Update Content - 2022-04-28

content/article/oomen18_advan_motion_contr_precis_mechat.md

@@ -1,7 +1,7 @@
 +++
 title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
 author = ["Dehaeze Thomas"]
-draft = true
+draft = false
 +++
 
 Tags
@@ -16,12 +16,171 @@ Author(s)
 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." >}}
 
 
-## Bibliography {#bibliography}
+## 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}
+
+## References
 
 <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>

content/zettels/feedback_control.md

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+title = "Feedback Control"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+Tags
+:
+
+## References
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/feedforward_control.md

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+title = "Feedforward Control"
+author = ["Dehaeze Thomas"]
+draft = false
++++
+
+Tags
+:
+
+## References
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/passive_damping.md

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+title = "Passive Damping"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+Tags
+:
+
+
+## Bibliography {#bibliography}
+
+## References
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>

content/zettels/tuned_mass_damper.md

@@ -23,16 +23,16 @@ The TMD then has large internal damping such that the energy is dissipated (i.e.
 
 The optimal parameters of the tuned mass damper can be roughly estimated as follows:
 
--   Choose the maximum mass of the TMD \\(m\\) and note:
-    \\[ \mu = m/M \\]
-    where \\(M\\) is the mass of the system to damp
+-   Choose the maximum acceptable mass of the TMD \\(m\_2\\) and note:
+    \\[ \mu = m\_2/m\_1 \\]
+    where \\(m\_1\\) is the mass of the system to damp
 -   The resonance frequency of the tuned mass damper should be chosen to be
     \\[ \nu = \frac{1}{1 + \mu} \approx 1 \\]
     As usually we have \\(\mu \ll 1\\) (i.e. TMD mass small compared to the structure mass, for instance few percent)
 -   This allows to compute the stiffness of the TMD:
-    \\[ k = \nu^2 K \mu = K \frac{\mu}{(1 + \mu)^2} \\]
+    \\[ k\_2 = \nu^2 k\_1 \mu = k\_1 \frac{\mu}{(1 + \mu)^2} \\]
 -   Finally, the optimal damping of the TMD is:
-    \\[ \xi = \sqrt{\frac{3\mu}{8 (1 + \mu)}} \Longrightarrow c = 2 \xi \sqrt{k m} \\]
+    \\[ \xi\_2 = \sqrt{\frac{3 \mu}{8 (1 + \mu)}} \Longrightarrow c\_2 = 2 \xi\_2 \sqrt{k\_2 m\_2} \\]
 
 
 ## Simple TMD model {#simple-tmd-model}