Add text about module margin

index.org

@@ -39,7 +39,18 @@
 #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
 :END:
 
-* TODO Introduction                                                   :ignore:
+* 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>>
@@ -53,7 +64,7 @@
   addpath('matlab')
 #+end_src
 
-* Introduction to the Control Methodology - Model Based Control
+* Introduction to Model Based Control
 <<sec:model_based_control>>
 
 ** Model Based Control - Methodology
@@ -101,7 +112,12 @@ It consists of three steps:
 
 In this document, we will mainly focus on steps 2 and 3.
 
-** Some Background: From Classical Control to Robust Control
+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
@@ -117,9 +133,8 @@ In this document, we will mainly focus on steps 2 and 3.
 |                         |       Phase and Gain margins       |                                      |                           Weighting Functions                           |
 |                         |                                    |                                      |                               Disk margin                               |
 |-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
-| *Control Architectures* | Proportional, Integral, Derivative |         Full State Feedback          |                      General Control Configuration                      |
-|                         |            Leads, Lags             |               LQR, LQG               |                                                                         |
-|                         |                                    |            Kalman Filters            |                                                                         |
+| *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                                   |
@@ -508,7 +523,7 @@ Let's take our example system and try to apply the Open-Loop shaping strategy to
 #+begin_exercice
 Using =SISOTOOL=, design a controller that fulfill the specifications.
 
-#+begin_src matlab :eval no
+#+begin_src matlab :eval no :tangle no
   sisotool(G)
 #+end_src
 #+end_exercice
@@ -583,7 +598,7 @@ The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$
 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
+#+begin_src matlab :eval no :tangle no
   K = loopsyn(G, Gd);
 #+end_src
 where:
@@ -852,7 +867,7 @@ It can indeed represent feedback as well as feedforward control architectures.
 | $u$      | Control signals                                 |
 #+end_important
 
-** The General Synthesis Problem Formulation
+** 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:
@@ -897,7 +912,7 @@ Note that the closed-loop transfer function from $w$ to $z$ is:
 \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
+#+begin_src matlab :eval no :tangoe no
   K = hinfsyn(P, nmeas, ncont);
 #+end_src
 where:
@@ -997,27 +1012,27 @@ Using Matlab, the generalized plant can be defined as follows:
 
 ** 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 design of controllers by hand.
+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.
 
-It is very easy to shape as it depends linearly on $K(s)$.
-Moreover, it gives information on important quantities such as the stability margins and the control bandwidth.
+$L(s)$ 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 determined by the *closed-loop transfer functions*.
+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
-- *Reference tracking* (Overshoot, Static error, Settling time, ...)
-  - From $r$ to $\epsilon$
-- *Disturbances 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 functions must be stable
-- *Robustness* (stability margins)
-  - Module margin (see Section [[sec:module_margin]])
+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}
@@ -1202,12 +1217,152 @@ Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we w
 ** Robustness: Module Margin
 <<sec:module_margin>>
 
-- [ ] Definition of Module margin
-- [ ] Why it represents robustness
-- [ ] Example
+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:
+#+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 that gives the loop gain shown in Figure [[fig:phase_gain_margin_model_plant]] where the following characteristics can be determined:
+- 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
+  Kt = 1.2e6*(s + w0)/s;
+#+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(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 =Gr= 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 loop gain is in Figure [[fig:phase_gain_margin_real_plant]].
+It is shown that 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 unstability.
+
+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 indicators to the robustness of the system to a *pure* change or gain or a *pure* change of phase but not a combination of both.
+#+end_exampl
+
+#+begin_seealso
+  To learn more about module (or disk) margin, you can check out [[https://www.youtube.com/watch?v=XazdN6eZF80][this]] video.
+#+end_seealso
+
+- [ ] Example why the module margin might be a good alternative
+- [ ] Stability => L does not does at the left of point -1
+- [ ] => ...
+
+
+#+begin_definition
+  The *module margin* $\Delta M$ is defined as the *minimum distance* between the point $-1$ and $L(j\omega) = G(j\omega) K(j\omega)$ in the complex plane.
+#+end_definition
+
+The module margin can be visually determined on the Nyquist plot ($L(j\omega)$ from $\omega = -\infty$ to $\infty$ in the complex plane).
+
+- [ ] Example of Nyquist plot and module margin
+- [ ] Express the module margin as a function of $\|S\|_\infty$
 
 \[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \]
 
+#+begin_important
+  Large peak value of $S$ indicate robustness problems.
+
+  The wanted robustness of the closed-loop system can there be specified by setting a maximum value on $\|S\|_\infty$.
+#+end_important
+
 ** How to *Shape* transfer function? Using of Weighting Functions!
 <<sec:weighting_functions>>