Tangle Matlab scripts
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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% Let's define the system parameters on Matlab.
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k = 1e6; % Stiffness [N/m]
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c = 4e2; % Damping [N/(m/s)]
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m = 10; % Mass [kg]
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% And now the system dynamics $G(s)$ and $G_d(s)$.
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G = 1/(m*s^2 + c*s + k); % Plant
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Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
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% #+HTML: <details><summary>Hint</summary>
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% #+LATEX: \tcbsubtitle{Hint}
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% You can follow this procedure:
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% 1. In order to have good disturbance rejection at low frequency, add a simple or double *integrator*
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% 2. In terms of the loop gain, the *bandwidth* can be defined at the frequency $\omega_c$ where $|l(j\omega_c)|$ first crosses 1 from above.
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% Therefore, adjust the *gain* such that $L(j\omega)$ crosses 1 at around 10Hz
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% 3. The roll-off at high frequency for noise attenuation should already be good enough.
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% If not, add a *low pass filter*
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% 4. Add a *Lead* centered around the crossover frequency (10 Hz) and tune it such that sufficient phase margin is added.
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% Verify that the gain margin is good enough.
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% #+HTML: </details>
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% #+end_exercice
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% Let's say we came up with the following controller.
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K = 14e8 * ... % Gain
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1/(s^2) * ... % Double Integrator
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1/(1 + s/2/pi/40) * ... % Low Pass Filter
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(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
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% The bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]] and we can verify that we have the wanted stability margins using the =margin= command:
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[Gm, Pm, ~, Wc] = margin(G*K);
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% #+RESULTS:
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% | Requirements | Manual Method |
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% |-----------------------------+---------------|
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% | Gain Margin $> 3$ [dB] | 3.1 |
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% | Phase Margin $> 30$ [deg] | 35.4 |
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% | Crossover $\approx 10$ [Hz] | 10.1 |
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freqs = logspace(0, 3, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-4, 1e1])
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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yticks(-360:90:360); ylim([-360, 0]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
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% <<sec:h_infinity_open_loop_shaping_example>>
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% To apply the $\mathcal{H}_\infty$ Loop Shaping Synthesis, the wanted shape of the loop gain should be determined from the specifications.
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% This is summarized in Table [[tab:open_loop_shaping_specifications]].
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% Such shape corresponds to the typical wanted Loop gain Shape shown in Figure [[fig:open_loop_shaping_shape]].
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% #+name: tab:open_loop_shaping_specifications
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% #+caption: Wanted Loop Shape corresponding to each specification
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% #+attr_latex: :environment tabularx :width \linewidth :align lXX
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% | | *Specification* | *Corresponding Loop Shape* |
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% |--------------+----------------------------------------+------------------------------------------|
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% | *Dist. Rej.* | Highest possible rejection below 1Hz | Slope of -40dB/dec at low frequency |
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% | *Pos. Speed* | Bandwidth of approximately 10Hz | $L$ crosses 1 at 10Hz |
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% | *Noise Att.* | Roll-off of -40dB/decade past 30Hz | Roll-off of -40dB/decade past 30Hz |
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% | *Robustness* | $\Delta G > 3dB$, $\Delta \phi > 30^o$ | Slope of -20dB/decade near the crossover |
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% Then, a (stable, minimum phase) transfer function $L_w(s)$ should be created that has the same gain as the wanted shape of the Loop gain.
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% For this example, a double integrator and a lead centered on 10Hz are used.
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% Then the gain is adjusted such that the $|L_w(j2 \pi 10)| = 1$.
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% Using Matlab, we have:
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Lw = 2.3e3 * ...
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1/(s^2) * ... % Double Integrator
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(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
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% The $\mathcal{H}_\infty$ open loop shaping synthesis is then performed using the =loopsyn= command:
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[K, ~, GAM] = loopsyn(G, Lw);
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% The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]] and matches the specified one by a factor $\gamma \approx 2$.
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freqs = logspace(0, 3, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
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plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz'))), 'k--', 'DisplayName', '$L_w(s)$');
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plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))*GAM, 'k-.', 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
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plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))/GAM, 'k-.', 'HandleVisibility', 'off');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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hold off;
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legend('location', 'northeast');
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ylim([1e-4, 1e2]);
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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yticks(-360:90:360); ylim([-360, 0]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:open_loop_shaping_hinf_L
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% #+caption: Obtained Open Loop Gain $L(s) = G(s) K(s)$ and comparison with the wanted Loop gain $L_w$
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% #+RESULTS:
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% [[file:figs/open_loop_shaping_hinf_L.png]]
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% #+begin_important
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% When using the $\mathcal{H}_\infty$ Synthesis, it is usually recommended to analyze the obtained controller.
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% This is usually done by breaking down the controller into simple elements such as low pass filters, high pass filters, notches, leads, etc.
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% #+end_important
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% Let's briefly analyze the obtained controller which bode plot is shown in Figure [[fig:open_loop_shaping_hinf_K]]:
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% - two integrators are used at low frequency to have the wanted low frequency high gain
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% - a lead is added centered with the crossover frequency to increase the phase margin
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% - 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)
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freqs = logspace(0, 3, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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plot(freqs, abs(squeeze(freqresp(K, freqs, 'Hz'))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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hold off;
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ax2 = nexttile;
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plot(freqs, 180/pi*angle(squeeze(freqresp(K, freqs, 'Hz'))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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yticks(-360:90:360); ylim([-180, 90]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:open_loop_shaping_hinf_K
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% #+caption: Obtained controller $K$ using the open-loop $\mathcal{H}_\infty$ shaping
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% #+RESULTS:
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% [[file:figs/open_loop_shaping_hinf_K.png]]
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% Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table [[tab:open_loop_shaping_compare]].
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[Gm_2, Pm_2, ~, Wc_2] = margin(G*K);
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