Add some examples and comments about loop shaping

figs/mixed_sensitivity_ref_tracking.svg

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index.org

@@ -38,7 +38,7 @@
 #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
 :END:
 
-* Introduction                                                        :ignore:
+* TODO Introduction                                                   :ignore:
 * 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>>
@@ -95,53 +95,36 @@ In this document, we will mainly focus on steps 2 and 3.
 
 ** Some Background: From Classical Control to Robust Control
 
-Classical Control (1930)
+#+name: tab:comparison_control_methods
-- Tools:
+#+caption: Table summurazing the main differences between classical, modern and robust control
-  - TF (input-output)
+|                         |        *Classical Control*         |           *Modern Control*           |                            *Robust Control*                             |
-  - Nyquist, Bode, Black, \ldots
+| <l>                     |                <c>                 |                 <c>                  |                                   <c>                                   |
-  - P-PI-PID, Phase lead-lag, \ldots
+|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
-- Advantages:
+| *Date*                  |               1930-                |                1960-                 |                                  1980-                                  |
-  - Stability
+|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
-  - Performances
+| *Tools*                 |         Transfer Functions         |       State Space formulation        | Systems and Signals Norms ($\mathcal{H}_\infty$, $\mathcal{H}_2$ Norms) |
-  - Robustness
+|                         |           Nyquist Plots            |          Riccati Equations           |                     Closed Loop Transfer Functions                      |
-- Disadvantages:
+|                         |             Bode Plots             |                                      |                        Open/Closed Loop Shaping                         |
-  - Manual Method
+|                         |       Phase and Gain margins       |                                      |                           Weighting Functions                           |
-  - Only SISO
+|                         |                                    |                                      |                               Disk margin                               |
-
+|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
-Modern Control (1960)
+| *Control Architectures* | Proportional, Integral, Derivative |         Full State Feedback          |                      General Control Configuration                      |
-- Tools:
+|                         |            Leads, Lags             |               LQR, LQG               |                                                                         |
-  - State Space
+|                         |                                    |            Kalman Filters            |                                                                         |
-  - Optimal Command
+|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
-  - LQR, LQG
+| *Advantages*            |          Study Stability           |         Automatic Synthesis          |                           Automatic Synthesis                           |
-- Advantages:
+|                         |               Simple               |                 MIMO                 |                                  MIMO                                   |
-  - Automatic Synthesis
+|                         |              Natural               |         Optimization Problem         |                          Optimization Problem                           |
-  - MIMO
+|                         |                                    |                                      |                          Guaranteed Robustness                          |
-  - Optimisation problem
+|                         |                                    |                                      |                   Easy specification of performances                    |
-- Disadvantages:
+|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
-  - Robustness
+| *Disadvantages*         |           Manual Method            |       No Guaranteed Robustness       |                  Required knowledge of specific tools                   |
-  - Rejection of Perturbations
+|                         |             Only SISO              | Difficult Rejection of Perturbations |               Need a reasonably good model of the system                |
-
-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
-
 
 ** Example System
 
-Let's consider the test-system shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
+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
@@ -187,9 +170,9 @@ The notations used are listed in Table [[tab:example_notations]].
 #+caption: Example system variables
 | *Notation*         | *Description*                                                  | *Value*        | *Unit*    |
 |--------------------+----------------------------------------------------------------+----------------+-----------|
-| $m$                | Payload's mass to position / isolate                           |                | [kg]      |
+| $m$                | Payload's mass to position / isolate                           | $10$           | [kg]      |
-| $k$                | Stiffness of the suspension system                             |                | [N/m]     |
+| $k$                | Stiffness of the suspension system                             | $10^6$         | [N/m]     |
-| $c$                | Damping coefficient of the suspension system                   |                | [N/(m/s)] |
+| $c$                | 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]       |
@@ -204,13 +187,22 @@ Derive the following open-loop transfer functions:
     G_d(s) &= \frac{y}{d}
   \end{align}
 
-*Hint:* You can follow this generic procedure:
+#+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
 
 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]].
@@ -239,75 +231,22 @@ Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure
 #+RESULTS:
 [[file:figs/classical_feedback_test_system.png]]
 
-#+begin_src matlab
+
+Let's define the system parameters on Matlab.
+#+begin_src matlab +n
   k = 1e6; % Stiffness [N/m]
   c = 4e2; % Damping [N/(m/s)]
-  m = 16; % Mass [kg]
+  m = 10; % Mass [kg]
 #+end_src
 
-#+begin_src matlab
+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]]).
-  G = 1/(m*s^2 + c*s + k);
+#+begin_src matlab +n -r
-  Gd = (c*s + k)/(m*s^2 + c*s + k);
+  G = 1/(m*s^2 + c*s + k); % Plant
+  Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
 #+end_src
 
-* Classical Open Loop Shaping
-** Introduction ot Open Loop Shaping
-Usually, the controller $K(s)$ is designed such that the loop gain $L(s)$ has desirable shape.
-This technique is called *Open Loop Shaping*.
-
-*************** TODO Explain why the Loop gain si an important "value"
-For instance example all the specifications can usually be explained in terms of the open loop gain.
-*************** END
-
-
-#+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
-#+RESULTS:
-[[file:figs/open_loop_shaping.png]]
-
-This is usually done manually has the loop gain $L(s)$ depends linearly of $K(s)$:
-\begin{equation}
-  L(s) = G(s) K(s)
-\end{equation}
-- where $L(s)$ is called the *Loop Gain Transfer Function*
-
-$K(s)$ then consists of a combination of leads, lags, notches, etc. such that its product with $G(s)$ has wanted shape.
-
-** Example of Open Loop Shaping
-
-
-#+begin_src matlab
-  k = 1e-6;
-  m = 10;
-  c = 10;
-
-  G =
-#+end_src
-
-
 #+begin_src matlab :exports none
-  freqs = logspace(1, 4, 1000);
+  freqs = logspace(0, 3, 1000);
 
   figure;
   tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
@@ -340,10 +279,87 @@ $K(s)$ then consists of a combination of leads, lags, notches, etc. such that it
 #+RESULTS:
 [[file:figs/bode_plot_example_afm.png]]
 
-Specifications:
+#+begin_src matlab :exports none
-- *Performance*: Bandwidth of approximately 50Hz
+  freqs = logspace(0, 3, 1000);
-- *Noise Attenuation*: Roll-off of -40dB/decade past 250Hz
+
-- *Robustness*: Gain margin > 5dB and Phase margin > 40 deg
+  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
+** Introduction to Open Loop Shaping
+
+#+begin_definition
+The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant (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
+
+#+begin_definition
+*Open Loop Shaping* refers to a control design technique where the controller $K(s)$ is designed such that the *Open Loop Gain* $L(s)$ has desirable shape.
+#+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 $L(s)$:
+- *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.
+
+** Example of Open Loop Shaping
+
+#+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.
@@ -357,13 +373,14 @@ In order to have the wanted Roll-off, two integrators are used, a lead is also a
 
 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 = 6e4 * ... % Gain
+  K = 14e8 * ... % Gain
       1/(s^2) * ... % Double Integrator
-      (1 + s/111)/(1 + s/888); % Lead
+      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(1, 4, 1000);
+  freqs = logspace(0, 3, 1000);
 
   figure;
   tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
@@ -375,7 +392,7 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
   hold off;
-  ylim([1e-5, 1e1])
+  ylim([1e-4, 1e1])
 
   ax2 = nexttile;
   hold on;
@@ -390,7 +407,7 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-  exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'tall');
+  exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
 #+name: fig:loop_gain_manual_afm
@@ -399,22 +416,23 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
 [[file:figs/loop_gain_manual_afm.png]]
 
 And we can verify that we have the wanted stability margins:
-#+begin_src matlab :results output replace
+#+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 [dB]', 'Phase Margin [deg]', 'Crossover [Hz]'}, {'Value'}, ' %.1f ');
+  data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Manual Method'}, ' %.1f ');
 #+end_src
 
 #+RESULTS:
-|                    | Value |
+|                             | Manual Method |
-|--------------------+-------|
+|-----------------------------+---------------|
-| Gain Margin [dB]   |   7.2 |
+| Gain Margin $> 3$ [dB]      |           3.1 |
-| Phase Margin [deg] |  48.1 |
+| Phase Margin $> 30$ [deg]   |          35.4 |
-| Crossover [Hz]     |  50.7 |
+| Crossover $\approx 10$ [Hz] |          10.1 |
 
 ** $\mathcal{H}_\infty$ Loop Shaping Synthesis
+
 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.
@@ -434,45 +452,88 @@ where:
 
 ** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
 
-Let's re-use the previous plant.
+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
-  G = tf(16,[1 0.16 16]);
+  Lw = 2.3e3 * ...
-
+       1/(s^2) * ... % Double Integrator
-  Gd = 3.7e4*1/s*(1 + s/2/pi/20)/(1 + s/2/pi/220)*1/(s + s/2/pi/500);
+       (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 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
-  bodeFig({Gd}, struct('phase', true))
+  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
+#+begin_src matlab :tangle no :exports results :results file replace
-  [K,CL,GAM,INFO] = loopsyn(G, Gd);
+  exportFig('figs/open_loop_shaping_hinf_K.pdf', 'width', 'wide', 'height', 'normal');
 #+end_src
 
-#+begin_src matlab
+#+name: fig:open_loop_shaping_hinf_K
-  bodeFig({K})
+#+caption: Obtained controller $K$ using the open-loop $\mathcal{H}_\infty$ shaping
-#+end_src
+#+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(1, 4, 1000);
+  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'))));
+  plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
-  plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))), 'k--');
+  plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz'))), 'k--', 'DisplayName', '$L_w(s)$');
-  plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz')))*GAM, 'k-.');
+  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(Gd, freqs, 'Hz')))/GAM, 'k-.');
+  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;
-  ylim([1e-5, 1e1])
+  legend('location', 'northeast');
+  ylim([1e-4, 1e2]);
 
   ax2 = nexttile;
   hold on;
@@ -486,8 +547,36 @@ Translate the specification into the wanted shape of the open loop gain.
   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
 
-* The $\mathcal{H}_\infty$ Norm
+#+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 Step in the $\mathcal{H}_\infty$ world
+** 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
@@ -502,31 +591,24 @@ Translate the specification into the wanted shape of the open loop gain.
 #+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
+: 7.9216e-06
 
 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);
+  freqs = logspace(0, 3, 1000);
   figure;
   hold on;
   plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
-  plot([0.5, 2], [hinfnorm(G) hinfnorm(G)], 'k--');
+  plot([20, 100], [hinfnorm(G) hinfnorm(G)], 'k--');
-  text(2, hinfnorm(G), '$\quad \|G\|_\infty$')
+  text(100, hinfnorm(G), '$\quad \|G\|_\infty$')
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   xlabel('Frequency [Hz]'); ylabel('Magnitude $|G(j\omega)|$');
@@ -543,7 +625,7 @@ The maximum value of the magnitude over all frequencies does correspond to the $
 [[file:figs/hinfinity_norm_siso_bode.png]]
 #+end_exampl
 
-* $\mathcal{H}_\infty$ Synthesis
+** $\mathcal{H}_\infty$ Synthesis
 
 *Optimization problem*:
 $\mathcal{H}_\infty$ 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 $\mathcal{H}_\infty$ norms of defined transfer functions are minimized.
@@ -560,7 +642,7 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
 - Fixed-Structure $\mathcal{H}_\infty$ Synthesis
 - Signal Based $\mathcal{H}_\infty$ Synthesis
 
-* The Generalized Plant
+** The Generalized Plant
 #+begin_src latex :file general_plant.pdf
   \begin{tikzpicture}
     \node[block={2.0cm}{2.0cm}] (P) {$P$};
@@ -598,45 +680,7 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
   \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
+** From a Classical Feedback Architecture to a Generalized Plant
-
-#+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}
@@ -672,7 +716,6 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
 | $d$        | 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
@@ -716,12 +759,12 @@ The procedure is:
 
     \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 $P(s)$};
+      \node[below] at (P.north) {Generalized Plant $P(s)$};
     \end{scope}
   \end{tikzpicture}
 #+end_src
 
-#+begin_exampl
+#+begin_exercice
   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
@@ -747,7 +790,46 @@ The procedure is:
          0  1;
          1 -G]
   #+end_src
-#+end_exampl
+#+end_exercice
+
+
+** The General Synthesis 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]]
+
 
 * Modern Interpretation of the Control Specifications
 ** Introduction