Add all bibliographic notes and zettels

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+title = "Books"
+author = ["Thomas Dehaeze"]
+type = "book"
+draft = false
+++
+
+Here is the list of books I took note about.
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+++
+title = "Modeling and control of vibration in mechanical systems"
+author = ["Dehaeze Thomas"]
+draft = true
+++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Du and Xie 2010</a>)
+
+Author(s)
+: Du, C., &amp; Xie, L.
+
+Year
+: 2010
+
+
+## 1. Mechanical Systems and Vibration {#1-dot-mechanical-systems-and-vibration}
+
+
+### 1.1 Magnetic recording system {#1-dot-1-magnetic-recording-system}
+
+
+### 1.2 Stewart platform {#1-dot-2-stewart-platform}
+
+
+### 1.3 Vibration sources and descriptions {#1-dot-3-vibration-sources-and-descriptions}
+
+
+### 1.4 Types of vibration {#1-dot-4-types-of-vibration}
+
+
+#### 1.4.1 Free and forced vibration {#1-dot-4-dot-1-free-and-forced-vibration}
+
+
+#### 1.4.2 Damped and undamped vibration {#1-dot-4-dot-2-damped-and-undamped-vibration}
+
+
+#### 1.4.3 Linear and nonlinear vibration {#1-dot-4-dot-3-linear-and-nonlinear-vibration}
+
+
+#### 1.4.4 Deterministic and random vibration {#1-dot-4-dot-4-deterministic-and-random-vibration}
+
+
+#### 1.4.5 Periodic and nonperiodic vibration {#1-dot-4-dot-5-periodic-and-nonperiodic-vibration}
+
+
+#### 1.4.6 Broad-band and narrow-band vibration {#1-dot-4-dot-6-broad-band-and-narrow-band-vibration}
+
+
+### 1.5 Random vibration {#1-dot-5-random-vibration}
+
+
+#### 1.5.1 Random process {#1-dot-5-dot-1-random-process}
+
+
+#### 1.5.2 Stationary random process {#1-dot-5-dot-2-stationary-random-process}
+
+
+#### 1.5.3 Gaussian random process {#1-dot-5-dot-3-gaussian-random-process}
+
+
+### 1.6 Vibration analysis {#1-dot-6-vibration-analysis}
+
+
+#### 1.6.1 Fourier transform and spectrum analysis {#1-dot-6-dot-1-fourier-transform-and-spectrum-analysis}
+
+
+#### 1.6.2 Relationship between the Fourier and Laplace transforms {#1-dot-6-dot-2-relationship-between-the-fourier-and-laplace-transforms}
+
+
+#### 1.6.3 Spectral analysis {#1-dot-6-dot-3-spectral-analysis}
+
+
+## 2. Modeling of Disk Drive System and Its Vibration {#2-dot-modeling-of-disk-drive-system-and-its-vibration}
+
+
+### 2.1 Introduction {#2-dot-1-introduction}
+
+
+### 2.2 System description {#2-dot-2-system-description}
+
+
+### 2.3 System modeling {#2-dot-3-system-modeling}
+
+
+#### 2.3.1 Modeling of a VCM actuator {#2-dot-3-dot-1-modeling-of-a-vcm-actuator}
+
+
+#### 2.3.2 Modeling of friction {#2-dot-3-dot-2-modeling-of-friction}
+
+
+#### 2.3.3 Modeling of a PZT microactuator {#2-dot-3-dot-3-modeling-of-a-pzt-microactuator}
+
+
+#### 2.3.4 An example {#2-dot-3-dot-4-an-example}
+
+
+### 2.4 Vibration modeling {#2-dot-4-vibration-modeling}
+
+
+#### 2.4.1 Spectrum-based vibration modeling {#2-dot-4-dot-1-spectrum-based-vibration-modeling}
+
+
+#### 2.4.2 Adaptive modeling of disturbance {#2-dot-4-dot-2-adaptive-modeling-of-disturbance}
+
+
+### 2.5 Conclusion {#2-dot-5-conclusion}
+
+
+## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms.md" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
+
+
+### 3.1 Introduction {#3-dot-1-introduction}
+
+
+### 3.2 System description and governing equations {#3-dot-2-system-description-and-governing-equations}
+
+
+### 3.3 Modeling using adaptive filtering approach {#3-dot-3-modeling-using-adaptive-filtering-approach}
+
+
+#### 3.3.1 Adaptive filtering theory {#3-dot-3-dot-1-adaptive-filtering-theory}
+
+
+#### 3.3.2 Modeling of a Stewart platform {#3-dot-3-dot-2-modeling-of-a-stewart-platform}
+
+
+### 3.4 Conclusion {#3-dot-4-conclusion}
+
+
+## 4. Classical Vibration Control {#4-dot-classical-vibration-control}
+
+
+### 4.1 Introduction {#4-dot-1-introduction}
+
+
+### 4.2 Passive control {#4-dot-2-passive-control}
+
+
+#### 4.2.1 Isolators {#4-dot-2-dot-1-isolators}
+
+
+#### 4.2.2 Absorbers {#4-dot-2-dot-2-absorbers}
+
+
+#### 4.2.3 Resonators {#4-dot-2-dot-3-resonators}
+
+
+#### 4.2.4 Suspension {#4-dot-2-dot-4-suspension}
+
+
+#### 4.2.5 An application example &amp;#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
+
+
+### 4.3 Self-adapting systems {#4-dot-3-self-adapting-systems}
+
+
+### 4.4 Active vibration control {#4-dot-4-active-vibration-control}
+
+
+#### 4.4.1 Actuators {#4-dot-4-dot-1-actuators}
+
+
+#### 4.4.2 Active systems {#4-dot-4-dot-2-active-systems}
+
+
+#### 4.4.3 Control strategy {#4-dot-4-dot-3-control-strategy}
+
+
+### 4.5 Conclusion {#4-dot-5-conclusion}
+
+
+## 5. Introduction to Optimal and Robust Control {#5-dot-introduction-to-optimal-and-robust-control}
+
+
+### 5.1 Introduction {#5-dot-1-introduction}
+
+
+### 5.2 H2 and H&amp;#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
+
+
+#### 5.2.1 H2 norm {#5-dot-2-dot-1-h2-norm}
+
+
+#### 5.2.2 H&amp;#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
+
+
+### 5.3 H2 optimal control {#5-dot-3-h2-optimal-control}
+
+
+#### 5.3.1 Continuous-time case {#5-dot-3-dot-1-continuous-time-case}
+
+
+#### 5.3.2 Discrete-time case {#5-dot-3-dot-2-discrete-time-case}
+
+
+### 5.4 H&amp;#8734; control {#5-dot-4-h-and-8734-control}
+
+
+#### 5.4.1 Continuous-time case {#5-dot-4-dot-1-continuous-time-case}
+
+
+#### 5.4.2 Discrete-time case {#5-dot-4-dot-2-discrete-time-case}
+
+
+### 5.5 Robust control {#5-dot-5-robust-control}
+
+
+### 5.6 Controller parametrization {#5-dot-6-controller-parametrization}
+
+
+### 5.7 Performance limitation {#5-dot-7-performance-limitation}
+
+
+#### 5.7.1 Bode integral constraint {#5-dot-7-dot-1-bode-integral-constraint}
+
+
+#### 5.7.2 Relationship between system gain and phase {#5-dot-7-dot-2-relationship-between-system-gain-and-phase}
+
+
+#### 5.7.3 Sampling {#5-dot-7-dot-3-sampling}
+
+
+### 5.8 Conclusion {#5-dot-8-conclusion}
+
+
+## 6. Mixed H2/H&amp;#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
+
+
+### 6.1 Introduction {#6-dot-1-introduction}
+
+
+### 6.2 Mixed H2/H&amp;#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
+
+
+### 6.3 Method 1: slack variable approach {#6-dot-3-method-1-slack-variable-approach}
+
+
+### 6.4 Method 2: an improved slack variable approach {#6-dot-4-method-2-an-improved-slack-variable-approach}
+
+
+### 6.5 Application in servo loop design for hard disk drives {#6-dot-5-application-in-servo-loop-design-for-hard-disk-drives}
+
+
+#### 6.5.1 Problem formulation {#6-dot-5-dot-1-problem-formulation}
+
+
+#### 6.5.2 Design results {#6-dot-5-dot-2-design-results}
+
+
+### 6.6 Conclusion {#6-dot-6-conclusion}
+
+
+## 7. Low-Hump Sensitivity Control Design for Hard Disk Drive Systems {#7-dot-low-hump-sensitivity-control-design-for-hard-disk-drive-systems}
+
+
+### 7.1 Introduction {#7-dot-1-introduction}
+
+
+### 7.2 Problem statement {#7-dot-2-problem-statement}
+
+
+### 7.3 Design in continuous-time domain {#7-dot-3-design-in-continuous-time-domain}
+
+
+#### 7.3.1 H&amp;#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
+
+
+#### 7.3.2 Application examples {#7-dot-3-dot-2-application-examples}
+
+
+#### 7.3.3 Implementation on a hard disk drive {#7-dot-3-dot-3-implementation-on-a-hard-disk-drive}
+
+
+### 7.4 Design in discrete-time domain {#7-dot-4-design-in-discrete-time-domain}
+
+
+#### 7.4.1 Synthesis method for low-hump sensitivity function {#7-dot-4-dot-1-synthesis-method-for-low-hump-sensitivity-function}
+
+
+#### 7.4.2 An application example {#7-dot-4-dot-2-an-application-example}
+
+
+#### 7.4.3 Implementation on a hard disk drive {#7-dot-4-dot-3-implementation-on-a-hard-disk-drive}
+
+
+### 7.5 Conclusion {#7-dot-5-conclusion}
+
+
+## 8. Generalized KYP Lemma-Based Loop Shaping Control Design {#8-dot-generalized-kyp-lemma-based-loop-shaping-control-design}
+
+
+### 8.1 Introduction {#8-dot-1-introduction}
+
+
+### 8.2 Problem description {#8-dot-2-problem-description}
+
+
+### 8.3 Generalized KYP lemma-based control design method {#8-dot-3-generalized-kyp-lemma-based-control-design-method}
+
+
+### 8.4 Peak filter {#8-dot-4-peak-filter}
+
+
+#### 8.4.1 Conventional peak filter {#8-dot-4-dot-1-conventional-peak-filter}
+
+
+#### 8.4.2 Phase lead peak filter {#8-dot-4-dot-2-phase-lead-peak-filter}
+
+
+#### 8.4.3 Group peak filter {#8-dot-4-dot-3-group-peak-filter}
+
+
+### 8.5 Application in high frequency vibration rejection {#8-dot-5-application-in-high-frequency-vibration-rejection}
+
+
+### 8.6 Application in mid-frequency vibration rejection {#8-dot-6-application-in-mid-frequency-vibration-rejection}
+
+
+### 8.7 Conclusion {#8-dot-7-conclusion}
+
+
+## 9. Combined H2 and KYP Lemma-Based Control Design {#9-dot-combined-h2-and-kyp-lemma-based-control-design}
+
+
+### 9.1 Introduction {#9-dot-1-introduction}
+
+
+### 9.2 Problem formulation {#9-dot-2-problem-formulation}
+
+
+### 9.3 Controller design for specific disturbance rejection and overall error minimization {#9-dot-3-controller-design-for-specific-disturbance-rejection-and-overall-error-minimization}
+
+
+#### 9.3.1 Q parametrization to meet specific specifications {#9-dot-3-dot-1-q-parametrization-to-meet-specific-specifications}
+
+
+#### 9.3.2 Q parametrization to minimize H2 performance {#9-dot-3-dot-2-q-parametrization-to-minimize-h2-performance}
+
+
+#### 9.3.3 Design steps {#9-dot-3-dot-3-design-steps}
+
+
+### 9.4 Simulation and implementation results {#9-dot-4-simulation-and-implementation-results}
+
+
+#### 9.4.1 System models {#9-dot-4-dot-1-system-models}
+
+
+#### 9.4.2 Rejection of specific disturbance and H2 performance minimization {#9-dot-4-dot-2-rejection-of-specific-disturbance-and-h2-performance-minimization}
+
+
+#### 9.4.3 Rejection of two disturbances with H[sub(2)] performance minimization {#9-dot-4-dot-3-rejection-of-two-disturbances-with-h-sub--2--performance-minimization}
+
+
+### 9.5 Conclusion {#9-dot-5-conclusion}
+
+
+## 10. Blending Control for Multi-Frequency Disturbance Rejection {#10-dot-blending-control-for-multi-frequency-disturbance-rejection}
+
+
+### 10.1 Introduction {#10-dot-1-introduction}
+
+
+### 10.2 Control blending {#10-dot-2-control-blending}
+
+
+#### 10.2.1 State feedback control blending {#10-dot-2-dot-1-state-feedback-control-blending}
+
+
+#### 10.2.2 Output feedback control blending {#10-dot-2-dot-2-output-feedback-control-blending}
+
+
+### 10.3 Control blending application in multi-frequency disturbance rejection {#10-dot-3-control-blending-application-in-multi-frequency-disturbance-rejection}
+
+
+#### 10.3.1 Problem formulation {#10-dot-3-dot-1-problem-formulation}
+
+
+#### 10.3.2 Controller design via the control blending technique {#10-dot-3-dot-2-controller-design-via-the-control-blending-technique}
+
+
+### 10.4 Simulation and experimental results {#10-dot-4-simulation-and-experimental-results}
+
+
+#### 10.4.1 Rejecting high-frequency disturbances {#10-dot-4-dot-1-rejecting-high-frequency-disturbances}
+
+
+#### 10.4.2 Rejecting a combined mid and high frequency disturbance {#10-dot-4-dot-2-rejecting-a-combined-mid-and-high-frequency-disturbance}
+
+
+### 10.5 Conclusion {#10-dot-5-conclusion}
+
+
+## 11. H&amp;#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
+
+
+### 11.1 Introduction {#11-dot-1-introduction}
+
+
+### 11.2 Conventional disturbance observer {#11-dot-2-conventional-disturbance-observer}
+
+
+### 11.3 A general form of disturbance observer {#11-dot-3-a-general-form-of-disturbance-observer}
+
+
+### 11.4 Application results {#11-dot-4-application-results}
+
+
+### 11.5 Conclusion {#11-dot-5-conclusion}
+
+
+## 12. Two-Dimensional H2 Control for Error Minimization {#12-dot-two-dimensional-h2-control-for-error-minimization}
+
+
+### 12.1 Introduction {#12-dot-1-introduction}
+
+
+### 12.2 2-D stabilization control {#12-dot-2-2-d-stabilization-control}
+
+
+### 12.3 2-D H2 control {#12-dot-3-2-d-h2-control}
+
+
+### 12.4 SSTW process and modeling {#12-dot-4-sstw-process-and-modeling}
+
+
+#### 12.4.1 SSTW servo loop {#12-dot-4-dot-1-sstw-servo-loop}
+
+
+#### 12.4.2 Two-dimensional model {#12-dot-4-dot-2-two-dimensional-model}
+
+
+### 12.5 Feedforward compensation method {#12-dot-5-feedforward-compensation-method}
+
+
+### 12.6 2-D control formulation for SSTW {#12-dot-6-2-d-control-formulation-for-sstw}
+
+
+### 12.7 2-D stabilization control for error propagation containment {#12-dot-7-2-d-stabilization-control-for-error-propagation-containment}
+
+
+#### 12.7.1 Simulation results {#12-dot-7-dot-1-simulation-results}
+
+
+### 12.8 2-D H2 control for error minimization {#12-dot-8-2-d-h2-control-for-error-minimization}
+
+
+#### 12.8.1 Simulation results {#12-dot-8-dot-1-simulation-results}
+
+
+#### 12.8.2 Experimental results {#12-dot-8-dot-2-experimental-results}
+
+
+### 12.9 Conclusion {#12-dot-9-conclusion}
+
+
+## 13. Nonlinearity Compensation and Nonlinear Control {#13-dot-nonlinearity-compensation-and-nonlinear-control}
+
+
+### 13.1 Introduction {#13-dot-1-introduction}
+
+
+### 13.2 Nonlinearity compensation {#13-dot-2-nonlinearity-compensation}
+
+
+### 13.3 Nonlinear control {#13-dot-3-nonlinear-control}
+
+
+#### 13.3.1 Design of a composite control law {#13-dot-3-dot-1-design-of-a-composite-control-law}
+
+
+#### 13.3.2 Experimental results in hard disk drives {#13-dot-3-dot-2-experimental-results-in-hard-disk-drives}
+
+
+### 13.4 Conclusion {#13-dot-4-conclusion}
+
+
+## 14. Quantization Effect on Vibration Rejection and Its Compensation {#14-dot-quantization-effect-on-vibration-rejection-and-its-compensation}
+
+
+### 14.1 Introduction {#14-dot-1-introduction}
+
+
+### 14.2 Description of control system with quantizer {#14-dot-2-description-of-control-system-with-quantizer}
+
+
+### 14.3 Quantization effect on error rejection {#14-dot-3-quantization-effect-on-error-rejection}
+
+
+#### 14.3.1 Quantizer frequency response measurement {#14-dot-3-dot-1-quantizer-frequency-response-measurement}
+
+
+#### 14.3.2 Quantization effect on error rejection {#14-dot-3-dot-2-quantization-effect-on-error-rejection}
+
+
+### 14.4 Compensation of quantization effect on error rejection {#14-dot-4-compensation-of-quantization-effect-on-error-rejection}
+
+
+### 14.5 Conclusion {#14-dot-5-conclusion}
+
+
+## 15. Adaptive Filtering Algorithms for Active Vibration Control {#15-dot-adaptive-filtering-algorithms-for-active-vibration-control}
+
+
+### 15.1 Introduction {#15-dot-1-introduction}
+
+
+### 15.2 Adaptive feedforward algorithm {#15-dot-2-adaptive-feedforward-algorithm}
+
+
+### 15.3 Adaptive feedback algorithm {#15-dot-3-adaptive-feedback-algorithm}
+
+
+### 15.4 Comparison between feedforward and feedback controls {#15-dot-4-comparison-between-feedforward-and-feedback-controls}
+
+
+### 15.5 Application in Stewart platform {#15-dot-5-application-in-stewart-platform}
+
+
+#### 15.5.1 Multi-channel adaptive feedback AVC system {#15-dot-5-dot-1-multi-channel-adaptive-feedback-avc-system}
+
+
+#### 15.5.2 Multi-channel adaptive feedback algorithm for hexapod platform {#15-dot-5-dot-2-multi-channel-adaptive-feedback-algorithm-for-hexapod-platform}
+
+
+#### 15.5.3 Simulation and implementation {#15-dot-5-dot-3-simulation-and-implementation}
+
+
+### 15.6 Conclusion {#15-dot-6-conclusion}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Lihua Xie. 2010. <i>Modeling and Control of Vibration in Mechanical Systems</i>. Automation and Control Engineering. CRC Press. doi:<a href="https://doi.org/10.1201/9781439817995">10.1201/9781439817995</a>.</div>
+</div>
@@ -0,0 +1,678 @@
+++
+title = "Multi-stage actuation systems and control"
+author = ["Dehaeze Thomas"]
+description = "Proposes a way to combine multiple actuators (short stroke and long stroke) for control."
+keywords = ["Control", "Mechatronics"]
+draft = false
+++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Du and Pang 2019</a>)
+
+Author(s)
+: Du, C., &amp; Pang, C. K.
+
+Year
+: 2019
+
+<div style="display: none;">
+\(
+\newcommand{\SI}[2]{#1\,#2}
+% Simulate SIunitx
+\newcommand{\ang}[1]{#1^{\circ}}
+\newcommand{\degree}{^{\circ}}
+\newcommand{\radian}{\text{rad}}
+\newcommand{\percent}{\%}
+\newcommand{\decibel}{\text{dB}}
+\newcommand{\per}{/}
+\)
+</div>
+
+
+## Mechanical Actuation Systems {#mechanical-actuation-systems}
+
+
+### Introduction {#introduction}
+
+When high bandwidth, high position accuracy and long stroke are required simultaneously: dual-stage systems composed of a coarse (or primary) actuator and a fine actuator working together are used.
+
+Popular choices for coarse actuator are:
+
+-   DC motor
+-   [Voice Coil Motors]({{< relref "voice_coil_actuators.md" >}}) (VCM)
+-   Permanent magnet stepper motor
+-   Permanent magnet linear synchronous motor
+
+As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}) are used.
+
+In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.
+
+
+### Actuators {#actuators}
+
+
+#### Primary Actuator {#primary-actuator}
+
+Without loss of generality, the VCM actuator is used as the primary actuator.
+When current passes through the coil, a force is produced which accelerates the actuator radially.
+The produced force is a function of the current \\(i\_c\\):
+\\[ f\_m = k\_t i\_c \\]
+where \\(k\_t\\) is a linearized nominal value called the torque constant.
+
+The resonance of the actuator is mainly due to the flexibility of the pivot bearing, arm, suspension.
+
+Then the bandwidth of the control loop is low and the resonances are not a limiting factor of the control design, the actuator model can be considered as follows:
+\\[ P\_v(s) = \frac{k\_{vcm}}{s^2} \\]
+
+When the bandwidth is high, the actuator resonances have to be considered in the control design since the flexible resonance modes will reduce the system stability and affect the control performance. Then the actuator model becomes
+\\[ P\_v(s) = \frac{k\_{vcm}}{s^2} P\_r(s) \\]
+which includes the resonance model
+\\[ P\_r(s) = \Pi\_{i=1}^{N} P\_{ri}(s) \\]
+and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following forms
+
+\begin{align\*}
+ P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
+ P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
+ P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
+\end{align\*}
+
+
+#### Secondary Actuators {#secondary-actuators}
+
+We here consider two types of secondary actuators: the PZT milliactuator ([Figure 1](#figure--fig:pzt-actuator)) and the microactuator.
+
+<a id="figure--fig:pzt-actuator"></a>
+
+{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="<span class='figure-number'>Figure 1: </span>A PZT-actuator suspension" >}}
+
+There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
+There characteristics are shown on [Table 1](#table--tab:microactuator).
+
+<a id="table--tab:microactuator"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:microactuator">Table 1</a>:</span>
+  Performance comparison of microactuators
+</div>
+
+|              | Elect.                                        | PZT                                           | Thermal                    |
+|--------------|-----------------------------------------------|-----------------------------------------------|----------------------------|
+| TF           | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
+| \\(\tau\\)   | \\(<\SI{0.1}{ms}\\)                           | \\(<\SI{0.05}{ms}\\)                          | \\(>\SI{0.1}{ms}\\)        |
+| \\(\omega\\) | \\(1-\SI{2}{kHz}\\)                           | \\(20-\SI{25}{kHz}\\)                         | \\(>\SI{15}{kHz}\\)        |
+
+
+### Single-Stage Actuation Systems {#single-stage-actuation-systems}
+
+A typical closed-loop control system is shown on [Figure 2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
+
+<a id="figure--fig:single-stage-control"></a>
+
+{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="<span class='figure-number'>Figure 2: </span>Block diagram of a single-stage actuation system" >}}
+
+
+### Dual-Stage Actuation Systems {#dual-stage-actuation-systems}
+
+Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
+The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
+
+<a id="figure--fig:dual-stage-control"></a>
+
+{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="<span class='figure-number'>Figure 3: </span>Block diagram of dual-stage actuation system" >}}
+
+
+### Three-Stage Actuation Systems {#three-stage-actuation-systems}
+
+Due to the limited allowed stroke of the microactuator, the control bandwidth has to be restricted and that limits the dual-stage disturbance rejection capability.
+
+A three-stage actuation system is therefore introduced to further increase the bandwidth.
+
+Typically, a VCM actuator is used as the primary actuator, PZT milliactuator as the second stage actuator and a third actuator more collocated is used.
+
+
+## High-Precision Positioning Control of Dual-Stage Actuation Systems {#high-precision-positioning-control-of-dual-stage-actuation-systems}
+
+
+### Introduction {#introduction}
+
+The sensitivity function of the closed-loop system has provided a straightforward view of its disturbance rejection capability.
+It is demanded that the sensitivity function magnitude in the low-frequency range be sufficiently low, while its hump in high-frequency range stays low enough.
+In view of this, the controller design for dual-stage actuation systems adopts a weighting function to shape the sensitivity function.
+
+
+### Control Schemes {#control-schemes}
+
+A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in [Figure 4](#figure--fig:decoupled-control).
+
+-   \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
+-   \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
+-   \\(d\_1\\) and \\(d\_2\\) denote internal disturbances
+-   \\(n\\) is the measurement noise
+-   \\(d\_u\\) stands for external vibration
+
+<a id="figure--fig:decoupled-control"></a>
+
+{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="<span class='figure-number'>Figure 4: </span>Decoupled control structure for the dual-stage actuation system" >}}
+
+The open-loop transfer function from \\(pes\\) to \\(y\\) is
+\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) + P\_v(z) C\_v(z) \hat{P}\_p(z) C\_p(z) \\]
+And the overall sensitivity function of the closed loop system from \\(r\\) to \\(pes\\) is
+\\[ S(z) = \frac{1}{1 + G(z)} \\]
+which is approximately
+\\[ S(z) = \frac{1}{[1 + P\_p(z) C\_p(z)] [1 + P\_v(z)C\_v(z)]} \\]
+since within a certain bandwidth
+\\[ \hat{P}\_p(z) \approx P\_p(z) \\]
+
+The sensitivity functions of the VCM loop and the secondary actuator loop are
+
+\begin{equation}
+  S\_v(z) = \frac{1}{1 + P\_v(z) C\_v(z)}, \quad S\_p(z) = \frac{1}{1 + P\_p(z) C\_p(z)}
+\end{equation}
+
+And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
+Thus, the dual-stage system control design can be decoupled into two independent controller designs.
+
+Another type of control scheme is the **parallel structure** as shown in [Figure 5](#figure--fig:parallel-control-structure).
+The open-loop transfer function from \\(pes\\) to \\(y\\) is
+\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
+
+The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
+\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
+
+<a id="figure--fig:parallel-control-structure"></a>
+
+{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="<span class='figure-number'>Figure 5: </span>Parallel control structure for the dual-stage actuator system" >}}
+
+Because of the limited displacement range of the secondary actuator, the control efforts for the two actuators should be distributed properly when designing respective controllers to meet the required performance, make the actuators not conflict with each other, as well as prevent the saturation of the secondary actuator.
+
+
+### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
+
+\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
+The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in [Figure 6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
+
+For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
+
+\begin{equation}
+  \\|T\_{zw}\\|\_\infty < 1
+\end{equation}
+
+is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
+
+<a id="figure--fig:h-inf-diagram"></a>
+
+{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class='figure-number'>Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
+
+Equation means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
+One form of \\(W(s)\\) is taken as
+
+\begin{equation}
+  W(s) = \frac{\frac{1}{M}s^2 + 2\xi\omega\frac{1}{\sqrt{M}}s + \omega^2}{s^2 + 2\omega\sqrt{\epsilon}s + \omega^2\epsilon}
+\end{equation}
+
+where \\(\omega\\) is the desired bandwidth, \\(\epsilon\\) is used to determine the desired low frequency level of sensitivity magnitude and \\(\xi\\) is the damping ratio.
+
+The controller can then be synthesis using the linear matrix inequality (LMI) approach.
+
+The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
+But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
+As seen in [Figure 7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
+
+<a id="figure--fig:dual-stage-loop-gain"></a>
+
+{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="<span class='figure-number'>Figure 7: </span>Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
+
+The sensitivity functions are shown in [Figure 8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
+This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
+
+<a id="figure--fig:dual-stage-sensitivity"></a>
+
+{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="<span class='figure-number'>Figure 8: </span>Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
+
+A basic requirement of the dual-stage actuation control system is to make the individual primary and secondary loops stable.
+It also required that the primary actuator path has a higher gain than the secondary actuator path at low frequency range and the secondary actuator path has a higher gain than the primary actuator path in high-frequency range.
+These can be achieve by choosing appropriate weighting function for the controllers design.
+
+
+### Conclusion {#conclusion}
+
+The controller design has been discussed for high-precision positioning control of the dual-stage actuation systems.
+The \\(\mathcal{H}\_\infty\\) loop shaping method has been applied and the design method has been presented.
+With the weighting functions, the desired sensitivity function can achieved.
+Such a design method can produce robust controllers with more disturbance rejection in the low frequency range and less disturbance amplification in the high-frequency range.
+
+
+## Modeling and Control of a Three-Stage Actuation System {#modeling-and-control-of-a-three-stage-actuation-system}
+
+
+### Introduction {#introduction}
+
+In view of the additional bandwidth requirement which is limited by stroke constraint and saturation of secondary actuators, three-stage actuation systems are thereby proposed to meet the demand of a higher bandwidth.
+In this section, a specific three-stage actuation system is presented and a controller strategy is proposed, which is based on a decoupled master-slave dual-stage control structure combined with a third stage actuation in parallel format.
+
+
+### Actuator and Vibration Modeling {#actuator-and-vibration-modeling}
+
+A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a PZT milliactuator as the second-stage actuator denoted by \\(P\_p(s)\\), and a thermal microactuator denoted by \\(P\_m(s)\\).
+
+
+### Control Strategy and Controller Design {#control-strategy-and-controller-design}
+
+[Figure 9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
+
+The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
+The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
+The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
+
+<a id="figure--fig:three-stage-control"></a>
+
+{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="<span class='figure-number'>Figure 9: </span>Control system for the three-stage actuation system" >}}
+
+The open-loop transfer function of the three-stage actuation system is derived as
+
+\begin{equation}
+  G(z) = G\_v(z) + G\_p(z) + G\_v(z) G\_p(z) + G\_m(z)
+\end{equation}
+
+with
+
+  \begin{align\*}
+  G\_v(z) &= P\_v(z) C\_v(z) \\\\
+  G\_p(z) &= P\_p(z) C\_p(z) \\\\
+  G\_m(z) &= P\_m(z) C\_m(z)
+\end{align\*}
+
+The overall sensitivity function is given by
+
+\begin{equation}
+  S(z) = \frac{1}{1 + G(z)}
+\end{equation}
+
+The VCM actuator \\(P\_v(s)\\) works in a low bandwidth below \\(\SI{1}{kHz}\\).
+The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandwidth up to \\(\SI{3}{kHz}\\).
+The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
+
+The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
+The open-loop frequency responses of the three stages are shown on [Figure 10](#figure--fig:open-loop-three-stage).
+
+<a id="figure--fig:open-loop-three-stage"></a>
+
+{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="<span class='figure-number'>Figure 10: </span>Frequency response of the open-loop transfer function" >}}
+
+The obtained sensitivity function is shown on [Figure 11](#figure--fig:sensitivity-three-stage).
+
+<a id="figure--fig:sensitivity-three-stage"></a>
+
+{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="<span class='figure-number'>Figure 11: </span>Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
+
+
+### Performance Evaluation {#performance-evaluation}
+
+External vibration from the system working environment is much higher than the internal disturbance, especially for ultra-high precision positioning systems.
+In the presence of external vibration, the actuators control effort is dominantly determined by the external vibration.
+But because the actuator input is constrained, the external vibration level has to be limited.
+Otherwise, saturation will occur in the control loop and the control system performance will be degraded.
+
+Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
+Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
+
+For the three-stage control architecture as shown on [Figure 9](#figure--fig:three-stage-control), the position error is
+\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
+The control signals and positions of the actuators are given by
+
+\begin{align\*}
+  u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\
+  u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\
+  u\_v &= C\_v ( 1 + \hat{P}\_pC\_p ) e,\ y\_v = P\_v ( u\_v + d\_1 )
+\end{align\*}
+
+The controller design for the microactuators with input constraints must take into account both external vibration requirements and actuators' stroke, based on which an appropriate bandwidth should be decided when designing the control system.
+Higher bandwidth/higher level of disturbance generally means high stroke needed.
+
+
+### Different Configurations of the Control System {#different-configurations-of-the-control-system}
+
+A decoupled control structure can be used for the three-stage actuation system (see [Figure 12](#figure--fig:three-stage-decoupled)).
+
+The overall sensitivity function is
+\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
+with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation and
+\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
+
+Denote the dual-stage open-loop transfer function as \\(G\_d\\)
+\\[ G\_d(z) = G\_v(z) + G\_p(z) + G\_v(z) G\_p(z) \\]
+
+The open-loop transfer function of the overall system is
+\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
+
+<a id="figure--fig:three-stage-decoupled"></a>
+
+{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="<span class='figure-number'>Figure 12: </span>Decoupled control structure for the three-stage actuation system" >}}
+
+The control signals and the positions of the three actuators are
+
+\begin{align\*}
+  u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\
+  u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\
+  u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
+\end{align\*}
+
+The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see [Figure 13](#figure--fig:three-stage-decoupled-loop-gain)).
+
+<a id="figure--fig:three-stage-decoupled-loop-gain"></a>
+
+{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="<span class='figure-number'>Figure 13: </span>Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
+
+
+### Conclusion {#conclusion}
+
+The relationship among the external vibration, the microactuator stroke, and the achievable control bandwidth has been discussed for being considered in the controller design.
+The discussion suggests that in addition to the traditional wisdom of just increasing the resonant frequency, adding more stroke to the microactuator will give more freedom to the loop shaping for the control system design.
+
+
+## Dual-Stage System Control Considering Secondary Actuator Stroke Limitation {#dual-stage-system-control-considering-secondary-actuator-stroke-limitation}
+
+
+### Introduction {#introduction}
+
+
+### More Freedom Loop Shaping for Microactuator Controller Design {#more-freedom-loop-shaping-for-microactuator-controller-design}
+
+
+### Dual-Stage System Control Design for 5 kHz Bandwidth {#dual-stage-system-control-design-for-5-khz-bandwidth}
+
+
+### Evaluation with the Consideration of External Vibration and Microactuator Stroke {#evaluation-with-the-consideration-of-external-vibration-and-microactuator-stroke}
+
+
+### Conclusion {#conclusion}
+
+
+## Saturation Control for Microactuators in Dual-Stage Actuation Systems {#saturation-control-for-microactuators-in-dual-stage-actuation-systems}
+
+
+### Introduction {#introduction}
+
+
+### Modeling and Feedback Control {#modeling-and-feedback-control}
+
+
+### Anti-Windup Compensation Design {#anti-windup-compensation-design}
+
+
+### Simulation and Experimental Results {#simulation-and-experimental-results}
+
+
+### Conclusion {#conclusion}
+
+
+## Time Delay and Sampling Rate Effect on Control Performance of Dual-Stage Actuation Systems {#time-delay-and-sampling-rate-effect-on-control-performance-of-dual-stage-actuation-systems}
+
+
+### Introduction {#introduction}
+
+
+### Modeling of Time Delay {#modeling-of-time-delay}
+
+
+### Dual-Stage Actuation System Modeling with Time Delay for Controller Design {#dual-stage-actuation-system-modeling-with-time-delay-for-controller-design}
+
+
+### Controller Design with Time Delay for the Dual-Stage Actuation Systems {#controller-design-with-time-delay-for-the-dual-stage-actuation-systems}
+
+
+### Time Delay Effect on Dual-Stage System Control Performance {#time-delay-effect-on-dual-stage-system-control-performance}
+
+
+### Sampling Rate Effect on Dual-Stage System Control Performance {#sampling-rate-effect-on-dual-stage-system-control-performance}
+
+
+### Conclusion {#conclusion}
+
+
+## PZT Hysteresis Modeling and Compensation {#pzt-hysteresis-modeling-and-compensation}
+
+
+### Introduction {#introduction}
+
+
+### Modeling of Hysteresis {#modeling-of-hysteresis}
+
+
+#### PI Model {#pi-model}
+
+
+#### GPI Model {#gpi-model}
+
+
+#### Inverse GPI Model {#inverse-gpi-model}
+
+
+### Application of GPI Model to a PZT-Actuated Structure {#application-of-gpi-model-to-a-pzt-actuated-structure}
+
+
+#### Modeling of the Hysteresis in the PZT-Actuated Structure {#modeling-of-the-hysteresis-in-the-pzt-actuated-structure}
+
+
+#### Hysteresis Compensator Design {#hysteresis-compensator-design}
+
+
+#### Experimental Verification {#experimental-verification}
+
+
+### Conclusion {#conclusion}
+
+
+## Seeking Control of Dual-Stage Actuation Systems with Trajectory Optimization {#seeking-control-of-dual-stage-actuation-systems-with-trajectory-optimization}
+
+
+### Introduction {#introduction}
+
+
+### Current Profile of VCM Primary Actuator {#current-profile-of-vcm-primary-actuator}
+
+
+#### PTOS Method {#ptos-method}
+
+
+#### A General Form of VCM Current Profiles {#a-general-form-of-vcm-current-profiles}
+
+
+### Control System Structure for the Dual-Stage Actuation System {#control-system-structure-for-the-dual-stage-actuation-system}
+
+
+### Design of VCM Current Profile a[sub(v)] and Dual-Stage Reference Trajectory r[sub(d)] {#design-of-vcm-current-profile-a-sub--v--and-dual-stage-reference-trajectory-r-sub--d}
+
+
+### Seeking within PZT Milliactuator Stroke {#seeking-within-pzt-milliactuator-stroke}
+
+
+### Seeking over PZT Milliactuator Stroke {#seeking-over-pzt-milliactuator-stroke}
+
+
+### Conclusion {#conclusion}
+
+
+## High-Frequency Vibration Control Using PZT Active Damping {#high-frequency-vibration-control-using-pzt-active-damping}
+
+
+### Introduction {#introduction}
+
+
+### Singular Perturbation Method-Based Controller Design {#singular-perturbation-method-based-controller-design}
+
+
+#### Singular Perturbation Control Topology {#singular-perturbation-control-topology}
+
+
+#### Identification of Fast Dynamics Using PZT as a Sensor {#identification-of-fast-dynamics-using-pzt-as-a-sensor}
+
+
+#### Design of Controllers {#design-of-controllers}
+
+
+##### Fast Subsystem Estimator  G[sub(v)][sup(\*)] {#fast-subsystem-estimator-g-sub--v--sup}
+
+
+##### Fast Controller C[sub(v)] {#fast-controller-c-sub--v}
+
+
+##### Slow Controller C[sub(v)] {#slow-controller-c-sub--v}
+
+
+#### Simulation and Experimental Results {#simulation-and-experimental-results}
+
+
+##### Frequency Responses {#frequency-responses}
+
+
+##### Time Responses {#time-responses}
+
+
+### H[sub(2)] Controller Design {#h-sub--2--controller-design}
+
+
+### Design of C[sub(d)](z) with H[sub(2)] Method and Notch Filters {#design-of-c-sub--d----z--with-h-sub--2--method-and-notch-filters}
+
+
+### Design of Mixed H[sub(2)]/H[sub(∞)] Controller C[sub(d)](z) {#design-of-mixed-h-sub--2--h-sub-----controller-c-sub--d----z}
+
+
+### Application Results {#application-results}
+
+
+#### System Modeling {#system-modeling}
+
+
+#### H[sub(2)] Active Damping Control {#h-sub--2--active-damping-control}
+
+
+#### Mixed H[sub(2)]/H[sub(∞)] Active Damping Control {#mixed-h-sub--2--h-sub-----active-damping-control}
+
+
+#### Experimental Results {#experimental-results}
+
+
+### Conclusion {#conclusion}
+
+
+## Self-Sensing Actuation of Dual-Stage Systems {#self-sensing-actuation-of-dual-stage-systems}
+
+
+### Introduction {#introduction}
+
+
+### Estimation of PZT Secondary Actuator’s Displacement y[sub(p)][sup(\*)] {#estimation-of-pzt-secondary-actuator-s-displacement-y-sub--p--sup}
+
+
+#### Self-Sensing Actuation and Bridge Circuit {#self-sensing-actuation-and-bridge-circuit}
+
+
+#### PZT Displacement Estimation Circuit H[sub(B)] {#pzt-displacement-estimation-circuit-h-sub--b}
+
+
+### Design of Controllers {#design-of-controllers}
+
+
+#### VCM Controller and Controller C[sub(D)] {#vcm-controller-and-controller-c-sub--d}
+
+
+#### PZT Controller {#pzt-controller}
+
+
+### Performance Evaluation {#performance-evaluation}
+
+
+#### Effectiveness of C[sub(D)] {#effectiveness-of-c-sub--d}
+
+
+#### Position Errors {#position-errors}
+
+
+### Conclusion {#conclusion}
+
+
+## Modeling and Control of a MEMS Micro X–Y Stage Media Platform {#modeling-and-control-of-a-mems-micro-x-y-stage-media-platform}
+
+
+### Introduction {#introduction}
+
+
+### MEMS Micro X–Y Stage {#mems-micro-x-y-stage}
+
+
+#### Design and Simulation of Micro X–Y Stage {#design-and-simulation-of-micro-x-y-stage}
+
+
+##### Static {#static}
+
+
+##### Dynamic {#dynamic}
+
+
+#### Modeling of Micro X–Y Stage {#modeling-of-micro-x-y-stage}
+
+
+#### Fabrication of the MEMS Micro X–Y Stage {#fabrication-of-the-mems-micro-x-y-stage}
+
+
+### Capacitive Self-Sensing Actuation {#capacitive-self-sensing-actuation}
+
+
+#### Design of CSSA Bridge Circuit {#design-of-cssa-bridge-circuit}
+
+
+#### Experimental Verification {#experimental-verification}
+
+
+### Robust Decoupling Controller Design {#robust-decoupling-controller-design}
+
+
+#### Choice of Pre-Shaping Filters {#choice-of-pre-shaping-filters}
+
+
+#### Controller Synthesis {#controller-synthesis}
+
+
+#### Frequency Responses {#frequency-responses}
+
+
+#### Time Responses {#time-responses}
+
+
+#### Robustness Analysis {#robustness-analysis}
+
+
+### Conclusion {#conclusion}
+
+
+## Conclusions {#conclusions}
+
+Many secondary actuators have been developed in addition to primary actuators in the field of mechanical actuation systems.
+The aim is to provide high performance such as high precision and fast response.
+Several types of secondary actuators have been introduced such as PZT milliactuator, electrostatic microactuator, PZT microactuator, and thermal microactuator.
+Comparison of these secondary actuators has been made, and these secondary actuators have made dual and multi-stage actuation mechanisms possible.
+
+Three-stage actuation systems have been proposed for the demand of wider bandwidth, to overcome the limitation by stroke constraint and saturation of secondary actuators.
+After the characteristics of the three-stage systems have been developed and the models have been identified, the control strategy and algorithm have been developed to deal with vibrations and meet different requirements.
+Particularly, for the three-stage actuation systems, the presented control strategies make it easy to further push the bandwidth and meet the performance requirement.
+The control of the thermal microactuator based dual-stage system has been discussed in detail, including linearization and controller design method.
+
+The developed advanced algorithms applied in the multi-stage systems include \\(\mathcal{H}\_\infty\\) loop shaping, anti-windup compensation, \\(\mathcal{H}\_2\\) control method,
+and mixed \\(\mathcal{H}\_2/\mathcal{H}\_\infty\\) control method.
+Typical problems of the milli and micro-actuators as the secondary actuators have been considered and appropriate solutions have been presented such as saturation compensation, hysteresis modeling and compensation, stroke limitation, and PZT self-sensing scheme.
+Time delay and sampling rate effect on the control performance have been analyzed to help select appropriate sampling rate and design suitable controllers.
+
+Specific usage of PZT elements has been produced for system performance improvement.
+Using PZT elements as a sensor to deal with high-frequency vibration beyond the bandwidth has been proposed and systematic controller design methods have been developed.
+As a more advanced concept, PZT elements being used as actuator and sensor simultaneously has also been addressed in this book with detailed scheme and controller design methodology for effective utilization.
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Chee Khiang Pang. 2019. <i>Multi-Stage Actuation Systems and Control</i>. Boca Raton, FL: CRC Press.</div>
+</div>
@@ -0,0 +1,858 @@
+++
+title = "Design, modeling and control of nanopositioning systems"
+author = ["Dehaeze Thomas"]
+description = "Talks about various topics related to nano-positioning systems."
+keywords = ["Control", "Metrology", "Flexible Joints"]
+draft = false
+++
+
+Tags
+: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>)
+
+Author(s)
+: Fleming, A. J., &amp; Leang, K. K.
+
+Year
+: 2014
+
+
+## Introduction to Nanotechnology {#introduction-to-nanotechnology}
+
+
+## Introduction to Nanopositioning {#introduction-to-nanopositioning}
+
+
+## Scanning Probe Microscopy {#scanning-probe-microscopy}
+
+
+## Challenges with Nanopositioning Systems {#challenges-with-nanopositioning-systems}
+
+
+### Hysteresis {#hysteresis}
+
+
+### Creep {#creep}
+
+
+### Thermal Drift {#thermal-drift}
+
+
+### Mechanical Resonance {#mechanical-resonance}
+
+
+## Control of Nanopositioning Systems {#control-of-nanopositioning-systems}
+
+
+### Feedback Control {#feedback-control}
+
+
+### Feedforward Control {#feedforward-control}
+
+
+## Book Summary {#book-summary}
+
+
+### Assumed Knowledge {#assumed-knowledge}
+
+
+### Content Summary {#content-summary}
+
+
+## References {#references}
+
+
+## The Piezoelectric Effect {#the-piezoelectric-effect}
+
+
+## Piezoelectric Compositions {#piezoelectric-compositions}
+
+
+## Manufacturing Piezoelectric Ceramics {#manufacturing-piezoelectric-ceramics}
+
+
+## Piezoelectric Transducers {#piezoelectric-transducers}
+
+
+## Application Considerations {#application-considerations}
+
+
+## Response of Piezoelectric Actuators {#response-of-piezoelectric-actuators}
+
+
+## Modeling Creep and Vibration in Piezoelectric Actuators {#modeling-creep-and-vibration-in-piezoelectric-actuators}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Piezoelectric Tube Nanopositioners {#piezoelectric-tube-nanopositioners}
+
+
+### 63mm Piezoelectric Tube {#63mm-piezoelectric-tube}
+
+
+### 40mm Piezoelectric Tube Nanopositioner {#40mm-piezoelectric-tube-nanopositioner}
+
+
+## Piezoelectric Stack Nanopositioners {#piezoelectric-stack-nanopositioners}
+
+
+### Phyisk Instrumente P-734 Nanopositioner {#phyisk-instrumente-p-734-nanopositioner}
+
+
+### Phyisk Instrumente P-733.3DD Nanopositioner {#phyisk-instrumente-p-733-dot-3dd-nanopositioner}
+
+
+### Vertical Nanopositioners {#vertical-nanopositioners}
+
+
+### Rotational Nanopositioners {#rotational-nanopositioners}
+
+
+### Low Temperature and UHV Nanopositioners {#low-temperature-and-uhv-nanopositioners}
+
+
+### Tilting Nanopositioners {#tilting-nanopositioners}
+
+
+### Optical Objective Nanopositioners {#optical-objective-nanopositioners}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Operating Environment {#operating-environment}
+
+
+## Methods for Actuation {#methods-for-actuation}
+
+
+## Flexure Hinges {#flexure-hinges}
+
+
+### Introduction {#introduction}
+
+
+### Types of Flexures {#types-of-flexures}
+
+
+### Flexure Hinge Compliance Equations {#flexure-hinge-compliance-equations}
+
+
+### Stiff Out-of-Plane Flexure Designs {#stiff-out-of-plane-flexure-designs}
+
+
+### Failure Considerations {#failure-considerations}
+
+
+### Finite Element Approach for Flexure Design {#finite-element-approach-for-flexure-design}
+
+
+## Material Considerations {#material-considerations}
+
+
+### Materials for Flexure and Platform Design {#materials-for-flexure-and-platform-design}
+
+
+### Thermal Stability of Materials {#thermal-stability-of-materials}
+
+
+## Manufacturing Techniques {#manufacturing-techniques}
+
+
+## Design Example: A High-Speed Serial-Kinematic Nanopositioner {#design-example-a-high-speed-serial-kinematic-nanopositioner}
+
+
+### State-of-the-Art Designs {#state-of-the-art-designs}
+
+
+### Tradeoffs and Limitations in Speed {#tradeoffs-and-limitations-in-speed}
+
+
+### Serial- Versus Parallel-Kinematic Configurations {#serial-versus-parallel-kinematic-configurations}
+
+
+### Piezoactuator Considerations {#piezoactuator-considerations}
+
+
+### Preloading Piezo-Stack Actuators {#preloading-piezo-stack-actuators}
+
+
+### Flexure Design for Lateral Positioning {#flexure-design-for-lateral-positioning}
+
+
+### Design of Vertical Stage {#design-of-vertical-stage}
+
+
+### Fabrication and Assembly {#fabrication-and-assembly}
+
+
+### Drive Electronics {#drive-electronics}
+
+\*\*\*\*0 Experimental Results
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Sensor Characteristics {#sensor-characteristics}
+
+
+### Calibration and Nonlinearity {#calibration-and-nonlinearity}
+
+
+### Drift and Stability {#drift-and-stability}
+
+
+### Bandwidth {#bandwidth}
+
+
+### Noise {#noise}
+
+
+### Resolution {#resolution}
+
+
+### Combining Errors {#combining-errors}
+
+
+### Metrological Traceability {#metrological-traceability}
+
+
+## Nanometer Position Sensors {#nanometer-position-sensors}
+
+
+### Resistive Strain Sensors {#resistive-strain-sensors}
+
+
+### Piezoresistive Strain Sensors {#piezoresistive-strain-sensors}
+
+
+### Piezoelectric Strain Sensors {#piezoelectric-strain-sensors}
+
+
+### Capacitive Sensors {#capacitive-sensors}
+
+
+### MEMs Capacitive and Thermal Sensors {#mems-capacitive-and-thermal-sensors}
+
+
+### Eddy-Current Sensors {#eddy-current-sensors}
+
+
+### Linear Variable Displacement Transformers {#linear-variable-displacement-transformers}
+
+
+### Laser Interferometers {#laser-interferometers}
+
+
+### Linear Encoders {#linear-encoders}
+
+
+## Comparison and Summary {#comparison-and-summary}
+
+
+## Outlook and Future Requirements {#outlook-and-future-requirements}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Shunt Circuit Modeling {#shunt-circuit-modeling}
+
+
+### Open-Loop {#open-loop}
+
+
+### Shunt Damping {#shunt-damping}
+
+
+## Implementation {#implementation}
+
+
+## Experimental Results {#experimental-results}
+
+
+### Tube Dynamics {#tube-dynamics}
+
+
+### Amplifier Performance {#amplifier-performance}
+
+
+### Shunt Damping Performance {#shunt-damping-performance}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Experimental Setup {#experimental-setup}
+
+
+## PI Control {#pi-control}
+
+
+## PI Control with Notch Filters {#pi-control-with-notch-filters}
+
+
+## PI Control with IRC Damping {#pi-control-with-irc-damping}
+
+
+## Performance Comparison {#performance-comparison}
+
+
+## Noise and Resolution {#noise-and-resolution}
+
+
+## Analog Implementation {#analog-implementation}
+
+
+## Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## References {#references}
+
+
+## Introduction {#introduction}
+
+
+## Modeling {#modeling}
+
+
+### Actuator Dynamics {#actuator-dynamics}
+
+
+### Sensor Dynamics {#sensor-dynamics}
+
+
+### Sensor Noise {#sensor-noise}
+
+
+### Mechanical Dynamics {#mechanical-dynamics}
+
+
+### System Properties {#system-properties}
+
+
+### Example System {#example-system}
+
+
+## Damping Control {#damping-control}
+
+
+## Tracking Control {#tracking-control}
+
+
+### Relationship Between Force and Displacement {#relationship-between-force-and-displacement}
+
+
+### Integral Displacement Feedback {#integral-displacement-feedback}
+
+
+### Direct Tracking Control {#direct-tracking-control}
+
+
+### Dual Sensor Feedback {#dual-sensor-feedback}
+
+
+### Low Frequency Bypass {#low-frequency-bypass}
+
+
+### Feedforward Inputs {#feedforward-inputs}
+
+
+### Higher-Order Modes {#higher-order-modes}
+
+
+## Experimental Results {#experimental-results}
+
+
+### Experimental Nanopositioner {#experimental-nanopositioner}
+
+
+### Actuators and Force Sensors {#actuators-and-force-sensors}
+
+
+### Control Design {#control-design}
+
+
+### Noise Performance {#noise-performance}
+
+
+## Chapter Summary {#chapter-summary}
+
+
+## References {#references}
+
+
+## Why Feedforward? {#why-feedforward}
+
+
+## Modeling for Feedforward Control {#modeling-for-feedforward-control}
+
+
+## Feedforward Control of Dynamics and Hysteresis {#feedforward-control-of-dynamics-and-hysteresis}
+
+
+### Simple DC-Gain Feedforward Control {#simple-dc-gain-feedforward-control}
+
+
+### An Inversion-Based Feedforward Approach  for Linear Dynamics {#an-inversion-based-feedforward-approach-for-linear-dynamics}
+
+
+### Frequency-Weighted Inversion: The Optimal Inverse {#frequency-weighted-inversion-the-optimal-inverse}
+
+
+### Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## Feedforward and Feedback Control {#feedforward-and-feedback-control}
+
+
+### Application to AFM Imaging {#application-to-afm-imaging}
+
+
+## Iterative Feedforward Control {#iterative-feedforward-control}
+
+
+### The ILC Problem {#the-ilc-problem}
+
+
+### Model-Based ILC {#model-based-ilc}
+
+
+### Nonlinear ILC {#nonlinear-ilc}
+
+
+### Conclusions {#conclusions}
+
+
+## References {#references}
+
+
+## 10.1 Introduction {#10-dot-1-introduction}
+
+
+### 10.1.1 Background {#10-dot-1-dot-1-background}
+
+
+### 10.1.2 The Optimal Periodic Input {#10-dot-1-dot-2-the-optimal-periodic-input}
+
+
+## 10.2 Signal Optimization {#10-dot-2-signal-optimization}
+
+
+## 10.3 Frequency Domain Cost Functions {#10-dot-3-frequency-domain-cost-functions}
+
+
+### 10.3.1 Background: Discrete Fourier Series {#10-dot-3-dot-1-background-discrete-fourier-series}
+
+
+### 10.3.2 Minimizing Signal Power {#10-dot-3-dot-2-minimizing-signal-power}
+
+
+### 10.3.3 Minimizing Frequency Weighted Power {#10-dot-3-dot-3-minimizing-frequency-weighted-power}
+
+
+### 10.3.4 Minimizing Velocity and Acceleration {#10-dot-3-dot-4-minimizing-velocity-and-acceleration}
+
+
+### 10.3.5 Single-Sided Frequency Domain Calculations {#10-dot-3-dot-5-single-sided-frequency-domain-calculations}
+
+
+## 10.4 Time Domain Cost Function {#10-dot-4-time-domain-cost-function}
+
+
+### 10.4.1 Minimum Velocity {#10-dot-4-dot-1-minimum-velocity}
+
+
+### 10.4.2 Minimum Acceleration {#10-dot-4-dot-2-minimum-acceleration}
+
+
+### 10.4.3 Frequency Weighted Objectives {#10-dot-4-dot-3-frequency-weighted-objectives}
+
+
+## 10.5 Application to Scan Generation {#10-dot-5-application-to-scan-generation}
+
+
+### 10.5.1 Choosing β and K {#10-dot-5-dot-1-choosing-β-and-k}
+
+
+### 10.5.2 Improving Feedback and Feedforward Controllers {#10-dot-5-dot-2-improving-feedback-and-feedforward-controllers}
+
+
+## 10.6 Comparison to Other Techniques {#10-dot-6-comparison-to-other-techniques}
+
+
+## 10.7 Experimental Application {#10-dot-7-experimental-application}
+
+
+## 10.8 Chapter Summary {#10-dot-8-chapter-summary}
+
+
+## References {#references}
+
+
+## 11.1 Introduction {#11-dot-1-introduction}
+
+
+## 11.2 Modeling Hysteresis {#11-dot-2-modeling-hysteresis}
+
+
+### 11.2.1 Simple Polynomial Model {#11-dot-2-dot-1-simple-polynomial-model}
+
+
+### 11.2.2 Maxwell Slip Model {#11-dot-2-dot-2-maxwell-slip-model}
+
+
+### 11.2.3 Duhem Model {#11-dot-2-dot-3-duhem-model}
+
+
+### 11.2.4 Preisach Model {#11-dot-2-dot-4-preisach-model}
+
+
+### 11.2.5 Classical Prandlt-Ishlinksii Model {#11-dot-2-dot-5-classical-prandlt-ishlinksii-model}
+
+
+## 11.3 Feedforward Hysteresis Compensation {#11-dot-3-feedforward-hysteresis-compensation}
+
+
+### 11.3.1 Feedforward Control Using the Presiach Model {#11-dot-3-dot-1-feedforward-control-using-the-presiach-model}
+
+
+### 11.3.2 Feedforward Control Using the Prandlt-Ishlinksii Model {#11-dot-3-dot-2-feedforward-control-using-the-prandlt-ishlinksii-model}
+
+
+## 11.4 Chapter Summary {#11-dot-4-chapter-summary}
+
+
+## References {#references}
+
+
+## 12.1 Introduction {#12-dot-1-introduction}
+
+
+## 12.2 Charge Drives {#12-dot-2-charge-drives}
+
+
+## 12.3 Application to Piezoelectric Stack Nanopositioners {#12-dot-3-application-to-piezoelectric-stack-nanopositioners}
+
+
+## 12.4 Application to Piezoelectric Tube Nanopositioners {#12-dot-4-application-to-piezoelectric-tube-nanopositioners}
+
+
+## 12.5 Alternative Electrode Configurations {#12-dot-5-alternative-electrode-configurations}
+
+
+### 12.5.1 Grounded Internal Electrode {#12-dot-5-dot-1-grounded-internal-electrode}
+
+
+### 12.5.2 Quartered Internal Electrode {#12-dot-5-dot-2-quartered-internal-electrode}
+
+
+## 12.6 Charge Versus Voltage {#12-dot-6-charge-versus-voltage}
+
+
+### 12.6.1 Advantages {#12-dot-6-dot-1-advantages}
+
+
+### 12.6.2 Disadvantages {#12-dot-6-dot-2-disadvantages}
+
+
+## 12.7 Impact on Closed-Loop Control {#12-dot-7-impact-on-closed-loop-control}
+
+
+## 12.8 Chapter Summary {#12-dot-8-chapter-summary}
+
+
+## References {#references}
+
+
+## 13.1 Introduction {#13-dot-1-introduction}
+
+
+## 13.2 Review of Random Processes {#13-dot-2-review-of-random-processes}
+
+
+### 13.2.1 Probability Distributions {#13-dot-2-dot-1-probability-distributions}
+
+
+### 13.2.2 Expected Value, Moments, Variance, and RMS {#13-dot-2-dot-2-expected-value-moments-variance-and-rms}
+
+
+### 13.2.3 Gaussian Random Variables {#13-dot-2-dot-3-gaussian-random-variables}
+
+
+### 13.2.4 Continuous Random Processes {#13-dot-2-dot-4-continuous-random-processes}
+
+
+### 13.2.5 Joint Density Functions and Stationarity {#13-dot-2-dot-5-joint-density-functions-and-stationarity}
+
+
+### 13.2.6 Correlation Functions {#13-dot-2-dot-6-correlation-functions}
+
+
+### 13.2.7 Gaussian Random Processes {#13-dot-2-dot-7-gaussian-random-processes}
+
+
+### 13.2.8 Power Spectral Density {#13-dot-2-dot-8-power-spectral-density}
+
+
+### 13.2.9 Filtered Random Processes {#13-dot-2-dot-9-filtered-random-processes}
+
+
+### 13.2.10 White Noise {#13-dot-2-dot-10-white-noise}
+
+
+### 13.2.11 Spectral Density in V/sqrtHz {#13-dot-2-dot-11-spectral-density-in-v-sqrthz}
+
+
+### 13.2.12 Single- and Double-Sided Spectra {#13-dot-2-dot-12-single-and-double-sided-spectra}
+
+
+## 13.3 Resolution and Noise {#13-dot-3-resolution-and-noise}
+
+
+## 13.4 Sources of Nanopositioning Noise {#13-dot-4-sources-of-nanopositioning-noise}
+
+
+### 13.4.1 Sensor Noise {#13-dot-4-dot-1-sensor-noise}
+
+
+### 13.4.2 External Noise {#13-dot-4-dot-2-external-noise}
+
+
+### 13.4.3 Amplifier Noise {#13-dot-4-dot-3-amplifier-noise}
+
+
+## 13.5 Closed-Loop Position Noise {#13-dot-5-closed-loop-position-noise}
+
+
+### 13.5.1 Noise Sensitivity Functions {#13-dot-5-dot-1-noise-sensitivity-functions}
+
+
+### 13.5.2 Closed-Loop Position Noise Spectral Density {#13-dot-5-dot-2-closed-loop-position-noise-spectral-density}
+
+
+### 13.5.3 Closed-Loop Noise Approximations with Integral Control {#13-dot-5-dot-3-closed-loop-noise-approximations-with-integral-control}
+
+
+### 13.5.4 Closed-Loop Position Noise Variance {#13-dot-5-dot-4-closed-loop-position-noise-variance}
+
+
+### 13.5.5 A Note on Units {#13-dot-5-dot-5-a-note-on-units}
+
+
+## 13.6 Simulation Examples {#13-dot-6-simulation-examples}
+
+
+### 13.6.1 Integral Controller Noise Simulation {#13-dot-6-dot-1-integral-controller-noise-simulation}
+
+
+### 13.6.2 Noise Simulation with Inverse Model Controller {#13-dot-6-dot-2-noise-simulation-with-inverse-model-controller}
+
+
+### 13.6.3 Feedback Versus Feedforward Control {#13-dot-6-dot-3-feedback-versus-feedforward-control}
+
+
+## 13.7 Practical Frequency Domain Noise Measurements {#13-dot-7-practical-frequency-domain-noise-measurements}
+
+
+### 13.7.1 Preamplification {#13-dot-7-dot-1-preamplification}
+
+
+### 13.7.2 Spectrum Estimation {#13-dot-7-dot-2-spectrum-estimation}
+
+
+### 13.7.3 Direct Measurement of Position Noise {#13-dot-7-dot-3-direct-measurement-of-position-noise}
+
+
+### 13.7.4 Measurement of the External Disturbance {#13-dot-7-dot-4-measurement-of-the-external-disturbance}
+
+
+## 13.8 Experimental Demonstration {#13-dot-8-experimental-demonstration}
+
+
+## 13.9 Time-Domain Noise Measurements {#13-dot-9-time-domain-noise-measurements}
+
+
+### 13.9.1 Total Integrated Noise {#13-dot-9-dot-1-total-integrated-noise}
+
+
+### 13.9.2 Estimating the Position Noise {#13-dot-9-dot-2-estimating-the-position-noise}
+
+
+### 13.9.3 Practical Considerations {#13-dot-9-dot-3-practical-considerations}
+
+
+### 13.9.4 Experimental Demonstration {#13-dot-9-dot-4-experimental-demonstration}
+
+
+## 13.10 A Simple Method for Measuring the Resolution  of Nanopositioning Systems {#13-dot-10-a-simple-method-for-measuring-the-resolution-of-nanopositioning-systems}
+
+
+## 13.11 Techniques for Improving Resolution {#13-dot-11-techniques-for-improving-resolution}
+
+
+## 13.12 Chapter Summary {#13-dot-12-chapter-summary}
+
+
+## References {#references}
+
+
+## Electrical Considerations {#electrical-considerations}
+
+
+### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
+
+<a id="figure--fig:fleming14-amplifier-model"></a>
+
+{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class='figure-number'>Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
+
+Consider the electrical circuit shown in [Figure 1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
+The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
+The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
+
+<div class="exampl">
+
+Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
+Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).
+
+</div>
+
+When considering the effects of both output impedance and cable inductance, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) is second-order low pass filter:
+
+\begin{equation}
+  \frac{V\_L(s)}{V\_s(s)} = \frac{1}{\frac{s^2}{\omega\_r^2} + 2 \xi \frac{s}{\omega\_r} + 1}
+\end{equation}
+
+with:
+
+-   \\(\omega\_r = \frac{1}{\sqrt{L C\_p}}\\)
+-   \\(\xi = \frac{R\_s \sqrt{L C\_p}}{2 L}\\)
+
+
+### Amplifier small-signal Bandwidth {#amplifier-small-signal-bandwidth}
+
+The most obvious bandwidth limitation is the small-signal bandwidth of the amplifier.
+
+If the inductance \\(L\\) is neglected, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) forms a first order filter with a cut-off frequency
+
+\begin{equation}
+  \omega\_c = \frac{1}{R\_s C\_p}
+\end{equation}
+
+This is thus highly dependent of the load.
+
+The high capacitive impedance nature of piezoelectric loads introduces phase-lag into the feedback path.
+A rule of thumb is that closed-loop bandwidth cannot exceed one-tenth the cut-off frequency of the pole formed by the amplifier output impedance \\(R\_s\\) and load capacitance \\(C\_p\\) (see [Table 1](#table--tab:piezo-limitation-Rs) for values).
+
+<a id="table--tab:piezo-limitation-Rs"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:piezo-limitation-Rs">Table 1</a>:</span>
+  Bandwidth limitation due to \(R_s\)
+</div>
+
+|              | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
+|--------------|-------------|-----------|------------|
+| Rs = 1 Ohm   | 1.6 MHz     | 160 kHz   | 16 kHz     |
+| Rs = 10 Ohm  | 160 kHz     | 16 kHz    | 1.6 kHz    |
+| Rs = 100 Ohm | 16 kHz      | 1.6 kHz   | 160 Hz     |
+
+The inductance \\(L\\) does also play a role in the amplifier bandwidth as it changes the resonance frequency.
+Ideally, low inductance cables should be used.
+It is however usually quite high compare to \\(\omega\_c\\) as shown in [Table 2](#table--tab:piezo-limitation-L).
+
+<a id="table--tab:piezo-limitation-L"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:piezo-limitation-L">Table 2</a>:</span>
+  Bandwidth limitation due to \(R_s\)
+</div>
+
+|             | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
+|-------------|-------------|-----------|------------|
+| L = 25 nH   | 3.2 MHz     | 1 MHz     | 320 kHz    |
+| L = 250 nH  | 1 MHz       | 320 kHz   | 100 kHz    |
+| L = 2500 nH | 320 kHz     | 100 kHz   | 32 kHz     |
+
+
+### Amplifier maximum slew rate {#amplifier-maximum-slew-rate}
+
+Further bandwidth restrictions are imposed by the maximum **slew rate** of the amplifier.
+This is the maximum rate at which the output voltage can change and is usually expressed in \\(V/\mu s\\).
+
+For sinusoidal signals, the amplifiers slew rate must exceed:
+\\[ SR\_{\text{sin}} > V\_{p-p} \pi f \\]
+where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.
+
+<div class="exampl">
+
+If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).
+
+</div>
+
+When dealing with capacitive loads, **the current limit is usually exceed well before the slew rate limit**.
+
+
+### Current and Power Limitations {#current-and-power-limitations}
+
+When driving the actuator off-resonance, the current delivered to a piezoelectric actuator is approximately:
+\\[ I\_L(s) = V\_L(s) C\_p s \\]
+
+For sinusoidal signals, the maximum positive and negative current is equal to:
+\\[ I\_L^\text{max} = V\_{p-p} \pi f C\_p \\]
+
+<a id="table--tab:piezo-required-current"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:piezo-required-current">Table 3</a>:</span>
+  Minimum current requirements for a 10V sinusoid
+</div>
+
+|             | Cp = 100 nF | Cp = 1 uF | Cp = 10 uF |
+|-------------|-------------|-----------|------------|
+| f = 30 Hz   | 0.19 mA     | 1.9 mA    | 19 mA      |
+| f = 3 kHz   | 19 mA       | 190 mA    | 1.9 A      |
+| f = 300 kHz | 1.9 A       | 19 A      | 190 A      |
+
+
+### Chapter Summary {#chapter-summary}
+
+The bandwidth limitations  of standard piezoelectric drives were identified as:
+
+-   High output impedance
+-   The presence of a ple in the voltage-feedback loop due to output impedance and load capacitance
+-   Insufficient current capacity due to power dissipation
+-   High cable and connector inductance
+
+
+### References {#references}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, A. J., and K. K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-06617-2">10.1007/978-3-319-06617-2</a>.</div>
+</div>
@@ -0,0 +1,26 @@
+++
+title = "The Art of Electronics - Third Edition"
+author = ["Dehaeze Thomas"]
+description = "One of the best book in electronics. Cover most topics (both analog and digital)."
+keywords = ["electronics"]
+draft = false
+++
+
+Tags
+: [Reference Books]({{< relref "reference_books.md" >}}), [Electronics]({{< relref "electronics.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Horowitz 2015</a>)
+
+Author(s)
+: Horowitz, P.
+
+Year
+: 2015
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Horowitz, Paul. 2015. <i>The Art of Electronics - Third Edition</i>. New York, NY, USA: Cambridge University Press.</div>
+</div>
@@ -0,0 +1,95 @@
+++
+title = "Fundamental principles of engineering nanometrology"
+author = ["Dehaeze Thomas"]
+keywords = ["Metrology"]
+draft = false
+++
+
+Tags
+: [Metrology]({{< relref "metrology.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Leach 2014</a>)
+
+Author(s)
+: Leach, R.
+
+Year
+: 2014
+
+
+## Measurement of angles {#measurement-of-angles}
+
+Unit:
+
+-   radian for plane angle
+-   steradian for solid angle
+
+\\(1 rad \approx 55.3deg\\)
+
+Instrument principles:
+
+-   subdivision: index tacle, angular gratings, polygons, ...
+-   ratio of two lengths: angular interferometers, sin cars, small angle generators, ...
+-   autocollimators with a flat mirror
+
+
+## Sources of error in displacement interferometry {#sources-of-error-in-displacement-interferometry}
+
+Two error sources:
+
+-   error sources that are proportional to the displacement being measured \\(L\\): cumulative errors
+-   error sources that are independent of the displacement being measured: non-cumulative errors
+
+
+### Thermal expansion of the metrology frame {#thermal-expansion-of-the-metrology-frame}
+
+
+### Deadpath length {#deadpath-length}
+
+Deadpath length, \\(d\\), is defined as the difference in distance in air between the reference and measurement reflectors and the beam splitter when the interferometer measurement is initiated.
+Deadpath error occurs when there is a non-zero deadpath and environmental conditions change during a measurement.
+
+
+### Cosine error {#cosine-error}
+
+\\(\Delta l = l(1-\cos(\theta))\\)
+
+For small angles: \\(\Delta l = \frac{l \theta^2}{2}\\)
+
+The cosine error is then a second-order effect, contrary to the Abbe error which is a first order effect.
+The second order nature means that cosine error quickly diminish as the alignment is improved.
+
+
+## Latest advances in displacement interferometry {#latest-advances-in-displacement-interferometry}
+
+Commercial interferometers
+=&gt; fused silica optics housed in Invar mounts
+=&gt; all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
+
+One advantage that homodyme systems have over heterodyne systems is their ability to readily have the source fibre delivered to the interferometer.
+
+
+### Spatially separated interferometers {#spatially-separated-interferometers}
+
+It uses heterodyne interferometer and one quadrant photodiode.
+By knowing the beam size and detector geometry, the measurement target's angle change can be determined by differencing matched pairs of measured phase from the quadrant photodiode while the displacement is determined from the average phase over the four quadrants.
+
+
+## Angular interferometers {#angular-interferometers}
+
+Determination of an angle by the ratio of two lengths.
+The angular optics is used to create two parallel beam paths between the angular interferometer and the angular reflector.
+
+The beam that illuminates the angular optics contains two frequencies, \\(f1\\) and \\(f2\\). A polarising beam splitter in the angular interferometer splits the frequencies that travel along separate paths.
+
+The measurement of angles is then relative.
+
+This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard. 2014. <i>Fundamental Principles of Engineering Nanometrology</i>. Elsevier. doi:<a href="https://doi.org/10.1016/c2012-0-06010-3">10.1016/c2012-0-06010-3</a>.</div>
+</div>
@@ -0,0 +1,25 @@
+++
+title = "Basics of precision engineering - 1st edition"
+author = ["Dehaeze Thomas"]
+keywords = ["Metrology", "Mechatronics"]
+draft = true
+++
+
+Tags
+: [Precision Engineering]({{< relref "precision_engineering.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Leach and Smith 2018</a>)
+
+Author(s)
+: Leach, R., &amp; Smith, S. T.
+
+Year
+: 2018
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard, and Stuart T. Smith. 2018. <i>Basics of Precision Engineering - 1st Edition</i>. CRC Press.</div>
+</div>
@@ -0,0 +1,538 @@
+++
+title = "Understanding Digital Signal Processing"
+author = ["Dehaeze Thomas"]
+draft = true
+++
+
+Tags
+: [IRR and FIR Filters]({{< relref "irr_and_fir_filters.md" >}}), [Digital Filters]({{< relref "digital_filters.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Lyons 2011</a>)
+
+Author(s)
+: Lyons, R.
+
+Year
+: 2011
+
+
+## Discrete Sequences And Systems {#discrete-sequences-and-systems}
+
+
+### Discrete Sequences And Their Notation {#discrete-sequences-and-their-notation}
+
+
+### Signal Amplitude, Magnitude, Power {#signal-amplitude-magnitude-power}
+
+
+### Signal Processing Operational Symbols {#signal-processing-operational-symbols}
+
+
+### Introduction To Discrete Linear Time-Invariant Systems {#introduction-to-discrete-linear-time-invariant-systems}
+
+
+### Discrete Linear Systems {#discrete-linear-systems}
+
+
+### Time-Invariant Systems {#time-invariant-systems}
+
+
+### The Commutative Property Of Linear Time-Invariant Systems {#the-commutative-property-of-linear-time-invariant-systems}
+
+
+### Analyzing Linear Time-Invariant Systems {#analyzing-linear-time-invariant-systems}
+
+<a id="figure--fig:lyons11-lti-impulse-response"></a>
+
+{{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="<span class='figure-number'>Figure 1: </span>LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \\(x(n)\\) and impulse reponse output sequence \\(y(n)\\)." >}}
+
+<a id="figure--fig:lyons11-moving-average"></a>
+
+{{< figure src="/ox-hugo/lyons11_moving_average.png" caption="<span class='figure-number'>Figure 2: </span>Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}}
+
+
+## Periodic Sampling {#periodic-sampling}
+
+
+### Aliasing: Signal Ambiguity In The Frequency Domain {#aliasing-signal-ambiguity-in-the-frequency-domain}
+
+<a id="figure--fig:lyons11-frequency-ambiguity"></a>
+
+{{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="<span class='figure-number'>Figure 3: </span>Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}}
+
+
+### Sampling Lowpass Signals {#sampling-lowpass-signals}
+
+<a id="figure--fig:lyons11-noise-spectral-replication"></a>
+
+{{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="<span class='figure-number'>Figure 4: </span>Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}}
+
+<a id="figure--fig:lyons11-lowpass-sampling"></a>
+
+{{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="<span class='figure-number'>Figure 5: </span>Low pass analog filtering prior to sampling at a rate of \\(f\_s\\) Hz." >}}
+
+
+## The Discrete Fourier Transform {#the-discrete-fourier-transform}
+
+\begin{equation}
+X(f) = \int\_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt
+\end{equation}
+
+\begin{equation}
+X(m) = \sum\_{n = 0}^{N-1} x(n) e^{-j2 \pi n m /N}
+\end{equation}
+
+
+### Understanding The Dft Equation {#understanding-the-dft-equation}
+
+
+### Dft Symmetry {#dft-symmetry}
+
+
+### Dft Linearity {#dft-linearity}
+
+
+### Dft Magnitudes {#dft-magnitudes}
+
+
+### Dft Frequency Axis {#dft-frequency-axis}
+
+
+### Dft Shifting Theorem {#dft-shifting-theorem}
+
+
+### Inverse Dft {#inverse-dft}
+
+
+### Dft Leakage {#dft-leakage}
+
+
+### Windows {#windows}
+
+
+### Dft Scalloping Loss {#dft-scalloping-loss}
+
+
+### Dft Resolution, Zero Padding, And Frequency-Domain Sampling {#dft-resolution-zero-padding-and-frequency-domain-sampling}
+
+
+### Dft Processing Gain {#dft-processing-gain}
+
+
+### The Dft Of Rectangular Functions {#the-dft-of-rectangular-functions}
+
+
+### Interpreting The Dft Using The Discrete-Time Fourier Transform {#interpreting-the-dft-using-the-discrete-time-fourier-transform}
+
+
+## The Fast Fourier Transform {#the-fast-fourier-transform}
+
+
+### Relationship Of The Fft To The Dft {#relationship-of-the-fft-to-the-dft}
+
+
+### Hints On Using Ffts In Practice {#hints-on-using-ffts-in-practice}
+
+
+### Derivation Of The Radix-2 Fft Algorithm {#derivation-of-the-radix-2-fft-algorithm}
+
+
+### Fft Input/Output Data Index Bit Reversal {#fft-input-output-data-index-bit-reversal}
+
+
+### Radix-2 Fft Butterfly Structures {#radix-2-fft-butterfly-structures}
+
+
+### Alternate Single-Butterfly Structures {#alternate-single-butterfly-structures}
+
+
+## Finite Impulse Response Filters {#finite-impulse-response-filters}
+
+
+### An Introduction To Finite Impulse Response (Fir) Filters {#an-introduction-to-finite-impulse-response--fir--filters}
+
+
+### Convolution In Fir Filters {#convolution-in-fir-filters}
+
+
+### Lowpass Fir Filter Design {#lowpass-fir-filter-design}
+
+
+### Bandpass Fir Filter Design {#bandpass-fir-filter-design}
+
+
+### Highpass Fir Filter Design {#highpass-fir-filter-design}
+
+
+### Parks-Mcclellan Exchange Fir Filter Design Method {#parks-mcclellan-exchange-fir-filter-design-method}
+
+
+### Half-Band Fir Filters {#half-band-fir-filters}
+
+
+### Phase Response Of Fir Filters {#phase-response-of-fir-filters}
+
+
+### A Generic Description Of Discrete Convolution {#a-generic-description-of-discrete-convolution}
+
+
+### Analyzing Fir Filters {#analyzing-fir-filters}
+
+
+## Infinite Impulse Response Filters {#infinite-impulse-response-filters}
+
+
+### An Introduction To Infinite Impulse Response Filters {#an-introduction-to-infinite-impulse-response-filters}
+
+
+### The Laplace Transform {#the-laplace-transform}
+
+
+### The Z-Transform {#the-z-transform}
+
+
+### Using The Z-Transform To Analyze Iir Filters {#using-the-z-transform-to-analyze-iir-filters}
+
+
+### Using Poles And Zeros To Analyze Iir Filters {#using-poles-and-zeros-to-analyze-iir-filters}
+
+
+### Alternate Iir Filter Structures {#alternate-iir-filter-structures}
+
+
+### Pitfalls In Building Iir Filters {#pitfalls-in-building-iir-filters}
+
+
+### Improving Iir Filters With Cascaded Structures {#improving-iir-filters-with-cascaded-structures}
+
+
+### Scaling The Gain Of Iir Filters {#scaling-the-gain-of-iir-filters}
+
+
+### Impulse Invariance Iir Filter Design Method {#impulse-invariance-iir-filter-design-method}
+
+
+### Bilinear Transform Iir Filter Design Method {#bilinear-transform-iir-filter-design-method}
+
+
+### Optimized Iir Filter Design Method {#optimized-iir-filter-design-method}
+
+
+### A Brief Comparison Of Iir And Fir Filters {#a-brief-comparison-of-iir-and-fir-filters}
+
+
+## Specialized Digital Networks And Filters {#specialized-digital-networks-and-filters}
+
+
+### Differentiators {#differentiators}
+
+
+### Integrators {#integrators}
+
+
+### Matched Filters {#matched-filters}
+
+
+### Interpolated Lowpass Fir Filters {#interpolated-lowpass-fir-filters}
+
+
+### Frequency Sampling Filters: The Lost Art {#frequency-sampling-filters-the-lost-art}
+
+
+## Quadrature Signals {#quadrature-signals}
+
+
+### Why Care About Quadrature Signals? {#why-care-about-quadrature-signals}
+
+
+### The Notation Of Complex Numbers {#the-notation-of-complex-numbers}
+
+
+### Representing Real Signals Using Complex Phasors {#representing-real-signals-using-complex-phasors}
+
+
+### A Few Thoughts On Negative Frequency {#a-few-thoughts-on-negative-frequency}
+
+
+### Quadrature Signals In The Frequency Domain {#quadrature-signals-in-the-frequency-domain}
+
+
+### Bandpass Quadrature Signals In The Frequency Domain {#bandpass-quadrature-signals-in-the-frequency-domain}
+
+
+### Complex Down-Conversion {#complex-down-conversion}
+
+
+### A Complex Down-Conversion Example {#a-complex-down-conversion-example}
+
+
+### An Alternate Down-Conversion Method {#an-alternate-down-conversion-method}
+
+
+## The Discrete Hilbert Transform {#the-discrete-hilbert-transform}
+
+
+### Hilbert Transform Definition {#hilbert-transform-definition}
+
+
+### Why Care About The Hilbert Transform? {#why-care-about-the-hilbert-transform}
+
+
+### Impulse Response Of A Hilbert Transformer {#impulse-response-of-a-hilbert-transformer}
+
+
+### Designing A Discrete Hilbert Transformer {#designing-a-discrete-hilbert-transformer}
+
+
+### Time-Domain Analytic Signal Generation {#time-domain-analytic-signal-generation}
+
+
+### Comparing Analytical Signal Generation Methods {#comparing-analytical-signal-generation-methods}
+
+
+## 10 Sample Rate Conversion {#10-sample-rate-conversion}
+
+
+### 10.1 Decimation {#10-dot-1-decimation}
+
+
+### 10.2 Two-Stage Decimation {#10-dot-2-two-stage-decimation}
+
+
+### 10.3 Properties Of Downsampling {#10-dot-3-properties-of-downsampling}
+
+
+### 10.4 Interpolation {#10-dot-4-interpolation}
+
+
+### 10.5 Properties Of Interpolation {#10-dot-5-properties-of-interpolation}
+
+
+### 10.6 Combining Decimation And Interpolation {#10-dot-6-combining-decimation-and-interpolation}
+
+
+### 10.7 Polyphase Filters {#10-dot-7-polyphase-filters}
+
+
+### 10.8 Two-Stage Interpolation {#10-dot-8-two-stage-interpolation}
+
+
+### 10.9 Z-Transform Analysis Of Multirate Systems {#10-dot-9-z-transform-analysis-of-multirate-systems}
+
+
+### 10.10 Polyphase Filter Implementations {#10-dot-10-polyphase-filter-implementations}
+
+
+### 10.11 Sample Rate Conversion By Rational Factors {#10-dot-11-sample-rate-conversion-by-rational-factors}
+
+
+### 10.12 Sample Rate Conversion With Half-Band Filters {#10-dot-12-sample-rate-conversion-with-half-band-filters}
+
+
+### 10.13 Sample Rate Conversion With Ifir Filters {#10-dot-13-sample-rate-conversion-with-ifir-filters}
+
+
+### 10.14 Cascaded Integrator-Comb Filters {#10-dot-14-cascaded-integrator-comb-filters}
+
+
+## 11 Signal Averaging {#11-signal-averaging}
+
+
+### 11.1 Coherent Averaging {#11-dot-1-coherent-averaging}
+
+
+### 11.2 Incoherent Averaging {#11-dot-2-incoherent-averaging}
+
+
+### 11.3 Averaging Multiple Fast Fourier Transforms {#11-dot-3-averaging-multiple-fast-fourier-transforms}
+
+
+### 11.4 Averaging Phase Angles {#11-dot-4-averaging-phase-angles}
+
+
+### 11.5 Filtering Aspects Of Time-Domain Averaging {#11-dot-5-filtering-aspects-of-time-domain-averaging}
+
+
+### 11.6 Exponential Averaging {#11-dot-6-exponential-averaging}
+
+
+## 12 Digital Data Formats And Their Effects {#12-digital-data-formats-and-their-effects}
+
+
+### 12.1 Fixed-Point Binary Formats {#12-dot-1-fixed-point-binary-formats}
+
+
+### 12.2 Binary Number Precision And Dynamic Range {#12-dot-2-binary-number-precision-and-dynamic-range}
+
+
+### 12.3 Effects Of Finite Fixed-Point Binary Word Length {#12-dot-3-effects-of-finite-fixed-point-binary-word-length}
+
+
+### 12.4 Floating-Point Binary Formats {#12-dot-4-floating-point-binary-formats}
+
+
+### 12.5 Block Floating-Point Binary Format {#12-dot-5-block-floating-point-binary-format}
+
+
+## 13 Digital Signal Processing Tricks {#13-digital-signal-processing-tricks}
+
+
+### 13.1 Frequency Translation Without Multiplication {#13-dot-1-frequency-translation-without-multiplication}
+
+
+### 13.2 High-Speed Vector Magnitude Approximation {#13-dot-2-high-speed-vector-magnitude-approximation}
+
+
+### 13.3 Frequency-Domain Windowing {#13-dot-3-frequency-domain-windowing}
+
+
+### 13.4 Fast Multiplication Of Complex Numbers {#13-dot-4-fast-multiplication-of-complex-numbers}
+
+
+### 13.5 Efficiently Performing The Fft Of Real Sequences {#13-dot-5-efficiently-performing-the-fft-of-real-sequences}
+
+
+### 13.6 Computing The Inverse Fft Using The Forward Fft {#13-dot-6-computing-the-inverse-fft-using-the-forward-fft}
+
+
+### 13.7 Simplified Fir Filter Structure {#13-dot-7-simplified-fir-filter-structure}
+
+
+### 13.8 Reducing A/D Converter Quantization Noise {#13-dot-8-reducing-a-d-converter-quantization-noise}
+
+
+### 13.9 A/D Converter Testing Techniques {#13-dot-9-a-d-converter-testing-techniques}
+
+
+### 13.10 Fast Fir Filtering Using The Fft {#13-dot-10-fast-fir-filtering-using-the-fft}
+
+
+### 13.11 Generating Normally Distributed Random Data {#13-dot-11-generating-normally-distributed-random-data}
+
+
+### 13.12 Zero-Phase Filtering {#13-dot-12-zero-phase-filtering}
+
+
+### 13.13 Sharpened Fir Filters {#13-dot-13-sharpened-fir-filters}
+
+
+### 13.14 Interpolating A Bandpass Signal {#13-dot-14-interpolating-a-bandpass-signal}
+
+
+### 13.15 Spectral Peak Location Algorithm {#13-dot-15-spectral-peak-location-algorithm}
+
+
+### 13.16 Computing Fft Twiddle Factors {#13-dot-16-computing-fft-twiddle-factors}
+
+
+### 13.17 Single Tone Detection {#13-dot-17-single-tone-detection}
+
+
+### 13.18 The Sliding Dft {#13-dot-18-the-sliding-dft}
+
+
+### 13.19 The Zoom Fft {#13-dot-19-the-zoom-fft}
+
+
+### 13.20 A Practical Spectrum Analyzer {#13-dot-20-a-practical-spectrum-analyzer}
+
+
+### 13.21 An Efficient Arctangent Approximation {#13-dot-21-an-efficient-arctangent-approximation}
+
+
+### 13.22 Frequency Demodulation Algorithms {#13-dot-22-frequency-demodulation-algorithms}
+
+
+### 13.23 Dc Removal {#13-dot-23-dc-removal}
+
+
+### 13.24 Improving Traditional Cic Filters {#13-dot-24-improving-traditional-cic-filters}
+
+
+### 13.25 Smoothing Impulsive Noise {#13-dot-25-smoothing-impulsive-noise}
+
+
+### 13.26 Efficient Polynomial Evaluation {#13-dot-26-efficient-polynomial-evaluation}
+
+
+### 13.27 Designing Very High-Order Fir Filters {#13-dot-27-designing-very-high-order-fir-filters}
+
+
+### 13.28 Time-Domain Interpolation Using The Fft {#13-dot-28-time-domain-interpolation-using-the-fft}
+
+
+### 13.29 Frequency Translation Using Decimation {#13-dot-29-frequency-translation-using-decimation}
+
+
+### 13.30 Automatic Gain Control (Agc) {#13-dot-30-automatic-gain-control--agc}
+
+
+### 13.31 Approximate Envelope Detection {#13-dot-31-approximate-envelope-detection}
+
+
+### 13.32 A Quadrature Oscillator {#13-dot-32-a-quadrature-oscillator}
+
+
+### 13.33 Specialized Exponential Averaging {#13-dot-33-specialized-exponential-averaging}
+
+
+### 13.34 Filtering Narrowband Noise Using Filter Nulls {#13-dot-34-filtering-narrowband-noise-using-filter-nulls}
+
+
+### 13.35 Efficient Computation Of Signal Variance {#13-dot-35-efficient-computation-of-signal-variance}
+
+
+### 13.36 Real-Time Computation Of Signal Averages And Variances {#13-dot-36-real-time-computation-of-signal-averages-and-variances}
+
+
+### 13.37 Building Hilbert Transformers From Half-Band Filters {#13-dot-37-building-hilbert-transformers-from-half-band-filters}
+
+
+### 13.38 Complex Vector Rotation With Arctangents {#13-dot-38-complex-vector-rotation-with-arctangents}
+
+
+### 13.39 An Efficient Differentiating Network {#13-dot-39-an-efficient-differentiating-network}
+
+
+### 13.40 Linear-Phase Dc-Removal Filter {#13-dot-40-linear-phase-dc-removal-filter}
+
+
+### 13.41 Avoiding Overflow In Magnitude Computations {#13-dot-41-avoiding-overflow-in-magnitude-computations}
+
+
+### 13.42 Efficient Linear Interpolation {#13-dot-42-efficient-linear-interpolation}
+
+
+### 13.43 Alternate Complex Down-Conversion Schemes {#13-dot-43-alternate-complex-down-conversion-schemes}
+
+
+### 13.44 Signal Transition Detection {#13-dot-44-signal-transition-detection}
+
+
+### 13.45 Spectral Flipping Around Signal Center Frequency {#13-dot-45-spectral-flipping-around-signal-center-frequency}
+
+
+### 13.46 Computing Missing Signal Samples {#13-dot-46-computing-missing-signal-samples}
+
+
+### 13.47 Computing Large Dfts Using Small Ffts {#13-dot-47-computing-large-dfts-using-small-ffts}
+
+
+### 13.48 Computing Filter Group Delay Without Arctangents {#13-dot-48-computing-filter-group-delay-without-arctangents}
+
+
+### 13.49 Computing A Forward And Inverse Fft Using A Single Fft {#13-dot-49-computing-a-forward-and-inverse-fft-using-a-single-fft}
+
+
+### 13.50 Improved Narrowband Lowpass Iir Filters {#13-dot-50-improved-narrowband-lowpass-iir-filters}
+
+
+### 13.51 A Stable Goertzel Algorithm {#13-dot-51-a-stable-goertzel-algorithm}
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Lyons, Richard. 2011. <i>Understanding Digital Signal Processing</i>. Upper Saddle River, NJ: Prentice Hall.</div>
+</div>
@@ -0,0 +1,24 @@
+++
+title = "System identification : a frequency domain approach"
+author = ["Dehaeze Thomas"]
+draft = true
+++
+
+Tags
+: [System Identification]({{< relref "system_identification.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Pintelon and Schoukens 2012</a>)
+
+Author(s)
+: Pintelon, R., &amp; Schoukens, J.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Pintelon, R., and J. Schoukens. 2012. <i>System Identification : a Frequency Domain Approach</i>. Hoboken, N.J. Piscataway, NJ: Wiley IEEE Press. doi:<a href="https://doi.org/10.1002/9781118287422">10.1002/9781118287422</a>.</div>
+</div>
@@ -0,0 +1,25 @@
+++
+title = "Mastering system identification in 100 exercises"
+author = ["Dehaeze Thomas"]
+draft = true
+++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Schoukens, Pintelon, and Rolain 2012</a>)
+
+Author(s)
+: Schoukens, J., Pintelon, R., &amp; Rolain, Y.
+
+Year
+: 2012
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Schoukens, Johan, Rik Pintelon, and Yves Rolain. 2012. <i>Mastering System Identification in 100 Exercises</i>. John Wiley &#38; Sons.</div>
+</div>
@@ -0,0 +1,25 @@
+++
+title = "Precision Machine Design"
+author = ["Dehaeze Thomas"]
+draft = true
+++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Slocum 1992</a>)
+
+Author(s)
+: Slocum, A. H.
+
+Year
+: 1992
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Slocum, Alexander H. 1992. <i>Precision Machine Design</i>. Society of Manufacturing Engineers.</div>
+</div>
@@ -0,0 +1,25 @@
+++
+title = "The scientist and engineer's guide to digital signal processing - second edition"
+author = ["Dehaeze Thomas"]
+keywords = ["Signal Processing"]
+draft = true
+++
+
+Tags
+: [Digital Signal Processing]({{< relref "digital_signal_processing.md" >}})
+
+Reference
+: (<a href="#citeproc_bib_item_1">Smith 1999</a>)
+
+Author(s)
+: Smith, S. W.
+
+Year
+: 1999
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Smith, Steven W. 1999. <i>The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition</i>. California Technical Publishing.</div>
+</div>
@@ -0,0 +1,146 @@
+++
+title = "Ultra Precision Bearings"
+author = ["Dehaeze Thomas"]
+draft = true
+++
+
+Tags
+:
+
+
+Reference
+: (<a href="#citeproc_bib_item_1">Wardle 2015</a>)
+
+Author(s)
+: Wardle, F.
+
+Year
+: 2015
+
+
+## Bearing motion error {#bearing-motion-error}
+
+Causes:
+
+-   Manufacturing Quality
+-   Bearing design
+-   External influences
+
+**Types of error motion**
+
+A distinction is made between (see Figure <fig:wardle15_synchronous_asynchronous_schematic>):
+
+-   motion errors that are harmonic of the basic rotor speed: _synchronous_ motion error
+-   those that are node: _asynchronous_ motion errors
+
+<a id="figure--fig:wardle15-synchronous-asynchronous-schematic"></a>
+
+{{< figure src="/ox-hugo/wardle15_synchronous_asynchronous_schematic.png" caption="<span class='figure-number'>Figure 1: </span>Effect of (a) synchronous, and (b) asynchronous motion error on surface form" >}}
+
+
+### Measurement of motion error {#measurement-of-motion-error}
+
+A capacitive sensor is typically used, and the measurement is performed over a time period corresponding to several (typically five) revolutions of the bearing.
+
+It is displayed as a polar plot of motion error amplitude versus angle of rotation (Figure <fig:wardle15_typical_error_plot>).
+
+<a id="figure--fig:wardle15-typical-error-plot"></a>
+
+{{< figure src="/ox-hugo/wardle15_typical_error_plot.png" caption="<span class='figure-number'>Figure 2: </span>Total error motion" >}}
+
+The Synchronous error motion (i.e. error that are harmonics of the rotational speed) can be extracted (Figure <fig:wardle15_synchronous_error_example>).
+
+<a id="figure--fig:wardle15-synchronous-error-example"></a>
+
+{{< figure src="/ox-hugo/wardle15_synchronous_error_example.png" caption="<span class='figure-number'>Figure 3: </span>Synchronous motion error" >}}
+
+It can then be separated into a "_fundamental error motion_" and a "_residual error motion_" (Figure <fig:wardle15_fundamental_and_residual_errors>).
+The fundamental error motion contains only one frequency corresponding to the speed of the rotation of the bearing.
+For radial measurements, it corresponds to the eccentricity, and is not always significant.
+
+<a id="figure--fig:wardle15-fundamental-and-residual-errors"></a>
+
+{{< figure src="/ox-hugo/wardle15_fundamental_and_residual_errors.png" caption="<span class='figure-number'>Figure 4: </span>(a) Fundamental error motion; and (b) residual synchronous error motion" >}}
+
+The Asynchronous error motion (Figure <fig:wardle15_asynchronous_error_motion_example>) contains all other motion error frequencies.
+
+<a id="figure--fig:wardle15-asynchronous-error-motion-example"></a>
+
+{{< figure src="/ox-hugo/wardle15_asynchronous_error_motion_example.png" caption="<span class='figure-number'>Figure 5: </span>Asynchronous motion error" >}}
+
+The measurements shown in previous figures may be quantified by a number of different parameters but it is commonplace to find the "Least Squares" best fit centre and then to place Maximum Inscribed and Minimum Circumscribed circles on the measurement.
+
+The radial separation of the centres of the circles then represents a "Peak to Peak" value of the error motion.
+
+In many cases, the displacement sensor is mounted over a rotating target surface attached to the shaft supported by the bearings.
+However, the displacement sensor now measures not only the motion error of the shaft but also any **geometrical errors present in the target surface**.
+For a radial error motion measurement, out of roundness of the target surface is recorded along with the shaft’s motion error.
+As the motion error of ultra precision bearings may be comparable in magnitude to the geometrical errors in the most accurately manufactured target surfaces then a correction must be made.
+A measurement procedure was proposed that involved two measurements, one with the target surface fixed at some angular position relative to the shaft and the second with it moved through 180 degrees.
+By adding or subtracting the two measurements, geometrical errors on the target surface can be separated from shaft motion errors.
+
+
+### Frequency Analysis {#frequency-analysis}
+
+In general, rotating systems will exhibit motion errors containing several series of harmonics, each of which relate to different aspects or components of the system.
+The main benefit of frequency analysis is therefore to obtain diagnostic information with which to identify the likely sources of motion error and to help reduce their amplitude should they be unacceptable.
+
+
+## Ball Bearings {#ball-bearings}
+
+Criterion used in this book to define ultra precision bearings: motion error of less than 100 nm peak to peak.
+
+Generally only the precision grades or low noise grades of ball bearing are
+likely to produce low motion errors. These types of ball bearing are widely
+used in high precision machine tools, quiet running electric motors,
+computer disc drives and instrumentation, where they provide good but
+not exceptional running accuracy at a competitive price.
+
+Single-row radial ball bearings are favoured in precision engineering
+applications such as computer disc drives and precision electric motors,
+where low motion errors or low noise are a primary requirement. Angular
+contact bearings, on the other hand, are widely used in precision applica-
+tions such as machine tool spindles and rotary tables where static stiffness
+is also important.
+
+
+### Motion Error {#motion-error}
+
+During the 1980s and 1990s, the computer disc drive industry emerged
+as a major application for ball bearings and motion error was recognised as
+a critical bearing performance parameter directly influencing disc capacity.
+Unlike the electric motor application, where bearings may operate under a
+diverse range of conditions, this application was focused on low cost,
+miniature bearings operating under specific conditions of light axial load
+and medium speed at near ambient temperatures. Early research work,
+performed mainly in Japan, developed an understanding of the factors that
+determine the radial motion error of disc drive ball bearings [34–38] and
+later focused specifically on reducing the ‘Non-Repeatable Run Out’
+(NRRO) [39–43]. Because in this application bearing speeds are moderate,
+the NRRO was found to be largely influenced by ball size variation.
+
+In terms of peak–peak motion error amplitudes, ball bearings can achieve
+a creditable performance. Amplitudes as low as 48 nm have been reported
+in scientific papers [41], for ball bearings used in computer hard disc drives.
+This is comparable to the motion error of some types of fluid film, but the
+**disadvantage of ball bearings is that the motion error is predominantly
+asynchronous whereas for fluid film bearings it is mostly synchronous**.
+
+The main reason is that for ball bearings, motion error frequencies relate to
+the orbital and spinning speeds of the balls and these can never be harmonic
+of shaft speed in a practical bearing design.
+
+Ball bearing motion error is
+influenced by a large number of parameters, some a function of the bearing
+design and manufacturing processes, others being dependent on application
+conditions. However, there are relatively few basic mechanisms by which
+motion error can be generated and by understanding these, the influence of
+different parameters can be more clearly defined and in many cases, even
+quantified.
+
+
+#### Dynamics model for estimating call bearing motion error {#dynamics-model-for-estimating-call-bearing-motion-error}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wardle, Frank. 2015. <i>Ultra Precision Bearings</i>. Elsevier.</div>
+</div>