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#+TITLE: A brief and practical introduction to $\mathcal{H}_\infty$ Control
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:shell :eval no-export
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :tangle no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* Introduction :ignore:
This document is structured as follows:
- As $\mathcal{H}_\infty$ Control is a /model based/ control technique, a short introduction to model based control is given in Section [[sec:model_based_control]]
- Classical /open/ loop shaping method is presented in Section [[sec:open_loop_shaping]].
It is also shown that $\mathcal{H}_\infty$ synthesis can be used for /open/ loop shaping.
- $\mathcal{H}_\infty$
Important concepts such as the $\mathcal{H}_\infty$ norm and the generalized plant are introduced.
- A
- Finally, an complete example of the
is performed in Section [[sec:h_infinity_mixed_sensitivity]].
* Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no
addpath('matlab')
#+end_src
* Introduction to Model Based Control
<<sec:model_based_control>>
** Model Based Control - Methodology
<<sec:model_based_control_methodology>>
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 $\mathcal{H}_\infty$ 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.
Step 2 will be discussed in Section [[sec:modern_interpretation_specification]].
There are two main methods for the controller synthesis (step 3):
- /open/ loop shaping discussed in Section [[sec:open_loop_shaping]]
- /closed/ loop shaping discussed in Sections [[sec:modern_interpretation_specification]] and [[sec:h_infinity_mixed_sensitivity]]
** From Classical Control to Robust Control
<<sec:comp_classical_modern_robust_control>>
#+name: tab:comparison_control_methods
#+caption: Table summurazing the main differences between classical, modern and robust control
| | *Classical Control* | *Modern Control* | *Robust Control* |
| <l> | <c> | <c> | <c> |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Date* | 1930- | 1960- | 1980- |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Tools* | Transfer Functions | State Space formulation | Systems and Signals Norms ($\mathcal{H}_\infty$, $\mathcal{H}_2$ Norms) |
| | Nyquist Plots | Riccati Equations | Closed Loop Transfer Functions |
| | Bode Plots | | Open/Closed Loop Shaping |
| | Phase and Gain margins | | Weighting Functions |
| | | | Disk margin |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Control Architectures* | Proportional, Integral, Derivative | Full State Feedback, LQR | General Control Configuration |
| | Leads, Lags | Kalman Filters, LQG | Generalized Plant |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Advantages* | Study Stability | Automatic Synthesis | Automatic Synthesis |
| | Simple | MIMO | MIMO |
| | Natural | Optimization Problem | Optimization Problem |
| | | | Guaranteed Robustness |
| | | | Easy specification of performances |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Disadvantages* | Manual Method | No Guaranteed Robustness | Required knowledge of specific tools |
| | Only SISO | Difficult Rejection of Perturbations | Need a reasonably good model of the system |
#+begin_src latex :file robustness_performance.pdf
\begin{tikzpicture}
% Scale
\def\yscale{0.8}
\def\xscale{1.0}
% Colors
\def\colorstart{blue}
\def\colorend{red}
% Axis
\draw [->] (-0.5,0) -- (10*\xscale,0) node[below left]{Robustness};
\draw [->] (0,-0.5) -- (0,10*\yscale) node[below left, rotate=90, anchor=south east]{Performance};
% Color Bar
\shade[draw, bottom color=\colorstart, top color=\colorend, fill opacity=0.5] (10*\xscale, 1*\yscale) rectangle (11*\xscale, 9*\yscale);
\node[rotate=90, above] at (10*\xscale, 5*\yscale) {Required information on plant};
\node[above] at (10.5*\xscale, 1*\yscale) {little};
\node[below] at (10.5*\xscale, 9*\yscale) {large};
% ===================================
% Classical Control
% ===================================
% Control Types
\node[align=center] (pid) at (7.0*\xscale, 1.2*\yscale) {PID\\Lead-Lag};
\begin{scope}[on background layer]
% Control Families
\node[ellipse, draw, dashed, minimum width=3.0*\xscale cm, minimum height=2.0*\yscale cm,
fill=\colorstart!90!\colorend, fill opacity=0.5, text opacity=1]
(classicalcontrol) at (pid) {};
\end{scope}
\node[above, align=center] at (classicalcontrol.north) {\textbf{Classical control} (1930)\\{\small SISO, Manual Method}};
% ===================================
% ===================================
% Modern Control
% ===================================
% Control Types
\node[align=center] (lqg) at (2.0*\xscale, 7.5*\yscale) {LQR\\LQG};
\begin{scope}[on background layer]
\node[ellipse, draw, dashed, minimum width=2.0*\xscale cm, minimum height=2.0*\yscale cm,
fill=\colorstart!20!\colorend, fill opacity=0.5, text opacity=1]
(moderncontrol) at (lqg) {};
\end{scope}
\node[above, align=center] at (moderncontrol.north) {\textbf{Modern control} (1960)\\{\small MIMO, Optimal}};
% ===================================
% ===================================
% Robust Control
% ===================================
% Control Types
\node[align=center] (hinf) at (4.5*\xscale, 4.8*\yscale) {$H_\infty$\\$H_2$};
\node[] (mu) at (5.5*\xscale, 4.8*\yscale) {$\mu$};
\begin{scope}[on background layer]
\node[ellipse, draw, dashed, minimum width=3.0*\xscale cm, minimum height=2.5*\yscale cm,
shade, left color=\colorstart!50!\colorend, right color=\colorstart!10!\colorend, fill opacity=0.5, text opacity=1]
(robustcontrol) at ($0.5*(hinf)+0.5*(mu)$) {};
\end{scope}
\node[above, align=center] at (robustcontrol.north) {\textbf{Robust control} (1990)\\{\small MIMO, Robust}};
% ===================================
\end{tikzpicture}
#+end_src
#+name: fig:robustness_performance
#+caption: Comparison of the performance and robustness of classical control methods, modern control methods and robust control methods. The required information on the plant to succesfuly apply each of the control methods are indicated by the colors.
#+RESULTS:
[[file:figs/robustness_performance.png]]
** Example System
<<sec:example_system>>
Let's consider the model shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
The notations used are listed in Table [[tab:example_notations]].
#+begin_src latex :file mech_sys_1dof_inertial_contr.pdf
\begin{tikzpicture}
% Parameters
\def\massw{3}
\def\massh{1}
\def\spaceh{1.8}
% Ground
\draw[] (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5](m){$m$};
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0, 0) -- ( 0, \spaceh) node[midway, left=0.3]{$c$};
\draw[actuator] ( 0.3*\massw, 0) -- (0.3*\massw, \spaceh) node[midway](F){};
% Displacements
\draw[dashed] (0.5*\massw, 0) -- ++(0.5, 0);
\draw[->] (0.6*\massw, 0) -- ++(0, 0.5) node[below right]{$d$};
% Inertial Sensor
\node[inertialsensor] (inertials) at (0.5*\massw, \spaceh+\massh){};
\node[addb={+}{-}{}{}{}, right=0.8 of inertials] (subf) {};
\node[block, below=0.4 of subf] (K){$K(s)$};
\draw[->] (inertials.east) node[above right]{$y$} -- (subf.west);
\draw[->] (subf.south) -- (K.north) node[above right]{$\epsilon$};
\draw[<-] (subf.north) -- ++(0, 0.6) node[below right]{$r$};
\draw[->] (K.south) |- (F.east) node[above right]{$u$};
\end{tikzpicture}
#+end_src
#+name: fig:mech_sys_1dof_inertial_contr
#+caption: Test System consisting of a payload with a mass $m$ on top of an active system with a stiffness $k$, damping $c$ and an actuator. A feedback controller $K(s)$ is added to position / isolate the payload.
#+RESULTS:
[[file:figs/mech_sys_1dof_inertial_contr.png]]
#+name: tab:example_notations
#+caption: Example system variables
| *Notation* | *Description* | *Value* | *Unit* |
|--------------------+----------------------------------------------------------------+----------------+-----------|
| $m$ | Payload's mass to position / isolate | $10$ | [kg] |
| $k$ | Stiffness of the suspension system | $10^6$ | [N/m] |
| $c$ | Damping coefficient of the suspension system | $400$ | [N/(m/s)] |
| $y$ | Payload absolute displacement (measured by an inertial sensor) | | [m] |
| $d$ | Ground displacement, it acts as a disturbance | | [m] |
| $u$ | Actuator force | | [N] |
| $r$ | Wanted position of the mass (the reference) | | [m] |
| $\epsilon = r - y$ | Position error | | [m] |
| $K$ | Feedback controller | to be designed | [N/m] |
#+begin_exercice
Derive the following open-loop transfer functions:
\begin{align}
G(s) &= \frac{y}{u} \\
G_d(s) &= \frac{y}{d}
\end{align}
#+HTML: <details><summary>Hint</summary>
You can follow this generic procedure:
1. List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), ...
2. Apply the Newton's Second Law on the payload
\[ m \ddot{y} = \Sigma F \]
3. Transform the differential equations into the Laplace domain:
\[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]
4. Write $y(s)$ as a function of $u(s)$ and $w(s)$
#+HTML: </details>
#+HTML: <details><summary>Results</summary>
\begin{align}
G(s) &= \frac{1}{m s^2 + cs + k} \\
G_d(s) &= \frac{cs + k}{m s^2 + cs + k}
\end{align}
#+HTML: </details>
#+end_exercice
Hi Musa,
Thank you very much for sharing this awesome package.
For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by <details> blocks.
For now, I am manually adding #+HTML: <details><summary>Code</summary> and #+HTML: </details> around the source blocks I want to hide...
This is a very simple solution, but not so elegent nor practical.
Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
Similarly, I would love to be able to export a <span> block with the name of the file corresponding to the source block.
For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
Thanks in advance
Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure [[fig:mech_sys_1dof_inertial_contr]] into a classical feedback form as shown in Figure [[fig:open_loop_shaping]].
#+begin_src latex :file classical_feedback_test_system.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){$G(s)$};
\node[addb={+}{}{}{}{}, right=0.8 of G] (addd){};
\node[block, above=0.5 of addd] (Gd){$G_d(s)$};
\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) -- (addd.west);
\draw[<-] (Gd.north) -- ++(0, 0.8) node[below right]{$d$};
\draw[->] (Gd.south) -- (addd.north);
\draw[->] (addd.east) -- ++(1.2, 0);
\draw[->] ($(addd.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -1.0) -| (addfb.south);
\end{tikzpicture}
#+end_src
#+name: fig:classical_feedback_test_system
#+caption: Block diagram corresponding to the example system
#+RESULTS:
[[file:figs/classical_feedback_test_system.png]]
Let's define the system parameters on Matlab.
#+begin_src matlab +n
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
#+end_src
And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in Figures [[fig:bode_plot_example_afm]] and [[fig:bode_plot_example_Gd]]).
#+begin_src matlab +n -r
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-270, 90]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_example_afm.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:bode_plot_example_afm
#+caption: Bode plot of the plant $G(s)$
#+RESULTS:
[[file:figs/bode_plot_example_afm.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_example_Gd.pdf', 'width', 'wide', 'height', 'small');
#+end_src
#+name: fig:bode_plot_example_Gd
#+caption: Magnitude of the disturbance transfer function $G_d(s)$
#+RESULTS:
[[file:figs/bode_plot_example_Gd.png]]
* Classical Open Loop Shaping
<<sec:open_loop_shaping>>
** Introduction to Loop Shaping
<<sec:open_loop_shaping_introduction>>
#+begin_definition
*Loop Shaping* refers to a design procedure that involves explicitly shaping the magnitude of the *Loop Transfer Function* $L(s)$.
#+end_definition
#+begin_definition
The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant ("Gain around the loop", see Figure [[fig:open_loop_shaping]]):
\begin{equation}
L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
\end{equation}
#+begin_src latex :file open_loop_shaping.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
\node[block, right=0.8 of addsub] (K) {$K(s)$};
\node[below] at (K.south) {Controller};
\node[block, right=0.8 of K] (G) {$G(s)$};
\node[below] at (G.south) {Plant};
\draw[<-] (addsub.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addsub) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(0.8, 0) node[above left]{$y$};
\draw[] ($(G.east) + (0.5, 0)$) -- ++(0, -1.4);
\draw[->] ($(G.east) + (0.5, -1.4)$) -| (addsub.south);
\draw [decoration={brace, raise=5pt}, decorate] (K.north west) -- node[above=6pt]{$L(s)$} (G.north east);
\end{tikzpicture}
#+end_src
#+name: fig:open_loop_shaping
#+caption: Classical Feedback Architecture
[[file:figs/open_loop_shaping.png]]
#+end_definition
This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$ such as:
- *Performance*: $L$ large
- *Good disturbance rejection*: $L$ large
- *Limitation of measurement noise on plant output*: $L$ small
- *Small magnitude of input signal*: $K$ and $L$ small
- *Nominal stability*: $L$ small (RHP zeros and time delays)
- *Robust stability*: $L$ small (neglected dynamics)
The Open Loop shape is usually done manually has the loop gain $L(s)$ depends linearly on $K(s)$ eqref:eq:loop_gain.
$K(s)$ then consists of a combination of leads, lags, notches, etc. such that $L(s)$ has the wanted shape (an example is shown in Figure [[fig:open_loop_shaping_shape]]).
#+begin_src latex :file open_loop_shaping_shape.pdf
\begin{tikzpicture}
% Phase Axis
\draw[->] (-0.3, -0.5) -- ++(8, 0) node[above]{$\omega$}; \draw[<-] (0, 0)
node[left]{$\angle L(j\omega)$} -- ++(0, -2.3);
% Gain Axis
\draw[->] (-0.3, 2) -- ++(8, 0) node[above]{$\omega$}; \draw[->] (0, 0.5) --
++(0, 3) node[left]{$\left|L(j\omega)\right|$};
% Gain Slopes
\draw[shift={(0,2)}] (0.5, 1.25) -- node[midway, above]{$-2$} (2, 0.5) --
node[midway, above]{$-1$} (6, -0.5) -- node[midway, below left]{$-2$} (7.5,
-1.25);
% Forbiden region
\path[shift={(0,1.8)}, fill=red!50!white] (0.5, 1.25) -- (2, 0.5) -| coordinate[near start](lfshaping) cycle;
\path[shift={(0,2.2)}, fill=red!50!white] (6, -0.5) -- (7.5, -1.25) |- coordinate[near end](hfshaping) cycle;
\draw[<-] (lfshaping) -- ++(0, -0.8) node[below, align=center]{Reference\\Tracking};
\draw[<-] (hfshaping) -- ++(0, 0.8) node[above, align=center]{Noise\\Rejection};
% Crossover frequency
\node[below] (wc) at (4,2){$\omega_c$};
\draw[<-] (wc.south) -- ++(0, -0.4) node[below, align=center]{Bandwidth};
% Phase
\draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4, -1.25)[out=0, in=-180] to
(6, -2) -- (7.5, -2); \draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4,
-1.25)[out=0, in=-180] to (6, -2) -- (7.5, -2);
% Phase Margin
\draw[->, dashed] (4, -2) -- (4, -1.25) node[above]{Phase Margin};
\draw[dashed] (0, -2) node[left]{$-\pi$} -- (7.5, -2);
\end{tikzpicture}
#+end_src
#+name: fig:open_loop_shaping_shape
#+caption: Typical Wanted Shape for the Loop Gain $L(s)$
#+RESULTS:
[[file:figs/open_loop_shaping_shape.png]]
** Example of Open Loop Shaping
<<sec:loop_shaping_example>>
#+begin_exampl
Let's take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
- *Performance*: Bandwidth of approximately 10Hz
- *Noise Attenuation*: Roll-off of -40dB/decade past 30Hz
- *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
#+end_exampl
#+begin_exercice
Using =SISOTOOL=, design a controller that fulfill the specifications.
#+begin_src matlab :eval no :tangle no
sisotool(G)
#+end_src
#+end_exercice
In order to have the wanted Roll-off, two integrators are used, a lead is also added to have sufficient phase margin.
The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]].
#+begin_src matlab
K = 14e8 * ... % Gain
1/(s^2) * ... % Double Integrator
1/(1 + s/2/pi/40) * ... % Low Pass Filter
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
#+end_src
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-4, 1e1])
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:loop_gain_manual_afm
#+caption: Bode Plot of the obtained Loop Gain $L(s) = G(s) K(s)$
#+RESULTS:
[[file:figs/loop_gain_manual_afm.png]]
And we can verify that we have the wanted stability margins:
#+begin_src matlab
[Gm, Pm, ~, Wc] = margin(G*K)
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Requirements', 'Manual Method'}, ' %.1f ');
#+end_src
#+RESULTS:
| Requirements | Manual Method |
|-----------------------------+---------------|
| Gain Margin $> 3$ [dB] | 3.1 |
| Phase Margin $> 30$ [deg] | 35.4 |
| Crossover $\approx 10$ [Hz] | 10.1 |
** $\mathcal{H}_\infty$ Loop Shaping Synthesis
<<sec:h_infinity_open_loop_shaping>>
The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.
Even though we will not go into details, we will provide one example.
Using Matlab, the $\mathcal{H}_\infty$ Loop Shaping Synthesis can be performed using the =loopsyn= command:
#+begin_src matlab :eval no :tangle no
K = loopsyn(G, Gd);
#+end_src
where:
- =G= is the (LTI) plant
- =Gd= is the wanted loop shape
- =K= is the synthesize controller
#+begin_seealso
Matlab documentation of =loopsyn= ([[https://www.mathworks.com/help/robust/ref/loopsyn.html][link]]).
#+end_seealso
** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
<<sec:h_infinity_open_loop_shaping_example>>
Let's reuse the previous plant.
Translate the specification into the wanted shape of the open loop gain.
- *Performance*: Bandwidth of approximately 10Hz: $|L_w(j2 \pi 10)| = 1$
- *Noise Attenuation*: Roll-off of -40dB/decade past 30Hz
- *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
#+begin_src matlab
Lw = 2.3e3 * ...
1/(s^2) * ... % Double Integrator
(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
#+end_src
The $\mathcal{H}_\infty$ optimal open loop shaping synthesis is performed using the =loopsyn= command:
#+begin_src matlab
[K, ~, GAM] = loopsyn(G, Lw);
#+end_src
The Bode plot of the obtained controller is shown in Figure [[fig:open_loop_shaping_hinf_K]].
#+begin_important
It is always important to analyze the controller after the synthesis is performed.
In the end, a synthesize controller is just a combination of low pass filters, high pass filters, notches, leads, etc.
#+end_important
Let's briefly analyze this controller:
- two integrators are used at low frequency to have the wanted low frequency high gain
- a lead is added centered with the crossover frequency to increase the phase margin
- a notch is added at the resonance of the plant to increase the gain margin (this is very typical of $\mathcal{H}_\infty$ controllers, and can be an issue, more info on that latter)
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
plot(freqs, abs(squeeze(freqresp(K, freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
plot(freqs, 180/pi*angle(squeeze(freqresp(K, freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-180, 90]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/open_loop_shaping_hinf_K.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:open_loop_shaping_hinf_K
#+caption: Obtained controller $K$ using the open-loop $\mathcal{H}_\infty$ shaping
#+RESULTS:
[[file:figs/open_loop_shaping_hinf_K.png]]
The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz'))), 'k--', 'DisplayName', '$L_w(s)$');
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))*GAM, 'k-.', 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))/GAM, 'k-.', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ylim([1e-4, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/open_loop_shaping_hinf_L.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:open_loop_shaping_hinf_L
#+caption: Obtained Open Loop Gain $L(s) = G(s) K(s)$ and comparison with the wanted Loop gain $L_w$
#+RESULTS:
[[file:figs/open_loop_shaping_hinf_L.png]]
Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table [[tab:open_loop_shaping_compare]].
#+begin_src matlab :exports none
[Gm_2, Pm_2, ~, Wc_2] = margin(G*K)
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Gm, Gm_2; Pm, Pm_2; Wc/2/pi, Wc_2/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Specifications', 'Manual Method', '$\mathcal{H}_\infty$ Method'}, ' %.1f ');
#+end_src
#+name: tab:open_loop_shaping_compare
#+caption: Comparison of the characteristics obtained with the two methods
#+RESULTS:
| Specifications | Manual Method | $\mathcal{H}_\infty$ Method |
|-----------------------------+---------------+-----------------------------|
| Gain Margin $> 3$ [dB] | 3.1 | 31.7 |
| Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
* First Steps in the $\mathcal{H}_\infty$ world
<<sec:h_infinity_introduction>>
** The $\mathcal{H}_\infty$ Norm
<<sec:h_infinity_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 compute the $\mathcal{H}_\infty$ norm of our test plant $G(s)$ using the =hinfnorm= function:
#+begin_src matlab :results value replace
hinfnorm(G)
#+end_src
#+RESULTS:
: 7.9216e-06
We can see that the $\mathcal{H}_\infty$ norm of $G(s)$ does corresponds to the peak value of $|G(j\omega)|$ as a function of frequency as shown in Figure [[fig:hinfinity_norm_siso_bode]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
plot([20, 100], [hinfnorm(G) hinfnorm(G)], 'k--');
text(100, 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
<<sec:h_infinity_synthesis>>
#+begin_definition
$\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller that stabilize the system and that *minimizes* the $\mathcal{H}_\infty$ norms of defined transfer functions.
#+end_definition
Why optimizing the $\mathcal{H}_\infty$ norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the $\mathcal{H}_\infty$ norm of transfer functions.
Then applying the $\mathcal{H}_\infty$ synthesis to a plant, the engineer work usually consists of the following steps
1. Write the problem as standard $\mathcal{H}_\infty$ problem
2. Translate the specifications as $\mathcal{H}_\infty$ norms of transfer functions
3. Make the synthesis and analyze the obtain controller
4. Reduce the order of the controller for implementation
Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
- Traditional $\mathcal{H}_\infty$ Synthesis (=hinfsyn= [[https://www.mathworks.com/help/robust/ref/hinfsyn.html][doc]])
- Open Loop Shaping $\mathcal{H}_\infty$ Synthesis (=loopsyn= [[https://www.mathworks.com/help/robust/ref/loopsyn.html][doc]])
- Mixed Sensitivity Loop Shaping (=mixsyn= [[https://www.mathworks.com/help/robust/ref/lti.mixsyn.html][doc]])
- Fixed-Structure $\mathcal{H}_\infty$ Synthesis (=hinfstruct= [[https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html][doc]])
- Signal Based $\mathcal{H}_\infty$ Synthesis
** The Generalized Plant
<<sec:generalized_plant>>
The first step when applying the $\mathcal{H}_\infty$ synthesis is usually to write the problem as a standard $\mathcal{H}_\infty$ problem.
This consist of deriving the *Generalized Plant* for the current problem.
It makes things much easier for the following steps.
The generalized plant, usually noted $P(s)$, is shown in Figure [[fig:general_plant]].
It has two inputs and two outputs (both could contains many signals).
The meaning of the inputs and outputs are summarized in Table [[tab:notation_general]].
Note that this generalized plant is as its name implies, quite /general/.
It can indeed represent feedback as well as feedforward control architectures.
\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}
#+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
#+name: fig:general_plant
#+caption: Inputs and Outputs of the generalized Plant
#+RESULTS:
[[file:figs/general_plant.png]]
#+begin_important
#+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 |
#+end_important
** The $\mathcal{H}_\infty$ Synthesis applied on the Generalized plant
<<sec:h_infinity_general_synthesis>>
Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis problem can be stated as follows:
#+begin_important
- $\mathcal{H}_\infty$ Synthesis applied on the generalized plant ::
Find a stabilizing controller $K$ that, using the sensed output $v$, generates a control signal $u$ such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is minimized.
After $K$ is found, the system is /robustified/ by adjusting the response around the unity gain frequency to increase stability margins.
#+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]]
Note that the closed-loop transfer function from $w$ to $z$ is:
\begin{equation}
\frac{z}{w} = P_{11} + P_{12} K \big( I - P_{22} K \big)^{-1} P_{21} \triangleq F_l(P, K)
\end{equation}
Using Matlab, the $\mathcal{H}_\infty$ Synthesis applied on a Generalized plant can be applied using the =hinfsyn= command ([[https://www.mathworks.com/help/robust/ref/hinfsyn.html][documentation]]):
#+begin_src matlab :eval no :tangoe no
K = hinfsyn(P, nmeas, ncont);
#+end_src
where:
- =P= is the generalized plant transfer function matrix
- =nmeas= is the number of sensed output (size of $v$)
- =ncont= is the number of control signals (size of $u$)
- =K= obtained controller that minimized the $\mathcal{H}_\infty$ norm from $w$ to $z$
** From a Classical Feedback Architecture to a Generalized Plant
<<sec:generalized_plant_derivation>>
The procedure to convert a typical control architecture as the one shown in Figure [[fig:classical_feedback_tracking]] to a generalized Plant is as follows:
1. Define signals ($w$, $z$, $u$ and $v$) of the generalized plant
2. Remove $K$ and rearrange the inputs and outputs to match the generalized configuration
#+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] at (P.north) {Generalized Plant $P(s)$};
\end{scope}
\end{tikzpicture}
#+end_src
#+begin_exercice
Compute 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]]
#+HTML: <details><summary>Hint</summary>
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.
#+HTML: </details>
#+HTML: <details><summary>Answer</summary>
The obtained 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]]
#+HTML: </details>
#+end_exercice
Using Matlab, the generalized plant can be defined as follows:
#+begin_src matlab :tangle no :eval no
P = [1 -G;
0 1;
1 -G]
P.InputName = {'w', 'u'};
P.OutputName = {'e', 'u', 'v'};
#+end_src
* Modern Interpretation of the Control Specifications
<<sec:modern_interpretation_specification>>
** Introduction
As shown in Section [[sec:open_loop_shaping]], the loop gain $L(s) = G(s) K(s)$ is a useful and easy tool for the manual design of controllers.
$L(s)$ is very easy to shape as it depends linearly on $K(s)$.
Moreover, important quantities such as the stability margins and the control bandwidth can be estimated from the shape/phase of $L(s)$.
However, the loop gain $L(s)$ does *not* directly give the performances of the closed-loop system.
The closed loop system behavior is indeed determined by the *closed-loop* transfer functions.
If we consider the feedback system shown in Figure [[fig:gang_of_four_feedback]], we can link to the following specifications to closed-loop transfer functions.
This is summarized in Table [[tab:spec_closed_loop_tf]].
#+name: tab:spec_closed_loop_tf
#+caption: Typical Specification and associated closed-loop transfer function
| Specification | Closed-Loop Transfer Function |
|--------------------------------+-----------------------------------------------|
| Reference Tracking | From $r$ to $\epsilon$ |
| Disturbance Rejection | From $d$ to $y$ |
| Measurement Noise Filtering | From $n$ to $y$ |
| Small Command Amplitude | From $n,r,d$ to $u$ |
| Stability | All closed-loop transfer function |
| Robustness (stability margins) | Module margin (see Section [[sec:module_margin]]) |
#+begin_src latex :file gang_of_four_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] (addd){};
\node[block, right=0.8 of addd] (G){$G(s)$};
\node[addb, below right=0.4 and 0.2 of G] (addn){};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (addd.west);
\draw[<-] (addd.north) -- ++(0, 0.6) node[below right]{$d$};
\draw[->] (addd.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(1.6, 0) node[above left]{$y$};
\draw[->] (G-|addn) node[branch]{} -- (addn.north);
\draw[<-] (addn.east) -- ++(0.8, 0) node[above left]{$n$};
\draw[->] (addn.west) -| (addfb.south);
\end{tikzpicture}
#+end_src
#+name: fig:gang_of_four_feedback
#+caption: Simple Feedback Architecture
#+RESULTS:
[[file:figs/gang_of_four_feedback.png]]
** Closed Loop Transfer Functions
<<sec:closed_loop_tf>>
As the performances of a controlled system depend on the *closed* loop transfer functions, it is very important to derive these closed-loop transfer functions as a function of the plant $G(s)$ and controller $K(s)$.
#+begin_exercice
Write the output signals $[\epsilon, u, y]$ as a function of the systems $K(s), G(s)$ and of the input signals $[r, d, n]$ as shown in Figure [[fig:gang_of_four_feedback]].
#+HTML: <details><summary>Hint</summary>
Take one of the output (e.g. $y$), and write it as a function of the inputs $[d, r, n]$ going step by step around the loop:
\begin{aligned}
y &= G u \\
&= G (d + K \epsilon) \\
&= G \big(d + K (r - n - y) \big) \\
&= G d + GK r - GK n - GK y
\end{aligned}
Isolate $y$ at the right hand side, and finally obtain:
\[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
Do the same procedure for $u$ and $\epsilon$
#+HTML: </details>
#+HTML: <details><summary>Anwser</summary>
The following equations should be obtained:
\begin{align}
y &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
\epsilon &= \frac{1 }{1 + GK} r - \frac{G}{1 + GK} d - \frac{G }{1 + GK} n \\
u &= \frac{K }{1 + GK} r - \frac{1}{1 + GK} d - \frac{K }{1 + GK} n
\end{align}
#+HTML: </details>
#+end_exercice
#+begin_important
We can see that they are 4 different transfer functions describing the behavior of the system in Figure [[fig:gang_of_four_feedback]].
These called the *Gang of Four*:
\begin{align}
S &= \frac{1 }{1 + GK}, \quad \text{the sensitivity function} \\
T &= \frac{GK}{1 + GK}, \quad \text{the complementary sensitivity function} \\
GS &= \frac{G }{1 + GK}, \quad \text{the load disturbance sensitivity function} \\
KS &= \frac{K }{1 + GK}, \quad \text{the noise sensitivity function}
\end{align}
#+end_important
#+begin_seealso
If a feedforward controller is included, a *Gang of Six* transfer functions can be defined.
More on that in this [[https://www.youtube.com/watch?v=b_8v8scghh8][short video]].
#+end_seealso
And we have:
\begin{align}
\epsilon &= S r - GS d - GS n \\
y &= T r + GS d - T n \\
u &= KS r - S d - KS n
\end{align}
Thus, for reference tracking, we want to shape the /closed-loop/ transfer function from $r$ to $\epsilon$, that is the sensitivity function $S(s)$.
Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we want to act on the complementary sensitivity function $T(s)$.
** Sensitivity Function
<<sec:sensitivity_transfer_functions>>
Suppose we have developed a "/reference/" controller $K_r(s)$ and made three small changes to obtained three controllers $K_1(s)$, $K_2(s)$ and $K_3(s)$.
The obtained sensitivity functions are shown in Figure [[fig:sensitivity_shape_effect]] and the corresponding step responses are shown in Figure [[fig:sensitivity_shape_effect_step]].
The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table [[tab:compare_sensitivity_shapes]].
#+name: tab:compare_sensitivity_shapes
#+caption: Comparison of the sensitivity function shape and the corresponding step response for the three controller variations
| Controller | Sensitivity Function Shape | Change of the Step Response |
|------------+----------------------------------------------------+----------------------------------|
| $K_1(s)$ | Larger bandwidth $\omega_b$ | Faster rise time |
| $K_2(s)$ | Larger peak value $\Vert S\Vert_\infty$ | Large overshoot and oscillations |
| $K_3(s)$ | Larger low frequency gain $\vert S(j\cdot 0)\vert$ | Larger static error |
#+begin_src matlab :exports none
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.02;
Kr = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
Kr = Kr/abs(evalfr(Kr*G, j*wc));
wc = 2*pi*2; L_w = 8; wi = 2*pi*0.045;
K1 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K1 = K1/abs(evalfr(K1*G, j*wc));
wc = 2*pi*1; L_w = 2; wi = 2*pi*0.03;
K2 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K2 = K2/abs(evalfr(K2*G, j*wc));
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.2;
K3 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K3 = K3/abs(evalfr(K3*G, j*wc));
Sr = 1/(1 + Kr*G);
S1 = 1/(1 + K1*G);
S2 = 1/(1 + K2*G);
S3 = 1/(1 + K3*G);
Tr = Kr*G/(1 + Kr*G);
T1 = K1*G/(1 + K1*G);
T2 = K2*G/(1 + K2*G);
T3 = K3*G/(1 + K3*G);
#+end_src
#+begin_src matlab :exports none
freqs = logspace(-2, 2, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$K_1(s)$');
plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$K_2(s)$');
plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$K_3(s)$');
plot(freqs, abs(squeeze(freqresp(Sr, freqs, 'Hz'))), 'k-', 'DisplayName', '$K_r(s)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude');
hold off;
legend('location', 'southeast');
ylim([1e-3, 3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/sensitivity_shape_effect.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:sensitivity_shape_effect
#+caption: Sensitivity function magnitude $|S(j\omega)|$ corresponding to the reference controller $K_r(s)$ and the three modified controllers $K_i(s)$
#+RESULTS:
[[file:figs/sensitivity_shape_effect.png]]
#+begin_src matlab :exports none
t = linspace(0, 5, 1000);
figure;
hold on;
plot(t, step(T1, t), 'DisplayName', '$K_1(s)$')
plot(t, step(T2, t), 'DisplayName', '$K_2(s)$')
plot(t, step(T3, t), 'DisplayName', '$K_3(s)$')
plot(t, step(Tr, t), 'k-', 'DisplayName', '$K_r(s)$')
hold off
xlabel('Time [s]'); ylabel('Step Response');
legend('location', 'northeast');
ylim([0, 1.7])
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/sensitivity_shape_effect_step.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:sensitivity_shape_effect_step
#+caption: Step response (response from $r$ to $y$) for the different controllers
#+RESULTS:
[[file:figs/sensitivity_shape_effect_step.png]]
#+begin_src latex :file h-infinity-spec-S.pdf
\begin{tikzpicture}
\begin{axis}[%
width=8cm,
height=4cm,
at={(0,0)},
xmode=log,
xmin=0.01,
xmax=10000,
ymin=-80,
ymax=40,
ylabel={Magnitude [dB]},
xlabel={Frequency [Hz]},
ytick={40, 20, 0, -20, -40, -60, -80},
xminorgrids,
yminorgrids
]
\addplot [thick, color=black, forget plot]
table[row sep=crcr]{%
0.01 -60\\
0.1 -60\\
190 6\\
10000 6\\
};
\draw[<-] (0.05, -60) -- (0.1, -70);
\draw (0.1, -70) -- (2, -70) node[right, fill=white, draw]{\footnotesize{Small static error}};
\draw[<-] (70, -3) -- (3, -3) node[left, fill=white, draw]{\footnotesize{Speed}};
\draw[<-] (300, 6) -- (200, 20);
\draw (200, 20) -- (10, 20) node[left, fill=white, draw]{\footnotesize{Robustness}};
\end{axis}
\end{tikzpicture}
#+end_src
#+begin_definition
- Closed-Loop Bandwidth ::
The closed-loop bandwidth $\omega_b$ is the frequency where $|S(j\omega)|$ first crosses $1/\sqrt{2} = -3dB$ from below.
In general, a large bandwidth corresponds to a faster rise time.
#+end_definition
#+begin_important
From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure [[fig:h-infinity-spec-S]].
The wanted characteristics on the magnitude of the sensitivity function are then:
- A small magnitude at low frequency to make the static errors small
- A wanted minimum closed-loop bandwidth in order to have fast rise time and good rejection of perturbations
- A small peak value in order to limit large overshoot and oscillations.
This generally means higher robustness.
This will become clear in the next section about the *module margin*.
#+name: fig:h-infinity-spec-S
#+caption: Typical wanted shape of the Sensitivity transfer function
[[file:figs/h-infinity-spec-S.png]]
#+end_important
** Robustness: Module Margin
<<sec:module_margin>>
Let's start by an example demonstrating why the phase and gain margins might not be good indicators of robustness.
#+begin_exampl
Let's consider the following plant $G_t(s)$:
#+begin_src matlab
w0 = 2*pi*100;
xi = 0.1;
k = 1e7;
Gt = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
#+end_src
Let's say we have designed a controller $K_t(s)$ that gives the loop gain shown in Figure [[fig:phase_gain_margin_model_plant]].
#+begin_src matlab
Kt = 1.2e6*(s + w0)/s;
#+end_src
The following characteristics can be determined from Figure [[fig:phase_gain_margin_model_plant]]:
- bandwidth of $\approx 10\, \text{Hz}$
- infinite gain margin (the phase of the loop-gain never reaches -180 degrees
- more than 90 degrees of phase margin
This might indicate very good robustness properties of the closed-loop system.
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gt*Kt, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-200, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/phase_gain_margin_model_plant.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:phase_gain_margin_model_plant
#+caption: Bode plot of the obtained Loop Gain $L(s)$
#+RESULTS:
[[file:figs/phase_gain_margin_model_plant.png]]
Now let's suppose the "real" plant $G_r(s)$ as a slightly lower damping factor:
#+begin_src matlab
xi = 0.03;
Gr = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
#+end_src
The obtained "real" loop gain is shown in Figure [[fig:phase_gain_margin_real_plant]].
At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 degrees while its magnitude is more than one which indicated instability.
It is confirmed by checking the stability of the closed loop system:
#+begin_src matlab :results value replace
isstable(feedback(Gp,K))
#+end_src
#+RESULTS:
: 0
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
plot(freqs, abs(squeeze(freqresp(Gr*Kt, freqs, 'Hz'))), 'DisplayName', '$L_r(s)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gt*Kt, freqs, 'Hz')))));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gr*Kt, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-200, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/phase_gain_margin_real_plant.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:phase_gain_margin_real_plant
#+caption: Bode plots of $L(s)$ (loop gain corresponding the nominal plant) and $L_r(s)$ (loop gain corresponding to the real plant)
#+RESULTS:
[[file:figs/phase_gain_margin_real_plant.png]]
Therefore, even a small change of the plant parameter makes the system unstable even though both the gain margin and the phase margin for the nominal plant are excellent.
This is due to the fact that the gain and phase margin are robustness indicators for a *pure* change or gain or a *pure* change of phase but not a combination of both.
#+end_exampl
Let's now determine a new robustness indicator based on the Nyquist Stability Criteria.
#+begin_definition
- Nyquist Stability Criteria (for stable systems) ::
If the open-loop transfer function $L(s)$ is stable, then the closed-loop system is unstable for any encirclement of the point $−1$ on the Nyquist plot.
- Nyquist Plot ::
The Nyquist plot shows the evolution of $L(j\omega)$ in the complex plane from $\omega = 0 \to \infty$.
#+end_definition
#+begin_seealso
For more information about the /general/ Nyquist Stability Criteria, you may want to look at [[https://www.youtube.com/watch?v=sof3meN96MA][this]] video.
#+end_seealso
From the Nyquist stability criteria, it is clear that we want $L(j\omega)$ to be as far away from the $-1$ point (called the /unstable point/) in the complex plane.
From this, we define the *module margin*.
#+begin_definition
- Module Margin ::
The Module Margin $\Delta M$ is defined as the *minimum distance* between the point $-1$ and the loop gain $L(j\omega)$ in the complex plane.
#+end_definition
A typical Nyquist plot is shown in Figure [[fig:module_margin_example]].
The gain, phase and module margins are graphically shown to have an idea of what they represent.
As expected from Figure [[fig:module_margin_example]], there is a close relationship between the module margin and the gain and phase margins.
We can indeed show that for a given value of the module margin $\Delta M$, we have:
\begin{equation}
\Delta G \ge \frac{\Delta M}{\Delta M - 1}; \quad \Delta \phi \ge \frac{1}{\Delta M}
\end{equation}
#+begin_src matlab :exports none
% Example Plant
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
G = 1/(m*s^2 + c*s + k); % Plant
% Example Controller
K = 14e8 * ... % Gain
1/(s^2) * ... % Double Integrator
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
L = G*K;
L_resp = squeeze(freqresp(L, freqs, 'Hz'));
% Module Margin
Dm = min(abs(1 + L_resp));
% Phase Gain Margin
[Gm, Pm, Wcg, Wcp] = margin(L);
freqs = logspace(0, 3, 1000);
figure;
hold on;
% Gain Margin
plot([-1, -1/Gm], [0, 0], '-', 'DisplayName', sprintf('$\\Delta G = %.1f$', Gm))
% Phase Margin
theta = -pi:0.01:-pi+Pm*pi/180;
plot(cos(theta), sin(theta), '-', 'DisplayName', sprintf('$\\Delta \\phi = %.1f^o$', Pm));
% Module Margin
theta = 0 : 0.01 : 2*pi;
plot(Dm*cos(theta)-1, Dm*sin(theta), '-', 'DisplayName', sprintf('$\\Delta M = %.1f$', Dm));
% Nyquist Plot
plot(real(L_resp), imag(L_resp), 'k-', 'DisplayName', '$L(j\omega)$')
plot(-1, 0, 'k*', 'HandleVisibility', 'off');
hold off;
xlabel('Real Axis'); ylabel('Imaginary Axis');
xlim([-1.5, 0.5]); ylim([-1, 1]);
axis equal;
legend('location', 'southeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/module_margin_example.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name: fig:module_margin_example
#+caption: Nyquist plot with visual indication of the Gain margin $\Delta G$, Phase margin $\Delta \phi$ and Module margin $\Delta M$
#+RESULTS:
[[file:figs/module_margin_example.png]]
Let's now try to express the Module margin $\Delta M$ as an $\mathcal{H}_\infty$ norm of a closed-loop transfer function:
\begin{align*}
\Delta M &= \text{minimum distance between } L(j\omega) \text{ and point } (-1) \\
&= \min_\omega |L(j\omega) - (-1)| \\
&= \min_\omega |1 + L(j\omega)| \\
&= \frac{1}{\max_\omega \frac{1}{|1 + L(j\omega)|}} \\
&= \frac{1}{\|S\|_\infty}
\end{align*}
#+begin_important
The $\mathcal{H}_\infty$ norm of the sensitivity function $\|S\|_\infty$ is a measure of the Module margin $\Delta M$ and therefore an indicator of the system robustness.
\begin{equation}
\Delta M = \frac{1}{\|S\|_\infty} \label{eq:module_margin_S}
\end{equation}
The wanted robustness of the closed-loop system can be specified by setting a maximum value on $\|S\|_\infty$.
#+end_important
Note that this is why large peak value of $|S(j\omega)|$ usually indicate robustness problems.
#+begin_exampl
Typical, we require $\|S\|_\infty < 2 (6dB)$ which implies $\Delta G \ge 2$ and $\Delta \phi \ge 29^o$
#+end_exampl
#+begin_seealso
To learn more about module/disk margin, you can check out [[https://www.youtube.com/watch?v=XazdN6eZF80][this]] video.
#+end_seealso
** How to *Shape* transfer function? Using of Weighting Functions!
<<sec:weighting_functions>>
- [ ] Maybe put this section in Previous chapter
Let's say we want to shape the sensitivity transfer function corresponding to the transfer function from $r$ to $\epsilon$ of the control architecture shown in Figure [[fig:loop_shaping_S_without_W]].
#+begin_src latex :file loop_shaping_S_without_W.pdf
\begin{tikzpicture}
\node[block] (G) {$G(s)$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\coordinate[above right=1.0 and 1.4 of addw] (epsilon);
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
\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[->] (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);
\begin{scope}[on background layer]
\node[fit={(G.south west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=12pt, draw, dashed, fill=black!20!white] (P) {};
\node[below] at (P.north) {Generalized Plant $P(s)$};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:loop_shaping_S_without_W
#+caption: Generalized Plant
#+RESULTS:
[[file:figs/loop_shaping_S_without_W.png]]
If the $\mathcal{H}_\infty$ synthesis is directly applied on the generalized plant $P(s)$ shown in Figure [[fig:loop_shaping_S_without_W]], if will minimize the $\mathcal{H}_\infty$ norm of transfer function from $r$ to $\epsilon$ (the sensitivity transfer function).
However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the transfer function's magnitude, it does not allow to *shape* the norm over all frequencies.
A /trick/ is to include a *weighting function* in the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
Applying the $\mathcal{H}_\infty$ synthesis to the /weighted/ generalized plant $\tilde{P}(s)$ (Figure [[fig:loop_shaping_S_with_W]]) will generate a controller $K(s)$ that minimizes the $\mathcal{H}_\infty$ norm between $r$ and $\tilde{\epsilon}$:
\begin{align}
& \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1)\nonumber \\
\Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\
\Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\
\Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping}
\end{align}
#+begin_important
As shown in Equation eqref:eq:sensitivity_shaping, the $\mathcal{H}_\infty$ synthesis allows to *shape* the magnitude of the sensitivity transfer function.
Therefore, the choice of the weighting function $W_s(s)$ is very important.
Its inverse magnitude will define the frequency dependent upper bound of the sensitivity transfer function magnitude.
#+end_important
#+begin_src latex :file loop_shaping_S_with_W.pdf
\begin{tikzpicture}
\node[block] (G) {$G(s)$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\node[block, above right=1.0 and 1.0 of addw] (Ws) {$W_s(s)$};
\coordinate[right=0.8 of Ws] (epsilon);
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
\begin{scope}[on background layer]
\node[fit={(G.south west) (Ws.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $\tilde{P}(s)$};
\end{scope}
\node[block, below=0.4 of P] (K) {$K(s)$};
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (Ws.west)node[above left]{$\epsilon$};
\draw[->] (Ws.east) -- (epsilon) node[above left](z1){$\tilde{\epsilon}$};
\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);
\end{tikzpicture}
#+end_src
#+name: fig:loop_shaping_S_with_W
#+caption: Weighted Generalized Plant
#+RESULTS:
[[file:figs/loop_shaping_S_with_W.png]]
Once the weighting function is designed, it should be added to the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
The weighted generalized plant can be defined in Matlab by either re-defining all the inputs or by pre-multiplying the (non-weighted) generalized plant by a block-diagonal MIMO transfer function containing the weights for the outputs $z$ and =1= for the outputs $v$.
#+begin_src matlab :tangle no :eval no
Pw = [Ws -Ws*G;
1 -G]
% Alternative
Pw = blkdiag(Ws, 1)*P;
#+end_src
** Design of Weighting Functions
<<sec:weighting_functions_design>>
Weighting function used must be *proper*, *stable* and *minimum phase* transfer functions.
- proper ::
more poles than zeros, this implies $\lim_{\omega \to \infty} |W(j\omega)| < \infty$
- stable ::
no poles in the right half plane
- minimum phase ::
no zeros in the right half plane
Matlab is providing the =makeweight= function that creates a first-order weights by specifying the low frequency gain, high frequency gain, and a gain at a specific frequency:
#+begin_src matlab :tangle no :eval no
W = makeweight(dcgain,[freq,mag],hfgain)
#+end_src
with:
- =dcgain=
- =freq=
- =mag=
- =hfgain=
#+begin_exampl
The Matlab code below produces a weighting function with a magnitude shape shown in Figure [[fig:first_order_weight]].
#+begin_src matlab
Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
#+end_src
#+begin_src matlab :exports none
freqs = logspace(-2, 2, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frquency [Hz]'); ylabel('Magnitude');
hold off;
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/first_order_weight.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:first_order_weight
#+caption: Obtained Magnitude of the Weighting Function
#+RESULTS:
[[file:figs/first_order_weight.png]]
#+end_exampl
#+begin_seealso
Quite often, higher orders weights are required.
In such case, the following formula can be used the design of these weights:
\begin{equation}
W(s) = \left( \frac{
\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
}{
\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
}\right)^n \label{eq:weight_formula_advanced}
\end{equation}
The parameters permit to specify:
- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
- the absolute slope between high and low frequency: $n$
A Matlab function implementing Equation eqref:eq:weight_formula_advanced is shown below:
#+name: lst:generateWeight
#+caption: Matlab Function that can be used to generate Weighting functions
#+begin_src matlab :tangle matlab/generateWeight.m :comments none :eval no
function [W] = generateWeight(args)
arguments
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2*pi
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
end
if (args.Gc <= args.G0 && args.Gc <= args.G1) || (args.Gc >= args.G0 && args.Gc >= args.G1)
eid = 'value:range';
msg = 'Gc must be between G0 and G1';
throwAsCaller(MException(eid,msg))
end
s = zpk('s');
W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
end
#+end_src
Let's use this function to generate three weights with the same high and low frequency gains, but but different slopes.
#+begin_src matlab
W1 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 1);
W2 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
W3 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 3);
#+end_src
The obtained shapes are shown in Figure [[fig:high_order_weight]].
#+begin_src matlab :exports none
freqs = logspace(-2, 2, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), ...
'DisplayName', '$n = 1$');
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), ...
'DisplayName', '$n = 2$');
plot(freqs, abs(squeeze(freqresp(W3, freqs, 'Hz'))), ...
'DisplayName', '$n = 3$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frquency [Hz]'); ylabel('Magnitude');
legend('location', 'northeast');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/high_order_weight.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:high_order_weight
#+caption: Higher order weights using Equation eqref:eq:weight_formula_advanced
#+RESULTS:
[[file:figs/high_order_weight.png]]
#+end_seealso
** Sensitivity Function Shaping - Example
<<sec:sensitivity_shaping_example>>
- Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
- Bandwidth:
- Slope of -2
First, the weighting functions is generated.
#+begin_src matlab
Ws = generateWeight('G0', 1e3, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
#+end_src
It is then added to the generalized plant.
#+begin_src matlab
Pw = blkdiag(Ws, 1)*P;
#+end_src
And the $\mathcal{H}_\infty$ synthesis is performed.
#+begin_src matlab :results output replace
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
#+end_src
#+RESULTS:
#+begin_example
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
Test bounds: 0.5 <= gamma <= 0.51
gamma X>=0 Y>=0 rho(XY)<1 p/f
5.05e-01 0.0e+00 0.0e+00 4.497e-28 p
Limiting gains...
5.05e-01 0.0e+00 0.0e+00 0.000e+00 p
5.05e-01 -1.8e+01 # -2.9e-15 1.514e-15 f
Best performance (actual): 0.504
#+end_example
The obtained $\gamma \approx 0.5$ means that it found a controller $K(s)$ that stabilize the closed-loop system, and such that:
\begin{aligned}
& \| W_s(s) S(s) \|_\infty < 0.5 \\
& \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
\end{aligned}
This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, 1./abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W_s|^{-1}$');
plot(freqs, abs(squeeze(freqresp(1/(1 + K*G), freqs, 'Hz'))), 'k-', 'DisplayName', '$|S|$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast', 'FontSize', 8);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/results_sensitivity_hinf.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:results_sensitivity_hinf
#+caption: Weighting function and obtained closed-loop sensitivity
#+RESULTS:
[[file:figs/results_sensitivity_hinf.png]]
** Complementary Sensitivity Function
** Summary
#+name: tab:specification_modern
#+caption: Table caption
| | Open-Loop Shaping | Closed-Loop Shaping |
|-----------------------------+--------------------+--------------------------------------------|
| Reference Tracking | $L$ large | $S$ small |
| Disturbance Rejection | $L$ large | $GS$ small |
| Measurement Noise Filtering | $L$ small | $T$ small |
| Small Command Amplitude | $K$ and $L$ small | $KS$ small |
| Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small |
#+begin_src latex :file h-infinity-4-blocs-constrains.pdf
\begin{tikzpicture}
\begin{scope}[shift={(0, 0)}]
\draw[] (2.5, 1.0) node[]{$S$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim 1$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\begin{scope}[shift={(6.4, 0)}]
\draw[] (2.5, 1.0) node[]{$GS$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim K^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim G$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (1, -1) to[out=45, in=180] (2.5,-0.2) to[out=0,in=180-45] (4,-1) to[out=-45,in=180-45] (5, -2);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\begin{scope}[shift={(0, -4.4)}]
\draw[] (2.5, 1.0) node[]{$KS$};
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -1.8) node[]{$\sim G^{-1}$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.3) node[]{$\sim K$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-1.5) to[out=45,in=180+45] (1, -0.5) to[out=45, in=180] (2.5,0.3) to[out=0,in=180-45] (4,-0.5) to[out=-45,in=180-45] (5, -1.5);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\begin{scope}[shift={(6.4, -4.4)}]
\draw[] (2.5, 1.0) node[]{$T$};
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim 1$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim GK$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:h-infinity-4-blocs-constrains
#+caption: Shaping the Gang of Four: Limitations
#+RESULTS:
[[file:figs/h-infinity-4-blocs-constrains.png]]
* $\mathcal{H}_\infty$ Mixed-Sensitivity Synthesis
<<sec:h_infinity_mixed_sensitivity>>
** Problem
** Typical Procedure
** Step 1 - Shaping of the Sensitivity Function
** Step 2 - Shaping of
** General Configuration for various shaping
*** S KS :ignore:
#+HTML: <details><summary>Shaping of S and KS</summary>
#+begin_src latex :file general_conf_shaping_S_KS.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\coordinate (Gin) at ($(G.west)+(-0.5, 0)$);
\begin{scope}[on background layer]
\node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.8 and 1.4 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.2, 0)$)node[branch]{} |- (W1.west);
\draw[->] (Gin)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
#+end_src
#+name: fig:general_conf_shaping_S_KS
#+caption: Generalized Plant to shape $S$ and $KS$
#+RESULTS:
[[file:figs/general_conf_shaping_S_KS.png]]
#+name: lst:general_plant_S_KS
#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_KS]]
#+begin_src matlab :eval no :tangle no
P = [W1 -G*W1
0 W2
1 -G];
#+end_src
- $W_1(s)$ is used to shape $S$
- $W_2(s)$ is used to shape $KS$
#+HTML: </details>
*** S T :ignore:
#+HTML: <details><summary>Shaping of S and T</summary>
#+begin_src latex :file general_conf_shaping_S_T.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\begin{scope}[on background layer]
\node[fit={(G.south west) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.8 and 0.8 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
#+end_src
#+name: fig:general_conf_shaping_S_T
#+caption: Generalized Plant to shape $S$ and $T$
#+RESULTS:
[[file:figs/general_conf_shaping_S_T.png]]
#+name: lst:general_plant_S_T
#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T]]
#+begin_src matlab :eval no :tangle no
P = [W1 -G*W1
0 G*W2
1 -G];
#+end_src
- $W_1$ is used to shape $S$
- $W_2$ is used to shape $T$
#+HTML: </details>
*** S T KS :ignore:
#+HTML: <details><summary>Shaping of S, T and KS</summary>
#+begin_src latex :file general_conf_shaping_S_T_KS.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {};
\node[block, above right=0.4 and 0.8 of addw] (W3) {$W_3$};
\node[block, above=0.2 of W3] (W2) {$W_2$};
\node[block, above=0.2 of W2] (W1) {$W_1$};
\coordinate (Gin) at ($(G.west)+(-0.5, 0)$);
\begin{scope}[on background layer]
\node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=0.8 of W1] (z);
\coordinate[above left=1.4 and 1.3 of G] (w);
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west);
\draw[->] (Gin)node[branch]{} |- (W2.west);
\draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W3.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$};
\draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west);
\draw[->] (w) node[above right]{$w$} -| (addw.north);
\end{tikzpicture}
#+end_src
#+name: fig:general_conf_shaping_S_T_KS
#+caption: Generalized Plant to shape $S$, $T$ and $KS$
#+RESULTS:
[[file:figs/general_conf_shaping_S_T_KS.png]]
#+name: lst:general_plant_S_T_KS
#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_KS]]
#+begin_src matlab :eval no :tangle no
P = [W1 -G*W1
0 W2
0 G*W3
1 -G];
#+end_src
- $W_1$ is used to shape $S$
- $W_2$ is used to shape $KS$
- $W_3$ is used to shape $T$
#+HTML: </details>
*** S T KS GS :ignore:
#+HTML: <details><summary>Shaping of S, T, KS and GS</summary>
#+begin_src latex :file general_conf_shaping_S_T_KS_GS.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$G$};
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addr) {};
\node[addb, left=0.6 of G] (addd) {};
\node[block, above right=0.4 and 0.8 of addr] (W2) {$W_2$};
\node[block, above=0.5 of W2] (W1) {$W_1$};
\node[block, above left=0.7 and 0.8 of addd] (W3) {$W_3$};
\node[block, above=0.5 of W3] (W4) {$1$};
\begin{scope}[on background layer]
\node[fit={(W3.west|-G.south) (W4.north -| W2.east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
\node[above] at (P.north) {Weighted Generalized Plant $P$};
\end{scope}
\node[block, below=0.6 of P] (K) {$K$};
\coordinate[right=1.0 of W1] (z);
\coordinate[left=1.0 of W3] (w);
% Connections
\draw[->] (G.east) -- (addr.west);
\draw[->] ($(addr.east)+( 0.2, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(addd.west)+(-0.4, 0)$)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$};
\draw[->] (addr.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} (addd-|w) -- (addd.west);
\draw[->] (addd.east) -- (G.west);
\draw[->] (W3.east) -| (addd.north);
\draw[->] (W4.east) -| (addr.north);
\draw[->] (w) node[above right]{$w_1$} -- (W3.west);
\draw[->] (w|-W4) node[above right]{$w_2$} -- (W4.west);
\end{tikzpicture}
#+end_src
#+name: fig:general_conf_shaping_S_T_KS_GS
#+caption: Generalized Plant to shape $S$, $T$, $KS$ and $GS$
#+RESULTS:
[[file:figs/general_conf_shaping_S_T_KS_GS.png]]
#+name: lst:general_plant_S_T_KS_GS
#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_KS_GS]]
#+begin_src matlab :eval no :tangle no
P = [ W1 -W1*G*W3 -G*W1
0 0 W2
1 -G*W3 -G];
#+end_src
- $W_1$ is used to shape $S$
- $W_2$ is used to shape $KS$
- $W_1W_3$ is used to shape $GS$
- $W_2W_3$ is used to shape $T$
#+HTML: </details>
* Conclusion
<<sec:conclusion>>
* Things to add :noexport:
** 2 blocs criterion - constrains
#+begin_src latex :file h-infinity-2-blocs-constrains.pdf
\begin{tikzpicture}
\begin{scope}[shift={(0, 0)}]
\draw[] (2.5, 1.0) node[]{$S$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim 1$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\begin{scope}[shift={(6.4, 0)}]
\draw[] (2.5, 1.0) node[]{$T$};
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim 1$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim GK$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\end{tikzpicture}
#+end_src
#+RESULTS:
[[file:figs/h-infinity-2-blocs-constrains.png]]
* Resources
yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin
yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o
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