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@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}}) : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
Reference Reference
: ([Preumont 2018](#org09bc150)) : ([Preumont 2018](#org2443fdb))
Author(s) Author(s)
: Preumont, A. : Preumont, A.
@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback} #### Feedback {#feedback}
<a id="org3a6239d"></a> <a id="org17539d6"></a>
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}} {{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#org3a6239d). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance. The principle of feedback is represented on figure [1](#org17539d6). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner. The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**. In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward} #### Feedforward {#feedforward}
<a id="org0c27bf0"></a> <a id="orgb6b4033"></a>
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}} {{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**. The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org0c27bf0). The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#orgb6b4033).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized. The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
### The Various Steps of the Design {#the-various-steps-of-the-design} ### The Various Steps of the Design {#the-various-steps-of-the-design}
<a id="org4371fad"></a> <a id="org0ed85b6"></a>
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}} {{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#org4371fad). The various steps of the design of a controlled structure are shown in figure [3](#org0ed85b6).
The **starting point** is: The **starting point** is:
@ -154,14 +154,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget} ### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#orgffc7bae)): From the block diagram of the control system (figure [4](#org90c3880)):
\begin{align\*} \begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\ y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
\end{align\*} \end{align\*}
<a id="orgffc7bae"></a> <a id="org90c3880"></a>
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}} {{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
@ -186,12 +186,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\). It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response. \\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3ce8916). A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3209437).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted. It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration. The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
<a id="org3ce8916"></a> <a id="org3209437"></a>
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}} {{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
@ -398,11 +398,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i} D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation} \end{equation}
<a id="org4219cc6"></a> <a id="org4377aea"></a>
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}} {{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#org4219cc6)). If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#org4377aea)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**: And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\] \\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@ -441,9 +441,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real: If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\] \\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgd4f4723)). \\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgd6e521d)).
<a id="orgd4f4723"></a> <a id="orgd6e521d"></a>
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}} {{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
</div> </div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org5748d4e). If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org3cc8875).
<a id="org5748d4e"></a> <a id="org3cc8875"></a>
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}} {{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div> </div>
By looking at figure [7](#orgd4f4723), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right. By looking at figure [7](#orgd6e521d), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="org0dd5dad"></a> <a id="org145a286"></a>
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}} {{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)} G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation} \end{equation}
The corresponding Bode plot is represented in figure [9](#org0dd5dad). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\). The corresponding Bode plot is represented in figure [9](#org145a286). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property. In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@ -508,12 +508,12 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa. A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#org576a7f9)): The system consists of (see figure [10](#org589d929)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap - A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially - A coil which is free to move axially
<a id="org576a7f9"></a> <a id="org589d929"></a>
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}} {{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
@ -551,9 +551,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator} #### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orge7047bc)). A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#org29d3a41)).
<a id="orge7047bc"></a> <a id="org29d3a41"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}} {{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
@ -583,9 +583,9 @@ with:
</div> </div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#orgf19efd8)). Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#orgfadd8ec)).
<a id="orgf19efd8"></a> <a id="orgfadd8ec"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}} {{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
@ -610,7 +610,7 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\). Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
<a id="org3d1f388"></a> <a id="orgfc5e619"></a>
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}} {{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti
### General Electromechanical Transducer {#general-electromechanical-transducer} ### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org6266d1c). The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org0f65711).
<a id="org6266d1c"></a> <a id="org0f65711"></a>
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}} {{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
@ -646,7 +646,7 @@ With:
Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals. Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained. Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#org37d6035) can be used. To do so, the bridge circuit as shown on figure [15](#orgf6f982f) can be used.
We can show that We can show that
@ -656,7 +656,7 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals. which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
<a id="org37d6035"></a> <a id="orgf6f982f"></a>
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}} {{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
### Smart Materials {#smart-materials} ### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature. Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org4f66ebd) lists various effects that are observed in materials in response to various inputs. Figure [16](#org4c1156c) lists various effects that are observed in materials in response to various inputs.
<a id="org4f66ebd"></a> <a id="org4c1156c"></a>
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}} {{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div> </div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org9f89282)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer: If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org9ace96d)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
\begin{equation} \begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix} \begin{bmatrix}Q\\\Delta\end{bmatrix}
@ -782,7 +782,7 @@ where
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\)) - \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\)) - \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="org9f89282"></a> <a id="org9ace96d"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer} #### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org1d36e1c). Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#orgb03700a).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
\end{equation} \end{equation}
<a id="org1d36e1c"></a> <a id="orgb03700a"></a>
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
@ -844,10 +844,10 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer} #### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#org645378f), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\). Consider the system of figure [19](#orga70a814), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\). The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="org645378f"></a> <a id="orga70a814"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
@ -866,9 +866,9 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2 \frac{z^2 - p^2}{z^2} = k^2
\end{equation} \end{equation}
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org68045c5)). Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#orgc632b09)).
<a id="org68045c5"></a> <a id="orgc632b09"></a>
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
@ -1566,7 +1566,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC. When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgb182ef8). The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgfaf8470).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages: This approach has the following advantages:
@ -1574,7 +1574,7 @@ This approach has the following advantages:
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin) - The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin) - The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="orgb182ef8"></a> <a id="orgfaf8470"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}} {{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
@ -1818,4 +1818,4 @@ This approach has the following advantages:
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org09bc150"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>. <a id="org2443fdb"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.

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title = "Singular Value Decomposition"
author = ["Thomas Dehaeze"]
draft = false
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Tags
:
## SVD of a MIMO system {#svd-of-a-mimo-system}
We are interested by the physical interpretation of the SVD when applied to the frequency response of a MIMO system \\(G(s)\\) with \\(m\\) inputs and \\(l\\) outputs.
\begin{equation}
G = U \Sigma V^H
\end{equation}
\\(\Sigma\\)
: is an \\(l \times m\\) matrix with \\(k = \min\\{l, m\\}\\) non-negative **singular values** \\(\sigma\_i\\), arranged in descending order along its main diagonal, the other entries are zero.
\\(U\\)
: is an \\(l \times l\\) unitary matrix. The columns of \\(U\\), denoted \\(u\_i\\), represent the **output directions** of the plant. They are orthonormal.
\\(V\\)
: is an \\(m \times m\\) unitary matrix. The columns of \\(V\\), denoted \\(v\_i\\), represent the **input directions** of the plant. They are orthonormal.
The input and output directions are related through the singular values:
\begin{equation}
G v\_i = \sigma\_i u\_i
\end{equation}
So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\).
Furthermore, since \\(\normtwo{v\_i}=1\\) and \\(\normtwo{u\_i}=1\\), we see that **the singular value \\(\sigma\_i\\) directly gives the gain of the matrix \\(G\\) in this direction**.
The **largest gain** for any input is equal to the **maximum singular value**:
\\[\maxsv(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_1}}{\normtwo{v\_1}} \\]
The **smallest gain** for any input direction is equal to the **minimum singular value**:
\\[\minsv(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_k}}{\normtwo{v\_k}} \\]
We define \\(u\_1 = \bar{u}\\), \\(v\_1 = \bar{v}\\), \\(u\_k=\ubar{u}\\) and \\(v\_k = \ubar{v}\\). Then is follows that:
\\[ G\bar{v} = \maxsv \bar{u} ; \quad G\ubar{v} = \minsv \ubar{u} \\]
## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices}
This is taken from [Preumont's book](preumont18_vibrat_contr_activ_struc_fourt_edition.md).
The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix:
\\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\]
With:
- \\(U\\) and \\(V\\) orthogonal matrices. The columns \\(u\_i\\) and \\(v\_i\\) of \\(U\\) and \\(V\\) are the eigenvectors of the square matrices \\(JJ^T\\) and \\(J^TJ\\) respectively
- \\(\Sigma\\) a rectangular diagonal matrix of dimension \\(m \times n\\) containing the square root of the common non-zero eigenvalues of \\(JJ^T\\) and \\(J^TJ\\)
- \\(r\\) is the number of non-zero singular values of \\(J\\)
The pseudo-inverse of \\(J\\) is:
\\[ J^+ = V\Sigma^+U^T = \sum\_{i=1}^r \frac{1}{\sigma\_i} v\_i u\_i^T \\]
The conditioning of the Jacobian is measured by the **condition number**:
\\[ c(J) = \frac{\sigma\_{max}}{\sigma\_{min}} \\]
When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum.
This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\).
<./biblio/references.bib>

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| Newport | [link](https://www.newport.com/search/?q1=hexapod%3Arelevance%3Acompatibility%3AMETRIC%3AisObsolete%3Afalse%3A-excludeCountries%3AFR%3AnpCategory%3Ahexapods&ajax&text=hexapod) | USA | | Newport | [link](https://www.newport.com/search/?q1=hexapod%3Arelevance%3Acompatibility%3AMETRIC%3AisObsolete%3Afalse%3A-excludeCountries%3AFR%3AnpCategory%3Ahexapods&ajax&text=hexapod) | USA |
| Symetrie | [link](https://symetrie.fr/en/hexapods-en/positioning-hexapods/) | France | | Symetrie | [link](https://symetrie.fr/en/hexapods-en/positioning-hexapods/) | France |
| CSA Engineering | [link](https://www.csaengineering.com/products-services/hexapod-positioning-systems/hexapod-models.html) | USA | | CSA Engineering | [link](https://www.csaengineering.com/products-services/hexapod-positioning-systems/hexapod-models.html) | USA |
| Aerotech | [link](https://www.aerotech.com/product-catalog/hexapods.aspx) | USA |
| SmarAct | [link](https://www.smaract.com/smarpod) | Germany |
| Gridbots | [link](https://www.gridbots.com/hexamove.html) | India |
## Stewart Platforms at ESRF {#stewart-platforms-at-esrf} ## Stewart Platforms at ESRF {#stewart-platforms-at-esrf}
@ -56,36 +59,36 @@ Tags
Papers by J.E. McInroy: Papers by J.E. McInroy:
- ([O’Brien et al. 1998](#org6990e82)) - ([O’Brien et al. 1998](#orgb1b8013))
- ([McInroy, O’Brien, and Neat 1999](#org2eaf165)) - ([McInroy, O’Brien, and Neat 1999](#org06dafcc))
- ([McInroy 1999](#org19d227f)) - ([McInroy 1999](#orga04f28d))
- ([McInroy and Hamann 2000](#org0de51ce)) - ([McInroy and Hamann 2000](#org61fde10))
- ([Chen and McInroy 2000](#orgfa121ee)) - ([Chen and McInroy 2000](#org4b76086))
- ([McInroy 2002](#org1ad3b68)) - ([McInroy 2002](#orgc3f19a5))
- ([Li, Hamann, and McInroy 2001](#orga0bd81f)) - ([Li, Hamann, and McInroy 2001](#org469fdd4))
- ([Lin and McInroy 2003](#org01eae42)) - ([Lin and McInroy 2003](#org52d73bd))
- ([Jafari and McInroy 2003](#org9490dc9)) - ([Jafari and McInroy 2003](#orga8e7146))
- ([Chen and McInroy 2004](#org3bd3a36)) - ([Chen and McInroy 2004](#org97f0e30))
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org3bd3a36"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>. <a id="org97f0e30"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
<a id="orgfa121ee"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="org4b76086"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="org9490dc9"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>. <a id="orga8e7146"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
<a id="org01eae42"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>. <a id="org52d73bd"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
<a id="orga0bd81f"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>. <a id="org469fdd4"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
<a id="org19d227f"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>. <a id="orga04f28d"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="org1ad3b68"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>. <a id="orgc3f19a5"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<a id="org0de51ce"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>. <a id="org61fde10"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
<a id="org2eaf165"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>. <a id="org06dafcc"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
<a id="org6990e82"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>. <a id="orgb1b8013"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.