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Thomas Dehaeze 2020-11-25 19:35:11 +01:00
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<h1 class="title">Robust Control - \(\mathcal{H}_\infty\) Synthesis</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgf9a4c6d">1. Introduction to the Control Methodology - Model Based Control</a></li>
<li><a href="#org3fca6e1">2. Some Background: From Classical Control to Robust Control</a></li>
<li><a href="#org2092347">3. The \(\mathcal{H}_\infty\) Norm</a></li>
<li><a href="#org2331e77">4. \(\mathcal{H}_\infty\) Synthesis</a></li>
<li><a href="#org9b8faac">5. The Generalized Plant</a></li>
<li><a href="#org8116ae5">6. Problem Formulation</a></li>
<li><a href="#org42f9bcf">7. Classical feedback control and closed loop transfer functions</a></li>
<li><a href="#orgf00feb8">8. From a Classical Feedback Architecture to a Generalized Plant</a></li>
<li><a href="#org7c94ee9">9. Modern Interpretation of the Control Specifications</a>
<ul>
<li><a href="#orgfcb96b5">9.1. Introduction</a></li>
</ul>
</li>
<li><a href="#orgc9ce5a6">10. Resources</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgf9a4c6d" class="outline-2">
<h2 id="orgf9a4c6d"><span class="section-number-2">1</span> Introduction to the Control Methodology - Model Based Control</h2>
<div class="outline-text-2" id="text-1">
<p>
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure <a href="#orgb1a2ae6">1</a>.
It consists of three steps:
</p>
<ol class="org-ol">
<li><b>Identification or modeling</b>: \(\Longrightarrow\) mathematical model</li>
<li><b>Translate the specifications into mathematical criteria</b>:
<ul class="org-ul">
<li><span class="underline">Specifications</span>: Response Time, Noise Rejection, Maximum input amplitude, Robustness, &#x2026;</li>
<li><span class="underline">Mathematical Criteria</span>: Cost Function, Shape of TF</li>
</ul></li>
<li><b>Synthesis</b>: research of \(K\) that satisfies the specifications for the model of the system</li>
</ol>
<div id="orgb1a2ae6" class="figure">
<p><img src="figs/control-procedure.png" alt="control-procedure.png" />
</p>
<p><span class="figure-number">Figure 1: </span>Typical Methodoly for Model Based Control</p>
</div>
<p>
In this document, we will mainly focus on steps 2 and 3.
</p>
</div>
</div>
<div id="outline-container-org3fca6e1" class="outline-2">
<h2 id="org3fca6e1"><span class="section-number-2">2</span> Some Background: From Classical Control to Robust Control</h2>
<div class="outline-text-2" id="text-2">
<p>
Classical Control (1930)
</p>
<ul class="org-ul">
<li>Tools:
<ul class="org-ul">
<li>TF (input-output)</li>
<li>Nyquist, Bode, Black, \ldots</li>
<li>P-PI-PID, Phase lead-lag, \ldots</li>
</ul></li>
<li>Advantages:
<ul class="org-ul">
<li>Stability</li>
<li>Performances</li>
<li>Robustness</li>
</ul></li>
<li>Disadvantages:
<ul class="org-ul">
<li>Manual Method</li>
<li>Only SISO</li>
</ul></li>
</ul>
<p>
Modern Control (1960)
</p>
<ul class="org-ul">
<li>Tools:
<ul class="org-ul">
<li>State Space</li>
<li>Optimal Command</li>
<li>LQR, LQG</li>
</ul></li>
<li>Advantages:
<ul class="org-ul">
<li>Automatic Synthesis</li>
<li>MIMO</li>
<li>Optimisation problem</li>
</ul></li>
<li>Disadvantages:
<ul class="org-ul">
<li>Robustness</li>
<li>Rejection of Perturbations</li>
</ul></li>
</ul>
<p>
Robust Control (1980)
</p>
<ul class="org-ul">
<li>Tools:
<ul class="org-ul">
<li>Disk Margin</li>
<li>Systems and Signals norms (\(\mathcal{H}_\infty\) and \(\mathcal{H}_2\) norms)</li>
<li>Closed Loop Transfer Functions</li>
<li>Loop Shaping</li>
</ul></li>
<li>Advantages:
<ul class="org-ul">
<li>Stability</li>
<li>Performances</li>
<li>Robustness</li>
<li>Automatic Synthesis</li>
<li>MIMO</li>
<li>Optimization Problem</li>
</ul></li>
<li>Disadvantages:
<ul class="org-ul">
<li>Requires the knowledge of specific tools</li>
<li>Need a reasonably good model of the system</li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-org2092347" class="outline-2">
<h2 id="org2092347"><span class="section-number-2">3</span> The \(\mathcal{H}_\infty\) Norm</h2>
<div class="outline-text-2" id="text-3">
<div class="definition" id="org7770f0d">
<p>
The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
</p>
\begin{equation}
\|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big)
\end{equation}
<p>
For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a function of frequency:
</p>
\begin{equation}
\|G(s)\|_\infty = \max_{\omega} |G(j\omega)| \label{eq:hinf_norm_siso}
\end{equation}
</div>
<div class="exampl" id="org5cdde3f">
<p>
Let&rsquo;s define a plant dynamics:
</p>
<div class="org-src-container">
<pre class="src src-matlab">w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span>; k = 1e6; xi = 0.04;
G = 1<span class="org-type">/</span>k<span class="org-type">/</span>(s<span class="org-type">^</span>2<span class="org-type">/</span>w0<span class="org-type">^</span>2 <span class="org-type">+</span> 2<span class="org-type">*</span>xi<span class="org-type">*</span>s<span class="org-type">/</span>w0 <span class="org-type">+</span> 1);
</pre>
</div>
<p>
And compute its \(\mathcal{H}_\infty\) norm using the <code>hinfnorm</code> function:
</p>
<div class="org-src-container">
<pre class="src src-matlab">hinfnorm(G)
</pre>
</div>
<pre class="example">
1.0013e-05
</pre>
<p>
The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure <a href="#orgd616903">2</a>.
The maximum value of the magnitude over all frequencies does correspond to the \(\mathcal{H}_\infty\) norm of \(G(s)\) as Equation \eqref{eq:hinf_norm_siso} implies.
</p>
<div id="orgd616903" class="figure">
<p><img src="figs/hinfinity_norm_siso_bode.png" alt="hinfinity_norm_siso_bode.png" />
</p>
<p><span class="figure-number">Figure 2: </span>Example of the \(\mathcal{H}_\infty\) norm of a SISO system</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org2331e77" class="outline-2">
<h2 id="org2331e77"><span class="section-number-2">4</span> \(\mathcal{H}_\infty\) Synthesis</h2>
<div class="outline-text-2" id="text-4">
<p>
<b>Optimization problem</b>:
\(\hinf\) synthesis is a method that uses an <b>algorithm</b> (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the \(\hinf\) norms of defined transfer functions are minimized.
</p>
<p>
<b>Engineer work</b>:
</p>
<ol class="org-ol">
<li>Write the problem as standard \(\hinf\) problem</li>
<li>Translate the specifications as \(\hinf\) norms</li>
<li>Make the synthesis and analyze the obtain controller</li>
<li>Reduce the order of the controller for implementation</li>
</ol>
<p>
<b>Many ways to use the \(\hinf\) Synthesis</b>:
</p>
<ul class="org-ul">
<li>Traditional \(\hinf\) Synthesis</li>
<li>Mixed Sensitivity Loop Shaping</li>
<li>Fixed-Structure \(\hinf\) Synthesis</li>
<li>Signal Based \(\hinf\) Synthesis</li>
</ul>
</div>
</div>
<div id="outline-container-org9b8faac" class="outline-2">
<h2 id="org9b8faac"><span class="section-number-2">5</span> The Generalized Plant</h2>
<div class="outline-text-2" id="text-5">
<div id="orgd112ab1" class="figure">
<p><img src="figs/general_plant.png" alt="general_plant.png" />
</p>
</div>
<table id="orgc823a19" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 1:</span> Notations for the general configuration</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Notation</th>
<th scope="col" class="org-left">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(P\)</td>
<td class="org-left">Generalized plant model</td>
</tr>
<tr>
<td class="org-left">\(w\)</td>
<td class="org-left">Exogenous inputs: commands, disturbances, noise</td>
</tr>
<tr>
<td class="org-left">\(z\)</td>
<td class="org-left">Exogenous outputs: signals to be minimized</td>
</tr>
<tr>
<td class="org-left">\(v\)</td>
<td class="org-left">Controller inputs: measurements</td>
</tr>
<tr>
<td class="org-left">\(u\)</td>
<td class="org-left">Control signals</td>
</tr>
</tbody>
</table>
\begin{equation}
\begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w \\ u \end{bmatrix}
\end{equation}
</div>
</div>
<div id="outline-container-org8116ae5" class="outline-2">
<h2 id="org8116ae5"><span class="section-number-2">6</span> Problem Formulation</h2>
<div class="outline-text-2" id="text-6">
<div class="important" id="org179b25f">
<p>
The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize
</p>
\begin{equation}
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
\end{equation}
</div>
<div id="orgcf0dd39" class="figure">
<p><img src="figs/general_control_names.png" alt="general_control_names.png" />
</p>
<p><span class="figure-number">Figure 4: </span>General Control Configuration</p>
</div>
</div>
</div>
<div id="outline-container-org42f9bcf" class="outline-2">
<h2 id="org42f9bcf"><span class="section-number-2">7</span> Classical feedback control and closed loop transfer functions</h2>
<div class="outline-text-2" id="text-7">
<div id="orgb1f039f" class="figure">
<p><img src="figs/classical_feedback.png" alt="classical_feedback.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Classical Feedback Architecture</p>
</div>
<table id="orgfaf4a42" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<caption class="t-above"><span class="table-number">Table 2:</span> Notations for the Classical Feedback Architecture</caption>
<colgroup>
<col class="org-left" />
<col class="org-left" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Notation</th>
<th scope="col" class="org-left">Meaning</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">\(G\)</td>
<td class="org-left">Plant model</td>
</tr>
<tr>
<td class="org-left">\(K\)</td>
<td class="org-left">Controller</td>
</tr>
<tr>
<td class="org-left">\(r\)</td>
<td class="org-left">Reference inputs</td>
</tr>
<tr>
<td class="org-left">\(y\)</td>
<td class="org-left">Plant outputs</td>
</tr>
<tr>
<td class="org-left">\(u\)</td>
<td class="org-left">Control signals</td>
</tr>
<tr>
<td class="org-left">\(d\)</td>
<td class="org-left">Input Disturbance</td>
</tr>
<tr>
<td class="org-left">\(\epsilon\)</td>
<td class="org-left">Tracking Error</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="outline-container-orgf00feb8" class="outline-2">
<h2 id="orgf00feb8"><span class="section-number-2">8</span> From a Classical Feedback Architecture to a Generalized Plant</h2>
<div class="outline-text-2" id="text-8">
<p>
The procedure is:
</p>
<ol class="org-ol">
<li>define signals of the generalized plant</li>
<li>Remove \(K\) and rearrange the inputs and outputs</li>
</ol>
<div class="exampl" id="orgd38b593">
<p>
Let&rsquo;s find the Generalized plant of corresponding to the tracking control architecture shown in Figure <a href="#orgaa07663">6</a>
</p>
<div id="orgaa07663" class="figure">
<p><img src="figs/classical_feedback_tracking.png" alt="classical_feedback_tracking.png" />
</p>
<p><span class="figure-number">Figure 6: </span>Classical Feedback Control Architecture (Tracking)</p>
</div>
<p>
First, define the signals of the generalized plant:
</p>
<ul class="org-ul">
<li>Exogenous inputs: \(w = r\)</li>
<li>Signals to be minimized: \(z_1 = \epsilon\), \(z_2 = u\)</li>
<li>Control signals: \(v = y\)</li>
<li>Control inputs: \(u\)</li>
</ul>
<p>
Then, Remove \(K\) and rearrange the inputs and outputs.
We obtain the generalized plant shown in Figure <a href="#orgb73ca0b">7</a>.
</p>
<div id="orgb73ca0b" class="figure">
<p><img src="figs/mixed_sensitivity_ref_tracking.png" alt="mixed_sensitivity_ref_tracking.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Generalized plant of the Classical Feedback Control Architecture (Tracking)</p>
</div>
<p>
Using Matlab, the generalized plant can be defined as follows:
</p>
<div class="org-src-container">
<pre class="src src-matlab">P = [1 <span class="org-type">-</span>G;
0 1;
1 <span class="org-type">-</span>G]
</pre>
</div>
</div>
</div>
</div>
<div id="outline-container-org7c94ee9" class="outline-2">
<h2 id="org7c94ee9"><span class="section-number-2">9</span> Modern Interpretation of the Control Specifications</h2>
<div class="outline-text-2" id="text-9">
</div>
<div id="outline-container-orgfcb96b5" class="outline-3">
<h3 id="orgfcb96b5"><span class="section-number-3">9.1</span> Introduction</h3>
<div class="outline-text-3" id="text-9-1">
<ul class="org-ul">
<li><b>Reference tracking</b> Overshoot, Static error, Setling time
<ul class="org-ul">
<li>\(S(s) = T_{r \rightarrow \epsilon}\)</li>
</ul></li>
<li><b>Disturbances rejection</b>
<ul class="org-ul">
<li>\(G(s) S(s) = T_{d \rightarrow \epsilon}\)</li>
</ul></li>
<li><b>Measurement noise filtering</b>
<ul class="org-ul">
<li>\(T(s) = T_{n \rightarrow \epsilon}\)</li>
</ul></li>
<li><b>Small command amplitude</b>
<ul class="org-ul">
<li>\(K(s) S(s) = T_{r \rightarrow u}\)</li>
</ul></li>
<li><b>Stability</b>
<ul class="org-ul">
<li>\(S(s)\), \(T(s)\), \(K(s)S(s)\), \(G(s)S(s)\)</li>
</ul></li>
<li><b>Robustness to plant uncertainty</b> (stability margins)</li>
<li><b>Controller implementation</b></li>
</ul>
<p>
**
</p>
</div>
</div>
</div>
<div id="outline-container-orgc9ce5a6" class="outline-2">
<h2 id="orgc9ce5a6"><span class="section-number-2">10</span> Resources</h2>
<div class="outline-text-2" id="text-10">
<p>
<iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin" frameborder="0" allowfullscreen></iframe>
</p>
<p>
<iframe width="1280" height="720" src="https://www.youtube.com/embed/?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o" frameborder="0" allowfullscreen></iframe>
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-11-25 mer. 19:34</p>
</div>
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#+TITLE: Robust Control - $\mathcal{H}_\infty$ Synthesis
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* Introduction to the Control Methodology - Model Based Control
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
It consists of three steps:
1. *Identification or modeling*: $\Longrightarrow$ mathematical model
2. *Translate the specifications into mathematical criteria*:
- _Specifications_: Response Time, Noise Rejection, Maximum input amplitude, Robustness, ...
- _Mathematical Criteria_: Cost Function, Shape of TF
# - Cost Function, Needed Bandwidth, Roll-off, ...
# - $\Longrightarrow$ We will use the $\hinf$ Norm
3. *Synthesis*: research of $K$ that satisfies the specifications for the model of the system
#+begin_src latex :file control-procedure.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
\node[block, right=1.5 of addsub] (controller) {Controller};
\node[block, right=1.5 of controller] (plant) {Plant};
\node[block, above=1 of controller] (controller_design) {Synthesis};
\node[block, above=1 of plant] (model_plant) {Model};
\draw[<-] (addsub.west) -- ++(-1, 0) node[above right]{$r$};
\draw[->] (addsub) -- (controller.west) node[above left]{$\epsilon$};
\draw[->] (controller) -- (plant.west) node[above left]{$u$};
\draw[->] (plant.east) -- ++(1, 0) node[above left]{$y$};
\draw[] ($(plant.east) + (0.5, 0)$) -- ++(0, -1);
\draw[->] ($(plant.east) + (0.5, -1)$) -| (addsub.south);
\draw[->, dashed] (plant) -- node[midway, right, labelc, solid]{1} (model_plant);
\draw[->, dashed] (controller_design) --node[midway, right, labelc, solid]{3} (controller);
\draw[->, dashed] (model_plant) -- (controller_design);
\draw[<-, dashed] (controller_design.west) -- node[midway, above, labelc, solid]{2} ++(-1, 0) node[left, style={align=center}]{Specifications};
\end{tikzpicture}
#+end_src
#+name: fig:control-procedure
#+caption: Typical Methodoly for Model Based Control
#+RESULTS:
[[file:figs/control-procedure.png]]
In this document, we will mainly focus on steps 2 and 3.
* Some Background: From Classical Control to Robust Control
Classical Control (1930)
- Tools:
- TF (input-output)
- Nyquist, Bode, Black, \ldots
- P-PI-PID, Phase lead-lag, \ldots
- Advantages:
- Stability
- Performances
- Robustness
- Disadvantages:
- Manual Method
- Only SISO
Modern Control (1960)
- Tools:
- State Space
- Optimal Command
- LQR, LQG
- Advantages:
- Automatic Synthesis
- MIMO
- Optimisation problem
- Disadvantages:
- Robustness
- Rejection of Perturbations
Robust Control (1980)
- Tools:
- Disk Margin
- Systems and Signals norms ($\mathcal{H}_\infty$ and $\mathcal{H}_2$ norms)
- Closed Loop Transfer Functions
- Loop Shaping
- Advantages:
- Stability
- Performances
- Robustness
- Automatic Synthesis
- MIMO
- Optimization Problem
- Disadvantages:
- Requires the knowledge of specific tools
- Need a reasonably good model of the system
* The $\mathcal{H}_\infty$ Norm
#+begin_definition
The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response
\begin{equation}
\|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big)
\end{equation}
For a SISO system $G(s)$, it is simply the peak value of $|G(j\omega)|$ as a function of frequency:
\begin{equation}
\|G(s)\|_\infty = \max_{\omega} |G(j\omega)| \label{eq:hinf_norm_siso}
\end{equation}
#+end_definition
#+begin_exampl
Let's define a plant dynamics:
#+begin_src matlab
w0 = 2*pi; k = 1e6; xi = 0.04;
G = 1/k/(s^2/w0^2 + 2*xi*s/w0 + 1);
#+end_src
And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
#+begin_src matlab :results value replace
hinfnorm(G)
#+end_src
#+RESULTS:
: 1.0013e-05
The magnitude $|G(j\omega)|$ of the plant $G(s)$ as a function of frequency is shown in Figure [[fig:hinfinity_norm_siso_bode]].
The maximum value of the magnitude over all frequencies does correspond to the $\mathcal{H}_\infty$ norm of $G(s)$ as Equation eqref:eq:hinf_norm_siso implies.
#+begin_src matlab :exports none
freqs = logspace(-1, 1, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
plot([0.5, 2], [hinfnorm(G) hinfnorm(G)], 'k--');
text(2, hinfnorm(G), '$\quad \|G\|_\infty$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude $|G(j\omega)|$');
ylim([1e-8, 2e-5]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/hinfinity_norm_siso_bode.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:hinfinity_norm_siso_bode
#+caption: Example of the $\mathcal{H}_\infty$ norm of a SISO system
#+RESULTS:
[[file:figs/hinfinity_norm_siso_bode.png]]
#+end_exampl
* $\mathcal{H}_\infty$ Synthesis
*Optimization problem*:
$\hinf$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the $\hinf$ norms of defined transfer functions are minimized.
*Engineer work*:
1. Write the problem as standard $\hinf$ problem
2. Translate the specifications as $\hinf$ norms
3. Make the synthesis and analyze the obtain controller
4. Reduce the order of the controller for implementation
*Many ways to use the $\hinf$ Synthesis*:
- Traditional $\hinf$ Synthesis
- Mixed Sensitivity Loop Shaping
- Fixed-Structure $\hinf$ Synthesis
- Signal Based $\hinf$ Synthesis
* The Generalized Plant
#+begin_src latex :file general_plant.pdf
\begin{tikzpicture}
\node[block={2.0cm}{2.0cm}] (P) {$P$};
\node[above] at (P.north) {Generalized Plant};
% Input and outputs coordinates
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
% Connections and labels
\draw[<-] (inputw) -- ++(-0.8, 0) node[above right]{$w$};
\draw[<-] (inputu) -- ++(-0.8, 0) node[above right]{$u$};
\draw[->] (outputz) -- ++(0.8, 0) node[above left]{$z$};
\draw[->] (outputv) -- ++(0.8, 0) node[above left]{$v$};
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/general_plant.png]]
#+name: tab:notation_general
#+caption: Notations for the general configuration
| Notation | Meaning |
|----------+-------------------------------------------------|
| $P$ | Generalized plant model |
| $w$ | Exogenous inputs: commands, disturbances, noise |
| $z$ | Exogenous outputs: signals to be minimized |
| $v$ | Controller inputs: measurements |
| $u$ | Control signals |
\begin{equation}
\begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w \\ u \end{bmatrix}
\end{equation}
* Problem Formulation
#+begin_important
The $\mathcal{H}_\infty$ Synthesis objective is to find all stabilizing controllers $K$ which minimize
\begin{equation}
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
\end{equation}
#+end_important
#+begin_src latex :file general_control_names.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {$P$};
\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
% Input and outputs coordinates
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
% Connections and labels
\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
\end{tikzpicture}
#+end_src
#+name: fig:general_control_names
#+caption: General Control Configuration
#+RESULTS:
[[file:figs/general_control_names.png]]
* Classical feedback control and closed loop transfer functions
#+begin_src latex :file classical_feedback.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[addb={+}{}{}{}{}, right=0.8 of K] (addu){};
\node[block, right=0.8 of addu] (G){$G(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (addu.west) node[above left]{$u$};
\draw[->] (addu.east) -- (G.west);
\draw[<-] (addu.north) -- ++(0, 0.8) node[below right]{$d$};
\draw[->] (G.east) -- ++(1.2, 0);
\draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south);
\end{tikzpicture}
#+end_src
#+name: fig:classical_feedback
#+caption: Classical Feedback Architecture
#+RESULTS:
[[file:figs/classical_feedback.png]]
#+name: table:notation_conventional
#+caption: Notations for the Classical Feedback Architecture
| Notation | Meaning |
|------------+-------------------|
| $G$ | Plant model |
| $K$ | Controller |
| $r$ | Reference inputs |
| $y$ | Plant outputs |
| $u$ | Control signals |
| $d$ | Input Disturbance |
| $\epsilon$ | Tracking Error |
* From a Classical Feedback Architecture to a Generalized Plant
The procedure is:
1. define signals of the generalized plant
2. Remove $K$ and rearrange the inputs and outputs
#+begin_src latex :file classical_feedback_tracking.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[block, right=0.8 of K] (G){$G(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(1.2, 0);
\draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south);
\end{tikzpicture}
#+end_src
#+begin_src latex :file mixed_sensitivity_ref_tracking.pdf
\begin{tikzpicture}
\node[block] (G) {$G(s)$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\coordinate[above right=0.6 and 1.4 of addw] (u);
\coordinate[above=0.6 of u] (epsilon);
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.4, 0)$);
\node[block, below left=0.8 and 0 of addw] (K) {$K(s)$};
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above left](z1){$\epsilon$};
\draw[->] ($(G.west)+(-0.4, 0)$)node[branch](start){} |- (u) node[above left](z2){$u$};
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
\draw [decoration={brace, raise=5pt}, decorate] (z1.north east) -- node[right=6pt]{$z$} (z2.south east);
\begin{scope}[on background layer]
\node[fit={(G.south-|start.west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=6pt, draw, dashed, fill=black!20!white] (P) {};
\node[below right] at (P.north west) {Generalized Plant $P(s)$};
\end{scope}
\end{tikzpicture}
#+end_src
#+begin_exampl
Let's find the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
#+name: fig:classical_feedback_tracking
#+caption: Classical Feedback Control Architecture (Tracking)
[[file:figs/classical_feedback_tracking.png]]
First, define the signals of the generalized plant:
- Exogenous inputs: $w = r$
- Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$
- Control signals: $v = y$
- Control inputs: $u$
Then, Remove $K$ and rearrange the inputs and outputs.
We obtain the generalized plant shown in Figure [[fig:mixed_sensitivity_ref_tracking]].
#+name: fig:mixed_sensitivity_ref_tracking
#+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking)
[[file:figs/mixed_sensitivity_ref_tracking.png]]
Using Matlab, the generalized plant can be defined as follows:
#+begin_src matlab :tangle no :eval no
P = [1 -G;
0 1;
1 -G]
#+end_src
#+end_exampl
* Modern Interpretation of the Control Specifications
** Introduction
- *Reference tracking* Overshoot, Static error, Setling time
- $S(s) = T_{r \rightarrow \epsilon}$
- *Disturbances rejection*
- $G(s) S(s) = T_{d \rightarrow \epsilon}$
- *Measurement noise filtering*
- $T(s) = T_{n \rightarrow \epsilon}$
- *Small command amplitude*
- $K(s) S(s) = T_{r \rightarrow u}$
- *Stability*
- $S(s)$, $T(s)$, $K(s)S(s)$, $G(s)S(s)$
- *Robustness to plant uncertainty* (stability margins)
- *Controller implementation*
**
* Resources
yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin
yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o