Finish section about module margin

index.org

@@ -1174,53 +1174,13 @@ Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we w
   xlabel('Time [s]'); ylabel('Step Response');
 #+end_src
 
-#+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]{$\SI{-60}{\decibel} \rightarrow$ \footnotesize{Static error}};
-      \draw[<-] (1, -40) -- (10, -40) node[right, fill=white, draw]{$\SI{20}{\decibel/dec} \rightarrow$ \footnotesize{Ref. track.}};
-      \draw[<-] (100, 0) -- (3, 0) node[left, fill=white, draw]{$\omega_c \rightarrow$ \footnotesize{Speed}};
-
-      \draw[<-] (300, 6) -- (200, 20);
-      \draw (200, 20) -- (10, 20) node[left, fill=white, draw]{$\SI{6}{\decibel} \rightarrow$ \footnotesize{Module margin}};
-    \end{axis}
-  \end{tikzpicture}
-#+end_src
-
-#+name: fig:h-infinity-spec-S
-#+caption: Typical wanted shape of the Sensitivity transfer function
-#+RESULTS:
-[[file:figs/h-infinity-spec-S.png]]
-
 ** 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:
+Let's consider the following plant $G_t(s)$:
 #+begin_src matlab
   w0 = 2*pi*100;
   xi = 0.1;
@@ -1229,17 +1189,19 @@ Let's consider the following plant:
   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:
+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
-  Kt = 1.2e6*(s + w0)/s;
-#+end_src
-
 #+begin_src matlab :exports none
   freqs = logspace(0, 3, 1000);
 
@@ -1277,14 +1239,14 @@ This might indicate very good robustness properties of the closed-loop system.
 [[file:figs/phase_gain_margin_model_plant.png]]
 
 
-Now let's suppose the "real" plant =Gr= as a slightly lower damping factor:
+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 loop gain is in Figure [[fig:phase_gain_margin_real_plant]].
+The obtained "real" loop gain is shown 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.
+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
@@ -1334,35 +1296,125 @@ It is confirmed by checking the stability of the closed loop system:
 
 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.
+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 (or disk) margin, you can check out [[https://www.youtube.com/watch?v=XazdN6eZF80][this]] video.
+  To learn more about module/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>>
 
@@ -1603,6 +1655,47 @@ The obtained shapes are shown in Figure [[fig:high_order_weight]].
 ** Sensitivity Function Shaping - Example
 <<sec:sensitivity_shaping_example>>
 
+#+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]{$\SI{-60}{\decibel} \rightarrow$ \footnotesize{Static error}};
+      \draw[<-] (1, -40) -- (10, -40) node[right, fill=white, draw]{$\SI{20}{\decibel/dec} \rightarrow$ \footnotesize{Ref. track.}};
+      \draw[<-] (100, 0) -- (3, 0) node[left, fill=white, draw]{$\omega_c \rightarrow$ \footnotesize{Speed}};
+
+      \draw[<-] (300, 6) -- (200, 20);
+      \draw (200, 20) -- (10, 20) node[left, fill=white, draw]{$\SI{6}{\decibel} \rightarrow$ \footnotesize{Module margin}};
+    \end{axis}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:h-infinity-spec-S
+#+caption: Typical wanted shape of the Sensitivity transfer function
+#+RESULTS:
+[[file:figs/h-infinity-spec-S.png]]
+
+
 - Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
 - Bandwidth:
 - Slope of -2