Update Content - 2021-05-02

2021-05-02 20:37:00 +02:00
parent 699e8a7905
commit f975b4bd90
77 changed files with 1212 additions and 666 deletions
--- a/content/book/albertos04_multiv_contr_system.md
+++ b/content/book/albertos04_multiv_contr_system.md
@@ -8,7 +8,7 @@ Tags
 : [Multivariable Control]({{< relref "multivariable_control" >}})

 Reference
-: ([Albertos and Antonio 2004](#org911b8ba))
+: ([Albertos and Antonio 2004](#orgb06343d))

 Author(s)
 : Albertos, P., & Antonio, S.
@@ -17,6 +17,116 @@ Year
 : 2004


+## Introduction to Multivariable Control {#introduction-to-multivariable-control}
+
+
+## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence}
+
+
+## Linear Systems Analysis {#linear-systems-analysis}
+
+
+## Solutions to the Control Problem {#solutions-to-the-control-problem}
+
+
+## Decentralised and Decoupled Control {#decentralised-and-decoupled-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" >}}) 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 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.
+
+**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.
+
+
+#### Feedback Decoupling {#feedback-decoupling}
+
+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" >}}) 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 [1](#org3d5b40c), 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="org3d5b40c"></a>
+
+{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: 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.
+
+
+## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control}
+
+
+## Optimisation-based Control {#optimisation-based-control}
+
+
+## Designing for Robustness {#designing-for-robustness}
+
+
+## Implementation and Other Issues {#implementation-and-other-issues}
+
+
+## Appendices {#appendices}
+
+
+### Summary of SISO System Analysis {#summary-of-siso-system-analysis}
+
+
+### Matrices {#matrices}
+
+
+### Signal and System Norms {#signal-and-system-norms}
+
+
+### Optimisation {#optimisation}
+
+
+### Multivariable Statistics {#multivariable-statistics}
+
+
+### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis}
+
+
+
 ## Bibliography {#bibliography}

-<a id="org911b8ba"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
+<a id="orgb06343d"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.
--- a/content/book/du10_model_contr_vibrat_mechan_system.md
+++ b/content/book/du10_model_contr_vibrat_mechan_system.md
@@ -8,7 +8,7 @@ Tags
 : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})

 Reference
-: ([Du and Xie 2010](#orge0a6379))
+: ([Du and Xie 2010](#org5093366))

 Author(s)
 : Du, C., & Xie, L.
@@ -16,9 +16,524 @@ Author(s)
 Year
 : 2010

-Read Chapter 1 and 3.
+
+## 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" >}}) {#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 &#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&#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&#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&#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&#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&#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&#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&#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}

-<a id="orge0a6379"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.
+<a id="org5093366"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.
--- a/content/book/hatch00_vibrat_matlab_ansys.md
+++ b/content/book/hatch00_vibrat_matlab_ansys.md
@@ -8,7 +8,7 @@ Tags
 : [Finite Element Model]({{< relref "finite_element_model" >}})

 Reference
-: ([Hatch 2000](#orgebf8ccb))
+: ([Hatch 2000](#orgf661cf4))

 Author(s)
 : Hatch, M. R.
@@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen

 ## Introduction {#introduction}

-<a id="org4115a7e"></a>
+<a id="org1a34f33"></a>

 The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.


 ### Modal Analysis {#modal-analysis}

-The diagram in Figure [1](#org242701e) shows the methodology for analyzing a lightly damped structure using normal modes.
+The diagram in Figure [1](#org3a8e4cc) shows the methodology for analyzing a lightly damped structure using normal modes.

 <div class="important">
  <div></div>
@@ -46,7 +46,7 @@ The steps are:

 </div>

-<a id="org242701e"></a>
+<a id="org3a8e4cc"></a>

 {{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}

@@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
 <div class="important">
  <div></div>

-Figure [2](#org398c443) shows such process, the steps are:
+Figure [2](#org0a7e8db) shows such process, the steps are:

 -   start with the finite element model
 -   compute the eigenvalues and eigenvectors (as many as dof in the model)
@@ -71,14 +71,14 @@ Figure [2](#org398c443) shows such process, the steps are:

 </div>

-<a id="org398c443"></a>
+<a id="org0a7e8db"></a>

 {{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}


 ### Notations {#notations}

-Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
+Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.

 <a id="table--tab:notations-modes-nodes"></a>
 <div class="table-caption">
@@ -127,22 +127,22 @@ Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3]

 ## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}

-<a id="orga20292c"></a>
+<a id="orgea88319"></a>
 The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.

 We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.

-Figure [3](#org02d84e8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
+Figure [3](#orgc82b5d8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
 The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).

-<a id="org02d84e8"></a>
+<a id="orgc82b5d8"></a>

 {{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}

 <div class="important">
  <div></div>

-([Miu 1993](#org39eead7)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
+([Miu 1993](#org849cfe4)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.

 The resonances of the "overhanging appendages" of the constrained system create the zeros.

@@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create

 ## State Space Analysis {#state-space-analysis}

-<a id="org24eb004"></a>
+<a id="org79a4830"></a>


 ## Modal Analysis {#modal-analysis}

-<a id="orgc6df38d"></a>
+<a id="orgb718e30"></a>

 Lightly damped structures are typically analyzed with the "normal mode" method described in this section.

@@ -196,9 +196,9 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and

 #### Equation of Motion {#equation-of-motion}

-Let's consider the model shown in Figure [4](#org0c2921d) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
+Let's consider the model shown in Figure [4](#orgebf4457) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).

-<a id="org0c2921d"></a>
+<a id="orgebf4457"></a>

 {{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}

@@ -297,17 +297,17 @@ One then find:
  \end{bmatrix}
 \end{equation}

-Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc90fe3a), [6](#orgfd8222c) and [7](#orgaf9cc36).
+Virtual interpretation of the eigenvectors are shown in Figures [5](#org520a99d), [6](#org722a9ff) and [7](#org9e25b28).

-<a id="orgc90fe3a"></a>
+<a id="org520a99d"></a>

 {{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}

-<a id="orgfd8222c"></a>
+<a id="org722a9ff"></a>

 {{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}

-<a id="orgaf9cc36"></a>
+<a id="org9e25b28"></a>

 {{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}

@@ -346,9 +346,9 @@ There are many options for change of basis, but we will show that **when eigenve
 The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
 The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.

-This procedure is schematically shown in Figure [8](#orgf9a2963).
+This procedure is schematically shown in Figure [8](#orgfbabf08).

-<a id="orgf9a2963"></a>
+<a id="orgfbabf08"></a>

 {{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}

@@ -696,7 +696,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of

 ## Frequency Response: Modal Form {#frequency-response-modal-form}

-<a id="orgfd97109"></a>
+<a id="org3f5ad6c"></a>

 The procedure to obtain the frequency response from a modal form is as follow:

@@ -704,9 +704,9 @@ The procedure to obtain the frequency response from a modal form is as follow:
 -   use Laplace transform to obtain the transfer functions in principal coordinates
 -   back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident

-This will be applied to the model shown in Figure [9](#org48b68a4).
+This will be applied to the model shown in Figure [9](#orge102983).

-<a id="org48b68a4"></a>
+<a id="orge102983"></a>

 {{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}

@@ -888,9 +888,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha

 </div>

-Figure [10](#org87763b9) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
+Figure [10](#org3024448) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).

-<a id="org87763b9"></a>
+<a id="org3024448"></a>

 {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}

@@ -899,16 +899,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f

 ## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}

-<a id="org031e9ac"></a>
+<a id="org6842a3c"></a>


 ### Introduction {#introduction}

-In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orga66d597).
+In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2292476).
 A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
 The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.

-<a id="orga66d597"></a>
+<a id="org2292476"></a>

 {{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}

@@ -987,7 +987,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the

 ## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}

-<a id="org9a76f4b"></a>
+<a id="orgc13e165"></a>


 ### Model Description {#model-description}
@@ -1001,25 +1001,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the

 ## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}

-<a id="orga8b2a2f"></a>
+<a id="org03aa6d1"></a>

-In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org94e126d)).
+In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org143e4e8)).


 ### Actuator Description {#actuator-description}

-<a id="org94e126d"></a>
+<a id="org143e4e8"></a>

 {{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}

-The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#org4a20950)).
+The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orgc294fc5)).

-<a id="org4a20950"></a>
+<a id="orgc294fc5"></a>

 {{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}

 For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
-Figure [13](#org4a20950) shows the nodes used for the reduced Matlab model.
+Figure [13](#orgc294fc5) shows the nodes used for the reduced Matlab model.
 The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
 The arrows at the nodes indicate the direction of forces.

@@ -1087,7 +1087,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\

 ## Balanced Reduction {#balanced-reduction}

-<a id="org56fcc2f"></a>
+<a id="org1f06bfa"></a>

 In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.

@@ -1202,14 +1202,14 @@ The **states to be kept are the states with the largest diagonal terms**.

 ## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}

-<a id="orga7cf69e"></a>
+<a id="orgfc560f8"></a>

-In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org1453e17)).
+In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org7003388)).


 ### Actuator Description {#actuator-description}

-<a id="org1453e17"></a>
+<a id="org7003388"></a>

 {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}

@@ -1231,9 +1231,9 @@ Since the same forces are being applied to both piezo elements, they represent t

 ### Ansys Model Description {#ansys-model-description}

-In Figure [15](#orge94bde1) are shown the principal nodes used for the model.
+In Figure [15](#org472d510) are shown the principal nodes used for the model.

-<a id="orge94bde1"></a>
+<a id="org472d510"></a>

 {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}

@@ -1352,11 +1352,11 @@ And we note:
  G = zn * Gp;
 ```

-<a id="org22a3db4"></a>
+<a id="orgfd8bb64"></a>

 {{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}

-<a id="org33c49a2"></a>
+<a id="org8f08d6b"></a>

 {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}

@@ -1454,13 +1454,13 @@ State Space Model

 ### Simple mode truncation {#simple-mode-truncation}

-Let's plot the frequency of the modes (Figure [18](#orga04e866)).
+Let's plot the frequency of the modes (Figure [18](#org6e52a4a)).

-<a id="orga04e866"></a>
+<a id="org6e52a4a"></a>

 {{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}

-<a id="org0c4b8bc"></a>
+<a id="org52d380e"></a>

 {{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}

@@ -1529,7 +1529,7 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
  [dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
 ```

-<a id="orga62ba4f"></a>
+<a id="org873b074"></a>

 {{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}

@@ -1873,7 +1873,7 @@ Then, we compute the controllability and observability gramians.

 And we plot the diagonal terms

-<a id="orgab7d0ba"></a>
+<a id="org295f621"></a>

 {{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}

@@ -1891,7 +1891,7 @@ We use `balreal` to rank oscillatory states.
  [G_b, G, T, Ti] = balreal(G_m);
 ```

-<a id="orgb25e36e"></a>
+<a id="org2867cfa"></a>

 {{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}

@@ -2134,8 +2134,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
 ```


+
 ## Bibliography {#bibliography}

-<a id="orgebf8ccb"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
+<a id="orgf661cf4"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.

-<a id="org39eead7"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
+<a id="org849cfe4"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
--- a/content/book/leach14_fundam_princ_engin_nanom.md
+++ b/content/book/leach14_fundam_princ_engin_nanom.md
@@ -8,7 +8,7 @@ Tags
 : [Metrology]({{< relref "metrology" >}})

 Reference
-: ([Leach 2014](#orgc132434))
+: ([Leach 2014](#org023e404))

 Author(s)
 : Leach, R.
@@ -87,6 +87,7 @@ The measurement of angles is then relative.
 This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).


+
 ## Bibliography {#bibliography}

-<a id="orgc132434"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
+<a id="org023e404"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
--- a/content/book/leach18_basic_precis_engin_edition.md
+++ b/content/book/leach18_basic_precis_engin_edition.md
@@ -8,7 +8,7 @@ Tags
 : [Precision Engineering]({{< relref "precision_engineering" >}})

 Reference
-: ([Leach and Smith 2018](#org50ae2e1))
+: ([Leach and Smith 2018](#orgdc805b5))

 Author(s)
 : Leach, R., & Smith, S. T.
@@ -17,6 +17,7 @@ Year
 : 2018


+
 ## Bibliography {#bibliography}

-<a id="org50ae2e1"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
+<a id="orgdc805b5"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
--- a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
+++ b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
@@ -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" >}})

 Reference
-: ([Preumont 2018](#orgd83c544))
+: ([Preumont 2018](#org29acb4a))

 Author(s)
 : Preumont, A.
@@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback

 #### Feedback {#feedback}

-<a id="orgda21dda"></a>
+<a id="orga09f785"></a>

 {{< 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](#orgda21dda). 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](#orga09f785). 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.

 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}

-<a id="orgf75c047"></a>
+<a id="org57ee378"></a>

 {{< 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 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](#orgf75c047).
+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](#org57ee378).

 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}

-<a id="org1939c0d"></a>
+<a id="org8ea735d"></a>

 {{< 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](#org1939c0d).
+The various steps of the design of a controlled structure are shown in figure [3](#org8ea735d).

 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}

-From the block diagram of the control system (figure [4](#orgaf01f6c)):
+From the block diagram of the control system (figure [4](#orga135390)):

 \begin{align\*}
 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
 \end{align\*}

-<a id="orgaf01f6c"></a>
+<a id="orga135390"></a>

 {{< 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\\).
 \\(\sigma\_z(0)\\) is then the global RMS response.

-A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org7ddcf2a).
+A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orge835b98).
 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.

-<a id="org7ddcf2a"></a>
+<a id="orge835b98"></a>

 {{< 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}
 \end{equation}

-<a id="orga618336"></a>
+<a id="orga21e5bb"></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\\)" >}}

-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](#orga618336)).
+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](#orga21e5bb)).

 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 \\]
@@ -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:
 \\[ 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](#orgecdb253)).
+\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org4ad84e0)).

-<a id="orgecdb253"></a>
+<a id="org4ad84e0"></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" >}}

@@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in

 </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](#org2e6ee6b).
+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](#org0d5b542).

-<a id="org2e6ee6b"></a>
+<a id="org0d5b542"></a>

 {{< 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>

-By looking at figure [7](#orgecdb253), 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](#org4ad84e0), 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="org8e5acfb"></a>
+<a id="org6f76f34"></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" >}}

@@ -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)}
 \end{equation}

-The corresponding Bode plot is represented in figure [9](#org8e5acfb). 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](#org6f76f34). 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.


@@ -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.

-The system consists of (see figure [10](#org5b9842b)):
+The system consists of (see figure [10](#orgc872907)):

 -   A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
 -   A coil which is free to move axially

-<a id="org5b9842b"></a>
+<a id="orgc872907"></a>

 {{< 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}

-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](#org608f53f)).
+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](#org4783db3)).

-<a id="org608f53f"></a>
+<a id="org4783db3"></a>

 {{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}

@@ -583,9 +583,9 @@ with:

 </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](#org21ce10b)).
+Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1a21332)).

-<a id="org21ce10b"></a>
+<a id="org1a21332"></a>

 {{< 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\\).

-<a id="orgb548c88"></a>
+<a id="orgb1e0d40"></a>

 {{< 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}

-The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org98492c9).
+The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgf2af0aa).

-<a id="org98492c9"></a>
+<a id="orgf2af0aa"></a>

 {{< 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.
 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](#org3b85763) can be used.
+To do so, the bridge circuit as shown on figure [15](#orgeb5cb84) can be used.

 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.

-<a id="org3b85763"></a>
+<a id="orgeb5cb84"></a>

 {{< 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 have the ability to respond significantly to stimuli of different physical nature.
-Figure [16](#org6279c77) lists various effects that are observed in materials in response to various inputs.
+Figure [16](#org1e5bcfc) lists various effects that are observed in materials in response to various inputs.

-<a id="org6279c77"></a>
+<a id="org1e5bcfc"></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" >}}

@@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect

 </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](#org8006b4a)) 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](#orgffdc1af)) 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{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\\))
 -   \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))

-<a id="org8006b4a"></a>
+<a id="orgffdc1af"></a>

 {{< 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}

-Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org7c30411).
+Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org890e9f3).

 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
 \end{equation}

-<a id="org7c30411"></a>
+<a id="org890e9f3"></a>

 {{< 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}

-Consider the system of figure [19](#org5060008), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
+Consider the system of figure [19](#org87aa6cd), 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}\\).

-<a id="org5060008"></a>
+<a id="org87aa6cd"></a>

 {{< 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
 \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](#org7f3b3bf)).
+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](#org12f8fb9)).

-<a id="org7f3b3bf"></a>
+<a id="org12f8fb9"></a>

 {{< 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.

-The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org62e1395).
+The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgca40454).
 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:

@@ -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 larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)

-<a id="org62e1395"></a>
+<a id="orgca40454"></a>

 {{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}

@@ -1816,6 +1816,7 @@ This approach has the following advantages:
 ### Problems {#problems}


+
 ## Bibliography {#bibliography}

-<a id="orgd83c544"></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="org29acb4a"></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/content/book/skogestad07_multiv_feedb_contr.md
+++ b/content/book/skogestad07_multiv_feedb_contr.md