexure-jointed hexapods using estimated joint-space mass-inertia matrix"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Decoupled Control]({{< relref "decoupled_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Chen and McInroy 2004</a>)
+
+Author(s)
+: Chen, Y., &amp; McInroy, J.
+
+Year
+: 2004
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Y., and J. E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” <i>IEEE Transactions on Control Systems Technology</i> 12 (3): 413–21. doi:<a href="https://doi.org/10.1109/tcst.2004.824339">10.1109/tcst.2004.824339</a>.</div>
+</div>

content/article/chesne16_enhan_dampin_flexib_struc_using_force_feedb.md

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+title = "Enhanced damping of flexible structures using force feedback"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Active Damping]({{< relref "active_damping.md" >}}), [Integral Force Feedback]({{< relref "integral_force_feedback.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Chesné, Milhomem, and Collette 2016</a>)
+
+Author(s)
+: Simon Chesné, Milhomem, A., &amp; Collette, C.
+
+Year
+: 2016
+
+One problem of Integral Force Feedback (IFF) is that the achievable damping decreases at high frequency.
+A modification of the IFF is proposed in order to significantly increase the damping of **a** selected mode.
+
+The test system is shown in [Figure 1](#figure--fig:chesne16-2dof-system).
+
+Classical IFF corresponds to:
+
+\begin{equation}
+H(s) = \frac{g}{s}
+\end{equation}
+
+<a id="figure--fig:chesne16-2dof-system"></a>
+
+{{< figure src="/ox-hugo/chesne16_2dof_system.png" caption="<span class='figure-number'>Figure 1: </span>Two DoF system representing a flexible structuer controlled by an active mount" >}}
+
+The proposed controller, called **alpha controller** is:
+
+\begin{equation}
+H(s) = g \frac{s + \alpha}{s^2}
+\end{equation}
+
+where \\(\alpha\\) is a parameter.
+
+A new pair of pole/zero has been introduced.
+The new pole is located at \\(s = 0\\) and the zeros at \\(s = -\alpha\\).
+
+For \\(\omega > \alpha\\) the controller is essentially an integrator.
+For \\(\omega < \alpha\\) the controller is a double integrator.
+
+Depending on the chosen \\(\alpha\\) we obtain different root locus as shown in [Figure 2](#figure--fig:chesne16-root-locus-alpha).
+There is an optimal gain \\(\alpha^\star\\) at which the attainable damping of the flexible mode is maximized.
+
+<a id="figure--fig:chesne16-root-locus-alpha"></a>
+
+{{< figure src="/ox-hugo/chesne16_root_locus_alpha.png" caption="<span class='figure-number'>Figure 2: </span>Root locus with the alpha controller for different values of \\(\alpha\\)" >}}
+
+The obtained transmissibility is shown without controller, for classical IFF and for \\(\alpha\\) controller in [Figure 3](#figure--fig:chesne16-transmissibility).
+
+Using the \\(\alpha\\) controller, the compliance is however degraded a lot.
+
+<a id="figure--fig:chesne16-transmissibility"></a>
+
+{{< figure src="/ox-hugo/chesne16_transmissibility.png" caption="<span class='figure-number'>Figure 3: </span>Transmissibility \\(x\_1/x\_0\\)" >}}
+
+In order to recover the compliance at low frequency, high pass filters can be added to the controller.
+
+\begin{equation}
+H(s) = g \frac{s + \alpha}{(s + \beta)^2}
+\end{equation}
+
+The condition for stability found here is:
+
+\begin{equation}
+\alpha \ge \beta/2
+\end{equation}
+
+<div class="sum">
+
+The active damping of flexible structures with collocated force sensor/actuator pairs have been reviewed in this Note.
+In the first part of the Note, two limitations of the integral force feedback (IFF) have been discussed, which are the limited damping of flexible modes and the loss of compliance.
+By slightly modifying the controller, it has been shown that the active damping of a target mode can be significantly increased.
+Analytical formulas of the optimal parameters have been derived.
+In the second part, the loss of compliance inherent to IFF has been addressed.
+It has been shown that, when a high-pass filter is inserted into the IFF controller, the compliance at low frequency can be recovered but the unconditional stability is lost.
+On the other side, with the new proposed control law, the stability is always guaranteed even when using a high-pass filter.
+
+</div>
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chesné, S., A. Milhomem, and C. Collette. 2016. “Enhanced Damping of Flexible Structures Using Force Feedback.” <i>Journal of Guidance, Control, and Dynamics</i> 39 (7): 1654–58. doi:<a href="https://doi.org/10.2514/1.g001620">10.2514/1.g001620</a>.</div>
+</div>

content/article/claeyssen07_amplif_piezoel_actuat.md

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++++
+title = "Amplified piezoelectric actuators: static & dynamic applications"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Claeyssen et al. 2007</a>)
+
+Author(s)
+: Claeyssen, F., Letty, R. L., Barillot, F., &amp; Sosnicki, O.
+
+Year
+: 2007
+
+The amplified piezo actuator APA is an external leveraged actuator based on a shell used both for the ceramic **pre stress** and for the ceramic **motion magnification**.
+
+It is based on low voltage multilayer piezoelectric ceramics (PZT type).
+In static conditions, their free strain \\(S\_p\\) is typically 0.1% when driven at 150 V.
+
+The displacement amplification effect is related in a first approximation to the ratio of the shell long axis length to the short axis height.
+The flatter is the actuator, the higher is the amplification.
+
+Piezoceramics can bear large compressive stress but they can not bear tensile forces with a good reliability.
+The usual way to solve this limitation consists in prestressing the ceramics by maintaining a compressive stress.
+This introduces another force limit: if the internal dynamic forces are above the prestress, the actuator is endangered because of the ceramic goes in tensile stress and also the ceramic stack looses contact with the shell interface.
+
+For many APA actuators, the amplitude of maximal applicable external force is close to half the actuator blocked force.
+
+The maximum dynamic force achievable by the actuator is determined by the prestress.
+The prestress design allows a peak force equal to half the blocked force.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Claeyssen, F., R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static &#38; Dynamic Applications.” <i>Ferroelectrics</i> 351 (1): 3–14. doi:<a href="https://doi.org/10.1080/00150190701351865">10.1080/00150190701351865</a>.</div>
+</div>

content/article/collette11_review_activ_vibrat_isolat_strat.md

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++++
+title = "Review of active vibration isolation strategies"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Collette, Janssens, and Artoos 2011</a>)
+
+Author(s)
+: Collette, C., Janssens, S., &amp; Artoos, K.
+
+Year
+: 2011
+
+
+## Background and Motivations {#background-and-motivations}
+
+
+### Passive Isolation Tradeoffs {#passive-isolation-tradeoffs}
+
+1DoF Equations:
+
+\begin{equation}
+\boxed{X(s) = \underbrace{\frac{cs + k}{ms^2 + cs + k}}\_{T\_{wx}(s)} W(s) + \underbrace{\frac{1}{ms^2 + cs + k}}\_{T\_{Fx}(s)} F(s)}
+\end{equation}
+
+-   \\(T\_{wx}(s)\\) is called the **transmissibility** of the isolator. It characterize the way seismic vibrations \\(w\\) are transmitted to the equipment.
+-   \\(T\_{Fx}(s)\\) is called the **compliance**. It characterize the capacity of disturbing forces \\(F\\) to create motion \\(x\\) of the equipment.
+
+In order to minimize the vibrations of a sensitive equipment, a general objective to design a good isolator is to minimize both \\(\abs{T\_{wx}}\\) and \\(\abs{T\_{Fx}}\\) in the frequency range of interest.
+
+To decrease the amplitude of the overshoot at the resonance frequency, **damping** can be increased.
+The price to pay is degradation of the isolation at high frequency (the roll off becomes \\(-1\\) instead of \\(-2\\)).
+
+**First Trade-off**: Trade-off between damping and isolation.
+
+To improve the transmissibility, the resonance frequency can be decreased.
+However, the systems becomes more sensitive to external force \\(F\\) applied on the equipment.
+
+**Second trade-off**: Trade-off between isolation and robustness to external force
+
+
+### Active Isolation {#active-isolation}
+
+We apply a feedback control.
+The general expression of the force delivered by the actuator is \\(f = g\_a \ddot{x} + g\_v \dot{x} + g\_p x\\). \\(g\_a\\), \\(g\_v\\) and \\(g\_p\\) are constant gains.
+
+<a id="table--table:active-isolation"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--table:active-isolation">Table 1</a>:</span>
+  Active isolation techniques
+</div>
+
+| **Feedback Signal** | **Effect**                               | **Applications** |
+|---------------------|------------------------------------------|------------------|
+| Acceleration        | Add virtual mass                         | Few              |
+| Velocity            | Add virtual dashpot connected to the sky | Sky-Hook Damping |
+| Position            | Add virtual spring connected to the sky  | Sky-Hook Spring  |
+
+
+## Practical Realizations {#practical-realizations}
+
+
+## Sensor Limitations {#sensor-limitations}
+
+
+## Conclusions {#conclusions}
+
+<a id="figure--fig:collette11-comp-isolation-strategies"></a>
+
+{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="<span class='figure-number'>Figure 1: </span>Comparison of Active Vibration Isolation Strategies" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., S. Janssens, and K. Artoos. 2011. “Review of Active Vibration Isolation Strategies.” <i>Recent Patents on Mechanical Engineeringe</i> 4 (3): 212–19. doi:<a href="https://doi.org/10.2174/2212797611104030212">10.2174/2212797611104030212</a>.</div>
+</div>

content/article/collette14_vibrat.md

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++++
+title = "Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Sensor Fusion]({{< relref "sensor_fusion.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Collette and Matichard 2014</a>)
+
+Author(s)
+: Collette, C., &amp; Matichard, F.
+
+Year
+: 2014
+
+
+## Introduction {#introduction}
+
+[Sensor Fusion]({{< relref "sensor_fusion.md" >}}) is used  to combine the benefits of different types of sensors:
+
+-   Relative sensor for DC positioning capability at low frequency
+-   Inertial sensors for isolation at high frequency
+-   Force sensor / collocated sensor to improve the robustness
+
+
+## Different types of sensors {#different-types-of-sensors}
+
+In this paper, three types of sensors are used. Their advantages and disadvantages are summarized [Table 1](#table--tab:sensors).
+
+> Several types of sensors can be used for the feedback control of vibration isolation systems:
+>
+> -   Feedback control based on **relative motion sensors** (inductive, capactive, ferromagnetic sensors...) typically permits to servo-position a system or platform relative to a reference (e.g. floor or support base), but does not provide isolation from the ground motion.
+> -   Feedback control based on **force sensors** typically lowers the effective natural frequency, and therefore increases the isolation, but sacrifices the systems compliance in doing so.
+> -   Feedback control based on **inertial sensors** (geophones, seismometers, accelerometers...) improves not only the vibration isolation but also the compliance. Inertial sensors are, however, AC coupled and noisy at low frequencies.
+
+<a id="table--tab:sensors"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:sensors">Table 1</a>:</span>
+  Types of sensors
+</div>
+
+| Sensors          | Advantages                       | Disadvantages                         |
+|------------------|----------------------------------|---------------------------------------|
+| Relative motion  | Servo-position                   | No isolation from ground motion       |
+| Force sensors    | Improve isolation                | Increase compliance                   |
+| Inertial sensors | Improve isolation and compliance | AC couple and noisy at high frequency |
+
+
+## Inertial Control and sensor fusion configurations {#inertial-control-and-sensor-fusion-configurations}
+
+For a simple 1DoF model, two fusion-sensor configuration are studied. The results are summarized [Table 2](#table--tab:fusion-trade-off).
+
+<a id="table--tab:fusion-trade-off"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:fusion-trade-off">Table 2</a>:</span>
+  Sensor fusion configurations
+</div>
+
+| Low freq. sensor | High freq. sensor | Transmissibility | Compliance | Trade-off                                          |
+|------------------|-------------------|------------------|------------|----------------------------------------------------|
+| Inertial         | Force sensor      | Unchanged        | Degraded   | Sensor noise filtering / compliance degradation    |
+| Inertial         | Relative sensor   | Degraded         | Unchanged  | Isolation in the bandwidth / amplification outside |
+
+
+## Flexible structure {#flexible-structure}
+
+Flexibility is added between the inertial sensor and the actuator.
+Now the sensor and actuator are not collocated anymore and the system is unstable because there is no zero between the two poles.
+We use sensor fusion to obtain stability at high frequency.
+
+
+### Inertial and small accelerometer {#inertial-and-small-accelerometer}
+
+The idea is to use a small accelerometer which is easier to locate near the actuator at high frequency.
+However, it is important to verify that the noise introduced by the accelerometer does not degrades too much the isolation performance.
+
+
+### Inertial and force sensor {#inertial-and-force-sensor}
+
+Here the advantage is that the deformation mode is almost not present in the open-loop transfer function.
+This simplifies the loop shaping of the controller.
+
+
+### Inertial and relative sensor {#inertial-and-relative-sensor}
+
+The relative sensor introduces coupling between both side of the actuator which induces degradation of the isolation at high frequency. However, the compliance remains unchanged at high frequency.
+
+
+## Conclusion {#conclusion}
+
+Fusion of inertial instruments with sensors collocated with the actuator permits to increase the feedback control bandwidth of active isolation systems.
+
+Three types of sensors have been considered for the high frequency part of the fusion:
+
+-   The fusion with a **relative sensor** improves the stability but compromises the transmissibility. It can be of interested for stiff suspension where high frequency isolation can be sacrified to improve stability.
+-   The fusion with an **accelerometre** is used to increase the loop gain. However, as the accelerometer is not dual with the actuator, there is no guaranty stability when the isolation stage is mounted on a flexible support.
+-   The fusion with a **force sensor** can be used to increase the loop gain with little effect on the compliance and passive isolation, provided that the blend is possible and that no active damping of flexible modes is required.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In <i>International Conference on Noise and Vibration Engineering (ISMA2014)</i>.</div>
+</div>

content/article/collette15_sensor_fusion_method_high_perfor.md

@@ -0,0 +1,32 @@
++++
+title = "Sensor fusion methods for high performance active vibration isolation systems"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Collette and Matichard 2015</a>)
+
+Author(s)
+: Collette, C., &amp; Matichard, F.
+
+Year
+: 2015
+
+In order to have good stability margins, it is common practice to collocate sensors and actuators. This ensures alternating poles and zeros along the imaginary axis. Then, each phase lag introduced by the poles is compensated by phase lead introduced by the zeroes. This guarantees stability and such system is referred to as **hyperstable**.
+
+In this paper, we study and compare different sensor fusion methods combining inertial sensors at low frequency with sensors adding stability at high frequency.
+The stability margins of the controller can be significantly increased with no or little effect on the low-frequency active isolation, provided that the two following conditions are fulfilled:
+
+-   the high frequency sensor and the actuator are dual
+-   there exists a bandwidth where we can superimpose the open loop transfer functions obtained with the two sensors.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” <i>Journal of Sound and Vibration</i> 342: 1–21. doi:<a href="https://doi.org/10.1016/j.jsv.2015.01.006">10.1016/j.jsv.2015.01.006</a>.</div>
+</div>

content/article/csencsics20_explor_paret_front_actuat_techn.md

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+title = "Exploring the pareto fronts of actuation technologies for high performance mechatronic systems"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Csencsics and Schitter 2020</a>)
+
+Author(s)
+: Csencsics, E., &amp; Schitter, G.
+
+Year
+: 2020
+
+
+## Abstract {#abstract}
+
+> This paper proposes a novel method for estimating the limitations of individual actuation technologies for a desired system class based on analytically obtained relations, which can be used to systematically trade off desired range and speed specifications in the design phase.
+> The method is presented along the example of **fast steering mirrors** with the tradeoff limit curves estimated for the established **piezoelectric**, **lorentz force** and **hybrid reluctance** actuation technologies.
+
+<a id="figure--fig:csencsics20-fsm-schematic"></a>
+
+{{< figure src="/ox-hugo/csencsics20_fsm_schematic.png" caption="<span class='figure-number'>Figure 1: </span>Fast Steering Mirror system. The main components are: mirror, actuators, position sensors and suspension system." >}}
+
+
+## Fast Steering Mirrors {#fast-steering-mirrors}
+
+
+### Application area and performance specification {#application-area-and-performance-specification}
+
+<a id="table--tab:fsm-requirements"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:fsm-requirements">Table 1</a>:</span>
+  FSM performance requirements for two application
+</div>
+
+| Application       | Pointing        | Scanning |
+|-------------------|-----------------|----------|
+| System Range      | large           | large    |
+| System Dimensions | arbitrary       | compact  |
+| Main objective    | dist. rejection | tracking |
+| Bandwidth         | high            | high     |
+| Motion amplitude  | small           | large    |
+| Mover inertia     | arbitrary       | small    |
+| Precision         | high            | high     |
+
+
+### Safe operating area {#safe-operating-area}
+
+The concept of the Safe Operating Area (SOA) relates the frequency of a sinusoidal reference to the maximum admissible scan amplitude that still stays within the limits of the system.
+
+From [Figure 2](#figure--fig:csencsics20-soa) we can already see that piezo are typically used for system with high bandwidth and small range.
+
+<a id="figure--fig:csencsics20-soa"></a>
+
+{{< figure src="/ox-hugo/csencsics20_soa.png" caption="<span class='figure-number'>Figure 2: </span>Measured safe operating area of closed-loop FSM systems with sinusoidal reference signals. Piezo actuated in blue, lorentz force actuated in red and hybrid reluctance actuated in green." >}}
+
+
+## Limitations of actuator technology {#limitations-of-actuator-technology}
+
+
+### Piezo actuation {#piezo-actuation}
+
+Piezo actuated FMS are in general **high stiffness** system, for which the **bandwidth limitation** for feedback control is typically given by the **first mechanical resonance**.
+
+<a id="figure--fig:csencsics20-typical-piezo-fsm"></a>
+
+{{< figure src="/ox-hugo/csencsics20_typical_piezo_fsm.png" caption="<span class='figure-number'>Figure 3: </span>Piezo actuated FSM cross section" >}}
+
+The angular range of the FSM is:
+
+\begin{equation}
+\phi = \frac{L/1000}{2 d}
+\end{equation}
+
+with \\(L\\) the length of the stack, and d the distance between the stacks and the center of rotation (the factor 1000 is linked to the fact that typical piezo stack have a store equal to 0.1% of their length).
+
+The first resonance frequency is:
+
+\begin{equation}
+f\_{PZA} = \frac{1}{2\pi L}\sqrt{\frac{3E}{\rho\_\text{piezo}}}
+\end{equation}
+
+with \\(E\\) the elastic modulus and \\(\rho\_\text{piezo}\\) the density of the piezo material.
+
+As the resonance limits the achievable bandwidth, we therefore have that \\(f\_{\text{max,PZA}} \propto 1/\phi\\).
+
+
+### Lorentz force actuation {#lorentz-force-actuation}
+
+Lorentz force actuated FSM are in general **low stiffness** systems, which typically have a control bandwidth beyond the suspension mode that is usually limited by the **internal modes of the moving part**.
+
+The mover's mass is dominating the dynamics of low stiffness systems beyond the suspension mode.
+
+<a id="figure--fig:csencsics20-typical-lorentz-fsm"></a>
+
+{{< figure src="/ox-hugo/csencsics20_typical_lorentz_fsm.png" caption="<span class='figure-number'>Figure 4: </span>Lorentz force actuator designs." >}}
+
+\begin{equation}
+f\_\text{max,LFA} = \frac{1}{2\pi} k\_\text{LFA} \sqrt{\frac{1}{\phi J\_\text{init} + \Delta\_J + 2 d \phi^2}}
+\end{equation}
+
+
+### Hybrid reluctance force actuation {#hybrid-reluctance-force-actuation}
+
+<a id="figure--fig:csencsics20-typical-hybrid-reluctance-fsm"></a>
+
+{{< figure src="/ox-hugo/csencsics20_typical_hybrid_reluctance_fsm.png" caption="<span class='figure-number'>Figure 5: </span>Hybrid reluctance actuator designs" >}}
+
+
+## Pareto front estimates for FSM systems {#pareto-front-estimates-for-fsm-systems}
+
+<a id="figure--fig:csencsics20-pareto-estimate"></a>
+
+{{< figure src="/ox-hugo/csencsics20_pareto_estimate.png" caption="<span class='figure-number'>Figure 6: </span>Two dimensional performance space for FSM systems showing the tradeoff between range and bandwidth. Commercially available (symbols) as well as academically reported systems (dots) actuated by piezo (blue), Lorentz force (red) and reluctance actuators (green) are depicted." >}}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Csencsics, Ernst, and Georg Schitter. 2020. “Exploring the Pareto Fronts of Actuation Technologies for High Performance Mechatronic Systems.” <i>IEEE/ASME Transactions on Mechatronics</i>. IEEE.</div>
+</div>

content/article/dasgupta00_stewar_platf_manip.md

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+title = "The stewart platform manipulator: a review"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Dasgupta and Mruthyunjaya 2000</a>)
+
+Author(s)
+: Dasgupta, B., &amp; Mruthyunjaya, T.
+
+Year
+: 2000
+
+<a id="table--tab:parallel-vs-serial-manipulators"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:parallel-vs-serial-manipulators">Table 1</a>:</span>
+  Parallel VS serial manipulators
+</div>
+
+|              | **Advantages**            | **Disadvantages**     |
+|--------------|---------------------------|-----------------------|
+| **Serial**   | Maneuverability           | Poor precision        |
+|              | Large workspace           | Bends under high load |
+|              |                           | Vibrate at high speed |
+| **Parallel** | High stiffness            | Small workspace       |
+|              | Good dynamic performances |                       |
+|              | Precise positioning       |                       |
+
+The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platform) connected through six extensible legs, each with spherical joints at both ends.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Dasgupta, B., and T. S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” <i>Mechanism and Machine Theory</i> 35 (1): 15–40. doi:<a href="https://doi.org/10.1016/s0094-114x(99)00006-3">10.1016/s0094-114x(99)00006-3</a>.</div>
+</div>

content/article/devasia07_survey_contr_issues_nanop.md

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+title = "A survey of control issues in nanopositioning"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Devasia, Eleftheriou, and Moheimani 2007</a>)
+
+Author(s)
+: Devasia, S., Eleftheriou, E., &amp; Moheimani, S. R.
+
+Year
+: 2007
+
+-   Talks about Scanning Tunneling Microscope (STM) and Scanning Probe Microscope (SPM)
+-   [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}): Creep, Hysteresis, Vibrations, Modeling errors
+-   Interesting analysis about Bandwidth-Precision-Range tradeoffs
+-   Control approaches for piezoelectric actuators: feedforward, Feedback, Iterative, Sensorless controls
+
+<a id="figure--fig:devasia07-piezoelectric-tradeoff"></a>
+
+{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="<span class='figure-number'>Figure 1: </span>Tradeoffs between bandwidth, precision and range" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” <i>IEEE Transactions on Control Systems Technology</i> 15 (5). IEEE: 802–23.</div>
+</div>

content/article/fleming10_nanop_system_with_force_feedb.md

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+title = "Nanopositioning system with force feedback for high-performance tracking and vibration control"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Force Sensors]({{< relref "force_sensors.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Fleming 2010</a>)
+
+Author(s)
+: Fleming, A.
+
+Year
+: 2010
+
+
+## Summary {#summary}
+
+-   The noise generated by a piezoelectric force sensor is much less than a capacitive sensor
+-   Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor
+-   Noise of a piezoelectric force sensor
+-   IFF with a piezoelectric stack actuator and piezoelectric force sensor
+-   A force sensor is used as a displacement sensor below the frequency of the first zero
+-   Sensor fusion architecture with a capacitive sensor and a force sensor and using complementary filters
+-   Virtual sensor fusion architecture (called low-frequency bypass)
+-   Analog implementation of the control strategies to avoid quantization noise, finite resolution and sampling delay
+
+
+## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
+
+<a id="figure--fig:fleming10-piezo-model"></a>
+
+{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="<span class='figure-number'>Figure 1: </span>Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
+
+The actuator experiences an internal stress in response to an applied voltage.
+This stress is represented by the voltage dependent force \\(F\_a\\) and is related to free displacement by
+\\[ \Delta L = \frac{F\_a}{k\_a} \\]
+
+-   \\(\Delta L\\) is the change in actuator length in [m]
+-   \\(k\_a\\) is the actuator stiffness in [N/m]
+
+The developed force \\(F\_a\\) is related to the applied voltage by:
+\\[ \Delta L = d\_{33} n V\_a \\]
+
+-   \\(d\_{33}\\) is the piezoelectric strain constant in [m/V]
+-   \\(n\\) is the number of layers
+-   \\(V\_a\\) is the applied voltage in [V]
+
+Combining the two equations, we obtain:
+\\[ F\_a = d\_{33} n k\_a V\_a \\]
+
+The ratio of the developed force to applied voltage is \\(d\_{33} n k\_a\\) in [N/V].
+We denote this constant by \\(g\_a\\) and:
+\\[ F\_a = g\_a V\_a, \quad g\_a = d\_{33} n k\_a \\]
+
+
+## Dynamics of a piezoelectric force sensor {#dynamics-of-a-piezoelectric-force-sensor}
+
+Piezoelectric force sensors provide a high sensitivity and bandwidth with low noise at high frequencies.
+
+If a **single wafer** of piezoelectric material is sandwiched between the actuator and platform:
+\\[ D = d\_{33} T \\]
+
+-   \\(D\\) is the amount of generated charge per unit area in \\([C/m^2]\\)
+-   \\(T\\) is the stress in \\([N/m^2]\\)
+-   \\(d\_{33}\\) is the piezoelectric strain constant in \\([m/V] = [C/N]\\)
+
+The generated charge is then
+\\[ q = d\_{33} F\_s \\]
+
+If an **n-layer** piezoelectric transducer is used as a force sensor, the generated charge is then:
+\\[ q = n d\_{33} F\_s \\]
+
+---
+
+We can use a **charge amplifier** to measure the force \\(F\_s\\).
+
+{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="<span class='figure-number'>Figure 2: </span>Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \\(q\\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
+
+The output voltage \\(V\_s\\) is equal to
+\\[ V\_s = -\frac{q}{C\_s} = -\frac{n d\_{33}F\_s}{C\_s} \\]
+that is, the scaling between the force and voltage is \\(-\frac{n d\_{33}F\_s}{C\_s}\ [V/N]\\) .
+
+---
+
+We can also use a voltage amplifier.
+In that case, the generated charge is deposited on the transducer's internal capacitance.
+
+The open-circuit voltage of a piezoelectric force sensor is:
+\\[ V\_s = \frac{n d\_{33} F\_s}{C} \\]
+
+-   \\(C\\) is the transducer capacitance defined by \\(C = n \epsilon\_T A / h\\) in [F]
+    -   \\(A\\) is the area in \\([m^2]\\)
+    -   \\(h\\) is the layer thickness in [m]
+    -   \\(\epsilon\_T\\) is the dielectric permittivity under a constant stress in \\([F/m]\\)
+
+We obtain
+\\[ V\_s = g\_s F\_s, \quad g\_s = \frac{n d\_{33}}{C} \\]
+
+
+## Noise of a piezoelectric force sensor {#noise-of-a-piezoelectric-force-sensor}
+
+As piezoelectric sensors have a capacitive source impedance, the sensor noise density \\(N\_{V\_s}(\omega)\\) is primarily due to current noise \\(i\_n\\) reacting the capacitive source impedance:
+\\[ N\_{V\_s}(\omega) = i\_n \frac{1}{C \omega} \\]
+
+-   \\(N\_{V\_s}\\) is the measured noise in \\(V/\sqrt{\text{Hz}}\\)
+-   \\(i\_n\\) is the current noise in \\(A/\sqrt{\text{Hz}}\\)
+-   \\(C\\) is the capacitance of the piezoelectric in \\(F\\)
+
+The current noise density of a general purpose LM833 FET-input op-amp is \\(0.5\ pA/\sqrt{\text{Hz}}\\).
+The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and \\(100 \mu F\\).
+
+
+## Tested feedback control strategies {#tested-feedback-control-strategies}
+
+<a id="figure--fig:fleming10-fb-control-strats"></a>
+
+{{< figure src="/ox-hugo/fleming10_fb_control_strats.png" caption="<span class='figure-number'>Figure 3: </span>Comparison of: (a) basic integral control. (b) direct tracking control. (c) dual-sensor feedback. (d) low frequency bypass" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 15 (3): 433–47. doi:<a href="https://doi.org/10.1109/tmech.2009.2028422">10.1109/tmech.2009.2028422</a>.</div>
+</div>

content/article/fleming12_estim.md

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+title = "Estimating the resolution of nanopositioning systems from frequency domain data"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Fleming 2012</a>)
+
+Author(s)
+: Fleming, A. J.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In <i>2012 IEEE International Conference on Robotics and Automation</i>. doi:<a href="https://doi.org/10.1109/icra.2012.6224850">10.1109/icra.2012.6224850</a>.</div>
+</div>

content/article/fleming13_review_nanom_resol_posit_sensor.md

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+title = "A review of nanometer resolution position sensors: operation and performance"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Position Sensors]({{< relref "position_sensors.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Fleming 2013</a>)
+
+Author(s)
+: Fleming, A. J.
+
+Year
+: 2013
+
+-   Define concise performance metric and provide expressions for errors sources (non-linearity, drift, noise)
+-   Review current position sensor technologies and compare their performance
+
+
+## Sensor Characteristics {#sensor-characteristics}
+
+
+### Calibration and nonlinearity {#calibration-and-nonlinearity}
+
+Usually quoted as a percentage of the fill-scale range (FSR):
+
+\begin{equation}
+  \text{mapping error (\\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
+\end{equation}
+
+With \\(e\_m(v)\\) is the mapping error.
+
+<a id="figure--fig:mapping-error"></a>
+
+{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="<span class='figure-number'>Figure 1: </span>The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
+
+
+### Drift and Stability {#drift-and-stability}
+
+If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.
+
+<a id="figure--fig:drift-stability"></a>
+
+{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="<span class='figure-number'>Figure 2: </span>The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
+
+
+### Bandwidth {#bandwidth}
+
+The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\\,dB\\).
+
+Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.
+
+The frequency domain position error is
+
+\begin{equation}
+  \begin{aligned}
+    e\_{bw}(s) &= x(s) - v(s) \\\\
+              &= x(s) (1 - P(s))
+  \end{aligned}
+\end{equation}
+
+If the actual position is a sinewave of peak amplitude \\(A = \text{FSR}/2\\):
+
+\begin{equation}
+  \begin{aligned}
+    e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\
+           &\approx \pm A n \frac{f}{f\_c}
+  \end{aligned}
+\end{equation}
+
+with \\(n\\) is the low pass filter order corresponding to the sensor dynamics and \\(f\_c\\) is the measurement bandwidth.
+
+Thus, the sensor bandwidth must be significantly higher than the operating frequency if dynamic errors are to be avoided.
+
+
+### Noise {#noise}
+
+In addition to the actual position signal, all sensors produce some additive measurement noise.
+In many types of sensor, the majority of noise arises from the thermal noise in resistors and the voltage and current noise in conditioning circuit transistors.
+These noise processes can usually be approximated by a Gaussian random process.<br />
+
+A Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.
+
+The autocorrelation function of a random process \\(\mathcal{X}\\) is
+
+\begin{equation}
+  R\_{\mathcal{X}}(\tau) = E[\mathcal{X}(t)\mathcal{X}(t + \tau)]
+\end{equation}
+
+where \\(E\\) is the expected value operator.
+
+The variance of the process is equal to \\(R\_\mathcal{X}(0)\\) and is the expected value of the varying part squared:
+
+\begin{equation}
+  \text{Var} \mathcal{X} = E \left[ (\mathcal{X} - E[\mathcal{X}])^2 \right]
+\end{equation}
+
+The standard deviation \\(\sigma\\) is the square root of the variance:
+
+\begin{equation}
+  \sigma\_\mathcal{X} = \sqrt{\text{Var} \mathcal{X}}
+\end{equation}
+
+The standard deviation is also the Root Mean Square (RMS) value of a zero-mean random process.
+
+The Power Spectral Density \\(S\_\mathcal{X}(f)\\) of a random process represents the distribution of power (or variance) across frequency \\(f\\).
+
+For example, if the random process under consideration was measured in volts, the power spectral density would have the units of \\(V^2/\text{Hz}\\).
+
+The Power Spectral Density can be obtained from the autocorrelation function from the Wiener-Khinchin relation:
+
+\begin{equation}
+  S\_{\mathcal{X}} = 2 \mathcal{F}\\{ R\_\mathcal{X}(\tau) \\} = 2 \int\_{-\infty}^{\infty} R\_\mathcal{X}(\tau) e^{-2j\pi f \tau} d\tau
+\end{equation}
+
+If the power Spectral Density is known, the variance of the generating process can be found from the area under the curve:
+
+\begin{equation}
+  \sigma\_\mathcal{X}^2 = E[\mathcal{X}^2(t)] = R\_\mathcal{X}(0) = \int\_0^\infty S\_\mathcal{X}(f) df
+\end{equation}
+
+Rather than plotting the frequency distribution of power, it is often convenient to plot the frequency distribution of the standard deviation, which is referred to as the spectral density.
+It is related to the power spectral density by a square root:
+
+\begin{equation}
+  \text{spectral density} = \sqrt{S\_\mathcal{X}(f)}
+\end{equation}
+
+The units of \\(\sqrt{S\_\mathcal{X}(f)}\\) are \\(\text{units}/\sqrt{Hz}\\).
+
+The spectral density if preferred in the electronics literature as the RMS value of a noise process can be determined directly from the noise density and effective bandwidth.
+
+
+### Resolution {#resolution}
+
+The random noise of a position sensor causes an uncertainty in the measured position.
+If the distance between two measured locations is smaller than the uncertainty, it is possible to mistake one point for the other.
+
+To characterize the resolution, we use the probability that the measured value is within a certain error bound.
+
+If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \\(\sigma\\) (RMS value).
+
+The empirical rule states that there is a \\(99.7\\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
+This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
+
+Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
+There is usually a trade-off between bandwidth and resolution ([Figure 3](#figure--fig:tradeoff-res-bandwidth)).
+
+<a id="figure--fig:tradeoff-res-bandwidth"></a>
+
+{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="<span class='figure-number'>Figure 3: </span>The resolution versus banwidth of a position sensor." >}}
+
+Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
+As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
+A convenient method for reporting this ratio is in parts-per-million (ppm):
+
+\begin{equation}
+  \text{DNR}\_{\text{ppm}} = 10^6 \frac{\text{full scale range}}{6\sigma\text{-resolution}}
+\end{equation}
+
+
+## Comparison and summary {#comparison-and-summary}
+
+<a id="table--tab:summary-position-sensors"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a>:</span>
+  Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
+</div>
+
+| Sensor Type    | Range                              | DNR     | Resolution | Max. BW     | Accuracy  |
+|----------------|------------------------------------|---------|------------|-------------|-----------|
+| Metal foil     | \\(10-500\\,\mu m\\)               | 230 ppm | 23 nm      | 1-10 kHz    | 1% FSR    |
+| Piezoresistive | \\(1-500\\,\mu m\\)                | 5 ppm   | 0.5 nm     | &gt;100 kHz | 1% FSR    |
+| Capacitive     | \\(10\\,\mu m\\) to \\(10\\,mm\\)  | 24 ppm  | 2.4 nm     | 100 kHz     | 0.1% FSR  |
+| Electrothermal | \\(10\\,\mu m\\) to \\(1\\,mm\\)   | 100 ppm | 10 nm      | 10 kHz      | 1% FSR    |
+| Eddy current   | \\(100\\,\mu m\\) to \\(80\\,mm\\) | 10 ppm  | 1 nm       | 40 kHz      | 0.1% FSR  |
+| LVDT           | \\(0.5-500\\,mm\\)                 | 10 ppm  | 5 nm       | 1 kHz       | 0.25% FSR |
+| Interferometer | Meters                             |         | 0.5 nm     | &gt;100kHz  | 1 ppm FSR |
+| Encoder        | Meters                             |         | 6 nm       | &gt;100kHz  | 5 ppm FSR |
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A. J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” <i>Sensors and Actuators a: Physical</i> 190: 106–26. doi:<a href="https://doi.org/10.1016/j.sna.2012.10.016">10.1016/j.sna.2012.10.016</a>.</div>
+</div>

content/article/fleming15_low_order_dampin_track_contr.md

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+title = "Low-order damping and tracking control for scanning probe systems"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Fleming, Teo, and Leang 2015</a>)
+
+Author(s)
+: Fleming, A. J., Teo, Y. R., &amp; Leang, K. K.
+
+Year
+: 2015
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” <i>Frontiers in Mechanical Engineering</i> 1. doi:<a href="https://doi.org/10.3389/fmech.2015.00014">10.3389/fmech.2015.00014</a>.</div>
+</div>

content/article/furutani04_nanom_cuttin_machin_using_stewar.md

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+title = "Nanometre-cutting machine using a stewart-platform parallel mechanism"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Furutani, Suzuki, and Kudoh 2004</a>)
+
+Author(s)
+: Furutani, K., Suzuki, M., &amp; Kudoh, R.
+
+Year
+: 2004
+
+-   Lever mechanism to amplify the motion of piezoelectric stack actuators
+-   Use of flexure joints
+-   Eddy current displacement sensors for control (decentralized)
+
+{{< figure src="/ox-hugo/furutani04_ctrl_arch.png" >}}
+
+-   Isotropic performance (cubic configuration even if not said so)
+
+Possible sources of error:
+
+-   position error of the link ends in assembly =&gt; simulation of position error and it is not significant
+-   Inaccurate modelling of the links
+-   insufficient generative force
+-   unwanted deformation of the links
+
+To minimize the errors, a calibration is done between the required leg length and the wanted platform pose.
+Then, it is fitted with 4th order polynomial and included in the control architecture.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Furutani, K., M. Suzuki, and R. Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” <i>Measurement Science and Technology</i> 15 (2): 467–74. doi:<a href="https://doi.org/10.1088/0957-0233/15/2/022">10.1088/0957-0233/15/2/022</a>.</div>
+</div>

content/article/gao15_measur_techn_precis_posit.md

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+title = "Measurement technologies for precision positioning"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Position Sensors]({{< relref "position_sensors.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Gao et al. 2015</a>)
+
+Author(s)
+: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
+
+Year
+: 2015
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” <i>CIRP Annals</i> 64 (2): 773–96. doi:<a href="https://doi.org/10.1016/j.cirp.2015.05.009">10.1016/j.cirp.2015.05.009</a>.</div>
+</div>

content/article/garg07_implem_chall_multiv_contr.md

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+title = "Implementation challenges for multivariable control: what you did not learn in school!"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Multivariable Control]({{< relref "multivariable_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Garg 2007</a>)
+
+Author(s)
+: Garg, S.
+
+Year
+: 2007
+
+Discusses:
+
+-   When to use multivariable control and when not to?
+-   Two major issues with implementing multivariable control: **gain scheduling** and **integrator wind up protection**
+
+> Inline simple gain and phase margin measured for SISO, "robustness" determination of multivariable control requires complex analyses using **singular value techniques** and **Monte Carlo** simulations.
+
+**When to use multivariable control**:
+
+-   System has high input/output coupling and not much separation between loop bandwidth
+-   System is complex with large number of states
+-   When sequential SISO loop closure will not meet performance requirements
+
+Importance of having a mechanism to limit the control rate in the synthesis process.
+The control rate should be weighted appropriately in order to not saturate the system and stay in the linearity regime.
+
+-   importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In <i>AIAA Guidance, Navigation and Control Conference and Exhibit</i>. doi:<a href="https://doi.org/10.2514/6.2007-6334">10.2514/6.2007-6334</a>.</div>
+</div>

content/article/garrido12_centr_multiv_contr_by_simpl_decoup.md

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+title = "Centralized Multivariable Control By Simplified Decoupling"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Decoupled Control]({{< relref "decoupled_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Garrido, Vázquez, and Morilla 2012</a>)
+
+Author(s)
+: Garrido, J., Francisco V\\'azquez, &amp; Morilla, F.
+
+Year
+: 2012
+
+
+## Introduction {#introduction}
+
+Most decoupling approaches use the conventional decoupling scheme in [Figure 1](#figure--fig:garrido12-decoupling-control-system) with:
+
+-   \\(G(s)\\) the process matrix
+-   \\(D(s)\\) the decoupler matrix
+-   \\(C(s)\\) the diagonal control matrix
+
+The design of the decoupler is obtained from:
+
+\begin{equation}
+D(s) = G^{-1} (s) \cdot Q(s)
+\end{equation}
+
+where \\(Q(s)\\) is the desired apparent process which is a diagonal matrix.
+
+The main problem of this methodology is the fact that the complexity of the decoupler elements increases for high dimensional MIMO processes, which may require model reductions.
+
+An alternative decoupling methods, called _inverted decoupling_, maintains very simple apparent processes and decoupler element independently of the system size.
+However, inverted decoupling cannot be applied to processes with multivariable [Right Half Plane Zeros]({{< relref "right_half_plane_zeros.md" >}}).
+
+<a id="figure--fig:garrido12-decoupling-control-system"></a>
+
+{{< figure src="/ox-hugo/garrido12_decoupling_control_system.png" caption="<span class='figure-number'>Figure 1: </span>Block diagram of a decoupling control system" >}}
+
+This work focuses on one of the most extended forms of conventional decoupling called simplified decoupling, in  which \\(n\\) elements of the decoupler are set to unity.
+When the system has two inputs and two outputs (TITO), the simplified decoupling \\(G(s)\\) is given by:
+
+\begin{equation}
+D(s) = \begin{bmatrix}
+1 & -g\_{12}(s)/g\_{11}(s) \\\\
+-g\_{21}(s)/g\_{22}(s) & 1
+\end{bmatrix}
+\end{equation}
+
+And the decoupled apparent process \\(Q(s)\\) is given by:
+
+\begin{equation}
+Q(s) = G(s) \cdot D(s) = \begin{bmatrix}
+g\_{11}(s) - \frac{g\_{21}(s g\_{12}(s))}{g\_{22}(s)} & 0 \\\\
+0 & g\_{22}(s) - \frac{g\_{21}(s)g\_{12}(s)}{g\_{11}(s)}
+\end{bmatrix}
+\end{equation}
+
+In cases where the system is larger than 2x2, the decoupler elements set to unity are always the diagonal ones as found using:
+
+\begin{equation}
+D(s) = G(s)^{-1} (\text{diag}(G(s)^{-1}))^{-1}
+\end{equation}
+
+In this work, a simplified decoupling strategy is proposed for stable processes with possibly RHP zeros and time delays.
+
+
+## Methodology {#methodology}
+
+Assuming that the process \\(G(s)\\) may have RHP zeros and time delays, but does not have any unstable poles, the decoupler matrix \\(D(s)\\) is obtained as follows (one of many possible configurations):
+
+\begin{equation}
+D(s) = \begin{bmatrix}
+1 & \frac{\text{adj}G\_{12}}{\text{adj}G\_{22}} & \dots & \frac{\text{adj}G\_{1n}}{\text{adj}\_{nn}} \\\\
+\frac{\text{adj}G\_{21}}{\text{adj}G\_{11}} & 1 & \dots & \frac{\text{adj}G\_{2n}}{\text{adj}\_{nn}} \\\\
+\vdots & \vdots & \ddots & \vdots \\\\
+\frac{\text{adj}G\_{n1}}{\text{adj}G\_{11}} & \frac{\text{adj}G\_{n2}}{\text{adj}G\_{22}} & \dots & 1
+\end{bmatrix}
+\end{equation}
+
+And the decoupled apparent plant is:
+
+\begin{equation}
+A(s) = \begin{bmatrix}
+\frac{|G|}{\text{adj}G\_{11}} & 0 & \dots & 0 \\\\
+0 & \frac{|G|}{\text{adj}G\_{22}} & \dots & 0 \\\\
+\vdots & \vdots & \ddots & \vdots \\\\
+0 & 0 & \dots & \frac{|G|}{\text{adj}G\_{nn}}
+\end{bmatrix}
+\end{equation}
+
+where \\(|G(s)|\\) is the determinant of \\(G(s)\\), \\(\text{adj}G(s)\\) is the adjugate matrix of \\(G(s)\\), that is, the transpose of the cofactor matrix of \\(G(s)\\).
+
+The proposed general simplified decoupling control is performed in three steps:
+
+1.  select a configuration: select the \\(n\\) elements of \\(D(s)\\) to be set to unity, one for each column
+2.  Compose the decoupler elements of \\(D(s)\\)
+3.  Design the \\(n\\) controllers of the diagonal control \\(C(s)\\) for the decoupled processes
+
+The realizability requirement for the decoupler is that all of its elements must be proper, causal and stable.
+For processes with time delays, non-minimum phase zeros or different relative degrees, direct calculation of the decoupler element can lead to elements with RHP poles or negative relative degrees.
+
+Several advice for the proper chose of the configuration are given in the paper.
+
+
+## Design and practical considerations {#design-and-practical-considerations}
+
+It is usually necessary to approximate the expressions of \\(|G(s)|\\) and \\(\text{adj}G(s)\\) as it usually give non-rational expressions.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Garrido, Juan, Francisco Vázquez, and Fernando Morilla. 2012. “Centralized Multivariable Control by Simplified Decoupling.” <i>Journal of Process Control</i> 22 (6): 1044–62. doi:<a href="https://doi.org/10.1016/j.jprocont.2012.04.008">10.1016/j.jprocont.2012.04.008</a>.</div>
+</div>

content/article/geng95_intel_contr_system_multip_degree.md

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++++
+title = "An intelligent control system for multiple degree-of-freedom vibration isolation"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Geng et al. 1995</a>)
+
+Author(s)
+: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., &amp; Garba, J. A.
+
+Year
+: 1995
+
+<a id="figure--fig:geng95-control-structure"></a>
+
+{{< figure src="/ox-hugo/geng95_control_structure.png" caption="<span class='figure-number'>Figure 1: </span>Local force feedback and adaptive acceleration feedback for active isolation" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Geng, Z. J., G. G. Pan, L. S. Haynes, B. K. Wada, and J. A. Garba. 1995. “An Intelligent Control System for Multiple Degree-of-Freedom Vibration Isolation.” <i>Journal of Intelligent Material Systems and Structures</i> 6 (6): 787–800. doi:<a href="https://doi.org/10.1177/1045389x9500600607">10.1177/1045389x9500600607</a>.</div>
+</div>

content/article/geraldes23_fly_scan_orien_motion_analy.md

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+title = "Fly-scan-oriented motion analyses and upgraded beamline integration architecture for the high-dynamic double-crystal monochromator at sirius/lnls"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Geraldes et al. 2023</a>)
+
+Author(s)
+: Geraldes, R. R., Luiz, S. A. L., Neto, J. L. d. B., Telles Ren\\'e Silva Soares, Reis, R. D. d., Calligaris, G. A., Witvoet, G., …
+
+Year
+: 2023
+
+
+## Effect of different d spacing {#effect-of-different-d-spacing}
+
+> Thus, if different d-spacings are found in the two crystals, an ideal energy matching for maximum flux would be related to slightly different \\(\theta\_B\\) in the crystals, such that the monochromatic beam would no longer be exactly parallel to the incoming beam, and **the magnitude of the deviation would be variable over the operational energy range**.
+
+
+## Effect of pitch error on source motion {#effect-of-pitch-error-on-source-motion}
+
+> Then, considering that variations of the virtual source are often proportionally related to shifts of the beam at the sample through the beamline optics, **a common requirement is having them small compared with the source size**.
+> With **X-ray source sizes of about 5 um** and **L commonly of the order of 30m** for modern beamlines, a typical budget of 10% pushes **pitch errors to the range of 10 nrad** only.
+
+
+## Correct pitch errors with gap adjustments {#correct-pitch-errors-with-gap-adjustments}
+
+> It can be seen that displacements in the virtual source related to pitch errors may be at least partly compensated by energy-dependent beam offset corrections via gap adjustments.
+
+
+## Allow some flux loss in order to have a more stable beam {#allow-some-flux-loss-in-order-to-have-a-more-stable-beam}
+
+> The angular boundaries for pitch around an ideal energy tuning, which might be already out or perfect parallelism due to d-spacing variations, can be derived as a fraction of the angular bandwidth of the Darwin width of the crystals.
+> This can be used, for example, to **evaluate acceptable flux losses in trying to keep the incoming and outgoing beam parallel despite thermal effects**.
+
+The pitch bandwidth for typical Si111 and Si311 can vary from 100urad at low energy to &lt;1urad at high energy.
+
+
+## Analytical effect of miss-cut on the change of beam height {#analytical-effect-of-miss-cut-on-the-change-of-beam-height}
+
+> This indicates that in reality the **gap motion range may need to be larger by a few percent than nominally expected**, that sensitivities at low angles may vary by more than one order of magnitude, that **calibrations for fixed exit may require more than the simpler trigonometric relation** of (2), and that the required velocities and accelerations related to the fly scan are in practice different from nominal ones.
+
+
+### Estimate the effect of the miss-cut on the beam error for our values of angles and miss-cut {#estimate-the-effect-of-the-miss-cut-on-the-beam-error-for-our-values-of-angles-and-miss-cut}
+
+
+## High dynamic range: low energy and high energy issues {#high-dynamic-range-low-energy-and-high-energy-issues}
+
+> Hence, **differences of three to four orders of magnitude occur for the gap velocity for a given energy variation rate** within the operational range of the HD-DCM.
+>
+> For a control-based instrument like the HD-DCM, these aspects place demanding specifications on metrology and acquisition hardware, since very high resolution and low noise are required for the lower angular (higher energy) range, whereas high rates are necessary at the opposite limit.
+>
+> For example, while the angular resolution in the Bragg angle quadrature encoder is 50nrad for high angular resolution and small control errors, for an energy scan of 1keV/s, the crystal angular speed requirements would be around 0.1deg/s at the high energy range and as much as 40deg/s at the low energy limit.
+> In the latter case, the counting rates would have to be higher than the current electronics capacity of 10 MHz.
+>
+> Similarly for the gap, with a resolution of 0.1 nm from the quadrature laser interferometers for the nanometre-level control performance, an equivalent energy rate scan speed with Si(111) crystals without a miscut would translate to about 0.8 mm/s and 20 mm/s at the high and low energy limits, respectively.
+> In the latter case, counting rates would need to reach 200 MHz.
+
+
+## Bragg control has a bandwidth of 20Hz {#bragg-control-has-a-bandwidth-of-20hz}
+
+
+## Crystal control has a bandwidth between 150Hz and 250Hz {#crystal-control-has-a-bandwidth-between-150hz-and-250hz}
+
+
+## They are using the Bragg angle reference signal to measure the wanted crystal distance {#they-are-using-the-bragg-angle-reference-signal-to-measure-the-wanted-crystal-distance}
+
+They are not using the encoder signal as we are doing.
+
+
+## Modes of operation {#modes-of-operation}
+
+1.  Standalone (similar as what we are using).
+2.  Follower: follows an encoder signal from the ID
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Geraldes, Renan Ramalho, Sergio Augusto Lordano Luiz, João Leandro de Brito Neto, Telles René Silva Soares, Ricardo Donizeth dos Reis, Guilherme A. Calligaris, Gert Witvoet, and J. P. M. B. Vermeulen. 2023. “Fly-Scan-Oriented Motion Analyses and Upgraded Beamline Integration Architecture for the High-Dynamic Double-Crystal Monochromator at Sirius/Lnls.” <i>Journal of Synchrotron Radiation</i> 30 (1): 90–110. doi:<a href="https://doi.org/10.1107/s1600577522010724">10.1107/s1600577522010724</a>.</div>
+</div>

content/article/hauge04_sensor_contr_space_based_six.md

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++++
+title = "Sensors and control of a space-based six-axis vibration isolation system"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Hauge and Campbell 2004</a>)
+
+Author(s)
+: Hauge, G., &amp; Campbell, M.
+
+Year
+: 2004
+
+**Discusses**:
+
+-   Choice of sensors and control architecture
+-   Predictability and limitations of the system dynamics
+-   Two-Sensor control architecture
+-   Vibration isolation using a Stewart platform
+-   Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
+
+<a id="figure--fig:hauge04-stewart-platform"></a>
+
+{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Hexapod for active vibration isolation" >}}
+
+**Stewart platform** ([Figure 1](#figure--fig:hauge04-stewart-platform)):
+
+-   Low corner frequency
+-   Large actuator stroke (\\(\pm5mm\\))
+-   Sensors in each strut ([Figure 2](#figure--fig:hauge05-struts)):
+    -   three-axis load cell
+    -   base and payload geophone in parallel with the struts
+    -   LVDT
+
+<a id="figure--fig:hauge05-struts"></a>
+
+{{< figure src="/ox-hugo/hauge05_struts.png" caption="<span class='figure-number'>Figure 2: </span>Strut" >}}
+
+> Force sensors typically work well because they are not as sensitive to payload and base dynamics, but are limited in performance by a low-frequency zero pair resulting from the cross-axial stiffness.
+
+**Performance Objective** (frequency domain metric):
+
+-   The transmissibility should be close to 1 between 0-1.5Hz
+    \\(-3dB < |T(\omega)| < 3db\\)
+-   The transmissibility should be below -20dB in the 5-20Hz range
+    \\(|T(\omega)| < -20db\\)
+
+With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and is used to obtain a scalar performance metric.
+
+**Challenge**:
+
+-   small frequency separation between the two requirements
+
+**Robustness**:
+
+-   minimization of the transmissibility amplification (Bode's "pop") outside the performance region
+
+**Model**:
+
+-   single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems
+
+<a id="figure--fig:hauge05-strut-model"></a>
+
+{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="<span class='figure-number'>Figure 3: </span>Strut model" >}}
+
+**Zero Pair when using a Force Sensor**:
+
+-   The frequency of the zero pair corresponds to the resonance frequency of the payload mass and the "parasitic" stiffness (sum of the cross-axial, suspension, wiring stiffnesses)
+-   This zero pair is usually not predictable nor repeatable
+-   In this Stewart platform, this zero pair uncertainty is due to the internal wiring of the struts
+
+**Control**:
+
+-   Single-axis controllers =&gt; combine them into a full six-axis controller =&gt; evaluate the full controller in terms of stability and robustness
+-   Sensitivity weighted LQG controller (SWLQG) =&gt; address robustness in flexible dynamic systems
+-   Three type of controller:
+    -   Force feedback (cell-based)
+    -   Inertial feedback (geophone-based)
+    -   Combined force/velocity feedback (load cell/geophone based)
+
+> The use of multivariable and robust control on the full 6x6 hexapod does not improve performance over single-axis designs.
+
+<a id="table--tab:hauge05-comp-load-cell-geophone"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a>:</span>
+  Typical characteristics of sensors used for isolation in hexapod systems
+</div>
+
+|                                         | **Load cell**                   | **Geophone**                        |
+|-----------------------------------------|---------------------------------|-------------------------------------|
+| Type                                    | Relative                        | Inertial                            |
+| Relationship with voice coil            | Collocated and Dual             | Non-Collocated and non-Dual         |
+| Open loop transfer function             | (+) Alternating poles/zeros     | (-) Large phase drop                |
+| Limitation from low-frequency zero pair | (-) Yes                         | (+) No                              |
+| Sensitive to payload/base dynamics      | (+) No                          | (-) Yes                             |
+| Best frequency range                    | High (low-freq zero limitation) | Low (high-freq toll-off limitation) |
+
+**Ability of a sensor-actuator pair to improve performance**:
+General system with input \\(u\\), performance \\(z\\), output \\(y\\) disturbance \\(u\\).
+
+Given a sensor \\(u\\) and actuator \\(y\\) and a controller \\(u = -K(s) y\\), the closed loop disturbance to performance transfer function can be written as:
+
+\\[ \left[ \frac{z}{w} \right]\_\text{CL} = \frac{G(s)\_{zw} + K(G(s)\_{zw} G(s)\_{yu} - G(s)\_{zu} G(s)\_{yw})}{1 + K G(s)\_{yu}} \\]
+
+In order to obtain a significant performance improvement is to use a high gain controller, _provided_ the term \\(G(s)\_{zw} + K(G(s)\_{zw} G(s)\_{yu} - G(s)\_{zu} G(s)\_{yw})\\) is small.
+
+We can compare the transfer function from \\(w\\) to \\(z\\) with and without a high gain controller.
+And we find that for \\(u\\) and \\(y\\) to be an acceptable pair for high gain control:
+\\[ \left| \frac{G(j\omega)\_{zw} G(j\omega)\_{yu} - G(j\omega)\_{zu} G(j\omega)\_{yw}}{K G(j\omega)\_{yu}} \right| \ll |G\_{zw}(j\omega)| \\]
+
+**Controllers**:
+
+**Force feedback**:
+
+-   Performance limited by the low frequency zero-pair
+-   It is desirable to separate the zero-pair and first most are separated by at least a decade in frequency
+-   This can be achieve by reducing the cross-axis stiffness
+-   If the low frequency zero pair is inverted, robustness is lost
+-   Thus, the force feedback controller should be designed to have combined performance and robustness at frequencies at least a decade above the zero pair
+-   The presented controller as a high pass filter at to reduce the gain below the zero-pair, a lag at low frequency to improve phase margin, and a low pass filter for roll off
+
+**Inertial feedback**:
+
+-   Non-Collocated =&gt; multiple phase drops that limit the bandwidth of the controller
+-   Good performance, but the transmissibility "pops" due to low phase margin and thus this indicates robustness problems
+
+**Combined force/velocity feedback**:
+
+-   Use the low frequency performance advantages of geophone sensor with the high robustness advantages of the load cell sensor
+-   A Single-Input-Multiple-Outputs (SIMO) controller is found using LQG
+-   The performance requirements are met
+-   Good robustness
+
+<a id="figure--fig:hauge04-obtained-transmissibility"></a>
+
+{{< figure src="/ox-hugo/hauge04_obtained_transmissibility.png" caption="<span class='figure-number'>Figure 4: </span>Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hauge, G. S., and M. E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” <i>Journal of Sound and Vibration</i> 269 (3-5): 913–31. doi:<a href="https://doi.org/10.1016/s0022-460x(03)00206-2">10.1016/s0022-460x(03)00206-2</a>.</div>
+</div>

content/article/heertjes11_minim_cross_talk_high_precis.md

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+title = "Minimizing cross-talk in high-precision motion systems using data-based dynamic decoupling"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Decoupled Control]({{< relref "decoupled_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Heertjes and van Engelen 2011</a>)
+
+Author(s)
+: Heertjes, M., &amp; Engelen, A. v.
+
+Year
+: 2011
+
+> In the field of high-precision motion control, a static decoupling control design is generally used to command motion in the directions of an orthogonal basis.
+> Around the center-of-gravity of the system it then usually suffices to apply single-input single-output control in each of these directions separately.
+> Among the advantages are robust stability and performance through straightforward control designs and loop shaping techniques.
+>
+> If the static decoupling part does not fully achieve desired decoupling of the underlying MIMO motion system, a multi-variable controller can be sought to replace the SISO controller part.
+> A more natural approach would therefore be to replace the MIMO static decoupling part by a dynamic part and leave the SISO controller part intact.
+
+<!--quoteend-->
+
+> The aim of the paper is to minimize directly the cross-talk outputs via data-based optimization.
+> The criterion to be optimized consists solely of time-domain signals taken from a performance-relevant time interval.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Heertjes, Marcel, and Arjan van Engelen. 2011. “Minimizing Cross-Talk in High-Precision Motion Systems Using Data-Based Dynamic Decoupling.” <i>Control Engineering Practice</i> 19 (12): 1423–32. doi:<a href="https://doi.org/10.1016/j.conengprac.2011.07.016">10.1016/j.conengprac.2011.07.016</a>.</div>
+</div>

content/article/herpen14_exploit_addit_actuat_sensor_nano.md

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+title = "Exploiting additional actuators and sensors for nano-positioning robust motion control"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Van Herpen et al. 2014</a>)
+
+Author(s)
+: Herpen, R. v., Oomen, T., Kikken, E., Wal, M. v. d., Aangenent, W., &amp; Steinbuch, M.
+
+Year
+: 2014
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Herpen, Robbert van, Tom Oomen, Edward Kikken, Marc van de Wal, Wouter Aangenent, and Maarten Steinbuch. 2014. “Exploiting Additional Actuators and Sensors for Nano-Positioning Robust Motion Control.” <i>Mechatronics</i> 24 (6): 619–31. doi:<a href="https://doi.org/10.1016/j.mechatronics.2014.03.008">10.1016/j.mechatronics.2014.03.008</a>.</div>
+</div>

content/article/holler12_instr_x_ray_nano_imagin.md

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++++
+title = "An instrument for 3d x-ray nano-imaging"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Holler et al. 2012</a>)
+
+Author(s)
+: Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., &amp; Bunk, O.
+
+Year
+: 2012
+
+Instrument similar to the NASS.
+Obtain position stability of 10nm (standard deviation).
+
+<a id="figure--fig:holler12-station"></a>
+
+{{< figure src="/ox-hugo/holler12_station.png" caption="<span class='figure-number'>Figure 1: </span>Schematic of the tomography setup" >}}
+
+-   **Limited resolution due to instrumentation**:
+    The resolution of ptychographic tomography remains above 100nm due to instabilities and drifts of the scanning systems.
+-   **Need of a Metrology System**:
+
+    > To achieve positioning accuracy and stability in the nanometer range, one cannot rely on the position encoders built into individual positioning stages.
+    > A precise exteroceptive measurement of the relative position of the optical elements with respect to the sample is mandatory.
+    > Thus, thermal drifts and parasitic motions can be measured and compensated for.
+-   **Interferometer System Concept**:
+    The sample is aligned with the X-ray with the XYZ piezo stage.
+    As a result, the metrology sphere will be usually off center with respect to the rotation axis of the spindle.
+    That implies that the laser will not propagate back to the interferometer at all rotation angles.
+    A position sensitive detector (PSD) is used, it provides a measurement of the position of the sphere in the plane perpendicular to the laser.
+    The interferometer is positionned on top of a translation stage. The PSD information is used to close the loop so that the interferometer follows the displacement of the metrology sphere.
+-   **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” <i>Review of Scientific Instruments</i> 83 (7): 073703. doi:<a href="https://doi.org/10.1063/1.4737624">10.1063/1.4737624</a>.</div>
+</div>

content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md

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+title = "Active damping based on decoupled collocated control"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Active Damping]({{< relref "active_damping.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Holterman and de Vries 2005</a>)
+
+Author(s)
+: Holterman, J., &amp; deVries, T.
+
+Year
+: 2005
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Holterman, J., and T. J. A. de Vries. 2005. “Active Damping Based on Decoupled Collocated Control.” <i>IEEE/ASME Transactions on Mechatronics</i> 10 (2): 135–45. doi:<a href="https://doi.org/10.1109/tmech.2005.844702">10.1109/tmech.2005.844702</a>.</div>
+</div>

content/article/ito16_compar_class_high_precis_actuat.md

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++++
+title = "Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Actuators]({{< relref "actuators.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Ito and Schitter 2016</a>)
+
+Author(s)
+: Ito, S., &amp; Schitter, G.
+
+Year
+: 2016
+
+
+## Classification of high-precision actuators {#classification-of-high-precision-actuators}
+
+<div class="table-caption">
+  <span class="table-number">Table 1:</span>
+  Zero/Low and High stiffness actuators
+</div>
+
+| **Categories** | **Pros**                  | **Cons**                    |
+|----------------|---------------------------|-----------------------------|
+| Zero stiffness | No vibration transmission | Large and Heavy             |
+| Low stiffness  | High vibration isolation  | Typically for low load      |
+| High Stiffness | High control bandwidth    | High vibration transmission |
+
+
+## Time Delay of Piezoelectric Electronics {#time-delay-of-piezoelectric-electronics}
+
+In this paper, the piezoelectric actuator/electronics adds a time delay which is much higher than the time delay added by the voice coil/electronics.
+
+
+## Definition of low-stiffness and high-stiffness actuator {#definition-of-low-stiffness-and-high-stiffness-actuator}
+
+-   **Low Stiffness** actuator is defined as the ones where the transmissibility stays below 0dB at all frequency
+-   **High Stiffness** actuator is defined as the ones where the transmissibility goes above 0dB at some frequency
+
+<a id="figure--fig:ito16-low-high-stiffness-actuators"></a>
+
+{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="<span class='figure-number'>Figure 1: </span>Definition of low-stiffness and high-stiffness actuator" >}}
+
+
+## Low-Stiffness / High-Stiffness characteristics {#low-stiffness-high-stiffness-characteristics}
+
+-   The low stiffness actuators achieve smooth transition from active isolation to passive isolation.
+-   The high stiffness actuators can have a gap between the passive and active isolation vibration where the vibrations are amplified in a certain frequency band.
+
+
+## Controller Design {#controller-design}
+
+<a id="figure--fig:ito16-transmissibility"></a>
+
+{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="<span class='figure-number'>Figure 2: </span>Obtained transmissibility" >}}
+
+
+## Discussion {#discussion}
+
+The stiffness requirement for low-stiffness actuators can be rephrased in the frequency domain as: "the cross-over frequency of the sensitivity function of the feedback system must be larger than \\(\sqrt{2} \omega\_r\\) with \\(\omega\_r\\) is the resonant frequency of the uncontrolled system".
+
+In practice, this is difficult to achieve with piezoelectric actuators as their first resonant frequency \\(\omega\_r\\) is **too close to other resonant frequencies to ensure close-loop stability**.
+In contrast, the frequency band between the first and the other resonances of Lorentz actuators can be broad by design making them more suitable to construct a low-stiffness actuators.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” <i>IEEE/ASME Transactions on Mechatronics</i> 21 (2): 1169–78. doi:<a href="https://doi.org/10.1109/tmech.2015.2478658">10.1109/tmech.2015.2478658</a>.</div>
+</div>

content/article/ito16_flexur_desig_precis_posit_using.md

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++++
+title = "Flexure design for precision positioning using low-stiffness actuators"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Ito et al. 2016</a>)
+
+Author(s)
+: Ito, S., Cigarini, F., Unger, S., &amp; Schitter, G.
+
+Year
+: 2016
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ito, Shingo, Francesco Cigarini, Severin Unger, and Georg Schitter. 2016. “Flexure Design for Precision Positioning Using Low-Stiffness Actuators.” <i>IFAC-PapersOnLine</i> 49 (21): 200–205. doi:<a href="https://doi.org/10.1016/j.ifacol.2016.10.548">10.1016/j.ifacol.2016.10.548</a>.</div>
+</div>

content/article/jiao18_dynam_model_exper_analy_stewar.md

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++++
+title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Jiao et al. 2018</a>)
+
+Author(s)
+: Jiao, J., Wu, Y., Yu, K., &amp; Zhao, R.
+
+Year
+: 2018
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Jiao, J., Y. Wu, K. Yu, and R. Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” <i>Journal of Vibration and Control</i> 25 (1): 151–71. doi:<a href="https://doi.org/10.1177/1077546318772474">10.1177/1077546318772474</a>.</div>
+</div>

content/article/kwakernaak93_robus_contr_h_tutor_paper.md

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++++
+title = "Robust control and H-Infinity optimization - Tutorial paper"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [H Infinity Control]({{< relref "h_infinity_control.md" >}}), [Weighting Functions]({{< relref "weighting_functions.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Kwakernaak 1993</a>)
+
+Author(s)
+: Kwakernaak, H.
+
+Year
+: 1993
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Kwakernaak, Huibert. 1993. “Robust Control and H$\Infty$-Optimization - Tutorial Paper.” <i>Automatica</i> 29 (2): 255–73. doi:<a href="https://doi.org/10.1016/0005-1098(93)90122-a">10.1016/0005-1098(93)90122-a</a>.</div>
+</div>

content/article/legnani12_new_isotr_decoup_paral_manip.md

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++++
+title = "A new isotropic and decoupled 6-dof parallel manipulator"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Legnani et al. 2012</a>)
+
+Author(s)
+: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., &amp; Tosi, D.
+
+Year
+: 2012
+
+-   Concepts of isotropy and decoupling for parallel manipulators
+-   **isotropy**: the kinetostatic properties (same applicable force, same possible velocity, same stiffness) are identical in all directions (e.g. cubic configuration for Stewart platform)
+-   **decoupling**: each DoF of the end effector can be controlled by a **single** actuator (not the case for the Stewart platform)
+
+Example of generated isotropic manipulator (not decoupled).
+
+<a id="figure--fig:legnani12-isotropy-gen"></a>
+
+{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="<span class='figure-number'>Figure 1: </span>Location of the leg axes using an isotropy generator" >}}
+
+<a id="figure--fig:legnani12-generated-isotropy"></a>
+
+{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="<span class='figure-number'>Figure 2: </span>Isotropic configuration" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” <i>Mechanism and Machine Theory</i> 58: 64–81. doi:<a href="https://doi.org/10.1016/j.mechmachtheory.2012.07.008">10.1016/j.mechmachtheory.2012.07.008</a>.</div>
+</div>

content/article/li01_simul_vibrat_isolat_point_contr.md

@@ -0,0 +1,26 @@
++++
+title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Li, Hamann, and McInroy 2001</a>)
+
+Author(s)
+: Li, X., Hamann, J. C., &amp; McInroy, J. E.
+
+Year
+: 2001
+
+-   if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In <i>Smart Structures and Materials 2001: Smart Structures and Integrated Systems</i>. doi:<a href="https://doi.org/10.1117/12.436521">10.1117/12.436521</a>.</div>
+</div>

content/article/lin06_distur_atten_precis_hexap_point.md

@@ -0,0 +1,25 @@
++++
+title = "Disturbance attenuation in precise hexapod pointing using positive force feedback"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Lin and McInroy 2006</a>)
+
+Author(s)
+: Lin, H., &amp; McInroy, J. E.
+
+Year
+: 2006
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lin, H., and J. E. McInroy. 2006. “Disturbance Attenuation in Precise Hexapod Pointing Using Positive Force Feedback.” <i>Control Engineering Practice</i> 14 (11): 1377–86. doi:<a href="https://doi.org/10.1016/j.conengprac.2005.10.002">10.1016/j.conengprac.2005.10.002</a>.</div>
+</div>

content/article/mcinroy00_desig_contr_flexur_joint_hexap.md

@@ -0,0 +1,25 @@
++++
+title = "Design and control of flexure jointed hexapods"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">McInroy and Hamann 2000</a>)
+
+Author(s)
+: McInroy, J., &amp; Hamann, J.
+
+Year
+: 2000
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J. E., and J. C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” <i>IEEE Transactions on Robotics and Automation</i> 16 (4): 372–81. doi:<a href="https://doi.org/10.1109/70.864229">10.1109/70.864229</a>.</div>
+</div>

content/article/mcinroy02_model_desig_flexur_joint_stewar.md

@@ -0,0 +1,274 @@
++++
+title = "Modeling and design of flexure jointed stewart platforms for control purposes"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_2">McInroy 2002</a>)
+
+Author(s)
+: McInroy, J.
+
+Year
+: 2002
+
+This short paper is very similar to (<a href="#citeproc_bib_item_1">McInroy 1999</a>).
+
+> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
+
+
+## Introduction {#introduction}
+
+> When pursuing micro-meter/micro-radian scale motion, two new phenomena become important:
+>
+> 1.  joint friction and backlash can cause extremely nonlinear micro-dynamics
+> 2.  base and/or payload vibrations become significant contributor to the motion
+
+<!--quoteend-->
+
+> If the spherical flexure is not properly matched to the particular application, it is shown that the complexity of the dynamics can greatly increase, thus limiting the control performance.
+
+
+## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
+
+<a id="figure--fig:mcinroy02-leg-model"></a>
+
+{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="<span class='figure-number'>Figure 1: </span>The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
+
+The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass ([Figure 1](#figure--fig:mcinroy02-leg-model)).
+
+Thus, **the strut does not output force directly, but rather outputs a mechanically filtered force**.
+
+The model of the strut are shown in [Figure 1](#figure--fig:mcinroy02-leg-model) with:
+
+-   \\(m\_{s\_i}\\) moving strut mass
+-   \\(k\_i\\) spring constant
+-   \\(b\_i\\) damping constant
+-   \\(f\_{m\_i}\\) force applied by the actuator
+-   \\(f\_{p\_i}\\) force exerted by the payload
+-   \\(p\_i\\) three dimensional position of the top
+-   \\(q\_i\\) three dimensional position of the bottom
+-   \\(l\_i\\) strut length
+-   \\(l\_{r\_i}\\) relaxed strut length
+
+In general, **the strut mass and spherical flexure stiffness will cause payload forces that are not perfectly aligned with the strut**.
+
+Applying Newton's second law and stacking the equations into a vector form gives:
+
+\begin{equation}
+  f\_p = f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2 \label{eq:strut\_dynamics\_vec}
+\end{equation}
+
+where:
+
+-   \\(\ddot{q}\_u = \left[ \hat{u}\_1^T \ddot{q}\_1 \ \dots \ \hat{u}\_6^T \ddot{q}\_6 \right]^T\\) notes the vector of base accelerations in the strut directions
+-   \\(g\_u\\) denotes the vector of gravity accelerations in the strut directions
+-   \\(Ms = \text{diag}([m\_1\ \dots \ m\_6])\\), \\(f\_p = [f\_{p\_1}\ \dots \ f\_{p\_6}]^T\\)
+-   \\(v\_2 = [ \dot{\hat{u}}\_1^T \dot{v}\_1 \ \dots \ \dot{\hat{u}}\_6^T \dot{v}\_6 ]^T\\) contains nonlinear Coriolis and centripetal accelerations
+
+
+### Payload Dynamics {#payload-dynamics}
+
+The payload is modeled as a rigid body:
+
+\begin{equation}
+\underbrace{\begin{bmatrix}
+m I\_3 & 0\_{3\times 3} \\\\
+0\_{3\times 3}   & {}^cI
+\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
+  0\_{3 \times 1} \\\ \omega \times {}^cI\omega
+\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
+\end{equation}
+
+where:
+
+-   \\(\ddot{\mathcal{X}}\\) is the \\(6 \times 1\\) generalized acceleration of the payload's center of mass
+-   \\(\omega\\) is the \\(3 \times 1\\) payload's angular velocity vector
+-   \\(\mathcal{F}\\) is the \\(6 \times 1\\) generalized force exerted on the payload
+-   \\(M\_x\\) is the combined mass/inertia matrix of the payload, written in the payload frame {P}
+-   \\(c(\omega)\\) represents the shown vector of Coriolis and centripetal terms
+
+Note \\(\dot{\mathcal{X}} = [\dot{p}^T\ \omega^T]^T\\) denotes the time derivative of the payload's combined position and orientation (or pose) with respect to a universal frame of reference {U}.
+
+First, consider the **generalized force due to struts**.
+Denoting this force as \\(\mathcal{F}\_s\\), it can be calculated form the strut forces as:
+
+\begin{equation}
+  \mathcal{F}\_s = {}^UJ^T f\_p = {}^U\_BR J^T f\_p
+\end{equation}
+
+where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation matrix from {B} to {U}.
+
+The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
+
+\begin{equation}
+  \mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
+\end{equation}
+
+where:
+
+-   \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
+-   \\(g\\) is the gravity vector
+
+By combining \ref{eq:strut\_dynamics\_vec}, \ref{eq:payload\_dynamics} and \ref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
+
+\begin{equation}
+  {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega) \label{eq:eom\_fjh}
+\end{equation}
+
+Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
+In the next section, a connection between the two will be found to complete the formulation
+
+
+## Direction of Payload Force {#direction-of-payload-force}
+
+Many prior hexapod dynamic formulations assume that the strut exerts force only along its direction of motion.
+
+The flexure joints Hexapods transmit forces (or torques) proportional to the deflection of the joints.
+This section establishes design guidelines for the spherical flexure joint to guarantee that the dynamics remain tractable for control.
+
+<a id="figure--fig:mcinroy02-model-strut-joint"></a>
+
+{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="<span class='figure-number'>Figure 2: </span>A simplified dynamic model of a strut and its joint" >}}
+
+[Figure 2](#figure--fig:mcinroy02-model-strut-joint) depicts a strut, along with the corresponding force diagram.
+The force diagram is obtained using standard finite element assumptions (\\(\sin \theta \approx \theta\\)).
+Damping terms are neglected.
+\\(k\_r\\) denotes the rotational stiffness of the spherical joint.
+
+From [Figure 2](#figure--fig:mcinroy02-model-strut-joint) (b), Newton's second law yields:
+
+\begin{equation}
+  f\_p = \begin{bmatrix}
+  -f\_m + m\_s \Delta \ddot{x} + k\Delta x \\\\
+  m\_s \Delta \ddot{y} + \frac{k\_r}{l^2} \Delta y \\\\
+  m\_s \Delta \ddot{z} + \frac{k\_r}{l^2} \Delta z
+\end{bmatrix}
+\end{equation}
+
+Note that the payload force is **not** in general aligned with the strut.
+The force is aligned perfectly with the strut only if \\(m\_s = 0\\) and \\(k\_r = 0\\) (i.e. the struts have negligible mass and the spherical joints have negligible rotational stiffness).
+
+To examine the passive behavior, let \\(f\_m = 0\\) and consider a sinusoidal motion:
+
+\begin{equation}
+  \begin{bmatrix} \Delta x \\\ \Delta y \\\ \Delta z \end{bmatrix} =
+  \begin{bmatrix} A\_x \cos \omega t \\\ A\_y \cos \omega t \\\  A\_z \cos \omega t \end{bmatrix}
+\end{equation}
+
+This yields:
+
+\begin{equation}
+  f\_p = \begin{bmatrix}
+    \Big( -m\_s \omega^2 + k \Big) A\_x \cos \omega t \\\\
+    \Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_y \cos \omega t \\\\
+    \Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_z \cos \omega t
+  \end{bmatrix}
+\end{equation}
+
+The direction of \\(f\_p\\) depends upon to motion specifications, leg inertia and control algorithm.
+
+The hypothesis that it is mostly along the strut direction can be tested by dividing the magnitude of the \\(x\\) component by the magnitude of the combined \\(y\\) and \\(z\\) components:
+
+\begin{equation}
+  x\_\text{gain} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|} \frac{|A\_x|}{\sqrt{A\_y^2 + A\_z^2}}
+\end{equation}
+
+Note that large \\(x\_\text{gain}\\) indicates \\(x\\) direction dominance.
+
+\\(x\_\text{gain}\\) is divided into two parts.
+The first part depends on the mechanical terms and the frequency of the movement:
+
+\begin{equation}
+  x\_{\text{gain}\_\omega} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|}
+\end{equation}
+
+<div class="important">
+
+In order to get dominance at low frequencies, the hexapod must be designed so that:
+
+\begin{equation}
+  \frac{k\_r}{l^2} \ll k \label{eq:cond\_stiff}
+\end{equation}
+
+</div>
+
+This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer.
+
+By satisfying \ref{eq:cond\_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
+\\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\]
+At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component:
+\\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\]
+
+<div class="important">
+
+To ensure that the control system acts only in the band of frequencies where dominance is retained, the control bandwidth can be selected so that:
+
+\begin{equation}
+  \text{control bandwidth} \ll \sqrt{\frac{k\_r}{m\_s l^2}} \label{eq:cond\_bandwidth}
+\end{equation}
+
+</div>
+
+The control bandwidth can be increase for hexapods that are designed so that \\(x\_{\text{gain}\_\omega} \gg 1\\) for \\(\omega \ll \sqrt{k/m\_s}\\).
+This can be achieve, for instance, by adding damping.
+In this case, it is reasonable to use:
+
+\begin{equation}
+  \text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
+\end{equation}
+
+<div class="important">
+
+By designing the flexure jointed hexapod and its controller so that both \ref{eq:cond\_stiff} and \ref{eq:cond\_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
+
+</div>
+
+
+## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
+
+Equation \ref{eq:eom\_fjh} is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
+
+This section will derive this implicit relationship.
+Let denote:
+
+-   \\(\mathcal{X}\_B\\) the pose of {B} with respect to {U}
+-   \\({}^B\mathcal{X}\_P\\) the pose of {P} with respect to {B}
+-   \\({}^Uq\_i = {}^U\_BR {}^Bq\_i + {}^UP\_{BORG}\\) the position of the ith base attachment point, expressed in the universal frame {U}
+-   \\(P\_{BORG}\\) the position of the origin of frame {B}
+
+Note that although \\({}^Bq\_i\\) is fixed, \\({}^Uq\_i\\) varies due to base motion.
+
+Differentiating twice and converting derivatives of rotation matrices into angular velocity cross products yields:
+
+\begin{equation}
+  {}^U\dot{q}\_i = \omega\_B \times {}^U\_BR {}^Bq\_i + \underbrace{{}^U\_BR {}^B\dot{q}\_i}\_{= 0} + v\_B
+\end{equation}
+
+\begin{equation}
+  {}^U\ddot{q}\_i = \dot{\omega}\_B \times {}^U\_BR {}^Bq\_i + \omega\_B \times \omega\_B \times {}^U\_BR {}^Bq\_i + \dot{v}\_B
+\end{equation}
+
+where:
+
+-   \\(\omega\_B\\) denotes the angular velocity of {B} with respect to {U}
+-   \\(v\_B = {}^U\dot{P}\_{BORG}\\) denotes the linear velocity of the origin of {B} with respect to {U}
+
+By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times a)\\) and putting the equation in a matrix form:
+
+\begin{equation}
+  {}^U \hat{u}\_i^T {}^U\ddot{q}\_i = \left[ {}^U\hat{u}\_i^T \left( {}^U\_BR {}^Bq\_i \times {}^U\hat{u}\_i \right)^T \right] \ddot{\mathcal{X}}\_B + {}^U\hat{u}\_i^T \left( \omega\_B \times \left[ \omega\_B \times {}^U\_BR {}^Bq\_i \right] \right)
+\end{equation}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J. E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>IEEE/ASME Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
+</div>

content/article/mcinroy99_dynam.md

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++++
+title = "Dynamic modeling of flexure jointed hexapods for control purposes"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">McInroy 1999</a>)
+
+Author(s)
+: McInroy, J.
+
+Year
+: 1999
+
+This conference paper has been further published in a journal as a short note (<a href="#citeproc_bib_item_2">McInroy 2002</a>).
+
+
+## Abstract {#abstract}
+
+> This paper presents a new dynamic model suitable for control of flexure jointed hexapods (FJH).
+>
+> Novel contributions include:
+>
+> 1.  Base acceleration inputs are included
+> 2.  The dynamic model is experimentally verified
+> 3.  The model is developed so that it is suitable for control
+> 4.  A decoupled force control is derived
+
+
+## Strut Dynamics {#strut-dynamics}
+
+The actuators for FJHs can be divided into two categories:
+
+1.  soft (voice coil), which employs a spring flexure mount
+2.  hard (piezoceramic or magnetostrictive), which employs a compressive load spring.
+
+<a id="figure--fig:mcinroy99-general-hexapod"></a>
+
+{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="<span class='figure-number'>Figure 1: </span>A general Stewart Platform" >}}
+
+Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in [Figure 2](#figure--fig:mcinroy99-strut-model).
+
+In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
+
+<a id="figure--fig:mcinroy99-strut-model"></a>
+
+{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="<span class='figure-number'>Figure 2: </span>The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
+
+<a id="table--tab:mcinroy99-strut-model"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a>:</span>
+  Definition of quantities on <a href="#orgef559a8">2</a>
+</div>
+
+| **Symbol**                   | **Meaning**                                |
+|------------------------------|--------------------------------------------|
+| \\(m\_i\\)                   | moving strut mass                          |
+| \\(k\_i\\)                   | spring constant                            |
+| \\(b\_i\\)                   | damping constant                           |
+| \\(f\_m\\)                   | force the actuator applies                 |
+| \\(f\_{p\_i}\\)              | forced exerted by the payload              |
+| \\(p\_i\\)                   | three dimensional position of the top      |
+| \\(q\_i\\)                   | three dimensional position of the bottom   |
+| \\(l\_i\\)                   | strut length                               |
+| \\(l\_{r\_i}\\)              | relaxed strut length                       |
+| \\(v\_i = p\_i - q\_i\\)     | vector pointing from the bottom to the top |
+| \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut                |
+
+It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in (<a href="#citeproc_bib_item_2">McInroy 2002</a>)).
+This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used.
+This allows to reduce considerably the complexity of the model.
+
+From [Figure 2](#figure--fig:mcinroy99-strut-model) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
+
+\begin{equation}
+  m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i
+\end{equation}
+
+The acceleration \\(\hat{u}\_i^T \ddot{p}\_i\\) can be written as:
+\\[ \hat{u}\_i^T \ddot{p}\_i = \ddot{l}\_i + \hat{u}\_i^T \ddot{q}\_i - \dot{\hat{u}}\_i^T \dot{v}\_i \\]
+
+-   [ ] Not sure how the last term is obtained
+
+Separating strut and base accelerations, and putting all six strut equations in a single vector yields:
+
+\begin{equation}
+  f\_p = f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u - M\_s g\_u + M\_s v\_2 \label{eq:strut\_dynamics\_vec}
+\end{equation}
+
+where:
+
+-   \\(\ddot{q}\_u = \left[ \hat{u}\_1^T \ddot{q}\_1 \ \dots \ \hat{u}\_6^T \ddot{q}\_6 \right]^T\\) notes the vector of base accelerations in the strut directions
+-   \\(g\_u\\) denotes the vector of gravity accelerations in the strut directions
+-   \\(Ms = \diag([m\_1\ \dots \ m\_6])\\), \\(f\_p = [f\_{p\_1}\ \dots \ f\_{p\_6}]^T\\)
+-   \\(v\_2 = [ \dot{\hat{u}}\_1^T \dot{v}\_1 \ \dots \ \dot{\hat{u}}\_6^T \dot{v}\_6 ]^T\\)
+
+
+## Payload Dynamics {#payload-dynamics}
+
+The payload is modeled as a rigid body:
+
+\begin{equation}
+\underbrace{\begin{bmatrix}
+m I\_3 & 0\_{3\times 3} \\\\
+0\_{3\times 3}   & {}^cI
+\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
+  0\_{3 \times 1} \\\ \omega \times {}^cI\omega
+\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
+\end{equation}
+
+where:
+
+-   \\(\ddot{\mathcal{X}}\\) is the \\(6 \times 1\\) generalized acceleration of the payload's center of mass
+-   \\(\omega\\) is the \\(3 \times 1\\) payload's angular velocity vector
+-   \\(\mathcal{F}\\) is the \\(6 \times 1\\) generalized force exerted on the payload
+-   \\(M\_x\\) is the combined mass/inertia matrix of the payload, written in the payload frame {P}
+-   \\(c(\omega)\\) represents the shown vector of Coriolis and centripetal terms
+
+Note \\(\dot{\mathcal{X}} = [\dot{p}^T\ \omega^T]^T\\) denotes the time derivative of the payload's combined position and orientation (or pose) with respect to a universal frame of reference {U}.
+
+First, consider the **generalized force due to struts**.
+Denoting this force as \\(\mathcal{F}\_s\\), it can be calculated form the strut forces as:
+
+\begin{equation}
+  \mathcal{F}\_s = {}^UJ^T f\_p = {}^U\_BR J^T f\_p
+\end{equation}
+
+where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation matrix from {B} to {U}.
+
+The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
+
+\begin{equation}
+  \mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
+\end{equation}
+
+where:
+
+-   \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
+-   \\(g\\) is the gravity vector
+
+By combining \ref{eq:strut\_dynamics\_vec}, \ref{eq:payload\_dynamics} and \ref{eq:generalized\_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
+
+\begin{aligned}
+  & {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\
+  & - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega)
+\end{aligned}
+
+Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
+In the next section, a connection between the two will be found to complete the formulation
+
+
+## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
+
+
+## Joint Space Dynamics {#joint-space-dynamics}
+
+
+## Control Example {#control-example}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, J. E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)</i>. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>IEEE/ASME Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
+</div>

content/article/nosova20_review_paral_struc_mechan_with_kinem_decoup.md

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++++
+title = "A review of the parallel structure mechanisms with kinematic decoupling"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Parallel Manipulators]({{< relref "parallel_manipulators.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Nosova 2020</a>)
+
+Author(s)
+: Nosova, N. Y.
+
+Year
+: 2020
+
+
+## Introduction {#introduction}
+
+Parallel mechanisms can be characterized by high speeds, since the engines are mounted on the base and the links have a relatively small mass.
+The disadvantages are: limited working space, the presence of singularities in the immediate vicinity of the workspace.
+
+The kinematic decoupling for a parallel structure manipulator consists in that one movement of the output platform is provided by only one input link or group of links of the kinematic chain.
+
+
+## Types of Kinematic Decoupling {#types-of-kinematic-decoupling}
+
+There are three different types of decoupling:
+
+1.  **strong coupling**: where each configuration parameter is a function of all joint variable (e.g. Stewart platform)
+2.  **complete decoupling**: each configuration parameter is a function of only one joint variable (e.g. Ortoglide)
+3.  **partial decoupling**: some configuration parameters are in function of only some joint variables
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Nosova, N. Yu. 2020. “A Review of the Parallel Structure Mechanisms with Kinematic Decoupling.” <i>Advanced Technologies in Robotics and Intelligent Systems</i>. Springer International Publishing, 247–55. doi:<a href="https://doi.org/10.1007/978-3-030-33491-8_30">10.1007/978-3-030-33491-8_30</a>.</div>
+</div>

content/article/oomen18_advan_motion_contr_precis_mechat.md

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+title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Motion Control]({{< relref "motion_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Oomen 2018</a>)
+
+Author(s)
+: Oomen, T.
+
+Year
+: 2018
+
+
+## Introduction {#introduction}
+
+Control of positioning systems is traditionally simplified by an excellent mechanical design.
+In particular, the mechanical design is such that the system is stiff and highly reproducible.
+In conjunction with moderate performance requirements, the control bandwidth is well-below the resonance frequency of the flexible mechanics as is shown in [Figure 1](#figure--fig:oomen18-next-gen-loop-gain) (a).
+As a result, the system can often be completely **decoupled** in the frequency range relevant for control.
+Consequently, the control design is divided into well-manageable SISO control loops.
+
+Although motion control design is well developed, presently available techniques mainly apply to positioning systems that behave as a rigid body in the relevant frequency range.
+On one hand, increasing performance requirements hamper the validity of this assumption, since the bandwidth has to increase, leading to flexible dynamics in the cross-over region, see [Figure 1](#figure--fig:oomen18-next-gen-loop-gain) (b).
+
+<a id="figure--fig:oomen18-next-gen-loop-gain"></a>
+
+{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="<span class='figure-number'>Figure 1: </span>Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}}
+
+
+## Traditional motion control {#traditional-motion-control}
+
+In the frequency range that is relevant for control, the dynamical behavior is mainly determined by the mechanics.
+In particular, the mechanics can typically be described as:
+
+\begin{equation}
+G\_m = \sum\_{i=1}^{n\_{RB}} \frac{c\_i b\_i^T}{s^2} + \sum\_{n\_{RB} + 1}^{n\_s} \frac{c\_i b\_i^T}{s^2 + 2\xi \omega\_i s + \omega\_i^2}
+\end{equation}
+
+where the first term refers to rigid body modes and the second term to flexible modes.
+
+-   \\(n\_{RB}\\) is the number of rigid body modes
+-   \\(c\_i \in \mathbb{R}^{n\_y}\\) and \\(b\_i \in \mathbb{R}^{n\_u}\\) are associated with the mode shapes
+-   \\(\xi\_i, \omega\_i \in \mathbb{R}\_+\\)
+
+In traditional positioning systems, the number of actuators \\(n\_u\\) and sensors \\(n\_y\\) equals the number of rigid body modes \\(n\_{RB}\\) and are positioned such that the matrix \\(\sum\_{i=1}^{n\_{RB}} c\_i b\_i^T\\) is invertible.
+In this case, matrices \\(T\_u\\) and \\(T\_y\\) can be selected such that:
+
+\begin{equation}
+G = T\_y G\_m T\_u = \frac{1}{s^2} I\_{n\_{RB}} + G\_{\text{flex}}
+\end{equation}
+
+A tradition motion control architecture is shown in [Figure 2](#figure--fig:oomen18-control-architecture).
+
+<a id="figure--fig:oomen18-control-architecture"></a>
+
+{{< figure src="/ox-hugo/oomen18_control_architecture.png" caption="<span class='figure-number'>Figure 2: </span>Traditional motion control architecture" >}}
+
+
+### Traditional feedforward design {#traditional-feedforward-design}
+
+[Feedforward Control]({{< relref "feedforward_control.md" >}}) can effectively compensate for reference induced error signals.
+In particular, \\(f\\) should be selected such that \\(r - G f\\) is minimized.
+In the low frequency range, the system is decoupled and \\(G\_{\text{flex}}\\) can be ignored, in which case \\(f = G^{-1} r\\).
+In practice, the feedforward signal is selected as \\(f = ms^2 r\\).
+
+
+### Traditional feedback design {#traditional-feedback-design}
+
+The [Feedback Controller]({{< relref "feedback_control.md" >}}) has to minimize \\((1 + GK)^{-1}(\delta - v)\\).
+The main idea is that rigid body decoupling of \\(G\\) enables the shaping of the diagonal elements of \\(K\\) through a decentralized feedback controller.
+As a result, each diagonal element of \\(K\\) may be tuned independently.
+Typically, a PID controller is tuned through manual loop-shaping, followed by notch filters to account the the flexible modes that hamper stability and/or performance.
+
+
+### Traditional design procedure {#traditional-design-procedure}
+
+Traditional motion control design divides the multi-variable control design problems into sub-problems that are manageable by manual control design.
+The traditional procedure consists of the following steps:
+
+-   identify an FRF of \\(G\_m\\)
+-   decouple the plant to obtain an FRF of \\(G\\)
+-   design \\(K\\) using manual loop-shaping, consisting of PID with notches
+-   tune a feedforward controller, e.g. \\(f = m s^2 r\\)
+
+
+## Precision motion control developments {#precision-motion-control-developments}
+
+
+### Challenges {#challenges}
+
+High performance mechatronic systems are becoming lighter and lighter.
+Such lightweight systems exhibit predominant flexible dynamical behavior, as well as an increased susceptibility to disturbances.
+
+This leads to several challenges for motion control design:
+
+-   **Unmeasured performance variables** due to spatio-temporal deformations.
+    In particular, the location where the performance is desired may not be directly measured.
+-   **Many additional inputs and outputs** can be exploited to actively control the flexible dynamical behavior.
+    Spatially distributed actuators can actively provide stiffness and damping to the mechanical deformations.
+-   **Position dependent behavior** is almost unavoidable.
+    For instance in gantry stage designs, mass distribution change due to motion, leading to additional position-dependent behavior.
+    A key challenge lies in handling the position dependence of future systems
+-   A **system-of-systems perspective** on motion control design provides a strong potential for performance enhancement of the overall system.
+    In particular, typical manufacturing machines and scientific instruments involves multiple controlled subsystems where the two subsystems have to move relative to each other.
+    Performance limitations in each subsystem will negatively impact the overall performance.
+    A joint design enables that individual subsystems will be able to compensate each other's limitations.
+    A main challenge lies in an increase of the complexity of the control problem.
+-   **Thermal dynamics**, in addition to mechanical deformations are expected to become substantially more important due to increasing performance specifications.
+-   **Vibrations**, such as flow induced vibrations of cooling liquids and floor vibrations, have to be attenuated.
+
+
+### Generalized plant approach {#generalized-plant-approach}
+
+A generalized plant framework allows for a systematic way to address the future challenges in advanced motion control.
+
+The generalized plant is depicted in [Figure 3](#figure--fig:oomen18-generalized-plant):
+
+-   \\(z\\) are the performance variables
+-   \\(y\\) and \\(u\\) are the measured variables and measured variables, respectively
+-   \\(w\\) contains the exogenous inputs, typically including both reference signals and disturbances.
+
+<a id="figure--fig:oomen18-generalized-plant"></a>
+
+{{< figure src="/ox-hugo/oomen18_generalized_plant.png" caption="<span class='figure-number'>Figure 3: </span>Generalized plant setup" >}}
+
+
+## Feedback and Identification for Control {#feedback-and-identification-for-control}
+
+Feedback control is essential to deal with uncertainty in the system dynamics \\(G\\) and disturbances \\(v\\).
+Indeed, the main goal of feedback si to render the system insensitive to such uncertainties.
+
+
+### Norm-based control {#norm-based-control}
+
+A model-based design is foreseen to be able to systematically address the above mentioned challenges.
+
+To specify the control goal, the criterion:
+
+\begin{equation}
+J(G, K) = \\| \mathcal{F}\_l(P(G), K) \\|
+\end{equation}
+
+is posed, where the goal is to compute:
+
+\begin{equation}
+K\_{\text{opt}} = \text{arg} \text{min}\_{K} J(G\_0, K)
+\end{equation}
+
+Where \\(\\| \cdot \\|\\) denotes a suitable norm, e.g. \\(\mathcal{H}\_2\\) or \\(\mathcal{H}\_\infty\\), and \\(\mathcal{F}\_l\\) denotes a lower linear fractional transformation.
+
+\\(G\_0\\) denotes the true system, which is generally unknown and represented by a model \\(\hat{G}\\).
+
+
+### Nominal modeling for control {#nominal-modeling-for-control}
+
+To arrive at a mathematically tractable optimization problem, knowledge of the true system is represented through a model \\(\hat{G}\\).
+The central question is how to obtain such a model that is suitable for controller design.
+[System Identification]({{< relref "system_identification.md" >}}) as opposed to first principles modeling, is an inexpensive, fast and accurate approach to obtain such a model.
+Indeed, the machine is often already built, enabling direct experimentation.
+
+The model \\(\hat{G}\\) that results from system identification is an approximation of the true system \\(G\_0\\) for several reasons:
+
+-   motion systems often contains an infinite number of modes \\(n\_s\\), while a model of limited complexity may be desirable from a control perspective
+-   parasitic non-linearities are present, including nonlinear damping
+-   identification experiments are based on finite time disturbed observations, leading to uncertainties on estimated parameters
+
+
+### Toward robust motion control {#toward-robust-motion-control}
+
+Doing a model based control design using an identified model may not work well due to a lack of robustness.
+Indeed, if \\(K(\hat{G})\\) is designed solely based on \\(\hat{G}\\), there is no reason to assume that it achieves a suitable level of performance on \\(G\_0\\).
+This motivates a robust control design, where the **model quality is explicitly addressed during controller synthesis**.
+
+
+## Feedforward and learning {#feedforward-and-learning}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” <i>IEEJ Journal of Industry Applications</i> 7 (2): 127–40. doi:<a href="https://doi.org/10.1541/ieejjia.7.127">10.1541/ieejjia.7.127</a>.</div>
+</div>

content/article/preumont02_force_feedb_versus_accel_feedb.md

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+title = "Force feedback versus acceleration feedback in active vibration isolation"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Preumont et al. 2002</a>)
+
+Author(s)
+: Preumont, A., A. Francois, Bossens, F., &amp; Abu-Hanieh, A.
+
+Year
+: 2002
+
+Summary:
+
+-   Compares the force feedback and acceleration feedback for active damping
+-   The use of a force sensor always give alternating poles and zeros in the open-loop transfer function between for force actuator and the force sensor which **guarantees the stability of the closed loop**
+-   Acceleration feedback produces alternating poles and zeros only when the flexible structure is stiff compared to the isolation system
+
+The force applied to a **rigid body** is proportional to its acceleration, thus sensing the total interface force gives a measured of the absolute acceleration of the solid body.
+Thus force feedback and acceleration feedback are equivalent for solid bodies.
+When there is a flexible payload, the two sensing options are not longer equivalent.
+
+-   For light payload ([Figure 1](#figure--fig:preumont02-force-acc-fb-light)), the acceleration feedback gives larger damping on the higher mode.
+-   For heavy payload ([Figure 2](#figure--fig:preumont02-force-acc-fb-heavy)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
+
+<a id="figure--fig:preumont02-force-acc-fb-light"></a>
+
+{{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="<span class='figure-number'>Figure 1: </span>Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
+
+<a id="figure--fig:preumont02-force-acc-fb-heavy"></a>
+
+{{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="<span class='figure-number'>Figure 2: </span>Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
+
+Guaranteed stability of the force feedback:
+
+> If two arbitrary flexible, undamped structures are connected with a single-axis soft isolator with force feedback, the poles and zeros of the open-loop transfer function from the force actuator to the force sensor alternate on the imaginary axis.
+
+The same is true for the transfer function from the force actuator to the relative displacement of the actuator.
+
+> According to physical interpretation of the zeros, they represent the resonances of the subsystem  constrained by the sensor and the actuator.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback versus Acceleration Feedback in Active Vibration Isolation.” <i>Journal of Sound and Vibration</i> 257 (4): 605–13. doi:<a href="https://doi.org/10.1006/jsvi.2002.5047">10.1006/jsvi.2002.5047</a>.</div>
+</div>

content/article/preumont07_six_axis_singl_stage_activ.md

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+title = "A six-axis single-stage active vibration isolator based on stewart platform"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Preumont et al. 2007</a>)
+
+Author(s)
+: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
+
+Year
+: 2007
+
+Summary:
+
+-   **Cubic** Stewart platform ([Figure 3](#figure--fig:preumont07-stewart-platform))
+    -   Provides uniform control capability
+    -   Uniform stiffness in all directions
+    -   minimizes the cross-coupling among actuators and sensors of different legs
+-   Flexible joints ([Figure 2](#figure--fig:preumont07-flexible-joints))
+-   Piezoelectric force sensors
+-   Voice coil actuators
+-   Decentralized feedback control approach for vibration isolation
+-   Effect of parasitic stiffness of the flexible joints on the IFF performance ([Figure 1](#figure--fig:preumont07-iff-effect-stiffness))
+-   The Stewart platform has 6 suspension modes at different frequencies.
+    Thus the gain of the IFF controller cannot be optimal for all the modes.
+    It is better if all the modes of the platform are near to each other.
+-   Discusses the design of the legs in order to maximize the natural frequency of the local modes.
+-   To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix
+
+<a id="figure--fig:preumont07-iff-effect-stiffness"></a>
+
+{{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="<span class='figure-number'>Figure 1: </span>Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
+
+<a id="figure--fig:preumont07-flexible-joints"></a>
+
+{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="<span class='figure-number'>Figure 2: </span>Flexible joints used for the Stewart platform" >}}
+
+<a id="figure--fig:preumont07-stewart-platform"></a>
+
+{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="<span class='figure-number'>Figure 3: </span>Stewart platform" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” <i>Journal of Sound and Vibration</i> 300 (3-5): 644–61. doi:<a href="https://doi.org/10.1016/j.jsv.2006.07.050">10.1016/j.jsv.2006.07.050</a>.</div>
+</div>

content/article/saxena12_advan_inter_model_contr_techn.md

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+title = "Advances in internal model control technique: a review and future prospects"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Complementary Filters]({{< relref "complementary_filters.md" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Saxena and Hote 2012</a>)
+
+Author(s)
+: Saxena, S., &amp; Hote, Y.
+
+Year
+: 2012
+
+
+## Proposed Filter \\(F(s)\\) {#proposed-filter-f--s}
+
+\begin{align\*}
+  F(s) &= \frac{1}{(\lambda s + 1)^n} \\\\
+  F(s) &= \frac{n \lambda + 1}{(\lambda s + 1)^n}
+\end{align\*}
+
+
+## Internal Model Control {#internal-model-control}
+
+Central concept in IMC: control can be acheive only if the control system involves, either implicitly or explicitly, some representation of the process to be controlled.
+
+
+### Basic IMC structure {#basic-imc-structure}
+
+IMC can be considered as a special case of classical feedback structure with plant \\(G(s)\\) and controller \\(C(s)\\).
+
+The plan model \\(G\_M(s)\\) is added and substracted into the feedback path of feedback controller.
+
+The structure can then be modified and we obtain a new controller \\(Q(s)\\).
+
+IMC is related to the classical controller through:
+
+\begin{align\*}
+Q(s) = \frac{C(s)}{1+G\_M(s)C(s)} \\\\
+C(s) = \frac{Q(s)}{1-G\_M(s)Q(s)}
+\end{align\*}
+
+Internal model control system is characterized by a control device consisting of the controller \\(Q(s)\\) and a predictive model \\(G\_M(s)\\) of the process (internal model).
+The internal model loop uses the difference between the outputs of the process \\(G(s)\\) to be controlled and the internal model.
+This difference \\(E(s)\\) represents the effect of disturbance and mismatch of the model.
+
+
+### Features of IMC Structure {#features-of-imc-structure}
+
+Three properties:
+
+-   **Dual stability**: assume that, if the plant model is perfect (\\(G\_M(s) = G(s)\\)) and disturbance is absent, the system becomes open-loop and the closed-loop stability is characterized by the stability of \\(G(s)\\) and \\(Q(s)\\)
+-   **Perfect control**: assume that, if the controller is equal to the model inverse (\\(Q(s) = G\_M^{-1}\\)) and \\(G(s) = G\_M(s)\\) with \\(G(s)\\) stable, then the system is perfectly controlled.
+-   **Zero Offset**: assume that, if the steady state gain of the controller is equal to the inverse of model gain, then offset free control is obtained for constant step of ramp type inputs and disturbances. As expected, the equivalent classical controller leads to integral action.
+
+Issues:
+
+-   the plant model is never perfect
+-   inverting the model can cause instability
+-   control signal may have large magnitude
+
+
+## Design procedure for IMC Compensator {#design-procedure-for-imc-compensator}
+
+1.  factorize the plant model as \\(G\_M(s) = G\_{M-}(s)G\_{M+}(s)\\) where \\(G\_{M-}(s)\\) is invertible and minimum phase and \\(G\_{M+}(s)\\) is non-invertible and contains all non-minimum phase elements (delays, RHP zeros). Then, the controller is the inverse of the invertible portion of the plant model: \\(Q\_1(s) = G\_{M-}^{-1}(s)\\).
+2.  Filter selection: to make the controller proper and robust against the plant-model mismatch, a low pass filter of the form \\(F(s) = \frac{n \lambda}{(\lambda s + 1)^n}\\) is augmented with the inverted model \\(Q\_1(s)\\): \\(Q(s) = Q\_1(s) F(s)\\). \\(\lambda\\) is a tuning parameter which has an inverse relationship with the speed of closed loop response, \\(n\\) is selected such that \\(Q(s)\\) becomes proper.
+
+
+## Issues in IMC {#issues-in-imc}
+
+
+### Filter selection and tuning guidelines {#filter-selection-and-tuning-guidelines}
+
+
+## Some advantages and future prospects {#some-advantages-and-future-prospects}
+
+
+## Conclusion {#conclusion}
+
+The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Saxena, S., and Y. V. Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” <i>IETE Technical Review</i> 29 (6): 461. doi:<a href="https://doi.org/10.4103/0256-4602.105001">10.4103/0256-4602.105001</a>.</div>
+</div>

content/article/schellekens98_desig_precis.md

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++++
+title = "Design for precision: current status and trends"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Precision Engineering]({{< relref "precision_engineering.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Schellekens et al. 1998</a>)
+
+Author(s)
+: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., &amp; Pril, W.
+
+Year
+: 1998
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” <i>Cirp Annals</i>, no. 2: 557–86. doi:<a href="https://doi.org/10.1016/s0007-8506(07)63243-0">10.1016/s0007-8506(07)63243-0</a>.</div>
+</div>

content/article/schroeck01_compen_desig_linear_time_invar.md

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++++
+title = "On compensator design for linear time-invariant dual-input single-output systems"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Schroeck, Messner, and McNab 2001</a>)
+
+Author(s)
+: Schroeck, S., Messner, W., &amp; McNab, R.
+
+Year
+: 2001
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” <i>IEEE/ASME Transactions on Mechatronics</i> 6 (1): 50–57. doi:<a href="https://doi.org/10.1109/3516.914391">10.1109/3516.914391</a>.</div>
+</div>

content/article/sebastian12_nanop_with_multip_sensor.md

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++++
+title = "Nanopositioning with multiple sensors: a case study in data storage"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Sebastian and Pantazi 2012</a>)
+
+Author(s)
+: Sebastian, A., &amp; Pantazi, A.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” <i>IEEE Transactions on Control Systems Technology</i> 20 (2): 382–94. doi:<a href="https://doi.org/10.1109/tcst.2011.2177982">10.1109/tcst.2011.2177982</a>.</div>
+</div>

content/article/souleille18_concep_activ_mount_space_applic.md

@@ -0,0 +1,100 @@
++++
+title = "A concept of active mount for space applications"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Active Damping]({{< relref "active_damping.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Souleille et al. 2018</a>)
+
+Author(s)
+: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, &amp; Collette, C.
+
+Year
+: 2018
+
+This article discusses the use of Integral Force Feedback with amplified piezoelectric stack actuators.
+
+> In the proposed configuration, it can also be noticed by the softening effect inherent to force control is limited by the metallic suspension.
+
+
+## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
+
+[Figure 1](#figure--fig:souleille18-model-piezo) shows a picture of the amplified piezoelectric stack.
+The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor.
+
+<a id="figure--fig:souleille18-model-piezo"></a>
+
+{{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="<span class='figure-number'>Figure 1: </span>Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
+
+<div class="table-caption">
+  <span class="table-number">Table 1:</span>
+  Parameters used for the model of the APA 100M
+</div>
+
+|            | Value                  | Meaning                                                        |
+|------------|------------------------|----------------------------------------------------------------|
+| \\(m\\)    | \\(1\\,[kg]\\)         | Payload mass                                                   |
+| \\(k\_e\\) | \\(4.8\\,[N/\mu m]\\)  | Stiffness used to adjust the pole of the isolator              |
+| \\(k\_1\\) | \\(0.96\\,[N/\mu m]\\) | Stiffness of the metallic suspension when the stack is removed |
+| \\(k\_a\\) | \\(65\\,[N/\mu m]\\)   | Stiffness of the actuator                                      |
+| \\(c\_1\\) | \\(10\\,[N/(m/s)]\\)   | Added viscous damping                                          |
+
+The dynamic equation of the system is:
+
+\begin{equation}
+  m \ddot{x}\_1 = \left( k\_1 + \frac{k\_ek\_a}{k\_e + k\_a} \right) ( w - x\_1) + c\_1 (\dot{w} - \dot{x}\_1) + F + \left( \frac{k\_e}{k\_e + k\_a} \right)f
+\end{equation}
+
+The expression of the force measured by the force sensor is:
+
+\begin{equation}
+  F\_s = \left( -\frac{k\_e k\_a}{k\_e + k\_a} \right) x\_1 + \left( \frac{k\_e k\_a}{k\_e + k\_a} \right) w + \left( \frac{k\_e}{k\_e + k\_a} \right) f
+\end{equation}
+
+and the control force is given by:
+
+\begin{equation}
+  f = F\_s G(s) = F\_s \frac{g}{s}
+\end{equation}
+
+The effect of the controller are shown in [Figure 2](#figure--fig:souleille18-tf-iff-result):
+
+-   the resonance peak is almost critically damped
+-   the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
+-   the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\)
+-   the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies
+
+<a id="figure--fig:souleille18-tf-iff-result"></a>
+
+{{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="<span class='figure-number'>Figure 2: </span>Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}}
+
+<a id="figure--fig:souleille18-root-locus"></a>
+
+{{< figure src="/ox-hugo/souleille18_root_locus.png" caption="<span class='figure-number'>Figure 3: </span>Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}}
+
+
+## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators}
+
+A heavy payload is mounted on a set of three isolators ([Figure 4](#figure--fig:souleille18-setup-flexible-payload)).
+The payload consists of two  masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz.
+
+<a id="figure--fig:souleille18-setup-flexible-payload"></a>
+
+{{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="<span class='figure-number'>Figure 4: </span>Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}}
+
+As shown in [Figure 5](#figure--fig:souleille18-result-damping-transmissibility), both the suspension modes and the flexible modes of the payload can be critically damped.
+
+<a id="figure--fig:souleille18-result-damping-transmissibility"></a>
+
+{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="<span class='figure-number'>Figure 5: </span>Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Souleille, A., T. Lampert, V. Lafarga, S. Hellegouarch, A. Rondineau, G. Rodrigues, and C. Collette. 2018. “A Concept of Active Mount for Space Applications.” <i>CEAS Space Journal</i> 10 (2). Springer: 157–65.</div>
+</div>

content/article/spanos95_soft_activ_vibrat_isolat.md

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+title = "A soft 6-axis active vibration isolator"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Spanos, Rahman, and Blackwood 1995</a>)
+
+Author(s)
+: Spanos, J., Rahman, Z., &amp; Blackwood, G.
+
+Year
+: 1995
+
+**Stewart Platform** ([Figure 1](#figure--fig:spanos95-stewart-platform)):
+
+-   Voice Coil
+-   Flexible joints (cross-blades)
+-   Force Sensors
+-   Cubic Configuration
+
+<a id="figure--fig:spanos95-stewart-platform"></a>
+
+{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Stewart Platform" >}}
+
+Total mass of the paylaod: 30kg
+Center of gravity is 9cm above the geometry center of the mount (cube's center?).
+
+Limitation of the **Decentralized Force Feedback**:
+
+-   high frequency pole due to internal resonances of the struts
+-   low frequency zero due to the rotational stiffness of the flexible joints
+
+After redesign of the struts:
+
+-   high frequency pole at 4.7kHz
+-   low frequency zero at 2.6Hz but non-minimum phase (not explained).
+    Small viscous damping material in the cross blade flexures made the zero minimum phase again.
+
+<a id="figure--fig:spanos95-iff-plant"></a>
+
+{{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="<span class='figure-number'>Figure 2: </span>Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
+
+The controller used consisted of:
+
+-   second order low pass filter to gain stabilize the plant at high frequencies and provide steep roll-off
+-   first order lead filter to provide adequate phase margin at the high frequency crossover
+-   first order lag filter to provide adequate phase margin at the low frequency crossover
+-   a first order high pass filter to attenuate the excess gain resulting from the low frequency zero
+
+The results in terms of transmissibility are shown in [Figure 3](#figure--fig:spanos95-results).
+
+<a id="figure--fig:spanos95-results"></a>
+
+{{< figure src="/ox-hugo/spanos95_results.png" caption="<span class='figure-number'>Figure 3: </span>Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In <i>Proceedings of 1995 American Control Conference - ACC’95</i>. doi:<a href="https://doi.org/10.1109/acc.1995.529280">10.1109/acc.1995.529280</a>.</div>
+</div>

content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md

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++++
+title = "Interferometric characterization of rotation stages for x-ray nanotomography"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}}), [Positioning Stations]({{< relref "positioning_stations.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Stankevic et al. 2017</a>)
+
+Author(s)
+: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
+
+Year
+: 2017
+
+-   Similar Station than the NASS
+-   Similar Metrology with fiber based interferometers and cylindrical reference mirror
+
+<a id="figure--fig:stankevic17-station"></a>
+
+{{< figure src="/ox-hugo/stankevic17_station.png" caption="<span class='figure-number'>Figure 1: </span>Positioning Station" >}}
+
+-   **Thermal expansion**: Stabilized down to \\(5mK/h\\) using passive water flow through the baseplate below the sample stage and in the interferometry reference frame.
+-   **Controller**: Two Independant PID loops
+-   Repeatable errors =&gt; feedforward (Look Up Table)
+-   Non-repeatable errors =&gt; feedback
+-   Result: 40nm runout error
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stankevic, T., C. Engblom, F. Langlois, F. Alves, A. Lestrade, N. Jobert, G. Cauchon, U. Vogt, and S. Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” <i>Review of Scientific Instruments</i> 88 (5): 053703. doi:<a href="https://doi.org/10.1063/1.4983405">10.1063/1.4983405</a>.</div>
+</div>

content/article/stein03_respec_unstab.md

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+title = "Respect the unstable"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Stein 2003</a>)
+
+Author(s)
+: Stein, G.
+
+Year
+: 2003
+
+
+## Introduction {#introduction}
+
+> The second trend has been evident at our conferences, and certainly in our journal, over the years.
+> This trend is the increasing worship of abstract mathematical results in control at the expense of more specific examinations of their practical, physical consequences.
+
+<div class="important">
+
+**Basic facts about unstable plants**:
+
+-   Unstable systems are fundamentally, and quantifiably more difficult to control than stable ones
+-   Controllers for unstable systems are operationally critical
+-   Closed-loop systems with unstable components are only locally stable
+
+</div>
+
+
+## The Bode Integrals {#the-bode-integrals}
+
+<div class="important">
+
+**Bode Integrals**:
+
+The first integral applies to stable plants and the second to unstable plants.
+They are valid for every stabilizing controller, assuming only that both plan and controller have finite bandwidths.
+In words, the integrals state that the log of magnitude of sensitivity function of a SISO feedback system, integrated over frequency, is constant.
+The constant is zero for stable plants, and it is positive for unstable ones.
+It becomes larger as the number of unstable poles increases and/or as the poles more farther into the right-half plane.
+
+\begin{align}
+  \int\_0^\infty \ln |S(j\omega)| d \omega & = 0 \label{eq:bode\_integral\_stable} \\\\
+  \int\_0^\infty \ln |S(j\omega)| d \omega & = \pi \sum\_{p \in P} \text{Re}(p) \label{eq:bode\_integral\_unstable}
+\end{align}
+
+</div>
+
+
+## A Bode Integral Interpretation {#a-bode-integral-interpretation}
+
+Bode integral can be thought as **conservation laws**.
+They state that a certain quantity, the integrated value of the log of the magnitude of the sensitivity function, is conserved under the action of feedback.
+The total amount of this quantity is always the same.
+It is equal to zero for stable plant/compensator pairs, and it is equal to some fixed positive amount for unstable ones.
+
+Since we are talking about the log of sensitivity magnitude, it follows that negative values are good, and positive values are bad.
+
+<div class="definition">
+
+It is curious, somehow, that our field has not adopted a name for this quantity being conserved (i.e. the integrated log of sensitivity magnitude).
+It is here proposed to call it **dirt**
+
+</div>
+
+The job of a serious control designer is then to more dirt from one place to another, using appropriate tools, without being able to get rid of any of it (illustrated in [Figure 1](#figure--fig:stein03-serious-design)).
+
+<a id="figure--fig:stein03-serious-design"></a>
+
+{{< figure src="/ox-hugo/stein03_serious_design.png" caption="<span class='figure-number'>Figure 1: </span>Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies" >}}
+
+In the same spirit, the job of a more academic control designer with more abstract tools such as LQG, \\(\mathcal{H}\_\infty\\), is to set parameters (weights) of a synthesis machine to adjust the contours of the machine's digging blades to get just the right shape for the sensitivity function ([Figure 2](#figure--fig:stein03-formal-design)).
+
+<a id="figure--fig:stein03-formal-design"></a>
+
+{{< figure src="/ox-hugo/stein03_formal_design.png" caption="<span class='figure-number'>Figure 2: </span>Sensitivity shaping automated by modern control tools" >}}
+
+
+## Available bandwidth {#available-bandwidth}
+
+An argument is sometimes made that the Bode integrals are not really restrictive because we only seek to dig holes over finite frequency bands.
+We then have an infinite frequency range left over into which to dump the dirt, so we can make the layer arbitrarily thin ([Figure 3](#figure--fig:stein03-spreading-it-thin)).
+
+<a id="figure--fig:stein03-spreading-it-thin"></a>
+
+{{< figure src="/ox-hugo/stein03_spreading_it_thin.png" caption="<span class='figure-number'>Figure 3: </span>It is possible to spead the increase of the sensitivity function over a larger frequency band" >}}
+
+The weakness of this argument is evident from standard classical theory.
+A thin layer, say with \\(\ln|S| = \epsilon\\) requires a loop transfer function whose Nyquist diagram falls on a near-unit circle, centered at \\((-1 + j 0)\\) with a radius \\(\approx (1-\epsilon)\\), over a wide frequency range.
+This means that the loop cannot simply attenuate at high frequencies but must attenuate in a very precise way.
+The loop must maintain very good frequency response fidelity over wide frequency ranges.
+
+But a key fact about physical systems is that they do not exhibit good frequency response fidelity beyond a certain bandwidth.
+This is due to uncertain or unmodeled dynamics in the plant, to digital control implementations, to power limits, to nonlinearities, and to many other factors.
+Let us call that bandwidth the available bandwidth" \\(\Omega\_a\\), to distinguish it from other bandwidths such as crossover or \\(3-dB\\) magnitude loss.
+The available bandwidth is the frequency up to which we can keep \\(G(j\omega) K(j\omega)\\) close to a nominal design and beyond which we can only guarantee that the actual loop magnitude will attenuate rapidly enough (e.g. \\(|G(j\omega) K(j\omeg\\))| &lt; &delta;/&omega;^2$).
+In today's popular robust control jargon, the available bandwidth is the frequency range over which the unstructured multiplicative perturbations are substantially less than unity.
+
+Note that the available bandwidth is not a function of the compensator or of the control design process.
+Rather, it is an a priori constraint imposed by the physical hardware we use in the control loop.
+Most importantly, the available bandwidth is always finite.
+
+Given all this, Bode's integrals really reduce to finite integrals over the range \\(0 \ge \omega \ge \Omega\_a\\):
+
+\begin{align}
+  \int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \delta \\\\
+  \int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \pi \sum\_{p \in P} \text{Re}(p) + \delta
+\end{align}
+
+All the action of the feedback design, the sensitivity improvements as well as the sensitivity deterioration, must occur within \\(0 \ge \omega \ge \Omega\_a\\).
+Only a small error \\(\delta\\) occurs outside that range, associated with the tail of the complete integrals.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stein, Gunter. 2003. “Respect the Unstable.” <i>IEEE Control Systems Magazine</i> 23 (4). IEEE: 12–25.</div>
+</div>

content/article/steinbuch21_motion_contr_mechat_desig_moores_law.md

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+title = "Motion control, mechatronics design, and moore's law"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Steinbuch, Oomen, and Vermeulen 2021</a>)
+
+Author(s)
+: Steinbuch, M., Oomen, T., &amp; Vermeulen, H.
+
+Year
+: 2021
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Steinbuch, Maarten, Tom Oomen, and Hans Vermeulen. 2021. “Motion Control, Mechatronics Design, and Moore’s Law.” <i>IEEJ Journal of Industry Applications</i>. The Institute of Electrical Engineers of Japan, 21006010.</div>
+</div>

content/article/steinbuch98_advan_motion_contr.md

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++++
+title = "Advanced motion control: an industrial perspective"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Steinbuch and Norg 1998</a>)
+
+Author(s)
+: Steinbuch, M., &amp; Norg, M.
+
+Year
+: 1998
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Steinbuch, M., and M.L. Norg. 1998. “Advanced Motion Control: An Industrial Perspective.” <i>European Journal of Control</i> 4 (4): 278–93. doi:<a href="https://doi.org/10.1016/s0947-3580(98)70121-9">10.1016/s0947-3580(98)70121-9</a>.</div>
+</div>

content/article/stoev17_tensor_method_mimo_decoup_contr.md

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++++
+title = "Tensor methods for mimo decoupling and control design using frequency response functions"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Decoupled Control]({{< relref "decoupled_control.md" >}}), [Multivariable Control]({{< relref "multivariable_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Stoev et al. 2017</a>)
+
+Author(s)
+: Stoev, J., Ertveldt, J., Oomen, T., &amp; Schoukens, J.
+
+Year
+: 2017
+
+
+## Introduction {#introduction}
+
+By appropriate system design, most systems are either decoupled or can be decoupled using static input-output transformations.
+Hence, most motion system and their motion software architecture use SISO control design method and solutions.
+
+The first step typically involves a FRF identification using specific excitation signals.
+Once the FRF is available, the controller \\(K\\) can be designed directly based on the FRF data.
+Many classical MIMO control design methods aim at decoupling the open loop function at some location in the feedback loop.
+Because their are strong non-intuitive aspect of MIMO loop-shaping, the following step-by-step approach is proposed, in which the design complexity is only increased if justified by the problem at hand:
+
+-   **[Interaction Analysis]({{< relref "interaction_analysis.md" >}})**.
+    The goal is to identify two sided interactions in the plant dynamics.
+    If there is no two sided interaction, then feedback design becomes a standard multi-loop SISO design problem.
+    Two measured of the plant interaction are [Relative Gain Array]({{< relref "relative_gain_array.md" >}}) and [Structured Singular Value]({{< relref "structured_singular_value.md" >}}).
+-   **Decoupling transformations**.
+    To reduce interaction, one may redefine the input and output of the plant using a decoupling transformation.
+    For motion systems, most transformations are found on the basis of **kinematic model**.
+    Herein, combinations of the actuators are defined so that actuator variables act in independent (orthogonal) directions at the center of gravity.
+    Similarly, combinations of the sensors are defined so that each translation and rotation of the center of gravity can be measured independently.
+    This, this basically amounts to the **inversion of a kinematic model** of the plant.
+-   Independent feedback control design
+-   Sequential feedback control design
+-   Norm based control design
+
+All steps, except for the last, can be performed with a non-parametric model of the plant (i.e. an identified FRF).
+
+
+## MIMO frequency response decomposition {#mimo-frequency-response-decomposition}
+
+The problem addressed in this paper is to decouple a given set of MIMO FRF.
+Such decoupled representation, if existing, would permit the MIMO FRF to be written as a linear combination of parallel SISO FRFs.
+The existing methods to convert the MIMO FRF into equivalent combination of SISO FRF fall into two groups:
+
+-   **matrix decomposition methods** use linear algebra based on eigen-value, or singular value decomposition which are able to diagonalize the FRF at a single frequency.
+-   **optimization methods** formulate the problem of simultaneous diagonalization of the FRF at multiple frequencies as an optimization problem.
+
+At each frequency \\(\omega\_i, i = 1 \dots N\_f\\), we have a square matrix \\(H(\omega\_i) \in \mathbb{C}^{N \times N}\\) with the complex response of the system relating the inputs and outputs.
+
+**MIMO decoupling of dyadic system**:
+
+\begin{align}
+H(\omega\_i) &= T\_y S(\omega\_i) T\_u + E(\omega\_i), \ i = 1 \dots N\_f \label{eq:decomposition} \\\\
+S(\omega\_i) &= \begin{bmatrix}
+S\_1(\omega\_i) & 0 & 0 \\\\
+0 & \ddots & 0 \\\\
+0 & 0 & S\_N(\omega\_i)
+\end{bmatrix}
+\end{align}
+
+where \\(S(\omega\_i)\\) is a diagonal matrix containing SISO FRFs \\(S\_k(\omega\_i) \in \mathbb{C}\\) on the main diagonal, \\(T\_y \in \mathbb{R}^{N \times N}\\), \\(T\_u \in \mathbb{R}^{N \times N}\\), \\(E(\omega\_i)\\) is the error.
+
+The approximate MIMO system decoupling is shown in [Figure 1](#figure--fig:stoev17-decoupled-system-schematic).
+
+In practical cases, the matrix \\(\hat{S}(\omega\_i) = T\_y^{-1} H(\omega\_i) T\_u^{-1}\\) will not be purely diagonal, but rather diagonally dominated.
+
+<a id="figure--fig:stoev17-decoupled-system-schematic"></a>
+
+{{< figure src="/ox-hugo/stoev17_decoupled_system_schematic.png" caption="<span class='figure-number'>Figure 1: </span>MIMO FRF decomposition in parallel branches" >}}
+
+The array \\(H(\omega\_i), i = 1 \dots N\_f\\) of complex matrices can be represented as a 3-dimensional sensor \\(\underline{H}\\).
+
+<div class="important">
+
+The core result of this paper is that the decomposition can be found by rephrasing \ref{eq:decomposition} as a "Canonical Polyadic Decomposition" (CPD).
+This is shown in [Figure 2](#figure--fig:stoev17-decompos-3d-tensor), where \\(T\_y,T\_u,S\_d\\) can be directly computed using a single Matlab function.
+
+</div>
+
+Mathematically equivalent form of CPD is shown in the lower part of [Figure 2](#figure--fig:stoev17-decompos-3d-tensor), where the tensor \\(\underline{S}\\) contains the rows of the matrix \\(S\_d\\) on each of its diagonals in the third dimension, which is exactly the problem of simultaneous diagonalization.
+
+The transformation effectively diagonalises the original frequency response tensor \\(\underline{H}\\) using two transformation matrices \\(T\_y, T\_u\\).
+This operation is closely related to the SVD on a single matrix, however in this case the diagonalisation occurs for a set of matrices, each describing the MIMO FRF at different frequency.
+
+<a id="figure--fig:stoev17-decompos-3d-tensor"></a>
+
+{{< figure src="/ox-hugo/stoev17_decompos_3d_tensor.png" caption="<span class='figure-number'>Figure 2: </span>Decomposition of 3D tensor" >}}
+
+The direct application of a CPD procedure on the above complex data tensor would result in complex solutions, including complex matrices \\(T\_y \in \mathbb{C}^{N \times N}\\), \\(T\_u \in \mathbb{C}^{N \times N}\\).
+This is not useful for a practical decoupling of physical systems as we require real solutions for \\(T\_y,T\_u\\).
+The direct solution we use for this is to take the imaginary and real part of the complex tensor \\(\underline{H} \in \mathbb{C}^{N \times N \times N\_f}\\), each of them a real tensor by itself, and stack them one behind the other in the dimension of the frequencies, thus getting an augmented real-valued tensor \\(\underline{\breve{H}} \in \mathbb{R}^{N \times N \times 2N\_f}\\).
+
+
+## Numerical Example {#numerical-example}
+
+Let's now make a Matlab example using the [Tensorlab](https://www.tensorlab.net/) toolbox.
+
+Let's define a 2x2 diagonal system:
+
+```matlab
+S = [4e3/(s^2 + 25*s + 4e3) 0
+     0 4e5/(s^2 + 250*s + 4e5)];
+```
+
+And coupled this system with two random matrices:
+
+```matlab
+Ty = [0.13 0.003
+      0.43 0.51];
+
+Tu = [0.32 0.67
+      0.95 0.006];
+```
+
+The couple system is defined:
+
+```matlab
+H = Ty * S * Tu;
+```
+
+Then, suppose with have the frequency response function of the coupled plant:
+
+```matlab
+freqs = logspace(0,3,1000);
+H_frf = freqresp(H, freqs, 'Hz');
+```
+
+<a id="figure--fig:stoev17-coupled-diagonal-plants"></a>
+
+{{< figure src="/ox-hugo/stoev17_coupled_diagonal_plants.png" caption="<span class='figure-number'>Figure 3: </span>Diagonal and coupled plants" >}}
+
+We take the real and imaginary part of the FRF and concatenate the two along the frequency dimension.
+
+```matlab
+H_frf_real = cat(3, real(H_frf), imag(H_frf));
+```
+
+Then random matrices are initialize the the CPD.
+
+```matlab
+U = cpd_rnd(size(H_frf_real), size(H_frf_real,1));
+```
+
+And the CPD is performed.
+
+```matlab
+[T, ~] = cpd3_sd(H_frf_real, U);
+```
+
+The obtained decoupling matrices are:
+
+```matlab
+Ty_est = T{1};
+```
+
+```text
+Ty_est =
+        -0.289402459385387      -0.00647742171539879
+        -0.957207509228524         -1.10111369041218
+```
+
+```matlab
+Tu_est = T{2};
+```
+
+```text
+Tu_est =
+         0.430893809258741         0.999980044721872
+         0.902402640256826       0.00631744869723862
+```
+
+And the decoupled plant using the estimated optimal decoupling matrices is:
+
+```matlab
+H_dec = inv(Ty_est) * H * inv(Tu_est);
+```
+
+<a id="figure--fig:stoev17-results-decoupling-example"></a>
+
+{{< figure src="/ox-hugo/stoev17_results_decoupling_example.png" caption="<span class='figure-number'>Figure 4: </span>Diagonal, coupled and decoupled plants" >}}
+
+
+## Conclusion {#conclusion}
+
+The paper presents an application for the tensor decomposition for the design of a static decoupling of a MIMO system.
+The results in this paper are obtained on a _non-parametric_ frequency domain model of the plant and indicate that the procedure is more robust that the eigen-value based decoupling.
+The advantages of this method with respect to some of th existing methods can be found when the FRF data available is disturbed by noise.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stoev, Julian, Julien Ertveldt, Tom Oomen, and Johan Schoukens. 2017. “Tensor Methods for Mimo Decoupling and Control Design Using Frequency Response Functions.” <i>Mechatronics</i> 45: 71–81. doi:<a href="https://doi.org/10.1016/j.mechatronics.2017.05.009">10.1016/j.mechatronics.2017.05.009</a>.</div>
+</div>

content/article/tang18_decen_vibrat_contr_voice_coil.md

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++++
+title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Tang, Cao, and Yu 2018</a>)
+
+Author(s)
+: Tang, J., Cao, D., &amp; Yu, T.
+
+Year
+: 2018
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tang, J., D. Cao, and T. Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” <i>Proceedings of the Institution of Mechanical Engineers, Part c: Journal of Mechanical Engineering Science</i> 233 (1): 132–45. doi:<a href="https://doi.org/10.1177/0954406218756941">10.1177/0954406218756941</a>.</div>
+</div>

content/article/thayer02_six_axis_vibrat_isolat_system.md

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++++
+title = "Six-axis vibration isolation system using soft actuators and multiple sensors"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Thayer et al. 2002</a>)
+
+Author(s)
+: Thayer, D., Campbell, M., Vagners, J., &amp; Flotow, A. v.
+
+Year
+: 2002
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Thayer, D., M. Campbell, J. Vagners, and A. von Flotow. 2002. “Six-Axis Vibration Isolation System Using Soft Actuators and Multiple Sensors.” <i>Journal of Spacecraft and Rockets</i> 39 (2): 206–12. doi:<a href="https://doi.org/10.2514/2.3821">10.2514/2.3821</a>.</div>
+</div>

content/article/thurner15_fiber_based_distan_sensin_inter.md

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++++
+title = "Fiber-Based Distance Sensing Interferometry"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Interferometers]({{< relref "interferometers.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Thurner et al. 2015</a>)
+
+Author(s)
+: Thurner, K., Quacquarelli, F. P., Braun, Pierre-Francois, Dal Savio, C., &amp; Karrai, K.
+
+Year
+: 2015
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” <i>Applied Optics</i> 54 (10). Optical Society of America: 3051–63.</div>
+</div>

content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md

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++++
+title = "Sensor fusion for active vibration isolation in precision equipment"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Tjepkema, van Dijk, and Soemers 2012</a>)
+
+Author(s)
+: Tjepkema, D., Dijk, J. v., &amp; Soemers, H.
+
+Year
+: 2012
+
+
+## Relative motion Control {#relative-motion-control}
+
+Control law: \\(f = -G(x-w)\\)
+
+\\[ \frac{x}{w} = \frac{k+G}{ms^2 + k+G} \\]
+\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
+
+
+## Force Control {#force-control}
+
+Control law: \\(f = -G F\_a = -G \left(f-k(x-w)\right)\\)
+
+\\[ \frac{x}{w} = \frac{k}{(1+G)ms^2 + k} \\]
+\\[ \frac{x}{F} = \frac{1+G}{(1+G)ms^2 + k} \\]
+
+
+## Inertial Control {#inertial-control}
+
+Control law: \\(f = -Gx\\)
+
+\\[ \frac{x}{w} = \frac{k}{ms^2 + k+G} \\]
+\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
+
+
+## Design constraints and control bandwidth {#design-constraints-and-control-bandwidth}
+
+Heavier sensor =&gt; lower noise but it is harder to maintain collocation with the actuator =&gt; that limits the bandwidth.
+There is a compromise between sensor noise and the influence of the sensor size on the system's design and on the control bandwidth.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Tjepkema, D., J. van Dijk, and H. M. J. R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” <i>Journal of Sound and Vibration</i> 331 (4): 735–49. doi:<a href="https://doi.org/10.1016/j.jsv.2011.09.022">10.1016/j.jsv.2011.09.022</a>.</div>
+</div>

content/article/vcech19_essen_chall_motion_contr_educat.md

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++++
+title = "Essential challenges in motion control education"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Ech et al. 2019</a>)
+
+Author(s)
+: M. \VCech, J. K\\"onigsmarkov\\'a, Goubej, M., Oomen, T., &amp; Visioli, A.
+
+Year
+: 2019
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>ech, M., J. Königsmarková, M. Goubej, T. Oomen, and A. Visioli. 2019. “Essential Challenges in Motion Control Education.” <i>IFAC-PapersOnLine</i> 52 (9): 200–205. doi:<a href="https://doi.org/10.1016/j.ifacol.2019.08.196">10.1016/j.ifacol.2019.08.196</a>.</div>
+</div>

content/article/wang12_autom_marker_full_field_hard.md

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++++
+title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Wang et al. 2012</a>)
+
+Author(s)
+: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., &amp; Feser, M.
+
+Year
+: 2012
+
+**Introduction of Markers**:
+That limits the type of samples that is studied
+
+There is a need for markerless nano-tomography
+=&gt; the key requirement is the precision and stability of the positioning stages.
+
+**Passive rotational run-out error system**:
+It uses calibrated metrology disc and capacitive sensors
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wang, J., Y.-c. K. Chen, Q. Yuan, A. Tkachuk, C. Erdonmez, B. Hornberger, and M. Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” <i>Applied Physics Letters</i> 100 (14): 143107. doi:<a href="https://doi.org/10.1063/1.3701579">10.1063/1.3701579</a>.</div>
+</div>

content/article/wang16_inves_activ_vibrat_isolat_stewar.md

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++++
+title = "Investigation on active vibration isolation of a stewart platform with piezoelectric actuators"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Wang et al. 2016</a>)
+
+Author(s)
+: Wang, C., Xie, X., Chen, Y., &amp; Zhang, Z.
+
+Year
+: 2016
+
+**Model of the Stewart platform**:
+
+-   Struts are treated as flexible beams
+-   Payload and the base are treated as flexible plates
+-   The FRF synthesis method permits to derive FRFs of the Stewart platform
+
+The model is compared with a Finite Element model and is shown to give the same results.
+The proposed model is thus effective.
+
+<a id="figure--fig:wang16-stewart-platform"></a>
+
+{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Stewart Platform" >}}
+
+**Control**:
+Combines:
+
+-   the FxLMS-based adaptive inverse control =&gt; suppress transmission of periodic vibrations
+-   direct feedback of integrated forces =&gt; dampen vibration of inherent modes and thus reduce random vibrations
+
+Force Feedback ([Figure 2](#figure--fig:wang16-force-feedback)).
+
+-   the force sensor is mounted **between the base and the strut**
+
+<a id="figure--fig:wang16-force-feedback"></a>
+
+{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="<span class='figure-number'>Figure 2: </span>Feedback of integrated forces in the platform" >}}
+
+Sorts of HAC-LAC control:
+
+-   LAC: Decentralized integral force feedback
+-   HAC: Inertial control using accelerometers. Use of the Jacobian to decouple the motion and then Fx-LMS based adaptive control is used
+
+**Experimental validation**:
+
+-   All 6 transfer function from actuator force to force sensors are almost the same (gain offset)
+-   Effectiveness of control methods are shown
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wang, C., X. Xie, Y. Chen, and Z. Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” <i>Journal of Sound and Vibration</i> 383. Elsevier BV: 1–19. doi:<a href="https://doi.org/10.1016/j.jsv.2016.07.021">10.1016/j.jsv.2016.07.021</a>.</div>
+</div>

content/article/yang19_dynam_model_decoup_contr_flexib.md

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++++
+title = "Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Yang et al. 2019</a>)
+
+Author(s)
+: Yang, X., Wu, H., Chen, B., Kang, S., &amp; Cheng, S.
+
+Year
+: 2019
+
+**Discusses**:
+
+-   flexible-rigid model of Stewart platform
+-   the impact of joint stiffness is compensated using a displacement sensor and a force sensor
+-   then the MIMO system is decoupled in modal space and 6 SISO controllers are applied for vibration isolation using force sensors
+
+The joint stiffness impose a limitation on the control performance using force sensors as it adds a zero at low frequency in the dynamics.
+Thus, this stiffness is taken into account in the dynamics and compensated for.
+
+**Stewart platform** ([Figure 1](#figure--fig:yang19-stewart-platform)):
+
+-   piezoelectric actuators
+-   flexible joints ([Figure 2](#figure--fig:yang19-flexible-joints))
+-   force sensors (used for vibration isolation)
+-   displacement sensors (used to decouple the dynamics)
+-   cubic (even though not said explicitly)
+
+<a id="figure--fig:yang19-stewart-platform"></a>
+
+{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="<span class='figure-number'>Figure 1: </span>Stewart Platform" >}}
+
+<a id="figure--fig:yang19-flexible-joints"></a>
+
+{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="<span class='figure-number'>Figure 2: </span>Flexible Joints" >}}
+
+The stiffness of the flexible joints ([Figure 2](#figure--fig:yang19-flexible-joints)) are computed with an FEM model and shown in [Table 1](#table--tab:yang19-stiffness-flexible-joints).
+
+<a id="table--tab:yang19-stiffness-flexible-joints"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:yang19-stiffness-flexible-joints">Table 1</a>:</span>
+  Stiffness of flexible joints obtained by FEM
+</div>
+
+| \\(k\_{\theta u},\ k\_{\psi u}\\) | \\(72 Nm/rad\\) |
+|-----------------------------------|-----------------|
+| \\(k\_{\theta s}\\)               | \\(51 Nm/rad\\) |
+| \\(k\_{\psi s}\\)                 | \\(62 Nm/rad\\) |
+| \\(k\_{\gamma s}\\)               | \\(64 Nm/rad\\) |
+
+**Dynamics**:
+If the bending and torsional stiffness of the flexible joints are neglected:
+\\[ M \ddot{x} + C \dot{x} + K x = J^T f \\]
+
+-   \\(M\\) is the mass matrix
+-   \\(C\\) is the damping matrix
+-   \\(K\\) is the stiffness matrix
+-   \\(x\\) is the generalized coordinates, representing the displacement and orientation of the payload plate
+-   \\(f\\) is the actuator forces
+-   \\(J\\) is the Jacobian matrix
+
+In this paper, the parasitic bending stiffness of the flexible joints are considered:
+\\[ M \ddot{x} + C \dot{x} + (K + K\_e) x = J^T f \\]
+where \\(K\_e\\) is the stiffness matrix induced by the parasitic stiffness of the flexible joints.
+
+Analytical expression for \\(K\_e\\) are derived in the paper.
+
+**Controller Design**:
+There is a strong coupling between the input forces and the state variables in the task space.
+The traditional modal decoupled control strategy cannot work with the flexible Stewart platform because it is impossible to achieve simultaneous diagonalization of the mass, damped and stiffness matrices.
+
+To make the six-dof system decoupled into six single-dof isolators, a controller based on the leg's force and position feedback is designed.
+
+> The idea is to synthesize the control force that can compensate the parasitic bending and torsional torques of the flexible joints and simultaneously achieve diagonalization of the matrices \\(M\\), \\(C\\) and \\(K\\)
+
+The force measured by the force sensors are:
+\\[ y = f - k J x - c J \dot{x} \\]
+The displacements measured by the position sensors are:
+\\[ z = [\Delta l\_1\ \dots\ \Delta l\_6]^T \\]
+
+Let's apply the feedback control based on both the force sensor and the position sensor:
+\\[ f = -H(s) y + (1 + H(s)) K\_{el} z \\]
+where \\(K\_{el} = J^{-T} K\_e J^T\\) is the stiffness matrix of the flexible joints expressed in joint space.
+
+We thus obtain:
+\\[ f = \frac{H(s)}{1 + H(s)} (k J x + c J \dot{x}) + J^{-T} K\_e x \\]
+
+If we substitute \\(f\\) in the dynamic equation, we obtain that the parasitic stiffness effect of the flexible joints has been compensated by the actuation forces and the system can now be decoupled in modal space \\(x = \Phi u\\).
+\\(\Phi\\) is the modal matrix selected such that \\(\Phi^T M \Phi = I\_6\\) and \\(k \Phi^T J^T J \Phi = \text{diag}(\omega\_1^2\ \dots\ \omega\_6^2)\\):
+\\[ s^2 + \frac{1}{1 + H(s)} \frac{c \omega\_i^2}{k} s + \frac{1}{1 + H(s)} \omega\_i^2 = 0, \quad i = 1,\ \dots,\ 6 \\]
+
+The six-dof system is now transformed into a six one-dof system where \\(H(s)\\) can be designed for control purpose.
+
+In order to apply this control strategy:
+
+-   A force sensor and displacement sensor are need in each strut
+-   The joint stiffness has to be known
+-   The jacobian has to be computed
+-   No information about modal matrix is needed
+
+The block diagram of the control strategy is represented in [Figure 3](#figure--fig:yang19-control-arch).
+
+<a id="figure--fig:yang19-control-arch"></a>
+
+{{< figure src="/ox-hugo/yang19_control_arch.png" caption="<span class='figure-number'>Figure 3: </span>Control Architecture used" >}}
+
+\\(H(s)\\) is designed as a proportional plus integral compensator:
+\\[ H(s) = k\_p + k\_i/s \\]
+
+Substituting \\(H(s)\\) in the equation of motion gives that:
+
+-   an increase of \\(k\_i\\) increase the damping and thus suppress the resonance peaks
+-   an increase of \\(k\_p\\) lowers the resonance frequency and thus the bandwidth of vibration isolation is examped
+
+**Experimental Validation**:
+An external Shaker is used to excite the base and accelerometers are located on the base and mobile platforms to measure their motion.
+The results are shown in [Figure 4](#figure--fig:yang19-results).
+In theory, the vibration performance can be improved, however in practice, increasing the gain causes saturation of the piezoelectric actuators and then the instability occurs.
+
+<a id="figure--fig:yang19-results"></a>
+
+{{< figure src="/ox-hugo/yang19_results.png" caption="<span class='figure-number'>Figure 4: </span>Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
+
+> A model-based controller is then designed based on the leg’s force and position feedback.
+> The position feedback compensates the effect of parasitic bending and torsional stiffness of the flexible joints.
+> The force feedback makes the six-DOF MIMO system decoupled into six SISO subsystems in modal space, where the control gains can be designed and analyzed more effectively and conveniently.
+> The proportional and integral gains in the sub-controller are used to separately regulate the vibration isolation bandwidth and active damping simultaneously for the six vibration modes.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yang, X., H. Wu, B. Chen, S. Kang, and S. Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” <i>Journal of Sound and Vibration</i> 439. Elsevier BV: 398–412. doi:<a href="https://doi.org/10.1016/j.jsv.2018.10.007">10.1016/j.jsv.2018.10.007</a>.</div>
+</div>

content/article/yong12_invit_review_artic.md

@@ -0,0 +1,22 @@
++++
+title = "Invited review article: high-speed flexure-guided nanopositioning: mechanical design and control issues"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Yong et al. 2012</a>)
+
+Author(s)
+: Yong, Y. K., Moheimani, S. O. R., Kenton, B. J., &amp; Leang, K. K.
+
+Year
+: 2012
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yong, Y. K., S. O. R. Moheimani, B. J. Kenton, and K. K. Leang. 2012. “Invited Review Article: High-Speed Flexure-Guided Nanopositioning: Mechanical Design and Control Issues.” <i>Review of Scientific Instruments</i> 83 (12): 121101. doi:<a href="https://doi.org/10.1063/1.4765048">10.1063/1.4765048</a>.</div>
+</div>

content/article/yun20_inves_two_stage_vibrat_suppr.md

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++++
+title = "Investigation on two-stage vibration suppression and precision pointing for space optical payloads"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Yun et al. 2020</a>)
+
+Author(s)
+: Yun, H., Liu, L., Li, Q., &amp; Yang, H.
+
+Year
+: 2020
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” <i>Aerospace Science and Technology</i> 96: 105543. doi:<a href="https://doi.org/10.1016/j.ast.2019.105543">10.1016/j.ast.2019.105543</a>.</div>
+</div>

content/article/zhang11_six_dof.md

@@ -0,0 +1,37 @@
++++
+title = "Six dof active vibration control using stewart platform with non-cubic configuration"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Zhang et al. 2011</a>)
+
+Author(s)
+: Zhang, Z., Liu, J., Mao, J., Guo, Y., &amp; Ma, Y.
+
+Year
+: 2011
+
+-   **Non-cubic** stewart platform
+-   **Flexible** joints
+-   Magnetostrictive actuators
+-   Strong coupled motions along different axes
+-   Non-cubic architecture =&gt; permits to have larger workspace which was required
+-   Structure parameters (radius of plates, length of struts) are determined by optimization of the condition number of the Jacobian matrix
+-   **Accelerometers** for active isolation
+-   Adaptive FIR filters for active isolation control
+
+<a id="figure--fig:zhang11-platform"></a>
+
+{{< figure src="/ox-hugo/zhang11_platform.png" caption="<span class='figure-number'>Figure 1: </span>Prototype of the non-cubic stewart platform" >}}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zhang, Z., J. Liu, J. Mao, Y. Guo, and Y. Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In <i>2011 6th IEEE Conference on Industrial Electronics and Applications</i>. doi:<a href="https://doi.org/10.1109/iciea.2011.5975679">10.1109/iciea.2011.5975679</a>.</div>
+</div>

content/book/_index.md

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++++
+title = "Books"
+author = ["Thomas Dehaeze"]
+type = "book"
+draft = false
++++
+
+Here is the list of books I took note about.

content/book/du10_model_contr_vibrat_mechan_system.md

@@ -0,0 +1,540 @@
++++
+title = "Modeling and control of vibration in mechanical systems"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Du and Xie 2010</a>)
+
+Author(s)
+: Du, C., &amp; Xie, L.
+
+Year
+: 2010
+
+
+## 1. Mechanical Systems and Vibration {#1-dot-mechanical-systems-and-vibration}
+
+
+### 1.1 Magnetic recording system {#1-dot-1-magnetic-recording-system}
+
+
+### 1.2 Stewart platform {#1-dot-2-stewart-platform}
+
+
+### 1.3 Vibration sources and descriptions {#1-dot-3-vibration-sources-and-descriptions}
+
+
+### 1.4 Types of vibration {#1-dot-4-types-of-vibration}
+
+
+#### 1.4.1 Free and forced vibration {#1-dot-4-dot-1-free-and-forced-vibration}
+
+
+#### 1.4.2 Damped and undamped vibration {#1-dot-4-dot-2-damped-and-undamped-vibration}
+
+
+#### 1.4.3 Linear and nonlinear vibration {#1-dot-4-dot-3-linear-and-nonlinear-vibration}
+
+
+#### 1.4.4 Deterministic and random vibration {#1-dot-4-dot-4-deterministic-and-random-vibration}
+
+
+#### 1.4.5 Periodic and nonperiodic vibration {#1-dot-4-dot-5-periodic-and-nonperiodic-vibration}
+
+
+#### 1.4.6 Broad-band and narrow-band vibration {#1-dot-4-dot-6-broad-band-and-narrow-band-vibration}
+
+
+### 1.5 Random vibration {#1-dot-5-random-vibration}
+
+
+#### 1.5.1 Random process {#1-dot-5-dot-1-random-process}
+
+
+#### 1.5.2 Stationary random process {#1-dot-5-dot-2-stationary-random-process}
+
+
+#### 1.5.3 Gaussian random process {#1-dot-5-dot-3-gaussian-random-process}
+
+
+### 1.6 Vibration analysis {#1-dot-6-vibration-analysis}
+
+
+#### 1.6.1 Fourier transform and spectrum analysis {#1-dot-6-dot-1-fourier-transform-and-spectrum-analysis}
+
+
+#### 1.6.2 Relationship between the Fourier and Laplace transforms {#1-dot-6-dot-2-relationship-between-the-fourier-and-laplace-transforms}
+
+
+#### 1.6.3 Spectral analysis {#1-dot-6-dot-3-spectral-analysis}
+
+
+## 2. Modeling of Disk Drive System and Its Vibration {#2-dot-modeling-of-disk-drive-system-and-its-vibration}
+
+
+### 2.1 Introduction {#2-dot-1-introduction}
+
+
+### 2.2 System description {#2-dot-2-system-description}
+
+
+### 2.3 System modeling {#2-dot-3-system-modeling}
+
+
+#### 2.3.1 Modeling of a VCM actuator {#2-dot-3-dot-1-modeling-of-a-vcm-actuator}
+
+
+#### 2.3.2 Modeling of friction {#2-dot-3-dot-2-modeling-of-friction}
+
+
+#### 2.3.3 Modeling of a PZT microactuator {#2-dot-3-dot-3-modeling-of-a-pzt-microactuator}
+
+
+#### 2.3.4 An example {#2-dot-3-dot-4-an-example}
+
+
+### 2.4 Vibration modeling {#2-dot-4-vibration-modeling}
+
+
+#### 2.4.1 Spectrum-based vibration modeling {#2-dot-4-dot-1-spectrum-based-vibration-modeling}
+
+
+#### 2.4.2 Adaptive modeling of disturbance {#2-dot-4-dot-2-adaptive-modeling-of-disturbance}
+
+
+### 2.5 Conclusion {#2-dot-5-conclusion}
+
+
+## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms.md" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
+
+
+### 3.1 Introduction {#3-dot-1-introduction}
+
+
+### 3.2 System description and governing equations {#3-dot-2-system-description-and-governing-equations}
+
+
+### 3.3 Modeling using adaptive filtering approach {#3-dot-3-modeling-using-adaptive-filtering-approach}
+
+
+#### 3.3.1 Adaptive filtering theory {#3-dot-3-dot-1-adaptive-filtering-theory}
+
+
+#### 3.3.2 Modeling of a Stewart platform {#3-dot-3-dot-2-modeling-of-a-stewart-platform}
+
+
+### 3.4 Conclusion {#3-dot-4-conclusion}
+
+
+## 4. Classical Vibration Control {#4-dot-classical-vibration-control}
+
+
+### 4.1 Introduction {#4-dot-1-introduction}
+
+
+### 4.2 Passive control {#4-dot-2-passive-control}
+
+
+#### 4.2.1 Isolators {#4-dot-2-dot-1-isolators}
+
+
+#### 4.2.2 Absorbers {#4-dot-2-dot-2-absorbers}
+
+
+#### 4.2.3 Resonators {#4-dot-2-dot-3-resonators}
+
+
+#### 4.2.4 Suspension {#4-dot-2-dot-4-suspension}
+
+
+#### 4.2.5 An application example &amp;#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
+
+
+### 4.3 Self-adapting systems {#4-dot-3-self-adapting-systems}
+
+
+### 4.4 Active vibration control {#4-dot-4-active-vibration-control}
+
+
+#### 4.4.1 Actuators {#4-dot-4-dot-1-actuators}
+
+
+#### 4.4.2 Active systems {#4-dot-4-dot-2-active-systems}
+
+
+#### 4.4.3 Control strategy {#4-dot-4-dot-3-control-strategy}
+
+
+### 4.5 Conclusion {#4-dot-5-conclusion}
+
+
+## 5. Introduction to Optimal and Robust Control {#5-dot-introduction-to-optimal-and-robust-control}
+
+
+### 5.1 Introduction {#5-dot-1-introduction}
+
+
+### 5.2 H2 and H&amp;#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
+
+
+#### 5.2.1 H2 norm {#5-dot-2-dot-1-h2-norm}
+
+
+#### 5.2.2 H&amp;#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
+
+
+### 5.3 H2 optimal control {#5-dot-3-h2-optimal-control}
+
+
+#### 5.3.1 Continuous-time case {#5-dot-3-dot-1-continuous-time-case}
+
+
+#### 5.3.2 Discrete-time case {#5-dot-3-dot-2-discrete-time-case}
+
+
+### 5.4 H&amp;#8734; control {#5-dot-4-h-and-8734-control}
+
+
+#### 5.4.1 Continuous-time case {#5-dot-4-dot-1-continuous-time-case}
+
+
+#### 5.4.2 Discrete-time case {#5-dot-4-dot-2-discrete-time-case}
+
+
+### 5.5 Robust control {#5-dot-5-robust-control}
+
+
+### 5.6 Controller parametrization {#5-dot-6-controller-parametrization}
+
+
+### 5.7 Performance limitation {#5-dot-7-performance-limitation}
+
+
+#### 5.7.1 Bode integral constraint {#5-dot-7-dot-1-bode-integral-constraint}
+
+
+#### 5.7.2 Relationship between system gain and phase {#5-dot-7-dot-2-relationship-between-system-gain-and-phase}
+
+
+#### 5.7.3 Sampling {#5-dot-7-dot-3-sampling}
+
+
+### 5.8 Conclusion {#5-dot-8-conclusion}
+
+
+## 6. Mixed H2/H&amp;#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
+
+
+### 6.1 Introduction {#6-dot-1-introduction}
+
+
+### 6.2 Mixed H2/H&amp;#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
+
+
+### 6.3 Method 1: slack variable approach {#6-dot-3-method-1-slack-variable-approach}
+
+
+### 6.4 Method 2: an improved slack variable approach {#6-dot-4-method-2-an-improved-slack-variable-approach}
+
+
+### 6.5 Application in servo loop design for hard disk drives {#6-dot-5-application-in-servo-loop-design-for-hard-disk-drives}
+
+
+#### 6.5.1 Problem formulation {#6-dot-5-dot-1-problem-formulation}
+
+
+#### 6.5.2 Design results {#6-dot-5-dot-2-design-results}
+
+
+### 6.6 Conclusion {#6-dot-6-conclusion}
+
+
+## 7. Low-Hump Sensitivity Control Design for Hard Disk Drive Systems {#7-dot-low-hump-sensitivity-control-design-for-hard-disk-drive-systems}
+
+
+### 7.1 Introduction {#7-dot-1-introduction}
+
+
+### 7.2 Problem statement {#7-dot-2-problem-statement}
+
+
+### 7.3 Design in continuous-time domain {#7-dot-3-design-in-continuous-time-domain}
+
+
+#### 7.3.1 H&amp;#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
+
+
+#### 7.3.2 Application examples {#7-dot-3-dot-2-application-examples}
+
+
+#### 7.3.3 Implementation on a hard disk drive {#7-dot-3-dot-3-implementation-on-a-hard-disk-drive}
+
+
+### 7.4 Design in discrete-time domain {#7-dot-4-design-in-discrete-time-domain}
+
+
+#### 7.4.1 Synthesis method for low-hump sensitivity function {#7-dot-4-dot-1-synthesis-method-for-low-hump-sensitivity-function}
+
+
+#### 7.4.2 An application example {#7-dot-4-dot-2-an-application-example}
+
+
+#### 7.4.3 Implementation on a hard disk drive {#7-dot-4-dot-3-implementation-on-a-hard-disk-drive}
+
+
+### 7.5 Conclusion {#7-dot-5-conclusion}
+
+
+## 8. Generalized KYP Lemma-Based Loop Shaping Control Design {#8-dot-generalized-kyp-lemma-based-loop-shaping-control-design}
+
+
+### 8.1 Introduction {#8-dot-1-introduction}
+
+
+### 8.2 Problem description {#8-dot-2-problem-description}
+
+
+### 8.3 Generalized KYP lemma-based control design method {#8-dot-3-generalized-kyp-lemma-based-control-design-method}
+
+
+### 8.4 Peak filter {#8-dot-4-peak-filter}
+
+
+#### 8.4.1 Conventional peak filter {#8-dot-4-dot-1-conventional-peak-filter}
+
+
+#### 8.4.2 Phase lead peak filter {#8-dot-4-dot-2-phase-lead-peak-filter}
+
+
+#### 8.4.3 Group peak filter {#8-dot-4-dot-3-group-peak-filter}
+
+
+### 8.5 Application in high frequency vibration rejection {#8-dot-5-application-in-high-frequency-vibration-rejection}
+
+
+### 8.6 Application in mid-frequency vibration rejection {#8-dot-6-application-in-mid-frequency-vibration-rejection}
+
+
+### 8.7 Conclusion {#8-dot-7-conclusion}
+
+
+## 9. Combined H2 and KYP Lemma-Based Control Design {#9-dot-combined-h2-and-kyp-lemma-based-control-design}
+
+
+### 9.1 Introduction {#9-dot-1-introduction}
+
+
+### 9.2 Problem formulation {#9-dot-2-problem-formulation}
+
+
+### 9.3 Controller design for specific disturbance rejection and overall error minimization {#9-dot-3-controller-design-for-specific-disturbance-rejection-and-overall-error-minimization}
+
+
+#### 9.3.1 Q parametrization to meet specific specifications {#9-dot-3-dot-1-q-parametrization-to-meet-specific-specifications}
+
+
+#### 9.3.2 Q parametrization to minimize H2 performance {#9-dot-3-dot-2-q-parametrization-to-minimize-h2-performance}
+
+
+#### 9.3.3 Design steps {#9-dot-3-dot-3-design-steps}
+
+
+### 9.4 Simulation and implementation results {#9-dot-4-simulation-and-implementation-results}
+
+
+#### 9.4.1 System models {#9-dot-4-dot-1-system-models}
+
+
+#### 9.4.2 Rejection of specific disturbance and H2 performance minimization {#9-dot-4-dot-2-rejection-of-specific-disturbance-and-h2-performance-minimization}
+
+
+#### 9.4.3 Rejection of two disturbances with H[sub(2)] performance minimization {#9-dot-4-dot-3-rejection-of-two-disturbances-with-h-sub--2--performance-minimization}
+
+
+### 9.5 Conclusion {#9-dot-5-conclusion}
+
+
+## 10. Blending Control for Multi-Frequency Disturbance Rejection {#10-dot-blending-control-for-multi-frequency-disturbance-rejection}
+
+
+### 10.1 Introduction {#10-dot-1-introduction}
+
+
+### 10.2 Control blending {#10-dot-2-control-blending}
+
+
+#### 10.2.1 State feedback control blending {#10-dot-2-dot-1-state-feedback-control-blending}
+
+
+#### 10.2.2 Output feedback control blending {#10-dot-2-dot-2-output-feedback-control-blending}
+
+
+### 10.3 Control blending application in multi-frequency disturbance rejection {#10-dot-3-control-blending-application-in-multi-frequency-disturbance-rejection}
+
+
+#### 10.3.1 Problem formulation {#10-dot-3-dot-1-problem-formulation}
+
+
+#### 10.3.2 Controller design via the control blending technique {#10-dot-3-dot-2-controller-design-via-the-control-blending-technique}
+
+
+### 10.4 Simulation and experimental results {#10-dot-4-simulation-and-experimental-results}
+
+
+#### 10.4.1 Rejecting high-frequency disturbances {#10-dot-4-dot-1-rejecting-high-frequency-disturbances}
+
+
+#### 10.4.2 Rejecting a combined mid and high frequency disturbance {#10-dot-4-dot-2-rejecting-a-combined-mid-and-high-frequency-disturbance}
+
+
+### 10.5 Conclusion {#10-dot-5-conclusion}
+
+
+## 11. H&amp;#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
+
+
+### 11.1 Introduction {#11-dot-1-introduction}
+
+
+### 11.2 Conventional disturbance observer {#11-dot-2-conventional-disturbance-observer}
+
+
+### 11.3 A general form of disturbance observer {#11-dot-3-a-general-form-of-disturbance-observer}
+
+
+### 11.4 Application results {#11-dot-4-application-results}
+
+
+### 11.5 Conclusion {#11-dot-5-conclusion}
+
+
+## 12. Two-Dimensional H2 Control for Error Minimization {#12-dot-two-dimensional-h2-control-for-error-minimization}
+
+
+### 12.1 Introduction {#12-dot-1-introduction}
+
+
+### 12.2 2-D stabilization control {#12-dot-2-2-d-stabilization-control}
+
+
+### 12.3 2-D H2 control {#12-dot-3-2-d-h2-control}
+
+
+### 12.4 SSTW process and modeling {#12-dot-4-sstw-process-and-modeling}
+
+
+#### 12.4.1 SSTW servo loop {#12-dot-4-dot-1-sstw-servo-loop}
+
+
+#### 12.4.2 Two-dimensional model {#12-dot-4-dot-2-two-dimensional-model}
+
+
+### 12.5 Feedforward compensation method {#12-dot-5-feedforward-compensation-method}
+
+
+### 12.6 2-D control formulation for SSTW {#12-dot-6-2-d-control-formulation-for-sstw}
+
+
+### 12.7 2-D stabilization control for error propagation containment {#12-dot-7-2-d-stabilization-control-for-error-propagation-containment}
+
+
+#### 12.7.1 Simulation results {#12-dot-7-dot-1-simulation-results}
+
+
+### 12.8 2-D H2 control for error minimization {#12-dot-8-2-d-h2-control-for-error-minimization}
+
+
+#### 12.8.1 Simulation results {#12-dot-8-dot-1-simulation-results}
+
+
+#### 12.8.2 Experimental results {#12-dot-8-dot-2-experimental-results}
+
+
+### 12.9 Conclusion {#12-dot-9-conclusion}
+
+
+## 13. Nonlinearity Compensation and Nonlinear Control {#13-dot-nonlinearity-compensation-and-nonlinear-control}
+
+
+### 13.1 Introduction {#13-dot-1-introduction}
+
+
+### 13.2 Nonlinearity compensation {#13-dot-2-nonlinearity-compensation}
+
+
+### 13.3 Nonlinear control {#13-dot-3-nonlinear-control}
+
+
+#### 13.3.1 Design of a composite control law {#13-dot-3-dot-1-design-of-a-composite-control-law}
+
+
+#### 13.3.2 Experimental results in hard disk drives {#13-dot-3-dot-2-experimental-results-in-hard-disk-drives}
+
+
+### 13.4 Conclusion {#13-dot-4-conclusion}
+
+
+## 14. Quantization Effect on Vibration Rejection and Its Compensation {#14-dot-quantization-effect-on-vibration-rejection-and-its-compensation}
+
+
+### 14.1 Introduction {#14-dot-1-introduction}
+
+
+### 14.2 Description of control system with quantizer {#14-dot-2-description-of-control-system-with-quantizer}
+
+
+### 14.3 Quantization effect on error rejection {#14-dot-3-quantization-effect-on-error-rejection}
+
+
+#### 14.3.1 Quantizer frequency response measurement {#14-dot-3-dot-1-quantizer-frequency-response-measurement}
+
+
+#### 14.3.2 Quantization effect on error rejection {#14-dot-3-dot-2-quantization-effect-on-error-rejection}
+
+
+### 14.4 Compensation of quantization effect on error rejection {#14-dot-4-compensation-of-quantization-effect-on-error-rejection}
+
+
+### 14.5 Conclusion {#14-dot-5-conclusion}
+
+
+## 15. Adaptive Filtering Algorithms for Active Vibration Control {#15-dot-adaptive-filtering-algorithms-for-active-vibration-control}
+
+
+### 15.1 Introduction {#15-dot-1-introduction}
+
+
+### 15.2 Adaptive feedforward algorithm {#15-dot-2-adaptive-feedforward-algorithm}
+
+
+### 15.3 Adaptive feedback algorithm {#15-dot-3-adaptive-feedback-algorithm}
+
+
+### 15.4 Comparison between feedforward and feedback controls {#15-dot-4-comparison-between-feedforward-and-feedback-controls}
+
+
+### 15.5 Application in Stewart platform {#15-dot-5-application-in-stewart-platform}
+
+
+#### 15.5.1 Multi-channel adaptive feedback AVC system {#15-dot-5-dot-1-multi-channel-adaptive-feedback-avc-system}
+
+
+#### 15.5.2 Multi-channel adaptive feedback algorithm for hexapod platform {#15-dot-5-dot-2-multi-channel-adaptive-feedback-algorithm-for-hexapod-platform}
+
+
+#### 15.5.3 Simulation and implementation {#15-dot-5-dot-3-simulation-and-implementation}
+
+
+### 15.6 Conclusion {#15-dot-6-conclusion}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Lihua Xie. 2010. <i>Modeling and Control of Vibration in Mechanical Systems</i>. Automation and Control Engineering. CRC Press. doi:<a href="https://doi.org/10.1201/9781439817995">10.1201/9781439817995</a>.</div>
+</div>

content/book/du19_multi_actuat_system_contr.md

@@ -0,0 +1,678 @@
++++
+title = "Multi-stage actuation systems and control"
+author = ["Dehaeze Thomas"]
+description = "Proposes a way to combine multiple actuators (short stroke and long stroke) for control."
+keywords = ["Control", "Mechatronics"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Du and Pang 2019</a>)
+
+Author(s)
+: Du, C., &amp; Pang, C. K.
+
+Year
+: 2019
+
+<div style="display: none;">
+\(
+\newcommand{\SI}[2]{#1\,#2}
+% Simulate SIunitx
+\newcommand{\ang}[1]{#1^{\circ}}
+\newcommand{\degree}{^{\circ}}
+\newcommand{\radian}{\text{rad}}
+\newcommand{\percent}{\%}
+\newcommand{\decibel}{\text{dB}}
+\newcommand{\per}{/}
+\)
+</div>
+
+
+## Mechanical Actuation Systems {#mechanical-actuation-systems}
+
+
+### Introduction {#introduction}
+
+When high bandwidth, high position accuracy and long stroke are required simultaneously: dual-stage systems composed of a coarse (or primary) actuator and a fine actuator working together are used.
+
+Popular choices for coarse actuator are:
+
+-   DC motor
+-   [Voice Coil Motors]({{< relref "voice_coil_actuators.md" >}}) (VCM)
+-   Permanent magnet stepper motor
+-   Permanent magnet linear synchronous motor
+
+As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}) are used.
+
+In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.
+
+
+### Actuators {#actuators}
+
+
+#### Primary Actuator {#primary-actuator}
+
+Without loss of generality, the VCM actuator is used as the primary actuator.
+When current passes through the coil, a force is produced which accelerates the actuator radially.
+The produced force is a function of the current \\(i\_c\\):
+\\[ f\_m = k\_t i\_c \\]
+where \\(k\_t\\) is a linearized nominal value called the torque constant.
+
+The resonance of the actuator is mainly due to the flexibility of the pivot bearing, arm, suspension.
+
+Then the bandwidth of the control loop is low and the resonances are not a limiting factor of the control design, the actuator model can be considered as follows:
+\\[ P\_v(s) = \frac{k\_{vcm}}{s^2} \\]
+
+When the bandwidth is high, the actuator resonances have to be considered in the control design since the flexible resonance modes will reduce the system stability and affect the control performance. Then the actuator model becomes
+\\[ P\_v(s) = \frac{k\_{vcm}}{s^2} P\_r(s) \\]
+which includes the resonance model
+\\[ P\_r(s) = \Pi\_{i=1}^{N} P\_{ri}(s) \\]
+and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following forms
+
+\begin{align\*}
+ P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
+ P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
+ P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
+\end{align\*}
+
+
+#### Secondary Actuators {#secondary-actuators}
+
+We here consider two types of secondary actuators: the PZT milliactuator ([Figure 1](#figure--fig:pzt-actuator)) and the microactuator.
+
+<a id="figure--fig:pzt-actuator"></a>
+
+{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="<span class='figure-number'>Figure 1: </span>A PZT-actuator suspension" >}}
+
+There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
+There characteristics are shown on [Table 1](#table--tab:microactuator).
+
+<a id="table--tab:microactuator"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:microactuator">Table 1</a>:</span>
+  Performance comparison of microactuators
+</div>
+
+|              | Elect.                                        | PZT                                           | Thermal                    |
+|--------------|-----------------------------------------------|-----------------------------------------------|----------------------------|
+| TF           | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
+| \\(\tau\\)   | \\(<\SI{0.1}{ms}\\)                           | \\(<\SI{0.05}{ms}\\)                          | \\(>\SI{0.1}{ms}\\)        |
+| \\(\omega\\) | \\(1-\SI{2}{kHz}\\)                           | \\(20-\SI{25}{kHz}\\)                         | \\(>\SI{15}{kHz}\\)        |
+
+
+### Single-Stage Actuation Systems {#single-stage-actuation-systems}
+
+A typical closed-loop control system is shown on [Figure 2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
+
+<a id="figure--fig:single-stage-control"></a>
+
+{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="<span class='figure-number'>Figure 2: </span>Block diagram of a single-stage actuation system" >}}
+
+
+### Dual-Stage Actuation Systems {#dual-stage-actuation-systems}
+
+Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
+The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
+
+<a id="figure--fig:dual-stage-control"></a>
+
+{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="<span class='figure-number'>Figure 3: </span>Block diagram of dual-stage actuation system" >}}
+
+
+### Three-Stage Actuation Systems {#three-stage-actuation-systems}
+
+Due to the limited allowed stroke of the microactuator, the control bandwidth has to be restricted and that limits the dual-stage disturbance rejection capability.
+
+A three-stage actuation system is therefore introduced to further increase the bandwidth.
+
+Typically, a VCM actuator is used as the primary actuator, PZT milliactuator as the second stage actuator and a third actuator more collocated is used.
+
+
+## High-Precision Positioning Control of Dual-Stage Actuation Systems {#high-precision-positioning-control-of-dual-stage-actuation-systems}
+
+
+### Introduction {#introduction}
+
+The sensitivity function of the closed-loop system has provided a straightforward view of its disturbance rejection capability.
+It is demanded that the sensitivity function magnitude in the low-frequency range be sufficiently low, while its hump in high-frequency range stays low enough.
+In view of this, the controller design for dual-stage actuation systems adopts a weighting function to shape the sensitivity function.
+
+
+### Control Schemes {#control-schemes}
+
+A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in [Figure 4](#figure--fig:decoupled-control).
+
+-   \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
+-   \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
+-   \\(d\_1\\) and \\(d\_2\\) denote internal disturbances
+-   \\(n\\) is the measurement noise
+-   \\(d\_u\\) stands for external vibration
+
+<a id="figure--fig:decoupled-control"></a>
+
+{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="<span class='figure-number'>Figure 4: </span>Decoupled control structure for the dual-stage actuation system" >}}
+
+The open-loop transfer function from \\(pes\\) to \\(y\\) is
+\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) + P\_v(z) C\_v(z) \hat{P}\_p(z) C\_p(z) \\]
+And the overall sensitivity function of the closed loop system from \\(r\\) to \\(pes\\) is
+\\[ S(z) = \frac{1}{1 + G(z)} \\]
+which is approximately
+\\[ S(z) = \frac{1}{[1 + P\_p(z) C\_p(z)] [1 + P\_v(z)C\_v(z)]} \\]
+since within a certain bandwidth
+\\[ \hat{P}\_p(z) \approx P\_p(z) \\]
+
+The sensitivity functions of the VCM loop and the secondary actuator loop are
+
+\begin{equation}
+  S\_v(z) = \frac{1}{1 + P\_v(z) C\_v(z)}, \quad S\_p(z) = \frac{1}{1 + P\_p(z) C\_p(z)}
+\end{equation}
+
+And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
+Thus, the dual-stage system control design can be decoupled into two independent controller designs.
+
+Another type of control scheme is the **parallel structure** as shown in [Figure 5](#figure--fig:parallel-control-structure).
+The open-loop transfer function from \\(pes\\) to \\(y\\) is
+\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
+
+The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
+\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
+
+<a id="figure--fig:parallel-control-structure"></a>
+
+{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="<span class='figure-number'>Figure 5: </span>Parallel control structure for the dual-stage actuator system" >}}
+
+Because of the limited displacement range of the secondary actuator, the control efforts for the two actuators should be distributed properly when designing respective controllers to meet the required performance, make the actuators not conflict with each other, as well as prevent the saturation of the secondary actuator.
+
+
+### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
+
+\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
+The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in [Figure 6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
+
+For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
+
+\begin{equation}
+  \\|T\_{zw}\\|\_\infty < 1
+\end{equation}
+
+is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
+
+<a id="figure--fig:h-inf-diagram"></a>
+
+{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class='figure-number'>Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
+
+Equation means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
+One form of \\(W(s)\\) is taken as
+
+\begin{equation}
+  W(s) = \frac{\frac{1}{M}s^2 + 2\xi\omega\frac{1}{\sqrt{M}}s + \omega^2}{s^2 + 2\omega\sqrt{\epsilon}s + \omega^2\epsilon}
+\end{equation}
+
+where \\(\omega\\) is the desired bandwidth, \\(\epsilon\\) is used to determine the desired low frequency level of sensitivity magnitude and \\(\xi\\) is the damping ratio.
+
+The controller can then be synthesis using the linear matrix inequality (LMI) approach.
+
+The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
+But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
+As seen in [Figure 7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
+
+<a id="figure--fig:dual-stage-loop-gain"></a>
+
+{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="<span class='figure-number'>Figure 7: </span>Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
+
+The sensitivity functions are shown in [Figure 8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
+This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
+
+<a id="figure--fig:dual-stage-sensitivity"></a>
+
+{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="<span class='figure-number'>Figure 8: </span>Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
+
+A basic requirement of the dual-stage actuation control system is to make the individual primary and secondary loops stable.
+It also required that the primary actuator path has a higher gain than the secondary actuator path at low frequency range and the secondary actuator path has a higher gain than the primary actuator path in high-frequency range.
+These can be achieve by choosing appropriate weighting function for the controllers design.
+
+
+### Conclusion {#conclusion}
+
+The controller design has been discussed for high-precision positioning control of the dual-stage actuation systems.
+The \\(\mathcal{H}\_\infty\\) loop shaping method has been applied and the design method has been presented.
+With the weighting functions, the desired sensitivity function can achieved.
+Such a design method can produce robust controllers with more disturbance rejection in the low frequency range and less disturbance amplification in the high-frequency range.
+
+
+## Modeling and Control of a Three-Stage Actuation System {#modeling-and-control-of-a-three-stage-actuation-system}
+
+
+### Introduction {#introduction}
+
+In view of the additional bandwidth requirement which is limited by stroke constraint and saturation of secondary actuators, three-stage actuation systems are thereby proposed to meet the demand of a higher bandwidth.
+In this section, a specific three-stage actuation system is presented and a controller strategy is proposed, which is based on a decoupled master-slave dual-stage control structure combined with a third stage actuation in parallel format.
+
+
+### Actuator and Vibration Modeling {#actuator-and-vibration-modeling}
+
+A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a PZT milliactuator as the second-stage actuator denoted by \\(P\_p(s)\\), and a thermal microactuator denoted by \\(P\_m(s)\\).
+
+
+### Control Strategy and Controller Design {#control-strategy-and-controller-design}
+
+[Figure 9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
+
+The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
+The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
+The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
+
+<a id="figure--fig:three-stage-control"></a>
+
+{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="<span class='figure-number'>Figure 9: </span>Control system for the three-stage actuation system" >}}
+
+The open-loop transfer function of the three-stage actuation system is derived as
+
+\begin{equation}
+  G(z) = G\_v(z) + G\_p(z) + G\_v(z) G\_p(z) + G\_m(z)
+\end{equation}
+
+with
+
+  \begin{align\*}
+  G\_v(z) &= P\_v(z) C\_v(z) \\\\
+  G\_p(z) &= P\_p(z) C\_p(z) \\\\
+  G\_m(z) &= P\_m(z) C\_m(z)
+\end{align\*}
+
+The overall sensitivity function is given by
+
+\begin{equation}
+  S(z) = \frac{1}{1 + G(z)}
+\end{equation}
+
+The VCM actuator \\(P\_v(s)\\) works in a low bandwidth below \\(\SI{1}{kHz}\\).
+The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandwidth up to \\(\SI{3}{kHz}\\).
+The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
+
+The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
+The open-loop frequency responses of the three stages are shown on [Figure 10](#figure--fig:open-loop-three-stage).
+
+<a id="figure--fig:open-loop-three-stage"></a>
+
+{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="<span class='figure-number'>Figure 10: </span>Frequency response of the open-loop transfer function" >}}
+
+The obtained sensitivity function is shown on [Figure 11](#figure--fig:sensitivity-three-stage).
+
+<a id="figure--fig:sensitivity-three-stage"></a>
+
+{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="<span class='figure-number'>Figure 11: </span>Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
+
+
+### Performance Evaluation {#performance-evaluation}
+
+External vibration from the system working environment is much higher than the internal disturbance, especially for ultra-high precision positioning systems.
+In the presence of external vibration, the actuators control effort is dominantly determined by the external vibration.
+But because the actuator input is constrained, the external vibration level has to be limited.
+Otherwise, saturation will occur in the control loop and the control system performance will be degraded.
+
+Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
+Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
+
+For the three-stage control architecture as shown on [Figure 9](#figure--fig:three-stage-control), the position error is
+\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
+The control signals and positions of the actuators are given by
+
+\begin{align\*}
+  u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\
+  u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\
+  u\_v &= C\_v ( 1 + \hat{P}\_pC\_p ) e,\ y\_v = P\_v ( u\_v + d\_1 )
+\end{align\*}
+
+The controller design for the microactuators with input constraints must take into account both external vibration requirements and actuators' stroke, based on which an appropriate bandwidth should be decided when designing the control system.
+Higher bandwidth/higher level of disturbance generally means high stroke needed.
+
+
+### Different Configurations of the Control System {#different-configurations-of-the-control-system}
+
+A decoupled control structure can be used for the three-stage actuation system (see [Figure 12](#figure--fig:three-stage-decoupled)).
+
+The overall sensitivity function is
+\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
+with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation and
+\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
+
+Denote the dual-stage open-loop transfer function as \\(G\_d\\)
+\\[ G\_d(z) = G\_v(z) + G\_p(z) + G\_v(z) G\_p(z) \\]
+
+The open-loop transfer function of the overall system is
+\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
+
+<a id="figure--fig:three-stage-decoupled"></a>
+
+{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="<span class='figure-number'>Figure 12: </span>Decoupled control structure for the three-stage actuation system" >}}
+
+The control signals and the positions of the three actuators are
+
+\begin{align\*}
+  u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\
+  u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\
+  u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
+\end{align\*}
+
+The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see [Figure 13](#figure--fig:three-stage-decoupled-loop-gain)).
+
+<a id="figure--fig:three-stage-decoupled-loop-gain"></a>
+
+{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="<span class='figure-number'>Figure 13: </span>Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
+
+
+### Conclusion {#conclusion}
+
+The relationship among the external vibration, the microactuator stroke, and the achievable control bandwidth has been discussed for being considered in the controller design.
+The discussion suggests that in addition to the traditional wisdom of just increasing the resonant frequency, adding more stroke to the microactuator will give more freedom to the loop shaping for the control system design.
+
+
+## Dual-Stage System Control Considering Secondary Actuator Stroke Limitation {#dual-stage-system-control-considering-secondary-actuator-stroke-limitation}
+
+
+### Introduction {#introduction}
+
+
+### More Freedom Loop Shaping for Microactuator Controller Design {#more-freedom-loop-shaping-for-microactuator-controller-design}
+
+
+### Dual-Stage System Control Design for 5 kHz Bandwidth {#dual-stage-system-control-design-for-5-khz-bandwidth}
+
+
+### Evaluation with the Consideration of External Vibration and Microactuator Stroke {#evaluation-with-the-consideration-of-external-vibration-and-microactuator-stroke}
+
+
+### Conclusion {#conclusion}
+
+
+## Saturation Control for Microactuators in Dual-Stage Actuation Systems {#saturation-control-for-microactuators-in-dual-stage-actuation-systems}
+
+
+### Introduction {#introduction}
+
+
+### Modeling and Feedback Control {#modeling-and-feedback-control}
+
+
+### Anti-Windup Compensation Design {#anti-windup-compensation-design}
+
+
+### Simulation and Experimental Results {#simulation-and-experimental-results}
+
+
+### Conclusion {#conclusion}
+
+
+## Time Delay and Sampling Rate Effect on Control Performance of Dual-Stage Actuation Systems {#time-delay-and-sampling-rate-effect-on-control-performance-of-dual-stage-actuation-systems}
+
+
+### Introduction {#introduction}
+
+
+### Modeling of Time Delay {#modeling-of-time-delay}
+
+
+### Dual-Stage Actuation System Modeling with Time Delay for Controller Design {#dual-stage-actuation-system-modeling-with-time-delay-for-controller-design}
+
+
+### Controller Design with Time Delay for the Dual-Stage Actuation Systems {#controller-design-with-time-delay-for-the-dual-stage-actuation-systems}
+
+
+### Time Delay Effect on Dual-Stage System Control Performance {#time-delay-effect-on-dual-stage-system-control-performance}
+
+
+### Sampling Rate Effect on Dual-Stage System Control Performance {#sampling-rate-effect-on-dual-stage-system-control-performance}
+
+
+### Conclusion {#conclusion}
+
+
+## PZT Hysteresis Modeling and Compensation {#pzt-hysteresis-modeling-and-compensation}
+
+
+### Introduction {#introduction}
+
+
+### Modeling of Hysteresis {#modeling-of-hysteresis}
+
+
+#### PI Model {#pi-model}
+
+
+#### GPI Model {#gpi-model}
+
+
+#### Inverse GPI Model {#inverse-gpi-model}
+
+
+### Application of GPI Model to a PZT-Actuated Structure {#application-of-gpi-model-to-a-pzt-actuated-structure}
+
+
+#### Modeling of the Hysteresis in the PZT-Actuated Structure {#modeling-of-the-hysteresis-in-the-pzt-actuated-structure}
+
+
+#### Hysteresis Compensator Design {#hysteresis-compensator-design}
+
+
+#### Experimental Verification {#experimental-verification}
+
+
+### Conclusion {#conclusion}
+
+
+## Seeking Control of Dual-Stage Actuation Systems with Trajectory Optimization {#seeking-control-of-dual-stage-actuation-systems-with-trajectory-optimization}
+
+
+### Introduction {#introduction}
+
+
+### Current Profile of VCM Primary Actuator {#current-profile-of-vcm-primary-actuator}
+
+
+#### PTOS Method {#ptos-method}
+
+
+#### A General Form of VCM Current Profiles {#a-general-form-of-vcm-current-profiles}
+
+
+### Control System Structure for the Dual-Stage Actuation System {#control-system-structure-for-the-dual-stage-actuation-system}
+
+
+### Design of VCM Current Profile a[sub(v)] and Dual-Stage Reference Trajectory r[sub(d)] {#design-of-vcm-current-profile-a-sub--v--and-dual-stage-reference-trajectory-r-sub--d}
+
+
+### Seeking within PZT Milliactuator Stroke {#seeking-within-pzt-milliactuator-stroke}
+
+
+### Seeking over PZT Milliactuator Stroke {#seeking-over-pzt-milliactuator-stroke}
+
+
+### Conclusion {#conclusion}
+
+
+## High-Frequency Vibration Control Using PZT Active Damping {#high-frequency-vibration-control-using-pzt-active-damping}
+
+
+### Introduction {#introduction}
+
+
+### Singular Perturbation Method-Based Controller Design {#singular-perturbation-method-based-controller-design}
+
+
+#### Singular Perturbation Control Topology {#singular-perturbation-control-topology}
+
+
+#### Identification of Fast Dynamics Using PZT as a Sensor {#identification-of-fast-dynamics-using-pzt-as-a-sensor}
+
+
+#### Design of Controllers {#design-of-controllers}
+
+
+##### Fast Subsystem Estimator  G[sub(v)][sup(\*)] {#fast-subsystem-estimator-g-sub--v--sup}
+
+
+##### Fast Controller C[sub(v)] {#fast-controller-c-sub--v}
+
+
+##### Slow Controller C[sub(v)] {#slow-controller-c-sub--v}
+
+
+#### Simulation and Experimental Results {#simulation-and-experimental-results}
+
+
+##### Frequency Responses {#frequency-responses}
+
+
+##### Time Responses {#time-responses}
+
+
+### H[sub(2)] Controller Design {#h-sub--2--controller-design}
+
+
+### Design of C[sub(d)](z) with H[sub(2)] Method and Notch Filters {#design-of-c-sub--d----z--with-h-sub--2--method-and-notch-filters}
+
+
+### Design of Mixed H[sub(2)]/H[sub(∞)] Controller C[sub(d)](z) {#design-of-mixed-h-sub--2--h-sub-----controller-c-sub--d----z}
+
+
+### Application Results {#application-results}
+
+
+#### System Modeling {#system-modeling}
+
+
+#### H[sub(2)] Active Damping Control {#h-sub--2--active-damping-control}
+
+
+#### Mixed H[sub(2)]/H[sub(∞)] Active Damping Control {#mixed-h-sub--2--h-sub-----active-damping-control}
+
+
+#### Experimental Results {#experimental-results}
+
+
+### Conclusion {#conclusion}
+
+
+## Self-Sensing Actuation of Dual-Stage Systems {#self-sensing-actuation-of-dual-stage-systems}
+
+
+### Introduction {#introduction}
+
+
+### Estimation of PZT Secondary Actuator’s Displacement y[sub(p)][sup(\*)] {#estimation-of-pzt-secondary-actuator-s-displacement-y-sub--p--sup}
+
+
+#### Self-Sensing Actuation and Bridge Circuit {#self-sensing-actuation-and-bridge-circuit}
+
+
+#### PZT Displacement Estimation Circuit H[sub(B)] {#pzt-displacement-estimation-circuit-h-sub--b}
+
+
+### Design of Controllers {#design-of-controllers}
+
+
+#### VCM Controller and Controller C[sub(D)] {#vcm-controller-and-controller-c-sub--d}
+
+
+#### PZT Controller {#pzt-controller}
+
+
+### Performance Evaluation {#performance-evaluation}
+
+
+#### Effectiveness of C[sub(D)] {#effectiveness-of-c-sub--d}
+
+
+#### Position Errors {#position-errors}
+
+
+### Conclusion {#conclusion}
+
+
+## Modeling and Control of a MEMS Micro X–Y Stage Media Platform {#modeling-and-control-of-a-mems-micro-x-y-stage-media-platform}
+
+
+### Introduction {#introduction}
+
+
+### MEMS Micro X–Y Stage {#mems-micro-x-y-stage}
+
+
+#### Design and Simulation of Micro X–Y Stage {#design-and-simulation-of-micro-x-y-stage}
+
+
+##### Static {#static}
+
+
+##### Dynamic {#dynamic}
+
+
+#### Modeling of Micro X–Y Stage {#modeling-of-micro-x-y-stage}
+
+
+#### Fabrication of the MEMS Micro X–Y Stage {#fabrication-of-the-mems-micro-x-y-stage}
+
+
+### Capacitive Self-Sensing Actuation {#capacitive-self-sensing-actuation}
+
+
+#### Design of CSSA Bridge Circuit {#design-of-cssa-bridge-circuit}
+
+
+#### Experimental Verification {#experimental-verification}
+
+
+### Robust Decoupling Controller Design {#robust-decoupling-controller-design}
+
+
+#### Choice of Pre-Shaping Filters {#choice-of-pre-shaping-filters}
+
+
+#### Controller Synthesis {#controller-synthesis}
+
+
+#### Frequency Responses {#frequency-responses}
+
+
+#### Time Responses {#time-responses}
+
+
+#### Robustness Analysis {#robustness-analysis}
+
+
+### Conclusion {#conclusion}
+
+
+## Conclusions {#conclusions}
+
+Many secondary actuators have been developed in addition to primary actuators in the field of mechanical actuation systems.
+The aim is to provide high performance such as high precision and fast response.
+Several types of secondary actuators have been introduced such as PZT milliactuator, electrostatic microactuator, PZT microactuator, and thermal microactuator.
+Comparison of these secondary actuators has been made, and these secondary actuators have made dual and multi-stage actuation mechanisms possible.
+
+Three-stage actuation systems have been proposed for the demand of wider bandwidth, to overcome the limitation by stroke constraint and saturation of secondary actuators.
+After the characteristics of the three-stage systems have been developed and the models have been identified, the control strategy and algorithm have been developed to deal with vibrations and meet different requirements.
+Particularly, for the three-stage actuation systems, the presented control strategies make it easy to further push the bandwidth and meet the performance requirement.
+The control of the thermal microactuator based dual-stage system has been discussed in detail, including linearization and controller design method.
+
+The developed advanced algorithms applied in the multi-stage systems include \\(\mathcal{H}\_\infty\\) loop shaping, anti-windup compensation, \\(\mathcal{H}\_2\\) control method,
+and mixed \\(\mathcal{H}\_2/\mathcal{H}\_\infty\\) control method.
+Typical problems of the milli and micro-actuators as the secondary actuators have been considered and appropriate solutions have been presented such as saturation compensation, hysteresis modeling and compensation, stroke limitation, and PZT self-sensing scheme.
+Time delay and sampling rate effect on the control performance have been analyzed to help select appropriate sampling rate and design suitable controllers.
+
+Specific usage of PZT elements has been produced for system performance improvement.
+Using PZT elements as a sensor to deal with high-frequency vibration beyond the bandwidth has been proposed and systematic controller design methods have been developed.
+As a more advanced concept, PZT elements being used as actuator and sensor simultaneously has also been addressed in this book with detailed scheme and controller design methodology for effective utilization.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Chee Khiang Pang. 2019. <i>Multi-Stage Actuation Systems and Control</i>. Boca Raton, FL: CRC Press.</div>
+</div>

content/book/fleming14_desig_model_contr_nanop_system.md

@@ -0,0 +1,858 @@
++++
+title = "Design, modeling and control of nanopositioning systems"
+author = ["Dehaeze Thomas"]
+description = "Talks about various topics related to nano-positioning systems."
+keywords = ["Control", "Metrology", "Flexible Joints"]
+draft = false
++++
+
+Tags
+: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>)
+
+Author(s)
+: Fleming, A. J., &amp; Leang, K. K.
+
+Year
+: 2014
+
+
+## Introduction to Nanotechnology {#introduction-to-nanotechnology}
+
+
+## Introduction to Nanopositioning {#introduction-to-nanopositioning}
+
+
+## Scanning Probe Microscopy {#scanning-probe-microscopy}
+
+
+## Challenges with Nanopositioning Systems {#challenges-with-nanopositioning-systems}
+
+
+### Hysteresis {#hysteresis}
+
+
+### Creep {#creep}
+
+
+### Thermal Drift {#thermal-drift}
+
+
+### Mechanical Resonance {#mechanical-resonance}
+
+
+## Control of Nanopositioning Systems {#control-of-nanopositioning-systems}
+
+
+### Feedback Control {#feedback-control}
+
+
+### Feedforward Control {#feedforward-control}
+
+
+## Book Summary {#book-summary}
+
+
+### Assumed Knowledge {#assumed-knowledge}
+
+
+### Content Summary {#content-summary}
+
+
+## References {#references}
+
+
+## The Piezoelectric Effect {#the-piezoelectric-effect}
+
+
+## Piezoelectric Compositions {#piezoelectric-compositions}
+
+
+## Manufacturing Piezoelectric Ceramics {#manufacturing-piezoelectric-ceramics}
+
+
+## Piezoelectric Transducers {#piezoelectric-transducers}
+
+
+## Application Considerations {#application-considerations}
+
+
+## Response of Piezoelectric Actuators {#response-of-piezoelectric-actuators}
+
+
+## Modeling Creep and Vibration in Piezoelectric Actuators {#modeling-creep-and-vibration-in-piezoelectric-actuators}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Piezoelectric Tube Nanopositioners {#piezoelectric-tube-nanopositioners}
+
+
+### 63mm Piezoelectric Tube {#63mm-piezoelectric-tube}
+
+
+### 40mm Piezoelectric Tube Nanopositioner {#40mm-piezoelectric-tube-nanopositioner}
+
+
+## Piezoelectric Stack Nanopositioners {#piezoelectric-stack-nanopositioners}
+
+
+### Phyisk Instrumente P-734 Nanopositioner {#phyisk-instrumente-p-734-nanopositioner}
+
+
+### Phyisk Instrumente P-733.3DD Nanopositioner {#phyisk-instrumente-p-733-dot-3dd-nanopositioner}
+
+
+### Vertical Nanopositioners {#vertical-nanopositioners}
+
+
+### Rotational Nanopositioners {#rotational-nanopositioners}
+
+
+### Low Temperature and UHV Nanopositioners {#low-temperature-and-uhv-nanopositioners}
+
+
+### Tilting Nanopositioners {#tilting-nanopositioners}
+
+
+### Optical Objective Nanopositioners {#optical-objective-nanopositioners}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Operating Environment {#operating-environment}
+
+
+## Methods for Actuation {#methods-for-actuation}
+
+
+## Flexure Hinges {#flexure-hinges}
+
+
+### Introduction {#introduction}
+
+
+### Types of Flexures {#types-of-flexures}
+
+
+### Flexure Hinge Compliance Equations {#flexure-hinge-compliance-equations}
+
+
+### Stiff Out-of-Plane Flexure Designs {#stiff-out-of-plane-flexure-designs}
+
+
+### Failure Considerations {#failure-considerations}
+
+
+### Finite Element Approach for Flexure Design {#finite-element-approach-for-flexure-design}
+
+
+## Material Considerations {#material-considerations}
+
+
+### Materials for Flexure and Platform Design {#materials-for-flexure-and-platform-design}
+
+
+### Thermal Stability of Materials {#thermal-stability-of-materials}
+
+
+## Manufacturing Techniques {#manufacturing-techniques}
+
+
+## Design Example: A High-Speed Serial-Kinematic Nanopositioner {#design-example-a-high-speed-serial-kinematic-nanopositioner}
+
+
+### State-of-the-Art Designs {#state-of-the-art-designs}
+
+
+### Tradeoffs and Limitations in Speed {#tradeoffs-and-limitations-in-speed}
+
+
+### Serial- Versus Parallel-Kinematic Configurations {#serial-versus-parallel-kinematic-configurations}
+
+
+### Piezoactuator Considerations {#piezoactuator-considerations}
+
+
+### Preloading Piezo-Stack Actuators {#preloading-piezo-stack-actuators}
+
+
+### Flexure Design for Lateral Positioning {#flexure-design-for-lateral-positioning}
+
+
+### Design of Vertical Stage {#design-of-vertical-stage}
+
+
+### Fabrication and Assembly {#fabrication-and-assembly}
+
+
+### Drive Electronics {#drive-electronics}
+
+\*\*\*\*0 Experimental Results
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Sensor Characteristics {#sensor-characteristics}
+
+
+### Calibration and Nonlinearity {#calibration-and-nonlinearity}
+
+
+### Drift and Stability {#drift-and-stability}
+
+
+### Bandwidth {#bandwidth}
+
+
+### Noise {#noise}
+
+
+### Resolution {#resolution}
+
+
+### Combining Errors {#combining-errors}
+
+
+### Metrological Traceability {#metrological-traceability}
+
+
+## Nanometer Position Sensors {#nanometer-position-sensors}
+
+
+### Resistive Strain Sensors {#resistive-strain-sensors}
+
+
+### Piezoresistive Strain Sensors {#piezoresistive-strain-sensors}
+
+
+### Piezoelectric Strain Sensors {#piezoelectric-strain-sensors}
+
+
+### Capacitive Sensors {#capacitive-sensors}
+
+
+### MEMs Capacitive and Thermal Sensors {#mems-capacitive-and-thermal-sensors}
+
+
+### Eddy-Current Sensors {#eddy-current-sensors}
+
+
+### Linear Variable Displacement Transformers {#linear-variable-displacement-transformers}
+
+
+### Laser Interferometers {#laser-interferometers}
+
+
+### Linear Encoders {#linear-encoders}
+
+
+## Comparison and Summary {#comparison-and-summary}
+
+
+## Outlook and Future Requirements {#outlook-and-future-requirements}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Shunt Circuit Modeling {#shunt-circuit-modeling}
+
+
+### Open-Loop {#open-loop}
+
+
+### Shunt Damping {#shunt-damping}
+
+
+## Implementation {#implementation}
+
+
+## Experimental Results {#experimental-results}
+
+
+### Tube Dynamics {#tube-dynamics}
+
+
+### Amplifier Performance {#amplifier-performance}
+
+
+### Shunt Damping Performance {#shunt-damping-performance}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Experimental Setup {#experimental-setup}
+
+
+## PI Control {#pi-control}
+
+
+## PI Control with Notch Filters {#pi-control-with-notch-filters}
+
+
+## PI Control with IRC Damping {#pi-control-with-irc-damping}
+
+
+## Performance Comparison {#performance-comparison}
+
+
+## Noise and Resolution {#noise-and-resolution}
+
+
+## Analog Implementation {#analog-implementation}
+
+
+## Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Modeling {#modeling}
+
+
+### Actuator Dynamics {#actuator-dynamics}
+
+
+### Sensor Dynamics {#sensor-dynamics}
+
+
+### Sensor Noise {#sensor-noise}
+
+
+### Mechanical Dynamics {#mechanical-dynamics}
+
+
+### System Properties {#system-properties}
+
+
+### Example System {#example-system}
+
+
+## Damping Control {#damping-control}
+
+
+## Tracking Control {#tracking-control}
+
+
+### Relationship Between Force and Displacement {#relationship-between-force-and-displacement}
+
+
+### Integral Displacement Feedback {#integral-displacement-feedback}
+
+
+### Direct Tracking Control {#direct-tracking-control}
+
+
+### Dual Sensor Feedback {#dual-sensor-feedback}
+
+
+### Low Frequency Bypass {#low-frequency-bypass}
+
+
+### Feedforward Inputs {#feedforward-inputs}
+
+
+### Higher-Order Modes {#higher-order-modes}
+
+
+## Experimental Results {#experimental-results}
+
+
+### Experimental Nanopositioner {#experimental-nanopositioner}
+
+
+### Actuators and Force Sensors {#actuators-and-force-sensors}
+
+
+### Control Design {#control-design}
+
+
+### Noise Performance {#noise-performance}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Why Feedforward? {#why-feedforward}
+
+
+## Modeling for Feedforward Control {#modeling-for-feedforward-control}
+
+
+## Feedforward Control of Dynamics and Hysteresis {#feedforward-control-of-dynamics-and-hysteresis}
+
+
+### Simple DC-Gain Feedforward Control {#simple-dc-gain-feedforward-control}
+
+
+### An Inversion-Based Feedforward Approach  for Linear Dynamics {#an-inversion-based-feedforward-approach-for-linear-dynamics}
+
+
+### Frequency-Weighted Inversion: The Optimal Inverse {#frequency-weighted-inversion-the-optimal-inverse}
+
+
+### Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## Feedforward and Feedback Control {#feedforward-and-feedback-control}
+
+
+### Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## Iterative Feedforward Control {#iterative-feedforward-control}
+
+
+### The ILC Problem {#the-ilc-problem}
+
+
+### Model-Based ILC {#model-based-ilc}
+
+
+### Nonlinear ILC {#nonlinear-ilc}
+
+
+### Conclusions {#conclusions}
+
+
+## References {#references}
+
+
+## 10.1 Introduction {#10-dot-1-introduction}
+
+
+### 10.1.1 Background {#10-dot-1-dot-1-background}
+
+
+### 10.1.2 The Optimal Periodic Input {#10-dot-1-dot-2-the-optimal-periodic-input}
+
+
+## 10.2 Signal Optimization {#10-dot-2-signal-optimization}
+
+
+## 10.3 Frequency Domain Cost Functions {#10-dot-3-frequency-domain-cost-functions}
+
+
+### 10.3.1 Background: Discrete Fourier Series {#10-dot-3-dot-1-background-discrete-fourier-series}
+
+
+### 10.3.2 Minimizing Signal Power {#10-dot-3-dot-2-minimizing-signal-power}
+
+
+### 10.3.3 Minimizing Frequency Weighted Power {#10-dot-3-dot-3-minimizing-frequency-weighted-power}
+
+
+### 10.3.4 Minimizing Velocity and Acceleration {#10-dot-3-dot-4-minimizing-velocity-and-acceleration}
+
+
+### 10.3.5 Single-Sided Frequency Domain Calculations {#10-dot-3-dot-5-single-sided-frequency-domain-calculations}
+
+
+## 10.4 Time Domain Cost Function {#10-dot-4-time-domain-cost-function}
+
+
+### 10.4.1 Minimum Velocity {#10-dot-4-dot-1-minimum-velocity}
+
+
+### 10.4.2 Minimum Acceleration {#10-dot-4-dot-2-minimum-acceleration}
+
+
+### 10.4.3 Frequency Weighted Objectives {#10-dot-4-dot-3-frequency-weighted-objectives}
+
+
+## 10.5 Application to Scan Generation {#10-dot-5-application-to-scan-generation}
+
+
+### 10.5.1 Choosing β and K {#10-dot-5-dot-1-choosing-β-and-k}
+
+
+### 10.5.2 Improving Feedback and Feedforward Controllers {#10-dot-5-dot-2-improving-feedback-and-feedforward-controllers}
+
+
+## 10.6 Comparison to Other Techniques {#10-dot-6-comparison-to-other-techniques}
+
+
+## 10.7 Experimental Application {#10-dot-7-experimental-application}
+
+
+## 10.8 Chapter Summary {#10-dot-8-chapter-summary}
+
+
+## References {#references}
+
+
+## 11.1 Introduction {#11-dot-1-introduction}
+
+
+## 11.2 Modeling Hysteresis {#11-dot-2-modeling-hysteresis}
+
+
+### 11.2.1 Simple Polynomial Model {#11-dot-2-dot-1-simple-polynomial-model}
+
+
+### 11.2.2 Maxwell Slip Model {#11-dot-2-dot-2-maxwell-slip-model}
+
+
+### 11.2.3 Duhem Model {#11-dot-2-dot-3-duhem-model}
+
+
+### 11.2.4 Preisach Model {#11-dot-2-dot-4-preisach-model}
+
+
+### 11.2.5 Classical Prandlt-Ishlinksii Model {#11-dot-2-dot-5-classical-prandlt-ishlinksii-model}
+
+
+## 11.3 Feedforward Hysteresis Compensation {#11-dot-3-feedforward-hysteresis-compensation}
+
+
+### 11.3.1 Feedforward Control Using the Presiach Model {#11-dot-3-dot-1-feedforward-control-using-the-presiach-model}
+
+
+### 11.3.2 Feedforward Control Using the Prandlt-Ishlinksii Model {#11-dot-3-dot-2-feedforward-control-using-the-prandlt-ishlinksii-model}
+
+
+## 11.4 Chapter Summary {#11-dot-4-chapter-summary}
+
+
+## References {#references}
+
+
+## 12.1 Introduction {#12-dot-1-introduction}
+
+
+## 12.2 Charge Drives {#12-dot-2-charge-drives}
+
+
+## 12.3 Application to Piezoelectric Stack Nanopositioners {#12-dot-3-application-to-piezoelectric-stack-nanopositioners}
+
+
+## 12.4 Application to Piezoelectric Tube Nanopositioners {#12-dot-4-application-to-piezoelectric-tube-nanopositioners}
+
+
+## 12.5 Alternative Electrode Configurations {#12-dot-5-alternative-electrode-configurations}
+
+
+### 12.5.1 Grounded Internal Electrode {#12-dot-5-dot-1-grounded-internal-electrode}
+
+
+### 12.5.2 Quartered Internal Electrode {#12-dot-5-dot-2-quartered-internal-electrode}
+
+
+## 12.6 Charge Versus Voltage {#12-dot-6-charge-versus-voltage}
+
+
+### 12.6.1 Advantages {#12-dot-6-dot-1-advantages}
+
+
+### 12.6.2 Disadvantages {#12-dot-6-dot-2-disadvantages}
+
+
+## 12.7 Impact on Closed-Loop Control {#12-dot-7-impact-on-closed-loop-control}
+
+
+## 12.8 Chapter Summary {#12-dot-8-chapter-summary}
+
+
+## References {#references}
+
+
+## 13.1 Introduction {#13-dot-1-introduction}
+
+
+## 13.2 Review of Random Processes {#13-dot-2-review-of-random-processes}
+
+
+### 13.2.1 Probability Distributions {#13-dot-2-dot-1-probability-distributions}
+
+
+### 13.2.2 Expected Value, Moments, Variance, and RMS {#13-dot-2-dot-2-expected-value-moments-variance-and-rms}
+
+
+### 13.2.3 Gaussian Random Variables {#13-dot-2-dot-3-gaussian-random-variables}
+
+
+### 13.2.4 Continuous Random Processes {#13-dot-2-dot-4-continuous-random-processes}
+
+
+### 13.2.5 Joint Density Functions and Stationarity {#13-dot-2-dot-5-joint-density-functions-and-stationarity}
+
+
+### 13.2.6 Correlation Functions {#13-dot-2-dot-6-correlation-functions}
+
+
+### 13.2.7 Gaussian Random Processes {#13-dot-2-dot-7-gaussian-random-processes}
+
+
+### 13.2.8 Power Spectral Density {#13-dot-2-dot-8-power-spectral-density}
+
+
+### 13.2.9 Filtered Random Processes {#13-dot-2-dot-9-filtered-random-processes}
+
+
+### 13.2.10 White Noise {#13-dot-2-dot-10-white-noise}
+
+
+### 13.2.11 Spectral Density in V/sqrtHz {#13-dot-2-dot-11-spectral-density-in-v-sqrthz}
+
+
+### 13.2.12 Single- and Double-Sided Spectra {#13-dot-2-dot-12-single-and-double-sided-spectra}
+
+
+## 13.3 Resolution and Noise {#13-dot-3-resolution-and-noise}
+
+
+## 13.4 Sources of Nanopositioning Noise {#13-dot-4-sources-of-nanopositioning-noise}
+
+
+### 13.4.1 Sensor Noise {#13-dot-4-dot-1-sensor-noise}
+
+
+### 13.4.2 External Noise {#13-dot-4-dot-2-external-noise}
+
+
+### 13.4.3 Amplifier Noise {#13-dot-4-dot-3-amplifier-noise}
+
+
+## 13.5 Closed-Loop Position Noise {#13-dot-5-closed-loop-position-noise}
+
+
+### 13.5.1 Noise Sensitivity Functions {#13-dot-5-dot-1-noise-sensitivity-functions}
+
+
+### 13.5.2 Closed-Loop Position Noise Spectral Density {#13-dot-5-dot-2-closed-loop-position-noise-spectral-density}
+
+
+### 13.5.3 Closed-Loop Noise Approximations with Integral Control {#13-dot-5-dot-3-closed-loop-noise-approximations-with-integral-control}
+
+
+### 13.5.4 Closed-Loop Position Noise Variance {#13-dot-5-dot-4-closed-loop-position-noise-variance}
+
+
+### 13.5.5 A Note on Units {#13-dot-5-dot-5-a-note-on-units}
+
+
+## 13.6 Simulation Examples {#13-dot-6-simulation-examples}
+
+
+### 13.6.1 Integral Controller Noise Simulation {#13-dot-6-dot-1-integral-controller-noise-simulation}
+
+
+### 13.6.2 Noise Simulation with Inverse Model Controller {#13-dot-6-dot-2-noise-simulation-with-inverse-model-controller}
+
+
+### 13.6.3 Feedback Versus Feedforward Control {#13-dot-6-dot-3-feedback-versus-feedforward-control}
+
+
+## 13.7 Practical Frequency Domain Noise Measurements {#13-dot-7-practical-frequency-domain-noise-measurements}
+
+
+### 13.7.1 Preamplification {#13-dot-7-dot-1-preamplification}
+
+
+### 13.7.2 Spectrum Estimation {#13-dot-7-dot-2-spectrum-estimation}
+
+
+### 13.7.3 Direct Measurement of Position Noise {#13-dot-7-dot-3-direct-measurement-of-position-noise}
+
+
+### 13.7.4 Measurement of the External Disturbance {#13-dot-7-dot-4-measurement-of-the-external-disturbance}
+
+
+## 13.8 Experimental Demonstration {#13-dot-8-experimental-demonstration}
+
+
+## 13.9 Time-Domain Noise Measurements {#13-dot-9-time-domain-noise-measurements}
+
+
+### 13.9.1 Total Integrated Noise {#13-dot-9-dot-1-total-integrated-noise}
+
+
+### 13.9.2 Estimating the Position Noise {#13-dot-9-dot-2-estimating-the-position-noise}
+
+
+### 13.9.3 Practical Considerations {#13-dot-9-dot-3-practical-considerations}
+
+
+### 13.9.4 Experimental Demonstration {#13-dot-9-dot-4-experimental-demonstration}
+
+
+## 13.10 A Simple Method for Measuring the Resolution  of Nanopositioning Systems {#13-dot-10-a-simple-method-for-measuring-the-resolution-of-nanopositioning-systems}
+
+
+## 13.11 Techniques for Improving Resolution {#13-dot-11-techniques-for-improving-resolution}
+
+
+## 13.12 Chapter Summary {#13-dot-12-chapter-summary}
+
+
+## References {#references}
+
+
+## Electrical Considerations {#electrical-considerations}
+
+
+### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
+
+<a id="figure--fig:fleming14-amplifier-model"></a>
+
+{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class='figure-number'>Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
+
+Consider the electrical circuit shown in [Figure 1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
+The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
+The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
+
+<div class="exampl">
+
+Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
+Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).
+
+</div>
+
+When considering the effects of both output impedance and cable inductance, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) is second-order low pass filter:
+
+\begin{equation}
+  \frac{V\_L(s)}{V\_s(s)} = \frac{1}{\frac{s^2}{\omega\_r^2} + 2 \xi \frac{s}{\omega\_r} + 1}
+\end{equation}
+
+with:
+
+-   \\(\omega\_r = \frac{1}{\sqrt{L C\_p}}\\)
+-   \\(\xi = \frac{R\_s \sqrt{L C\_p}}{2 L}\\)
+
+
+### Amplifier small-signal Bandwidth {#amplifier-small-signal-bandwidth}
+
+The most obvious bandwidth limitation is the small-signal bandwidth of the amplifier.
+
+If the inductance \\(L\\) is neglected, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) forms a first order filter with a cut-off frequency
+
+\begin{equation}
+  \omega\_c = \frac{1}{R\_s C\_p}
+\end{equation}
+
+This is thus highly dependent of the load.
+
+The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path.
+A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see [Table 1](#table--tab:piezo-limitation-Rs) for values).
+
+<a id="table--tab:piezo-limitation-Rs"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a>:</span>
+  Bandwidth limitation due to \(R_s\)
+</div>
+
+|              | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
+|--------------|-------------|-----------|------------|
+| Rs = 1 Ohm   | 1.6 MHz     | 160 kHz   | 16 kHz     |
+| Rs = 10 Ohm  | 160 kHz     | 16 kHz    | 1.6 kHz    |
+| Rs = 100 Ohm | 16 kHz      | 1.6 kHz   | 160 Hz     |
+
+The inductance \\(L\\) does also play a role in the amplifier bandwidth as it changes the resonance frequency.
+Ideally, low inductance cables should be used.
+It is however usually quite high compare to \\(\omega\_c\\) as shown in [Table 2](#table--tab:piezo-limitation-L).
+
+<a id="table--tab:piezo-limitation-L"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a>:</span>
+  Bandwidth limitation due to \(R_s\)
+</div>
+
+|             | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
+|-------------|-------------|-----------|------------|
+| L = 25 nH   | 3.2 MHz     | 1 MHz     | 320 kHz    |
+| L = 250 nH  | 1 MHz       | 320 kHz   | 100 kHz    |
+| L = 2500 nH | 320 kHz     | 100 kHz   | 32 kHz     |
+
+
+### Amplifier maximum slew rate {#amplifier-maximum-slew-rate}
+
+Further bandwidth restrictions are imposed by the maximum **slew rate** of the amplifier.
+This is the maximum rate at which the output voltage can change and is usually expressed in \\(V/\mu s\\).
+
+For sinusoidal signals, the amplifiers slew rate must exceed:
+\\[ SR\_{\text{sin}} > V\_{p-p} \pi f \\]
+where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.
+
+<div class="exampl">
+
+If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).
+
+</div>
+
+When dealing with capacitive loads, **the current limit is usually exceed well before the slew rate limit**.
+
+
+### Current and Power Limitations {#current-and-power-limitations}
+
+When driving the actuator off-resonance, the current delivered to a piezoelectric actuator is approximately:
+\\[ I\_L(s) = V\_L(s) C\_p s \\]
+
+For sinusoidal signals, the maximum positive and negative current is equal to:
+\\[ I\_L^\text{max} = V\_{p-p} \pi f C\_p \\]
+
+<a id="table--tab:piezo-required-current"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a>:</span>
+  Minimum current requirements for a 10V sinusoid
+</div>
+
+|             | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
+|-------------|-------------|-----------|------------|
+| f = 30 Hz   | 0.19 mA     | 1.9 mA    | 19 mA      |
+| f = 3 kHz   | 19 mA       | 190 mA    | 1.9 A      |
+| f = 300 kHz | 1.9 A       | 19 A      | 190 A      |
+
+
+### Chapter Summary {#chapter-summary}
+
+The bandwidth limitations  of standard piezoelectric drives were identified as:
+
+-   High output impedance
+-   The presence of a ple in the voltage-feedback loop due to output impedance and load capacitance
+-   Insufficient current capacity due to power dissipation
+-   High cable and connector inductance
+
+
+### References {#references}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A. J., and K. K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-06617-2">10.1007/978-3-319-06617-2</a>.</div>
+</div>

content/book/horowitz15_art_of_elect_third_edition.md

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+title = "The Art of Electronics - Third Edition"
+author = ["Dehaeze Thomas"]
+description = "One of the best book in electronics. Cover most topics (both analog and digital)."
+keywords = ["electronics"]
+draft = false
++++
+
+Tags
+: [Reference Books]({{< relref "reference_books.md" >}}), [Electronics]({{< relref "electronics.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Horowitz 2015</a>)
+
+Author(s)
+: Horowitz, P.
+
+Year
+: 2015
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Horowitz, Paul. 2015. <i>The Art of Electronics - Third Edition</i>. New York, NY, USA: Cambridge University Press.</div>
+</div>

content/book/leach14_fundam_princ_engin_nanom.md

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+title = "Fundamental principles of engineering nanometrology"
+author = ["Dehaeze Thomas"]
+keywords = ["Metrology"]
+draft = false
++++
+
+Tags
+: [Metrology]({{< relref "metrology.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Leach 2014</a>)
+
+Author(s)
+: Leach, R.
+
+Year
+: 2014
+
+
+## Measurement of angles {#measurement-of-angles}
+
+Unit:
+
+-   radian for plane angle
+-   steradian for solid angle
+
+\\(1 rad \approx 55.3deg\\)
+
+Instrument principles:
+
+-   subdivision: index tacle, angular gratings, polygons, ...
+-   ratio of two lengths: angular interferometers, sin cars, small angle generators, ...
+-   autocollimators with a flat mirror
+
+
+## Sources of error in displacement interferometry {#sources-of-error-in-displacement-interferometry}
+
+Two error sources:
+
+-   error sources that are proportional to the displacement being measured \\(L\\): cumulative errors
+-   error sources that are independent of the displacement being measured: non-cumulative errors
+
+
+### Thermal expansion of the metrology frame {#thermal-expansion-of-the-metrology-frame}
+
+
+### Deadpath length {#deadpath-length}
+
+Deadpath length, \\(d\\), is defined as the difference in distance in air between the reference and measurement reflectors and the beam splitter when the interferometer measurement is initiated.
+Deadpath error occurs when there is a non-zero deadpath and environmental conditions change during a measurement.
+
+
+### Cosine error {#cosine-error}
+
+\\(\Delta l = l(1-\cos(\theta))\\)
+
+For small angles: \\(\Delta l = \frac{l \theta^2}{2}\\)
+
+The cosine error is then a second-order effect, contrary to the Abbe error which is a first order effect.
+The second order nature means that cosine error quickly diminish as the alignment is improved.
+
+
+## Latest advances in displacement interferometry {#latest-advances-in-displacement-interferometry}
+
+Commercial interferometers
+=&gt; fused silica optics housed in Invar mounts
+=&gt; all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
+
+One advantage that homodyme systems have over heterodyne systems is their ability to readily have the source fibre delivered to the interferometer.
+
+
+### Spatially separated interferometers {#spatially-separated-interferometers}
+
+It uses heterodyne interferometer and one quadrant photodiode.
+By knowing the beam size and detector geometry, the measurement target's angle change can be determined by differencing matched pairs of measured phase from the quadrant photodiode while the displacement is determined from the average phase over the four quadrants.
+
+
+## Angular interferometers {#angular-interferometers}
+
+Determination of an angle by the ratio of two lengths.
+The angular optics is used to create two parallel beam paths between the angular interferometer and the angular reflector.
+
+The beam that illuminates the angular optics contains two frequencies, \\(f1\\) and \\(f2\\). A polarising beam splitter in the angular interferometer splits the frequencies that travel along separate paths.
+
+The measurement of angles is then relative.
+
+This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard. 2014. <i>Fundamental Principles of Engineering Nanometrology</i>. Elsevier. doi:<a href="https://doi.org/10.1016/c2012-0-06010-3">10.1016/c2012-0-06010-3</a>.</div>
+</div>

content/book/leach18_basic_precis_engin_edition.md

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+title = "Basics of precision engineering - 1st edition"
+author = ["Dehaeze Thomas"]
+keywords = ["Metrology", "Mechatronics"]
+draft = true
++++
+
+Tags
+: [Precision Engineering]({{< relref "precision_engineering.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Leach and Smith 2018</a>)
+
+Author(s)
+: Leach, R., &amp; Smith, S. T.
+
+Year
+: 2018
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard, and Stuart T. Smith. 2018. <i>Basics of Precision Engineering - 1st Edition</i>. CRC Press.</div>
+</div>

content/book/lyons11_under_digit_signal_proces.md

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++++
+title = "Understanding Digital Signal Processing"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [IRR and FIR Filters]({{< relref "irr_and_fir_filters.md" >}}), [Digital Filters]({{< relref "digital_filters.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Lyons 2011</a>)
+
+Author(s)
+: Lyons, R.
+
+Year
+: 2011
+
+
+## Discrete Sequences And Systems {#discrete-sequences-and-systems}
+
+
+### Discrete Sequences And Their Notation {#discrete-sequences-and-their-notation}
+
+
+### Signal Amplitude, Magnitude, Power {#signal-amplitude-magnitude-power}
+
+
+### Signal Processing Operational Symbols {#signal-processing-operational-symbols}
+
+
+### Introduction To Discrete Linear Time-Invariant Systems {#introduction-to-discrete-linear-time-invariant-systems}
+
+
+### Discrete Linear Systems {#discrete-linear-systems}
+
+
+### Time-Invariant Systems {#time-invariant-systems}
+
+
+### The Commutative Property Of Linear Time-Invariant Systems {#the-commutative-property-of-linear-time-invariant-systems}
+
+
+### Analyzing Linear Time-Invariant Systems {#analyzing-linear-time-invariant-systems}
+
+<a id="figure--fig:lyons11-lti-impulse-response"></a>
+
+{{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="<span class='figure-number'>Figure 1: </span>LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \\(x(n)\\) and impulse reponse output sequence \\(y(n)\\)." >}}
+
+<a id="figure--fig:lyons11-moving-average"></a>
+
+{{< figure src="/ox-hugo/lyons11_moving_average.png" caption="<span class='figure-number'>Figure 2: </span>Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}}
+
+
+## Periodic Sampling {#periodic-sampling}
+
+
+### Aliasing: Signal Ambiguity In The Frequency Domain {#aliasing-signal-ambiguity-in-the-frequency-domain}
+
+<a id="figure--fig:lyons11-frequency-ambiguity"></a>
+
+{{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="<span class='figure-number'>Figure 3: </span>Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}}
+
+
+### Sampling Lowpass Signals {#sampling-lowpass-signals}
+
+<a id="figure--fig:lyons11-noise-spectral-replication"></a>
+
+{{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="<span class='figure-number'>Figure 4: </span>Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}}
+
+<a id="figure--fig:lyons11-lowpass-sampling"></a>
+
+{{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="<span class='figure-number'>Figure 5: </span>Low pass analog filtering prior to sampling at a rate of \\(f\_s\\) Hz." >}}
+
+
+## The Discrete Fourier Transform {#the-discrete-fourier-transform}
+
+\begin{equation}
+X(f) = \int\_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt
+\end{equation}
+
+\begin{equation}
+X(m) = \sum\_{n = 0}^{N-1} x(n) e^{-j2 \pi n m /N}
+\end{equation}
+
+
+### Understanding The Dft Equation {#understanding-the-dft-equation}
+
+
+### Dft Symmetry {#dft-symmetry}
+
+
+### Dft Linearity {#dft-linearity}
+
+
+### Dft Magnitudes {#dft-magnitudes}
+
+
+### Dft Frequency Axis {#dft-frequency-axis}
+
+
+### Dft Shifting Theorem {#dft-shifting-theorem}
+
+
+### Inverse Dft {#inverse-dft}
+
+
+### Dft Leakage {#dft-leakage}
+
+
+### Windows {#windows}
+
+
+### Dft Scalloping Loss {#dft-scalloping-loss}
+
+
+### Dft Resolution, Zero Padding, And Frequency-Domain Sampling {#dft-resolution-zero-padding-and-frequency-domain-sampling}
+
+
+### Dft Processing Gain {#dft-processing-gain}
+
+
+### The Dft Of Rectangular Functions {#the-dft-of-rectangular-functions}
+
+
+### Interpreting The Dft Using The Discrete-Time Fourier Transform {#interpreting-the-dft-using-the-discrete-time-fourier-transform}
+
+
+## The Fast Fourier Transform {#the-fast-fourier-transform}
+
+
+### Relationship Of The Fft To The Dft {#relationship-of-the-fft-to-the-dft}
+
+
+### Hints On Using Ffts In Practice {#hints-on-using-ffts-in-practice}
+
+
+### Derivation Of The Radix-2 Fft Algorithm {#derivation-of-the-radix-2-fft-algorithm}
+
+
+### Fft Input/Output Data Index Bit Reversal {#fft-input-output-data-index-bit-reversal}
+
+
+### Radix-2 Fft Butterfly Structures {#radix-2-fft-butterfly-structures}
+
+
+### Alternate Single-Butterfly Structures {#alternate-single-butterfly-structures}
+
+
+## Finite Impulse Response Filters {#finite-impulse-response-filters}
+
+
+### An Introduction To Finite Impulse Response (Fir) Filters {#an-introduction-to-finite-impulse-response--fir--filters}
+
+
+### Convolution In Fir Filters {#convolution-in-fir-filters}
+
+
+### Lowpass Fir Filter Design {#lowpass-fir-filter-design}
+
+
+### Bandpass Fir Filter Design {#bandpass-fir-filter-design}
+
+
+### Highpass Fir Filter Design {#highpass-fir-filter-design}
+
+
+### Parks-Mcclellan Exchange Fir Filter Design Method {#parks-mcclellan-exchange-fir-filter-design-method}
+
+
+### Half-Band Fir Filters {#half-band-fir-filters}
+
+
+### Phase Response Of Fir Filters {#phase-response-of-fir-filters}
+
+
+### A Generic Description Of Discrete Convolution {#a-generic-description-of-discrete-convolution}
+
+
+### Analyzing Fir Filters {#analyzing-fir-filters}
+
+
+## Infinite Impulse Response Filters {#infinite-impulse-response-filters}
+
+
+### An Introduction To Infinite Impulse Response Filters {#an-introduction-to-infinite-impulse-response-filters}
+
+
+### The Laplace Transform {#the-laplace-transform}
+
+
+### The Z-Transform {#the-z-transform}
+
+
+### Using The Z-Transform To Analyze Iir Filters {#using-the-z-transform-to-analyze-iir-filters}
+
+
+### Using Poles And Zeros To Analyze Iir Filters {#using-poles-and-zeros-to-analyze-iir-filters}
+
+
+### Alternate Iir Filter Structures {#alternate-iir-filter-structures}
+
+
+### Pitfalls In Building Iir Filters {#pitfalls-in-building-iir-filters}
+
+
+### Improving Iir Filters With Cascaded Structures {#improving-iir-filters-with-cascaded-structures}
+
+
+### Scaling The Gain Of Iir Filters {#scaling-the-gain-of-iir-filters}
+
+
+### Impulse Invariance Iir Filter Design Method {#impulse-invariance-iir-filter-design-method}
+
+
+### Bilinear Transform Iir Filter Design Method {#bilinear-transform-iir-filter-design-method}
+
+
+### Optimized Iir Filter Design Method {#optimized-iir-filter-design-method}
+
+
+### A Brief Comparison Of Iir And Fir Filters {#a-brief-comparison-of-iir-and-fir-filters}
+
+
+## Specialized Digital Networks And Filters {#specialized-digital-networks-and-filters}
+
+
+### Differentiators {#differentiators}
+
+
+### Integrators {#integrators}
+
+
+### Matched Filters {#matched-filters}
+
+
+### Interpolated Lowpass Fir Filters {#interpolated-lowpass-fir-filters}
+
+
+### Frequency Sampling Filters: The Lost Art {#frequency-sampling-filters-the-lost-art}
+
+
+## Quadrature Signals {#quadrature-signals}
+
+
+### Why Care About Quadrature Signals? {#why-care-about-quadrature-signals}
+
+
+### The Notation Of Complex Numbers {#the-notation-of-complex-numbers}
+
+
+### Representing Real Signals Using Complex Phasors {#representing-real-signals-using-complex-phasors}
+
+
+### A Few Thoughts On Negative Frequency {#a-few-thoughts-on-negative-frequency}
+
+
+### Quadrature Signals In The Frequency Domain {#quadrature-signals-in-the-frequency-domain}
+
+
+### Bandpass Quadrature Signals In The Frequency Domain {#bandpass-quadrature-signals-in-the-frequency-domain}
+
+
+### Complex Down-Conversion {#complex-down-conversion}
+
+
+### A Complex Down-Conversion Example {#a-complex-down-conversion-example}
+
+
+### An Alternate Down-Conversion Method {#an-alternate-down-conversion-method}
+
+
+## The Discrete Hilbert Transform {#the-discrete-hilbert-transform}
+
+
+### Hilbert Transform Definition {#hilbert-transform-definition}
+
+
+### Why Care About The Hilbert Transform? {#why-care-about-the-hilbert-transform}
+
+
+### Impulse Response Of A Hilbert Transformer {#impulse-response-of-a-hilbert-transformer}
+
+
+### Designing A Discrete Hilbert Transformer {#designing-a-discrete-hilbert-transformer}
+
+
+### Time-Domain Analytic Signal Generation {#time-domain-analytic-signal-generation}
+
+
+### Comparing Analytical Signal Generation Methods {#comparing-analytical-signal-generation-methods}
+
+
+## 10 Sample Rate Conversion {#10-sample-rate-conversion}
+
+
+### 10.1 Decimation {#10-dot-1-decimation}
+
+
+### 10.2 Two-Stage Decimation {#10-dot-2-two-stage-decimation}
+
+
+### 10.3 Properties Of Downsampling {#10-dot-3-properties-of-downsampling}
+
+
+### 10.4 Interpolation {#10-dot-4-interpolation}
+
+
+### 10.5 Properties Of Interpolation {#10-dot-5-properties-of-interpolation}
+
+
+### 10.6 Combining Decimation And Interpolation {#10-dot-6-combining-decimation-and-interpolation}
+
+
+### 10.7 Polyphase Filters {#10-dot-7-polyphase-filters}
+
+
+### 10.8 Two-Stage Interpolation {#10-dot-8-two-stage-interpolation}
+
+
+### 10.9 Z-Transform Analysis Of Multirate Systems {#10-dot-9-z-transform-analysis-of-multirate-systems}
+
+
+### 10.10 Polyphase Filter Implementations {#10-dot-10-polyphase-filter-implementations}
+
+
+### 10.11 Sample Rate Conversion By Rational Factors {#10-dot-11-sample-rate-conversion-by-rational-factors}
+
+
+### 10.12 Sample Rate Conversion With Half-Band Filters {#10-dot-12-sample-rate-conversion-with-half-band-filters}
+
+
+### 10.13 Sample Rate Conversion With Ifir Filters {#10-dot-13-sample-rate-conversion-with-ifir-filters}
+
+
+### 10.14 Cascaded Integrator-Comb Filters {#10-dot-14-cascaded-integrator-comb-filters}
+
+
+## 11 Signal Averaging {#11-signal-averaging}
+
+
+### 11.1 Coherent Averaging {#11-dot-1-coherent-averaging}
+
+
+### 11.2 Incoherent Averaging {#11-dot-2-incoherent-averaging}
+
+
+### 11.3 Averaging Multiple Fast Fourier Transforms {#11-dot-3-averaging-multiple-fast-fourier-transforms}
+
+
+### 11.4 Averaging Phase Angles {#11-dot-4-averaging-phase-angles}
+
+
+### 11.5 Filtering Aspects Of Time-Domain Averaging {#11-dot-5-filtering-aspects-of-time-domain-averaging}
+
+
+### 11.6 Exponential Averaging {#11-dot-6-exponential-averaging}
+
+
+## 12 Digital Data Formats And Their Effects {#12-digital-data-formats-and-their-effects}
+
+
+### 12.1 Fixed-Point Binary Formats {#12-dot-1-fixed-point-binary-formats}
+
+
+### 12.2 Binary Number Precision And Dynamic Range {#12-dot-2-binary-number-precision-and-dynamic-range}
+
+
+### 12.3 Effects Of Finite Fixed-Point Binary Word Length {#12-dot-3-effects-of-finite-fixed-point-binary-word-length}
+
+
+### 12.4 Floating-Point Binary Formats {#12-dot-4-floating-point-binary-formats}
+
+
+### 12.5 Block Floating-Point Binary Format {#12-dot-5-block-floating-point-binary-format}
+
+
+## 13 Digital Signal Processing Tricks {#13-digital-signal-processing-tricks}
+
+
+### 13.1 Frequency Translation Without Multiplication {#13-dot-1-frequency-translation-without-multiplication}
+
+
+### 13.2 High-Speed Vector Magnitude Approximation {#13-dot-2-high-speed-vector-magnitude-approximation}
+
+
+### 13.3 Frequency-Domain Windowing {#13-dot-3-frequency-domain-windowing}
+
+
+### 13.4 Fast Multiplication Of Complex Numbers {#13-dot-4-fast-multiplication-of-complex-numbers}
+
+
+### 13.5 Efficiently Performing The Fft Of Real Sequences {#13-dot-5-efficiently-performing-the-fft-of-real-sequences}
+
+
+### 13.6 Computing The Inverse Fft Using The Forward Fft {#13-dot-6-computing-the-inverse-fft-using-the-forward-fft}
+
+
+### 13.7 Simplified Fir Filter Structure {#13-dot-7-simplified-fir-filter-structure}
+
+
+### 13.8 Reducing A/D Converter Quantization Noise {#13-dot-8-reducing-a-d-converter-quantization-noise}
+
+
+### 13.9 A/D Converter Testing Techniques {#13-dot-9-a-d-converter-testing-techniques}
+
+
+### 13.10 Fast Fir Filtering Using The Fft {#13-dot-10-fast-fir-filtering-using-the-fft}
+
+
+### 13.11 Generating Normally Distributed Random Data {#13-dot-11-generating-normally-distributed-random-data}
+
+
+### 13.12 Zero-Phase Filtering {#13-dot-12-zero-phase-filtering}
+
+
+### 13.13 Sharpened Fir Filters {#13-dot-13-sharpened-fir-filters}
+
+
+### 13.14 Interpolating A Bandpass Signal {#13-dot-14-interpolating-a-bandpass-signal}
+
+
+### 13.15 Spectral Peak Location Algorithm {#13-dot-15-spectral-peak-location-algorithm}
+
+
+### 13.16 Computing Fft Twiddle Factors {#13-dot-16-computing-fft-twiddle-factors}
+
+
+### 13.17 Single Tone Detection {#13-dot-17-single-tone-detection}
+
+
+### 13.18 The Sliding Dft {#13-dot-18-the-sliding-dft}
+
+
+### 13.19 The Zoom Fft {#13-dot-19-the-zoom-fft}
+
+
+### 13.20 A Practical Spectrum Analyzer {#13-dot-20-a-practical-spectrum-analyzer}
+
+
+### 13.21 An Efficient Arctangent Approximation {#13-dot-21-an-efficient-arctangent-approximation}
+
+
+### 13.22 Frequency Demodulation Algorithms {#13-dot-22-frequency-demodulation-algorithms}
+
+
+### 13.23 Dc Removal {#13-dot-23-dc-removal}
+
+
+### 13.24 Improving Traditional Cic Filters {#13-dot-24-improving-traditional-cic-filters}
+
+
+### 13.25 Smoothing Impulsive Noise {#13-dot-25-smoothing-impulsive-noise}
+
+
+### 13.26 Efficient Polynomial Evaluation {#13-dot-26-efficient-polynomial-evaluation}
+
+
+### 13.27 Designing Very High-Order Fir Filters {#13-dot-27-designing-very-high-order-fir-filters}
+
+
+### 13.28 Time-Domain Interpolation Using The Fft {#13-dot-28-time-domain-interpolation-using-the-fft}
+
+
+### 13.29 Frequency Translation Using Decimation {#13-dot-29-frequency-translation-using-decimation}
+
+
+### 13.30 Automatic Gain Control (Agc) {#13-dot-30-automatic-gain-control--agc}
+
+
+### 13.31 Approximate Envelope Detection {#13-dot-31-approximate-envelope-detection}
+
+
+### 13.32 A Quadrature Oscillator {#13-dot-32-a-quadrature-oscillator}
+
+
+### 13.33 Specialized Exponential Averaging {#13-dot-33-specialized-exponential-averaging}
+
+
+### 13.34 Filtering Narrowband Noise Using Filter Nulls {#13-dot-34-filtering-narrowband-noise-using-filter-nulls}
+
+
+### 13.35 Efficient Computation Of Signal Variance {#13-dot-35-efficient-computation-of-signal-variance}
+
+
+### 13.36 Real-Time Computation Of Signal Averages And Variances {#13-dot-36-real-time-computation-of-signal-averages-and-variances}
+
+
+### 13.37 Building Hilbert Transformers From Half-Band Filters {#13-dot-37-building-hilbert-transformers-from-half-band-filters}
+
+
+### 13.38 Complex Vector Rotation With Arctangents {#13-dot-38-complex-vector-rotation-with-arctangents}
+
+
+### 13.39 An Efficient Differentiating Network {#13-dot-39-an-efficient-differentiating-network}
+
+
+### 13.40 Linear-Phase Dc-Removal Filter {#13-dot-40-linear-phase-dc-removal-filter}
+
+
+### 13.41 Avoiding Overflow In Magnitude Computations {#13-dot-41-avoiding-overflow-in-magnitude-computations}
+
+
+### 13.42 Efficient Linear Interpolation {#13-dot-42-efficient-linear-interpolation}
+
+
+### 13.43 Alternate Complex Down-Conversion Schemes {#13-dot-43-alternate-complex-down-conversion-schemes}
+
+
+### 13.44 Signal Transition Detection {#13-dot-44-signal-transition-detection}
+
+
+### 13.45 Spectral Flipping Around Signal Center Frequency {#13-dot-45-spectral-flipping-around-signal-center-frequency}
+
+
+### 13.46 Computing Missing Signal Samples {#13-dot-46-computing-missing-signal-samples}
+
+
+### 13.47 Computing Large Dfts Using Small Ffts {#13-dot-47-computing-large-dfts-using-small-ffts}
+
+
+### 13.48 Computing Filter Group Delay Without Arctangents {#13-dot-48-computing-filter-group-delay-without-arctangents}
+
+
+### 13.49 Computing A Forward And Inverse Fft Using A Single Fft {#13-dot-49-computing-a-forward-and-inverse-fft-using-a-single-fft}
+
+
+### 13.50 Improved Narrowband Lowpass Iir Filters {#13-dot-50-improved-narrowband-lowpass-iir-filters}
+
+
+### 13.51 A Stable Goertzel Algorithm {#13-dot-51-a-stable-goertzel-algorithm}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lyons, Richard. 2011. <i>Understanding Digital Signal Processing</i>. Upper Saddle River, NJ: Prentice Hall.</div>
+</div>

content/book/pintelon12_system_ident.md

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+title = "System identification : a frequency domain approach"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+: [System Identification]({{< relref "system_identification.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Pintelon and Schoukens 2012</a>)
+
+Author(s)
+: Pintelon, R., &amp; Schoukens, J.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Pintelon, R., and J. Schoukens. 2012. <i>System Identification : a Frequency Domain Approach</i>. Hoboken, N.J. Piscataway, NJ: Wiley IEEE Press. doi:<a href="https://doi.org/10.1002/9781118287422">10.1002/9781118287422</a>.</div>
+</div>

content/book/schoukens12_master.md

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+title = "Mastering system identification in 100 exercises"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Schoukens, Pintelon, and Rolain 2012</a>)
+
+Author(s)
+: Schoukens, J., Pintelon, R., &amp; Rolain, Y.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schoukens, Johan, Rik Pintelon, and Yves Rolain. 2012. <i>Mastering System Identification in 100 Exercises</i>. John Wiley &#38; Sons.</div>
+</div>

content/book/slocum92_precis_machin_desig.md

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+title = "Precision Machine Design"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Slocum 1992</a>)
+
+Author(s)
+: Slocum, A. H.
+
+Year
+: 1992
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Slocum, Alexander H. 1992. <i>Precision Machine Design</i>. Society of Manufacturing Engineers.</div>
+</div>

content/book/smith99_scien_engin_guide_digit_signal.md

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+title = "The scientist and engineer's guide to digital signal processing - second edition"
+author = ["Dehaeze Thomas"]
+keywords = ["Signal Processing"]
+draft = true
++++
+
+Tags
+: [Digital Signal Processing]({{< relref "digital_signal_processing.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Smith 1999</a>)
+
+Author(s)
+: Smith, S. W.
+
+Year
+: 1999
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Smith, Steven W. 1999. <i>The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition</i>. California Technical Publishing.</div>
+</div>

content/book/wardle15_ultra_precis_bearin.md

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+title = "Ultra Precision Bearings"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Wardle 2015</a>)
+
+Author(s)
+: Wardle, F.
+
+Year
+: 2015
+
+
+## Bearing motion error {#bearing-motion-error}
+
+Causes:
+
+-   Manufacturing Quality
+-   Bearing design
+-   External influences
+
+**Types of error motion**
+
+A distinction is made between (see Figure <fig:wardle15_synchronous_asynchronous_schematic>):
+
+-   motion errors that are harmonic of the basic rotor speed: _synchronous_ motion error
+-   those that are node: _asynchronous_ motion errors
+
+<a id="figure--fig:wardle15-synchronous-asynchronous-schematic"></a>
+
+{{< figure src="/ox-hugo/wardle15_synchronous_asynchronous_schematic.png" caption="<span class='figure-number'>Figure 1: </span>Effect of (a) synchronous, and (b) asynchronous motion error on surface form" >}}
+
+
+### Measurement of motion error {#measurement-of-motion-error}
+
+A capacitive sensor is typically used, and the measurement is performed over a time period corresponding to several (typically five) revolutions of the bearing.
+
+It is displayed as a polar plot of motion error amplitude versus angle of rotation (Figure <fig:wardle15_typical_error_plot>).
+
+<a id="figure--fig:wardle15-typical-error-plot"></a>
+
+{{< figure src="/ox-hugo/wardle15_typical_error_plot.png" caption="<span class='figure-number'>Figure 2: </span>Total error motion" >}}
+
+The Synchronous error motion (i.e. error that are harmonics of the rotational speed) can be extracted (Figure <fig:wardle15_synchronous_error_example>).
+
+<a id="figure--fig:wardle15-synchronous-error-example"></a>
+
+{{< figure src="/ox-hugo/wardle15_synchronous_error_example.png" caption="<span class='figure-number'>Figure 3: </span>Synchronous motion error" >}}
+
+It can then be separated into a "_fundamental error motion_" and a "_residual error motion_" (Figure <fig:wardle15_fundamental_and_residual_errors>).
+The fundamental error motion contains only one frequency corresponding to the speed of the rotation of the bearing.
+For radial measurements, it corresponds to the eccentricity, and is not always significant.
+
+<a id="figure--fig:wardle15-fundamental-and-residual-errors"></a>
+
+{{< figure src="/ox-hugo/wardle15_fundamental_and_residual_errors.png" caption="<span class='figure-number'>Figure 4: </span>(a) Fundamental error motion; and (b) residual synchronous error motion" >}}
+
+The Asynchronous error motion (Figure <fig:wardle15_asynchronous_error_motion_example>) contains all other motion error frequencies.
+
+<a id="figure--fig:wardle15-asynchronous-error-motion-example"></a>
+
+{{< figure src="/ox-hugo/wardle15_asynchronous_error_motion_example.png" caption="<span class='figure-number'>Figure 5: </span>Asynchronous motion error" >}}
+
+The measurements shown in previous figures may be quantified by a number of different parameters but it is commonplace to find the "Least Squares" best fit centre and then to place Maximum Inscribed and Minimum Circumscribed circles on the measurement.
+
+The radial separation of the centres of the circles then represents a "Peak to Peak" value of the error motion.
+
+In many cases, the displacement sensor is mounted over a rotating target surface attached to the shaft supported by the bearings.
+However, the displacement sensor now measures not only the motion error of the shaft but also any **geometrical errors present in the target surface**.
+For a radial error motion measurement, out of roundness of the target surface is recorded along with the shaft’s motion error.
+As the motion error of ultra precision bearings may be comparable in magnitude to the geometrical errors in the most accurately manufactured target surfaces then a correction must be made.
+A measurement procedure was proposed that involved two measurements, one with the target surface fixed at some angular position relative to the shaft and the second with it moved through 180 degrees.
+By adding or subtracting the two measurements, geometrical errors on the target surface can be separated from shaft motion errors.
+
+
+### Frequency Analysis {#frequency-analysis}
+
+In general, rotating systems will exhibit motion errors containing several series of harmonics, each of which relate to different aspects or components of the system.
+The main benefit of frequency analysis is therefore to obtain diagnostic information with which to identify the likely sources of motion error and to help reduce their amplitude should they be unacceptable.
+
+
+## Ball Bearings {#ball-bearings}
+
+Criterion used in this book to define ultra precision bearings: motion error of less than 100 nm peak to peak.
+
+Generally only the precision grades or low noise grades of ball bearing are
+likely to produce low motion errors. These types of ball bearing are widely
+used in high precision machine tools, quiet running electric motors,
+computer disc drives and instrumentation, where they provide good but
+not exceptional running accuracy at a competitive price.
+
+Single-row radial ball bearings are favoured in precision engineering
+applications such as computer disc drives and precision electric motors,
+where low motion errors or low noise are a primary requirement. Angular
+contact bearings, on the other hand, are widely used in precision applica-
+tions such as machine tool spindles and rotary tables where static stiffness
+is also important.
+
+
+### Motion Error {#motion-error}
+
+During the 1980s and 1990s, the computer disc drive industry emerged
+as a major application for ball bearings and motion error was recognised as
+a critical bearing performance parameter directly influencing disc capacity.
+Unlike the electric motor application, where bearings may operate under a
+diverse range of conditions, this application was focused on low cost,
+miniature bearings operating under specific conditions of light axial load
+and medium speed at near ambient temperatures. Early research work,
+performed mainly in Japan, developed an understanding of the factors that
+determine the radial motion error of disc drive ball bearings [34–38] and
+later focused specifically on reducing the ‘Non-Repeatable Run Out’
+(NRRO) [39–43]. Because in this application bearing speeds are moderate,
+the NRRO was found to be largely influenced by ball size variation.
+
+In terms of peak–peak motion error amplitudes, ball bearings can achieve
+a creditable performance. Amplitudes as low as 48 nm have been reported
+in scientific papers [41], for ball bearings used in computer hard disc drives.
+This is comparable to the motion error of some types of fluid film, but the
+**disadvantage of ball bearings is that the motion error is predominantly
+asynchronous whereas for fluid film bearings it is mostly synchronous**.
+
+The main reason is that for ball bearings, motion error frequencies relate to
+the orbital and spinning speeds of the balls and these can never be harmonic
+of shaft speed in a practical bearing design.
+
+Ball bearing motion error is
+influenced by a large number of parameters, some a function of the bearing
+design and manufacturing processes, others being dependent on application
+conditions. However, there are relatively few basic mechanisms by which
+motion error can be generated and by understanding these, the influence of
+different parameters can be more clearly defined and in many cases, even
+quantified.
+
+
+#### Dynamics model for estimating call bearing motion error {#dynamics-model-for-estimating-call-bearing-motion-error}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wardle, Frank. 2015. <i>Ultra Precision Bearings</i>. Elsevier.</div>
+</div>

content/inbook/albertos04_decen_decoup_contr.md

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+title = "Decentralized and decoupled control"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+: [Multivariable Control]({{< relref "multivariable_control.md" >}}), [Decoupled Control]({{< relref "decoupled_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Albertos and Antonio 2004</a>)
+
+Author(s)
+: Albertos, P., &amp; Antonio, S.
+
+Year
+: 2004
+
+
+## Introduction {#introduction}
+
+Decentralized control is decomposed into two steps:
+
+1.  decoupled the plant into several subsystems
+2.  control the subsystems
+
+The initial effort of decoupling the system results in subsequent easier design, implementation and tuning.
+
+Decentralized control tries to control multivariable plants by a suitable decomposition into SISO control loops.
+If the process has strong coupling or conditioning problems, centralized control may be required.
+It however requires the availability of a precise model.
+
+Two approaches can be used to control a coupled system with SISO techniques:
+
+-   **decentralized control** tries to divide the plant and design _independent_ controllers for each subsystems.
+    Two alternative arise:
+    -   neglect the coupling
+    -   carry out a _decoupling_ operation by "canceling" coupling by transforming the system into a diagonal or triangular structure bia a transformation matrix
+-   **cascade control**
+
+
+## Mutli-Loop Control, Pairing Selection {#mutli-loop-control-pairing-selection}
+
+The strategy called _multi-loop control_ consists of first proper input/output pairing, and then design of several SISO controllers.
+In this way, a complex control problem is divided into several simpler ones.
+
+The multi-loop control may not work in strongly coupled systems.
+Therefore, a methodology the access the degree of interaction between the loops is needed.
+
+
+### [Relative Gain Array]({{< relref "relative_gain_array.md" >}}) {#relative-gain-array--relative-gain-array-dot-md}
+
+The Relative Gain Array (RGA) \\(\Lambda(s)\\) is defined as:
+
+\begin{equation}
+\Lambda(s) = G(s) \times (G(s)^T)^{-1}
+\end{equation}
+
+The RGA is scaling-independent and controller-independent.
+These coefficients can be interpreted as the ratio between the open-loop SISO static gain and the gain with "perfect" control on the rest of the loops.
+
+For demanding control specifications, the values of \\(\Lambda\\) car be drawn as a function of frequency.
+In this case, at frequencies important for control stability robustness (around the peak of the sensitivity transfer function), if \\(\Lambda(j\omega)\\) approaches the identity matrix, stability problems are avoided in multi-loop control.
+
+
+## Decoupling {#decoupling}
+
+In cases when multi-loop control is not effective in reaching the desired specifications, a possible strategy for tackling the MIMO control could be to transform the transfer function matrix into a diagonal dominant one.
+This strategy is called **decoupling**.
+
+[Decoupled Control]({{< relref "decoupled_control.md" >}}) can be achieved in two ways:
+
+-   feedforward cancellation of the cross-coupling terms
+-   based on state measurements, via a feedback law
+
+
+### Feedforward Decoupling {#feedforward-decoupling}
+
+A pre-compensator ([Figure 1](#figure--fig:albertos04-pre-compensator-decoupling)) can be added to transform the open-loop characteristics into a new one as chosen by the designer.
+This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
+
+<a id="figure--fig:albertos04-pre-compensator-decoupling"></a>
+
+{{< figure src="/ox-hugo/albertos04_pre_compensator_decoupling.png" caption="<span class='figure-number'>Figure 1: </span>Decoupler pre-compensator" >}}
+
+**Approximate decoupling**:
+To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state.
+If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used.
+
+Although at first glance, decoupling seems an appealing idea, there are some drawbacks:
+
+-   as decoupling is achieved via the coordination of sensors and actuators to achieve an "apparent" diagonal behavior, the failure of one the actuators may heavily affects all loops.
+-   a decoupling design (inverse-based controller) may not be desirable for all disturbance-rejection tasks.
+-   many MIMO non-minimum phase systems, when feedforward decoupled, increase the RHP-zero multiplicity so performance limitations due to its presence are exacerbated.
+-   decoupling may be very sensitive to modeling errors, specially for ill-conditionned plants
+-   feedback decoupling needs full state measurements
+
+
+### SVD Decoupling {#svd-decoupling}
+
+A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition.md" >}}) as:
+
+\begin{equation}
+M = U \Sigma V^T
+\end{equation}
+
+where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
+
+The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
+In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
+
+If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in [Figure 2](#figure--fig:albertos04-svd-decoupling), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
+Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth.
+
+<a id="figure--fig:albertos04-svd-decoupling"></a>
+
+{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="<span class='figure-number'>Figure 2: </span>SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
+
+The transformed sensor-actuator pair corresponding to the maximum singular value is the direction with biggest "gain" on the plant, that is, the combination of variables being "easiest to control".
+
+In ill-conditioned plants, the ratio between the biggest and lower singular value is large (for reference, greater than 20).
+They are very sensitive to input uncertainty as some "input directions" have much bigger gain than other ones.
+
+SVD decoupling produces the most suitable combinations for independent "multi-loop" control in the transformed variables, so its performance may be better than RGA-based design (at the expense of losing physical interpretability).
+If some of the vectors in \\(V\\) (input directions) have a significant component on a particular input, and the corresponding output direction is also significantly pointing to a particular output, that combination is a good candidate for an independent multi-loop control.
+
+
+## Conclusions {#conclusions}
+
+In this chapter, the control of systems with multiple inputs and outputs is discussed using SISO-based tools, either directly or after some multivariable decoupling transformations.
+
+Multi-loop strategies, if suitable, may present th advantages of fault tolerance, as well as simplicity.
+However, in some cases, tuning may be difficult and coupling may severely limit their performance.
+
+Decoupling is based on mathematical transformations of the system models into diagonal form.
+Feedforward decoupling can be used in many cases.
+Feedback decoupling achieves its objective if state is measurable and system is minimum-phase.
+However, decoupling may be very sensitive to modelling errors and it is not the optimal strategy for disturbance rejection.
+
+Cascade control is widely used in industry to improve the behaviour of basic SISO loops via the addition of extra sensors and actuators.
+However, ease of tuning requires that different time constants are involved in different subsystems.
+In general, addition of extra sensors and actuators in a SISO or MIMO loop, will improve achievable performance and/or tolerance to modelling errors.
+The level of improvement must be traded off against the cost of additional instrumentation.
+
+
+## Implementation and Other Issues {#implementation-and-other-issues}
+
+There are two main categories for the implementation of MIMO control:
+
+-   Decentralized, Decoupled, Cascade
+-   Centralized, optimization based
+
+A fundamental reason to use cascade and decentralized control in most practical applications is because they require less modelling effort.
+Other advantages of cascade and decentralized control are:
+
+-   its behaviour can be easily understood
+-   standard equipment can be used (PID controllers, etc.)
+-   their decoupled behavior enables easier tuning with model-free strategies
+-   decentralized implementation tends to be more fault-tolerant, as individual loops will try to keep their set-points even in the case some other components have failed.
+
+
+### [Anti-Windup Control]({{< relref "anti_windup_control.md" >}}) {#anti-windup-control--anti-windup-control-dot-md}
+
+In practice, it is possible that an actuator saturate.
+In such case, the feedback path is broken, and this has several implications:
+
+-   unstable processes: the process output might go out of control
+-   multi-loop and centralized control: even with stable plants, opening a feedback path may cause the overall loop to become unstable
+
+The wind-up problem can appear with integral action regulators: during significative step changes in the set point, the integral of the error keeps accumulation and when reaching the desired set-point the accumulated integral action produces a significant overshoot increment.
+In SISO PID regulators, anti-windup schemes are implemented by either stopping integration if the actuator is saturated or by implementing the following control law:
+
+\begin{equation}
+u = K(r - y) - K T\_D \frac{dy}{dt} + \int K T\_i^{-1} (r - y) + T\_t^{-1} (u\_m - u) dt \label{eq:antiwindup\_pid}
+\end{equation}
+
+where \\(u\\) is the calculated control action and \\(u\_m\\) is the actual control action applied to the plant.
+In non-saturated behaviour, \\(u=u\_m\\) and the equation is the ordinary PID.
+In saturation, \\(u\_m\\) is a constant and the resulting equations drive \\(u\\) down towards \\(u\_m\\) dynamically, with time constant \\(T\_T\\).
+
+
+### [Bumpless Transfer]({{< relref "bumpless_transfer.md" >}}) {#bumpless-transfer--bumpless-transfer-dot-md}
+
+When switching on the regulator, significant transient behavior can be seen and the controller may saturate the actuators.
+The solution is similar to that of the wind-up phenomenon: the regulator should be always on, carrying out calculations by using \ref{eq:antiwindup\_pid}.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Albertos, P., and S. Antonio. 2004. “Decentralized and Decoupled Control.” In <i>Multivariable Control Systems: An Engineering Approach</i>, 125–62. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. doi:<a href="https://doi.org/10.1007/b97506">10.1007/b97506</a>.</div>
+</div>

content/inbook/steinbuch11_advan_motion_contr_desig.md

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+title = "Advanced Motion Control Design"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Steinbuch et al. 2011</a>)
+
+Author(s)
+: Steinbuch, M., Merry, R., Boerlage, M., Ronde, M., &amp; Molengraft, M.
+
+Year
+: 2011
+
+
+## Introduction {#introduction}
+
+The industrial state of the art control of motion systems can be summarized as follows.
+Most systems, by design, are either decoupled, or can be decoupled using static input-output transformations.
+Hence, most motion systems and their motion software architecture use SISO control design methods and solutions.
+
+Feedback design is mostly done in the frequency domain, using [Loop-Shaping]({{< relref "loop_shaping.md" >}}) techniques.
+A typical motion controller has a PID structure, with a low pass at high frequencies and one or two notch filters to compensate flexible dynamics.
+In addition to the feedback controller, a feedforward controller is applied with acceleration, velocity from the reference signal.
+
+The setpoint itself is a result of a setpoint generator with jerk limitation profiles (see [Trajectory Generation]({{< relref "trajectory_generation.md" >}})).
+If the requirements increase, the dynamic coupling between the various DOFs can no longer be neglected and more advanced MIMO control is required.
+
+<div class="definition">
+
+[Centralized control]({{< relref "decoupled_control.md" >}})
+: the transfer function matrix of the controller is allowed to have any structure
+
+Decentralized control
+: diagonal controller transfer function, but constant decoupling manipulations of inputs and outputs are allowed
+
+Independent decentralized control
+: a single loop is designed without taking into account the effect of earlier or later designed loops
+
+Sequential decentralized control
+: a single loop is designed with taking into account the effect of all earlier closed loops
+
+</div>
+
+
+## Motion Systems {#motion-systems}
+
+Here, we focus on the control of linear time invariant electromechanical motion systems that have the same number of actuators and sensors as Rigid Body modes.
+The dynamics of such systems are often dominated by the mechanics, such that:
+
+\begin{equation}
+G\_p(s) = \sum\_{i=1}^{N\_{rb}} \frac{c\_i b\_i^T}{s^2} + \sum\_{i=N\_{rb} + 1}^{N} \frac{c\_ib\_i^T}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
+\end{equation}
+
+with \\(N\_{rb}\\) is the number of rigid body modes.
+The vectors \\(c\_i,b\_i\\) span the directions of the ith mode shapes.
+
+If the resonance frequencies \\(\omega\_i\\) are high enough, the plant can be approximately decoupled using static input/output transformations \\(T\_u,T\_y\\) so that:
+
+\begin{equation}
+G\_{yu} = T\_y G\_p(s) T\_u = \frac{1}{s^2} \begin{bmatrix}
+m      & 0       & & \dots  &     & 0 \\\\
+0      & m       & &        &     & \\\\
+       & &       m & \ddots &     & \vdots \\\\
+\vdots & & \ddots & I\_x      &     & \\\\
+       & &       &          & I\_y & 0 \\\\
+0      & & \dots &          & 0   & I\_z
+\end{bmatrix} + G\_{\text{flex}}(s)
+\end{equation}
+
+
+## Feedback Control Design {#feedback-control-design}
+
+
+### [Loop-Shaping]({{< relref "loop_shaping.md" >}}) - The SISO case {#loop-shaping--loop-shaping-dot-md--the-siso-case}
+
+The key idea of loopshaping is the modification of the controller such that the open-loop is made according to specifications.
+The reason this works well is that the controller enters linearly into the open-loop transfer function \\(L(s) = G(s)K(s)\\).
+However, in practice all specifications are of course given in terms of the final system performance, that is, as _closed-loop_ specifications.
+So we should convert the closed-loop specifications into specifications on the open-loop.
+
+Take as an example the simple case of a disturbance being a sinusoid of known amplitude and frequency.
+If we know the specifications on the error amplitude, we can derive the requirement on the process sensitivity at that frequency.
+Since at low frequency the sensitivity can be approximated as the inverse of the open-loop, we can translate this into a specification of the open-loop at that frequency.
+Because we know that the slope of the open-loop of a well tuned motion system will be between -2 and -1, we can estimate the required crossover frequency.
+
+
+### Loop-Shaping - The MIMO case {#loop-shaping-the-mimo-case}
+
+In MIMO systems, it is much less trivial to apply loopshaping.
+The stability is determined by the closed-loop polynomial, \\(\det(I + L(s))\\), and the characteristic loci (eigenvalues of the FRF \\(L(j\omega)\\) in the complex plane) can be used for this graphically.
+A system with N inputs and N outputs has N characteristic loci.
+
+If each eigen value locus does not encircle the point (-1,0), the MIMO system is closed-loop stable.
+The shaping of these eigenvalue loci is not straightforward if the plant has large off-diagonal elements.
+In that case, a single element of the controller will affect more eigenvalue loci.
+
+The strong non-intuitive aspect of MIMO loopshaping and the fact that SISO loopshaping is used often, are major obstacles in application of modern design tools in industrial motion systems.
+
+<div class="important">
+
+For that reason, the step-by-step approach is proposed:
+
+1.  [Interaction Analysis]({{< relref "interaction_analysis.md" >}})
+2.  Decoupling Transformations
+3.  Independent SISO design
+4.  Sequential SISO design
+5.  Norm-based MIMO design
+
+</div>
+
+
+#### Interaction Analysis {#interaction-analysis}
+
+The goal of the interaction analysis is to identify two-sided interactions in the plant dynamics.
+Two measured for plant interactions can be used:
+
+-   [Relative Gain Array]({{< relref "relative_gain_array.md" >}}) (RGA) per frequency
+
+    <div class="definition">
+
+    The frequency dependent relative gain array is calculated as:
+
+    \begin{equation}
+    \text{RGA}(G(j\omega)) = G(j\omega) \times (G(j\omega)^{-1})^{T}
+    \end{equation}
+
+    where \\(\times\\) denotes element wise multiplication.
+
+    </div>
+-   [Structure Singular Value]({{< relref "structured_singular_value.md" >}}) (SSV) of interaction as multiplicative output uncertainty
+
+    <div class="definition">
+
+    The structured singular value interaction measure is the following condition:
+
+    \begin{equation}
+    \mu\_D(E\_T(j\omega)) < \frac{1}{2}, \forall \omega
+    \end{equation}
+
+    with \\(E\_T(j\omega) = G\_{nd}(j\omega) G\_d^{-1}(j\omega)\\), \\(\mu\_D\\) is the structured singular value, with respect to the diagonal structure of the feedback controller.
+    \\(G\_d(s)\\) are the diagonal terms of the transfer function matrix, and \\(G\_{nd}(s) = G(s) - G\_d(s)\\).
+
+    If a diagonal transfer function matrix is used, controllers gains must be small at frequencies where this condition is not met.
+
+    </div>
+
+
+#### Decoupling Transformations {#decoupling-transformations}
+
+A common method to reduce plant interaction is to redefine the input and output of the plant.
+One can combine several inputs or outputs to control the system in more decoupled coordinates.
+For motion systems most of these transformations are found on the basis of _kinematic models_.
+Herein, combinations of the actuators are defined so that actuator variables act in independent (orthogonal) directions at the center of gravity.
+Likewise, combinations of the sensors are defined so that each translation and rotation of the center of gravity can be measured independently.
+This is basically the inversion of a kinematic model of the plant.
+
+As motion systems are often designed to be light and stiff, kinematic decoupling is often sufficient to achieve acceptable decoupling at the crossover frequency.
+
+
+#### Independent SISO design {#independent-siso-design}
+
+For systems where interaction is low, or the decoupling is almost successful, one can design a _diagonal_ controller by closing each control loop independently.
+The residual interaction can be accounted for in the analysis.
+
+For this, we make use of the following decomposition:
+
+\begin{equation}
+\det(I + GK) = \det(I + E\_T T\_d) \det(I + G\_d K)
+\end{equation}
+
+with \\(T\_d = G\_d K (I + G\_d K)^{-1}\\).
+\\(G\_d(s)\\) is defined to be only the diagonal terms of the plant transfer function matrix.
+The effect of the non-diagonal terms of the plant \\(G\_{nd}(s) = G(s) - G\_d(s)\\) is accounted for in \\(E\_T(s)\\).
+
+<div class="important">
+
+Then the MIMO closed-loop stability assessment can be slit up in two assessments:
+
+-   the first for stability of N non-interacting loops, namely \\(\det(I + G\_d(s)K(s))\\)
+-   the second for stability of \\(\det(I + E\_T(s)T\_d(s))\\)
+
+</div>
+
+If \\(G(s)\\) and \\(T\_d(s)\\) are stable, one can use the _small gain theorem_ to find a sufficient condition of stability of \\(\det(I + E\_TT\_d)\\) as
+
+\begin{equation}
+\rho(E\_T(j\omega) T\_d(j\omega)) < 1, \forall \omega
+\end{equation}
+
+where \\(\rho\\) is the spectral radius.
+
+Due to the fact that a sufficient condition is used, independent loop closing usually leads to conservative designs.
+
+
+#### Sequential SISO design {#sequential-siso-design}
+
+If the interaction is larger, the sequential loop closing method is appropriate.
+The controller is still a diagonal transfer function matrix, but each control designs are now dependent.
+In principle, one starts with the open-loop FRF of the MIMO Plant.
+Then one loop is closed using SISO loopshaping.
+The controller is taken into the plant description, and a new FRF is obtained with one input and output less.
+Then, the next loop is designed and so on.
+
+The multivariable system is nominally closed-loop stable if in each design step the system is closed-loop stable.
+However, the robustness margins in each design step do not guarantee robust stability of the final multivariable system.
+
+Drawbacks of sequential design are:
+
+-   the ordering of the design steps may have great impact on the achievable performance.
+    There is no general approach to determine the best sequence.
+-   there are no guarantees that robustness margins in earlier loops are preserved.
+-   as each design step usually considers only a single output, the responses in earlier designed loops may degrade.
+
+
+#### Norm-based MIMO design {#norm-based-mimo-design}
+
+If sequential SISO design is not successful, the next step is to start norm-based control design.
+This method requires a parametric model and weighting filters to express the control problem in terms of an operator norm like \\(H\_2\\) or \\(H\_\infty\\).
+
+Parametric models are usually build up step-by-step, first considering the unmodeled dynamics as (unstructured) uncertainty.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Steinbuch, Maarten, Roel Merry, Matthijs Boerlage, Michael Ronde, and Marinus Molengraft. 2011. “Advanced Motion Control Design.” In <i>Control System Applications</i>, 651–76. CRC Press.</div>
+</div>

content/inproceedings/abramovitch23_tutor_real_time_comput_issues_contr_system.md

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+title = "A tutorial on real-time computing issues for control systems"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Abramovitch et al. 2023</a>)
+
+Author(s)
+: Abramovitch, D. Y., Andersson, S., Leang, K. K., Nagel, W., &amp; Ruben, S.
+
+Year
+: 2023
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Abramovitch, D. Y., S. Andersson, K. K. Leang, W. Nagel, and S. Ruben. 2023. “A Tutorial on Real-Time Computing Issues for Control Systems.” In <i>2023 American Control Conference (ACC)</i>, 3751–68. doi:<a href="https://doi.org/10.23919/acc55779.2023.10156102">10.23919/acc55779.2023.10156102</a>.</div>
+</div>

content/inproceedings/henein10_flexur.md

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+title = "Flexures: simply subtle"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Henein 2010</a>)
+
+Author(s)
+: Henein, S.
+
+Year
+: 2010
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Henein, Simon. 2010. “Flexures: Simply Subtle.” In <i>Diamond Light Source Proceedings, MEDSI 2010</i>. Cambridge University Press.</div>
+</div>

content/inproceedings/mcinroy03_proper_stewar.md

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+title = "Properties of orthogonal stewart platform"
+author = ["Dehaeze Thomas"]
+draft = true
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">McInroy 2003</a>)
+
+Author(s)
+: McInroy, J. E.
+
+Year
+: 2003
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>McInroy, John E. 2003. “Properties of Orthogonal Stewart Platform.” In <i>Smart Structures and Materials 2003: Smart Structures and Integrated Systems</i>. doi:<a href="https://doi.org/10.1117/12.483460">10.1117/12.483460</a>.</div>
+</div>

content/phdthesis/afzali-far16_vibrat_dynam_isotr_hexap_analy_studies.md

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+title = "Vibrations and dynamic isotropy in hexapods-analytical studies"
+author = ["Dehaeze Thomas"]
+draft = false
+ref_author = "Afzali-Far, B."
+ref_year = 2016
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Isotropy of Parallel Manipulator]({{< relref "isotropy_of_parallel_manipulator.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Afzali-Far 2016</a>)
+
+Author(s)
+: Afzali-Far, B.
+
+Year
+: 2016
+
+
+## Abstract {#abstract}
+
+> The present work was initiated based on an industrial demand for designing a **high-bandwidth** hexapod of an advanced large optical telescope.
+> In this dissertation, we have generalized this industrial problem to fully-parametric models of the hexapod vibrations as well as analytical studies on dynamic isotropy in parallel robots, which can be directly used in any hexapod applications.
+>
+> This work firstly establishes a comprehensive and fully parametric model for the vibrations in hexapods at symmetric configurations.
+> We have developed three models:
+>
+> -   Cartesian-space formulation
+> -   joint-space formulation
+> -   refined model taking into account the inertia of the struts
+>
+> Kinematics of the hexapod are derived parametrically based on the Jacobian.
+> Inertia, stiffness and damping matrices are also parametrically formulated.
+> The eigenvectors and eigenfrequencies are then established in both the cartesian and joint spaces.
+> By introducing the inertia of the struts, despite the apparent symmetric geometry, the equivalent inertia matrix in the cartesian space turns out to be non-diagonal matrix.
+> In addition, the decoupled vibrations are analytically investigated where it is shown that the consideration of the strut inertia may lead to significant changes of the decoupling conditions.
+>
+> The problem of dynamic isotropy, as an optimal design solution for hexapods, is also addressed in this dissertation.
+> Dynamic isotropy is a condition in which all eigenfrequencies of a robot are equal.
+> This is a powerful tool in order to obtain dynamically optimized architectures for parallel robots.
+> We analytically present the conditions of dynamic isotropy in hexapods with and without the consideration of the strut inertia.
+
+
+## Introduction {#introduction}
+
+The design variables of a hexapod (i.e. geometry, stiffness, damping and inertia properties) can be optimized based upon the requirements on the modal behavior (i.e. eigenfrequencies and eigenvectors of the system).
+To do so, the following is performed parametrically:
+
+-   parametric model
+-   kinematics
+-   linearized equations of motion
+-   modal analysis
+
+The linearized equations of motion are identified by stiffness, damping and inertia matrices.
+These matrices can be expressed in terms of the **cartesian-space** or the **joint-space** coordinates.
+In the cartesian space, the stiffness matrix is a function of the flexibility of the struts as well as the geometrical variables.
+However, in the joint space, the stiffness matrix is not a function of geometrical variables.
+The inertia matrix is a function of inertia properties as well as the geometrical variables.
+
+Dynamic isotropy is an effective tool to avoid scattered eigenfrequencies in a system.
+In a dynamic isotropy condition, all the eigenfrequencies of a system are equal.
+Is is practically almost impossible to obtain dynamic isotropy based on the standard hexapod architecture.
+
+> Hence, due to the fact that the control bandwidth of a hexapod is mechanically restricted by its natural frequencies, the optimization of the natural frequencies is of great importance.
+
+
+## Parametric Modeling of Vibrations {#parametric-modeling-of-vibrations}
+
+
+## Analytical Studies on Dynamics Isotropy {#analytical-studies-on-dynamics-isotropy}
+
+<div class="definition">
+
+(complete) Dynamic isotropy is defined by:
+
+\begin{equation}
+M^{-1} K = \sigma I
+\end{equation}
+
+where \\(\sigma I\\) is a scaled identity matrix.
+This implies that the eigenfrequencies of the matrix \\(M^{-1} K\\) are all equal:
+
+\begin{equation}
+\omega\_1 = \dots = \omega\_6 = \sqrt{\sigma}
+\end{equation}
+
+</div>
+
+Dynamic isotropy for the Stewart platform leads to a series of restrictive conditions and a unique eigenfrequency:
+
+\begin{equation}
+\omega\_i = \sqrt{\frac{2k}{m\_p}}
+\end{equation}
+
+When considering inertia of the struts, conditions are becoming more complex.
+
+<a id="figure--fig:afzali-far16-isotropic-hexapod-example"></a>
+
+{{< figure src="/ox-hugo/afzali-far16_isotropic_hexapod_example.png" caption="<span class='figure-number'>Figure 1: </span>Architecture of the obtained dynamically isotropic hexapod" >}}
+
+<div class="definition">
+
+Static isotropy can be defined by:
+
+\begin{equation}
+K\_C = J^T K\_J J = \sigma I
+\end{equation}
+
+where \\(\sigma I\\) is a scaled identity matrix.
+
+</div>
+
+The isotropic constrain of the standard hexapod imposes special inertia of the top platform which may not be wanted in practice (\\(I\_{zz} = 4 I\_{yy} = 4 I\_{xx}\\)).
+
+A class of generalized Gough-Stewart platforms are proposed to eliminate the above constrains.
+[Figure 2](#figure--fig:afzali-far16-proposed-generalized-hexapod) shows a schematic of proposed generalized hexapod.
+
+<a id="figure--fig:afzali-far16-proposed-generalized-hexapod"></a>
+
+{{< figure src="/ox-hugo/afzali-far16_proposed_generalized_hexapod.png" caption="<span class='figure-number'>Figure 2: </span>Parametrization of the proposed generalized hexapod" >}}
+
+
+## Conclusions {#conclusions}
+
+<div class="sum">
+
+The main findings of this dissertation are:
+
+-   Comprehensive and fully parametric model of the hexapod for symmetric configurations are established both in the Cartesian and joint space.
+-   Inertia of the struts are taken into account to refine the model.
+-   A novel approach in order to obtain dynamically isotropic hexapods is proposed.
+-   A novel architecture of hexapod is introduced ([Figure 2](#figure--fig:afzali-far16-proposed-generalized-hexapod)) which is dynamically isotropic for a wide range of inertia properties.
+
+</div>
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.</div>
+</div>

content/phdthesis/babakhani12_activ_dampin_vibrat_high_precis_motion_system.md

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++++
+title = "Active damping of vibrations in high-precision motion systems"
+author = ["Dehaeze Thomas"]
+draft = false
+ref_author = "Babakhani, B."
+ref_year = 2012
++++
+
+Tags
+: [Active Damping]({{< relref "active_damping.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Babakhani 2012</a>)
+
+Author(s)
+: Babakhani, B.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. doi:<a href="https://doi.org/10.3990/1.9789036534642">10.3990/1.9789036534642</a>.</div>
+</div>

content/phdthesis/bishop02_devel_precis_point_contr_vibrat.md

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++++
+title = "Development of precision pointing controllers with and without vibration suppression for the NPS precision pointing hexapod"
+author = ["Dehaeze Thomas"]
+draft = true
+ref_author = "Bishop Jr, R. M."
+ref_year = 2002
++++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Bishop Jr 2002</a>)
+
+Author(s)
+: Bishop Jr, R. M.
+
+Year
+: 2002
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Bishop Jr, Ronald M. 2002. “Development of Precision Pointing Controllers with and without Vibration Suppression for the NPS Precision Pointing Hexapod.” Naval Postgraduate School, Monterey, California.</div>
+</div>

content/phdthesis/hanieh03_activ_stewar.md

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++++
+title = "Active isolation and damping of vibrations via stewart platform"
+author = ["Dehaeze Thomas"]
+draft = true
+ref_author = "Hanieh, A. A."
+ref_year = 2003
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Active Damping]({{< relref "active_damping.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Abu Hanieh 2003</a>)
+
+Author(s)
+: Hanieh, A. A.
+
+Year
+: 2003
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Abu Hanieh, A. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.</div>
+</div>

content/phdthesis/jabben07_mechat.md

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++++
+title = "Mechatronic design of a magnetically suspended rotating platform"
+author = ["Dehaeze Thomas"]
+draft = false
+ref_author = "Jabben, L."
+ref_year = 2007
++++
+
+Tags
+: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Jabben 2007</a>)
+
+Author
+: Jabben, L.
+
+Year
+: 2007
+
+
+## Dynamic Error Budgeting {#dynamic-error-budgeting}
+
+
+### Introduction {#introduction}
+
+A large class of mechatronic machines have specifications based on their _standstill_ performance.
+The standstill performance is then limited by the (stochastic) disturbances action on the closed loop.
+
+The difficulty in calculation with stochastic signals and Bode plots, is that, instead of calculating with the complex response at one frequency, the **area** over a frequency range should be taken into account.
+
+The **error budgeting** is often used to estimate how much each component contributes to the total error
+
+Since many of the disturbances have a stochastic nature, they can be modelled with their **Power Spectral Densities**.
+
+The PSD of the performance measure in the closed loop system is the weigted sum of PSDs of the contributions of each disturbance to the performance channel.
+This approach allows frequency dependent error budgeting, which is why it is referred to as **Dynamic Error Budgeting**.
+
+
+### Common Mechatronics Disturbances {#common-mechatronics-disturbances}
+
+
+#### Ground vibrations {#ground-vibrations}
+
+
+#### [Electronic Noise]({{< relref "electronic_noise.md" >}}) {#electronic-noise--electronic-noise-dot-md}
+
+**Thermal Noise** (or Johnson noise).
+This noise can be modeled as a voltage source in series with the system impedance.
+The noise source has a PSD given by:
+\\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
+with \\(k = 1.38 \cdot 10^{-23} \\,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
+
+<div class="exampl">
+
+A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \mu V\\) from zero up to one kHz.
+
+</div>
+
+**Shot Noise**.
+Seen with junctions in a transistor.
+It has a white spectral density:
+\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
+with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{dc}\\) the average current [A].
+
+<div class="exampl">
+
+A current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
+
+</div>
+
+**Excess Noise** (or \\(1/f\\) noise).
+It results from fluctuating conductivity due to imperfect contact between two materials.
+The PSD of excess noise increases when the frequency decreases:
+\\[ S\_E = \frac{K\_f}{f^\alpha}\ [V^2/Hz] \\]
+where \\(K\_f\\) is dependent on the average voltage drop over the resistor and the index \\(\alpha\\) is usually between 0.8 and 1.4, and often set to unity for approximate calculation.
+
+**Signal to Noise Ration**
+Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
+In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
+The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
+**The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.**
+Usually, the SNR is specified up to a certain cut-off frequency.
+If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
+\\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
+with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
+
+
+#### AD and DA converters {#ad-and-da-converters}
+
+ADC and DAC add quantization noise to the signal.
+The variance can be calculated to be:
+\\[ \sigma^2 = \frac{q^2}{12} \\]
+with \\(q\\) the quantization interval.
+
+The corresponding PSD is white up to the Nyquist frequency:
+\\[ S\_Q = \frac{q^2}{12 f\_N} \\]
+with \\(f\_N\\) the Nyquist frequency [Hz].
+
+<div class="exampl">
+
+Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
+Let's suppose the ADC is used to measure a position over a range of 1 mm.
+
+-   ADC quantization noise: it has 16 bits over the 1 mm range.
+    The standard deviation from the quantization is:
+    \\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^{16}}{\sqrt{12}} = 4.4\\,[nm] \\]
+-   ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\\,[mm]\\).
+    With a SNR of 80dB, the electronic noise from the ADC becomes:
+    \\[ \sigma\_{ADn} = 35\\,[nm] \\]
+
+Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\\,[nm]\\).
+
+The PSD of this digitalized sensor noise is:
+\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\\,[nm]\\]
+from which the PSD of the total sensor noise \\(S\_s\\) is calculated:
+\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\\,[nm^2/Hz] \\]
+with \\(f\_N\\) is the Nyquist frequency of 1kHz.
+
+</div>
+
+
+#### Acoustic Noise {#acoustic-noise}
+
+This can be a big error source in high precision machines, especially when the surface is big compare to the mass.
+
+The disturbance force acting on a body, is the **difference of pressure between the front and the back times the surface**.
+To have a pressure difference, the body must have a certain minimum dimension, depending on the wave length of the sound.
+For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
+
+<div class="exampl">
+
+Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal \\(2.2 \cdot 10^{-2}\\,[N^2/Hz]\\)
+
+</div>
+
+
+#### Brownian Noise {#brownian-noise}
+
+This is due to thermal effects and it notable where a small mass needs positioning.
+
+
+#### Turbulence {#turbulence}
+
+Rotation of the spindle introduces and air flow in which turbulence is cause by sharp angles on the rotor and stator.
+
+
+### Optimal Control {#optimal-control}
+
+
+#### The use of Optimal Control in DEB {#the-use-of-optimal-control-in-deb}
+
+Three factors influence the performance:
+
+-   the disturbances: often a given value
+-   the plant: can be costly to redesign
+-   the controller
+
+The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
+
+The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
+In [Figure 1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
+By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
+
+<a id="figure--fig:jabben07-general-plant"></a>
+
+{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="<span class='figure-number'>Figure 1: </span>Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
+
+
+#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
+
+Since disturbances are generally not white, the system of [Figure 1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
+
+A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
+
+This is illustrated in [Figure 2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
+
+The generalized plant framework also allows to include **weighting filters for the performance channels**.
+This is useful for three reasons:
+
+-   the performance channels might have different dimensions, which require scaling in order to compare
+-   some performance channels may be of more importance than others
+-   by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
+
+<a id="figure--fig:jabben07-weighting-functions"></a>
+
+{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="<span class='figure-number'>Figure 2: </span>Control system with the generalized plant \\(G\\) and weighting functions" >}}
+
+The weighting filters should be stable transfer functions.
+
+**Obtaining the weighting filters**:
+
+If the PSD is given as a function \\(S\_x(j\omega)\\), the disturbance filter can be using **spectral factorization**:
+
+> Given a positive even function \\(S\_x(f)\\) of finite area, find a minimum-phase stable function \\(L(s)\\), such that \\(|L(j2\pi f)|^2 = S(s)\\)
+
+**Harmonic signals** can be approximately modeled by filtering white noise with a badly damped second order system, having a \\(+1\\) slope below the resonance frequency and a \\(-1\\) slope above the resonance frequency:
+\\[ V\_h = \frac{s}{s^2 + 2 \xi \omega\_h + \omega\_h^2} \\]
+with \\(\xi\\) the relative damping and \\(\omega\_h\\) the resonance frequency [rad/s].
+By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of the harmonic signal, the propagation of the disturbance to the performance channel can be well approximated.
+
+
+#### Balancing Control Effort vs Performance {#balancing-control-effort-vs-performance}
+
+IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
+So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
+By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
+\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
+
+By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in [Figure 3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
+
+<a id="figure--fig:jabben07-pareto-curve-H2"></a>
+
+{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="<span class='figure-number'>Figure 3: </span>An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
+
+
+## Conclusion {#conclusion}
+
+> Using the DEB analysis during the design helped to formulate the specifications of the several subcomponents, such as:
+>
+> -   The target bandwidth of the decentralized closed loops, which is very important for the mechanical design, as mechanical resonances can severely limit the bandwidth.
+>     This value was also used to specify the current loop bandwidth of the custom designed power amplifiers for the RTAs and other components such as sensors and filters.
+> -   The target value of the stiffness of the actuators was derived at 1000 N/m.
+>     It was shown that the stiffness of a motor with back-iron is too much for the separated frame concept.
+> -   The analysis pinpointed the most limiting component in the final design to be the Analogue-to-Digital Converter (ADC).
+
+<!--quoteend-->
+
+> In the DEB-framework there are three distinct factors which determine the performance.
+> These are the plant, the controller and the disturbances.
+> Synthesizing optimal controllers, such as H2-control, in the design helps to eliminate the controller out of the equation.
+> If the performance specifications are not met with an optimal controller, it is certain that a redesign of the system is required.
+> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
+> To derive such models, spectral factorization is used.
+> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.

content/phdthesis/li01_simul_fault_vibrat_isolat_point.md

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+title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
+author = ["Dehaeze Thomas"]
+draft = false
+ref_author = "Li, X."
+ref_year = 2001
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}}), [Multivariable Control]({{< relref "multivariable_control.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Li 2001</a>)
+
+Author(s)
+: Li, X.
+
+Year
+: 2001
+
+
+## Introduction {#introduction}
+
+
+### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
+
+A general flexible jointed hexapod is shown in [Figure 1](#figure--fig:li01-flexure-hexapod-model).
+
+<a id="figure--fig:li01-flexure-hexapod-model"></a>
+
+{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="<span class='figure-number'>Figure 1: </span>A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
+
+Flexure jointed hexapods have been developed to meet two needs illustrated in [Figure 2](#figure--fig:li01-quet-dirty-box).
+
+<a id="figure--fig:li01-quet-dirty-box"></a>
+
+{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="<span class='figure-number'>Figure 2: </span>(left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
+
+Since only small movements are considered in flexure jointed hexapod, the Jacobian matrix, which relates changes in the Cartesian pose to changes in the strut lengths, can be considered constant.
+Thus a static kinematic decoupling algorithm can be implemented for both vibration isolation and pointed controls on flexible jointed hexapods.
+
+On the other hand, the flexures add some complexity to the hexapod dynamics.
+Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
+Moreover, base and/or payload vibrations become significant contributors to the motion.
+
+The University of Wyoming hexapods (example in [Figure 3](#figure--fig:li01-stewart-platform)) are:
+
+-   Cubic (mutually orthogonal)
+-   Flexure Jointed
+
+<a id="figure--fig:li01-stewart-platform"></a>
+
+{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="<span class='figure-number'>Figure 3: </span>Flexure jointed Stewart platform used for analysis and control" >}}
+
+The objectives of the hexapods are:
+
+-   Precise pointing in two axes (sub micro-radians)
+-   simultaneously, providing both passive and active vibration isolation in six axes
+
+
+### Jacobian matrix, Dynamic model, and decoupling algorithms {#jacobian-matrix-dynamic-model-and-decoupling-algorithms}
+
+
+#### Jacobian Matrix {#jacobian-matrix}
+
+The Jacobian matrix \\(J\\) relates changes in the cartesian pose \\(\mathcal{X}\\) to changes in the strut lengths \\(l\\):
+
+\begin{equation}
+\delta l = J \delta \mathcal{X}
+\end{equation}
+
+where \\(\mathcal{X}\\) is a 6x1 vector of payload plate translations and rotations
+
+\begin{equation}
+\mathcal{X} = \begin{bmatrix}
+p\_x & p\_y & p\_z & \theta\_x & \theta\_y & \theta\_z
+\end{bmatrix}
+\end{equation}
+
+\\(J\\) is given by:
+
+\begin{equation}
+J = \begin{bmatrix}
+{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\
+\vdots & \vdots \\\\
+{}^B\hat{u}\_6^T & [({}^B\_PR^P p\_6) \times {}^B\hat{u}\_6]^T
+\end{bmatrix}
+\end{equation}
+
+where (see [Figure 1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
+To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
+Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
+
+
+#### Dynamic model of flexure jointed hexapods {#dynamic-model-of-flexure-jointed-hexapods}
+
+The dynamics of a flexure jointed hexapod can be written in joint space:
+
+\begin{equation} \label{eq:hexapod\_eq\_motion}
+\begin{split}
+& \left( J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T \cdot J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\
+&\quad f\_m - \left( M\_s + J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + J^{-T} \cdot {}^U\_BR^T(\mathcal{F}\_e + \mathcal{G} + \mathcal{C})
+\end{split}
+\end{equation}
+
+where:
+
+-   \\({}^PM\_x\\) is the 6x6 mass/inertia matrix of the payload, found with respect to the payload frame {P}, whose origin is at the hexapod payload's center of mass
+-   \\({}^U\_BR\\) is the 6x6 rotation matrix from the base frame {B} to the inertial frame of reference {U} (it consists of two identical 3x3 rotation matrices forming a block diagonal 6x6 matrix).
+    Similarly, \\({}^B\_PR\\) is the rotation matrix from the payload frame to the base frame, and \\({}^U\_PR = {}^U\_BR {}^B\_PR\\)
+-   \\(J\\) is the 6x6 Jacobian matrix relating payload cartesian movements to strut length changes
+-   \\(M\_s\\) is a diagonal 6x6 matrix containing the moving mass of each strut
+-   \\(l\\) is the 6x1 vector of strut lengths
+-   \\(B\\) and \\(K\\) are 6x6 diagonal matrices containing the damping and stiffness, respectively, of each strut
+-   \\(l\_r\\) is the constant vector of relaxed strut lengths
+-   \\(f\_m\\) is the vector of strut motor force
+-   \\(J\_c\\) and \\(J\_b\\) are 6x6 Jacobian matrices capturing base motion
+-   \\(\ddot{q}\_u\\) is a 6x1 vector of base acceleration along each strut
+-   \\(\mathcal{F}\_r\\) is a vector of payload exogenous generalized forces
+-   \\(\mathcal{C}\\) is a vector containing all the Coriolis and centripetal terms
+-   \\(\mathcal{G}\\) is a vector containing all gravity terms
+
+
+#### Decoupling {#decoupling}
+
+Two decoupling algorithms are proposed by combining static input-output transformations with hexapod geometric design.
+
+Define a new input and a new output:
+
+\begin{equation}
+u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
+\end{equation}
+
+Equation \ref{eq:hexapod\_eq\_motion} can be rewritten as:
+
+\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
+\begin{split}
+& \left( {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T + J^T \cdot M\_s \cdot J \right) \cdot \ddot{y} + J^T \cdot B J \dot{y} + J^T \cdot K \cdot J y = \\\\
+&\quad u\_1 - \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
+\end{split}
+\end{equation}
+
+If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled ([Figure 4](#figure--fig:li01-decoupling-conf)).
+
+<a id="figure--fig:li01-decoupling-conf"></a>
+
+{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="<span class='figure-number'>Figure 4: </span>Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
+
+Alternatively, a new set of inputs and outputs can be defined:
+
+\begin{equation}
+u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
+\end{equation}
+
+And another decoupled plant is found ([Figure 5](#figure--fig:li01-decoupling-conf-bis)):
+
+\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
+\begin{split}
+& \left( J^{-1} \cdot J^{-T} \cdot {}^BM\_x + M\_s \right) \cdot \ddot{y} + B \dot{y} + K y = \\\\
+&\quad u\_2 - J^{-1} \cdot J^{-T} \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
+\end{split}
+\end{equation}
+
+<a id="figure--fig:li01-decoupling-conf-bis"></a>
+
+{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="<span class='figure-number'>Figure 5: </span>Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
+
+<div class="important">
+
+These decoupling algorithms have two constraints:
+
+1.  the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\))
+2.  the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen
+
+For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
+
+</div>
+
+
+## Simultaneous Vibration Isolation and Pointing Control {#simultaneous-vibration-isolation-and-pointing-control}
+
+Many applications require simultaneous vibration isolation and precision pointing.
+
+The basic idea to achieve such objective is to use:
+
+-   acceleration feedback to provide high-frequency vibration isolation
+-   cartesian pointing feedback to provide low-frequency pointing
+
+The compensation is divided in frequency because:
+
+-   pointing sensors often have low bandwidth
+-   acceleration sensors often have a poor low frequency response
+
+The control bandwidth is divided as follows:
+
+-   low-frequency disturbances are attenuated and tracking is accomplished by feedback from low bandwidth pointing sensors
+-   mid-frequency disturbances are attenuated by feedback from band-pass sensors like accelerometer or load cells
+-   high-frequency disturbances are attenuated by passive isolation techniques
+
+
+### Vibration Isolation {#vibration-isolation}
+
+The system is decoupled into six independent SISO subsystems using the architecture shown in [Figure 7](#figure--fig:li01-vibration-isolation-control).
+
+<a id="figure--fig:li01-vibration-isolation-control"></a>
+
+{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="<span class='figure-number'>Figure 6: </span>Vibration isolation control strategy" >}}
+
+One of the subsystem plant transfer function is shown in [Figure 7](#figure--fig:li01-vibration-isolation-control)
+
+<a id="figure--fig:li01-vibration-isolation-control"></a>
+
+{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="<span class='figure-number'>Figure 7: </span>Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
+
+Each compensator is designed using simple loop-shaping techniques.
+A typical compensator consists of the following elements:
+
+-   first order lag-lead filter to provide adequate phase margin a the low frequency crossover
+-   a second order lag-lead filter to increase the gain between crossovers and provide adequate phase margin at the high frequency crossover
+-   a second order notch filter to cancel the mode at 150Hz
+-   a second order low pass filter to provide steep roll-off and gain stabilize the plant at high frequency
+-   a first order high pass filter to eliminate DC signals
+
+The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**, so the vibration isolation loop works as a band-pass filter.
+
+<div class="important">
+
+Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling.
+Only about 20 dB separation is achieve between the diagonal and off-diagonal responses.
+
+</div>
+
+<div class="note">
+
+Severe phase delay exists in the actual transfer function.
+This is due to the limited sample frequency and sensor bandwidth limitation.
+
+The zero at around 130Hz is non-minimum phase which limits the control bandwidth.
+The reason is not explained.
+
+</div>
+
+
+### Pointing Control Techniques {#pointing-control-techniques}
+
+A block diagram of the pointing control system is shown in [Figure 8](#figure--fig:li01-pointing-control).
+
+<a id="figure--fig:li01-pointing-control"></a>
+
+{{< figure src="/ox-hugo/li01_pointing_control.png" caption="<span class='figure-number'>Figure 8: </span>Figure caption" >}}
+
+The plant is decoupled into two independent SISO subsystems.
+The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
+
+[Figure 9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
+
+<a id="figure--fig:li01-transfer-function-angle"></a>
+
+{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="<span class='figure-number'>Figure 9: </span>Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
+
+A typical compensator consists of the following elements:
+
+-   a first order low pass filter to increase the low frequency loop gain and provide a slope of -20dB/decade for the magnitude curve at the crossover
+-   two complex zeros with high \\(Q\\) to provide adequate phase margin at the crossover
+-   a pole after the zeros to decrease the excess gain caused by these zeros
+-   a second order notch filter to cancel the mode at 150Hz
+-   a second order low pass filter to provide steep roll off and gain stabilize the plant at high frequency
+
+The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
+
+A feedforward control is added as shown in [Figure 10](#figure--fig:li01-feedforward-control).
+\\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
+Ideally, the feedforward compensator is an invert of the plant dynamics.
+
+<a id="figure--fig:li01-feedforward-control"></a>
+
+{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="<span class='figure-number'>Figure 10: </span>Feedforward control" >}}
+
+
+### Simultaneous Control {#simultaneous-control}
+
+The simultaneous vibration isolation and pointing control is approached in two ways:
+
+1.  **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
+2.  **Closing the pointing loop first**: Reverse order.
+
+[Figure 11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
+
+<a id="figure--fig:li01-parallel-control"></a>
+
+{{< figure src="/ox-hugo/li01_parallel_control.png" caption="<span class='figure-number'>Figure 11: </span>A parallel scheme" >}}
+
+<div class="important">
+
+The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled.
+The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation.
+
+However, the interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability.
+
+</div>
+
+The dynamic interaction effect:
+
+-   Only happens in the unity bandwidth of the loop transmission of the first closed loop.
+-   Affect the closed loop transmission of the loop first closed (see [Figure 12](#figure--fig:li01-closed-loop-pointing) and [Figure 13](#figure--fig:li01-closed-loop-vibration))
+
+As shown in [Figure 12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
+The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
+
+<a id="figure--fig:li01-closed-loop-pointing"></a>
+
+{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="<span class='figure-number'>Figure 12: </span>Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
+
+The same happens when first closing the vibration isolation loop and after the pointing loop ([Figure 13](#figure--fig:li01-closed-loop-vibration)).
+The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
+
+<a id="figure--fig:li01-closed-loop-vibration"></a>
+
+{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="<span class='figure-number'>Figure 13: </span>Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
+
+<div class="important">
+
+From the analysis above, it is hard to say which loop has more significant affect on the other loop, but the isolation loop adds a second resonance peak at its high frequency crossover in the pointing closed loop transfer function, which may cause instability.
+Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed.
+
+</div>
+
+
+### Experimental results {#experimental-results}
+
+Two hexapods are stacked ([Figure 14](#figure--fig:li01-test-bench)):
+
+-   the bottom hexapod is used to generate disturbances matching candidate applications
+-   the top hexapod provide simultaneous vibration isolation and pointing control
+
+<a id="figure--fig:li01-test-bench"></a>
+
+{{< figure src="/ox-hugo/li01_test_bench.png" caption="<span class='figure-number'>Figure 14: </span>Stacked Hexapods" >}}
+
+First, the vibration isolation and pointing controls were implemented separately.
+Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in [Figure 15](#figure--fig:li01-vibration-isolation-control-results).
+
+<a id="figure--fig:li01-vibration-isolation-control-results"></a>
+
+{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="<span class='figure-number'>Figure 15: </span>Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
+
+The simultaneous control is of dual use:
+
+-   it provide simultaneous pointing and isolation control
+-   it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
+
+The results of simultaneous control is shown in [Figure 16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
+
+<a id="figure--fig:li01-simultaneous-control-results"></a>
+
+{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="<span class='figure-number'>Figure 16: </span>Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
+
+
+### Summary and Conclusion {#summary-and-conclusion}
+
+<div class="sum">
+
+A parallel control scheme is proposed in this chapters.
+This scheme is suitable for simultaneous vibration isolation and pointing control.
+Part of this scheme involves closing one loop first, then re-identifying and designing the new control before closed the other loop.
+
+An investigation into the interaction between loops shows that the order of closing loops is not important.
+However, only two channels need to be re-designed or adjusted for the pointing loop if the isolation loop is closed first.
+Experiments show that this scheme takes advantage of the bandwidths of both pointing and vibration sensors, and provides vibration isolation and pointing controls over a broad band.
+
+</div>
+
+
+## Future research areas {#future-research-areas}
+
+<div class="sum">
+
+Proposed future research areas include:
+
+-   **Include base dynamics in the control**:
+    The base dynamics is here neglected since the movements of the base are very small.
+    The base dynamics could be measured by mounting accelerometers at the bottom of each strut or by using force sensors.
+    It then could be included in the feedforward path.
+-   **Robust control and MIMO design**
+-   **New decoupling method**:
+    The proposed decoupling algorithm do not produce the expected decoupling, despite a reasonably good match between the modeled and the measured transfer functions.
+    Incomplete decoupling increases the difficulty in designing the controller.
+    New decoupling methods are needed.
+    These methods must be static in order to be implemented practically on precision hexapods
+-   **Identification**:
+    Many advanced control methods require a more accurate model or identified plant.
+    A closed-loop identification method is propose to solve some problems with the current identification methods used.
+-   **Other possible sensors**:
+    Many sensors can be used to expand the utility of the Stewart platform:
+    -   **3-axis load cells** to investigate the Coriolis and centripetal terms and new decoupling methods
+    -   **LVDT** to provide differential position of the hexapod payload with respect to the base
+    -   **Geophones** to provide payload and base velocity information
+
+</div>
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, X. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.</div>
+</div>