Tangle Matlab scripts

2026-10-12 11:16:59 +02:00
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+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+
+
+% Let's define the system parameters on Matlab.
+
+k = 1e6; % Stiffness [N/m]
+c = 4e2; % Damping [N/(m/s)]
+m = 10; % Mass [kg]
+
+
+
+% And now the system dynamics $G(s)$ and $G_d(s)$.
+
+G = 1/(m*s^2 + c*s + k); % Plant
+Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
+
+
+% with:
+% - =dcgain=: low frequency gain
+% - =[freq,mag]=: frequency =freq= at which the gain is =mag=
+% - =hfgain=: high frequency gain
+
+% #+begin_exampl
+% The Matlab code below produces a weighting function with the following characteristics (Figure [[fig:first_order_weight]]):
+% - Low frequency gain of 100
+% - Gain of 1 at 10Hz
+% - High frequency gain of 0.5
+
+
+Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
+
+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('Frequency [Hz]'); ylabel('Magnitude');
+hold off;
+
+
+
+% Let's use this function to generate three weights with the same high and low frequency gains, but but different slopes.
+
+
+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);
+
+
+
+% The obtained shapes are shown in Figure [[fig:high_order_weight]].
+
+
+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('Frequency [Hz]'); ylabel('Magnitude');
+legend('location', 'northeast');
+
+
+% #+HTML: </details>
+% #+end_exercice
+
+% Let's say we came up with the following weighting function:
+
+Ws = generateWeight('G0', 1e3, ...
+                    'G1', 1/2, ...
+                    'Gc', sqrt(2), 'wc', 2*pi*10, ...
+                    'n', 2);
+
+
+
+% The weighting function is then added to the generalized plant.
+
+P = [1 -G;
+     1 -G];
+Pw = blkdiag(Ws, 1)*P;
+
+
+
+% And the $\mathcal{H}_\infty$ synthesis is performed on the /weighted/ generalized plant.
+
+K = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+
+
+% #+RESULTS:
+% #+begin_example
+%   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     3.000e-16     p
+%   Limiting gains...
+%   5.05e-01     0.0e+00     0.0e+00     3.461e-16     p
+%   5.05e-01    -3.5e+01 #  -4.9e-14     1.732e-26     f
+
+%   Best performance (actual): 0.503
+% #+end_example
+
+% $\gamma \approx 0.5$ means that the $\mathcal{H}_\infty$ synthesis generated a controller $K(s)$ that stabilizes the closed-loop system, and such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is less than $\gamma$:
+% \begin{align*}
+%   & \| W_s(s) S(s) \|_\infty \approx 0.5 \\
+%   & \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
+% \end{align*}
+
+% This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
+
+% #+begin_important
+% Obtaining $\gamma < 1$ means that the $\mathcal{H}_\infty$ synthesis found a controller such that the specified closed-loop transfer functions are bellow the specified upper bounds.
+
+% Yet, obtaining a $\gamma$ slightly above one does not necessary means the synthesis is unsuccessful.
+% It just means that at some frequency, one of the closed-loop transfer functions is above the specified upper bound by a factor $\gamma$.
+% #+end_important
+
+
+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);