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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% #+name: fig:classical_feedback_test_system_bis
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% #+caption: Block diagram corresponding to the example system
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% #+RESULTS:
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% [[file:figs/ex_test_system_feedback.png]]
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% The two transfer functions present in the system are derived and defined below:
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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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% Control Plant
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G = 1/(m*s^2 + c*s + k);
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% Disturbance dynamics
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Gd = (c*s + k)/(m*s^2 + c*s + k);
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% We also define the inputs signals that will be used for time domain simulations.
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% They are graphically shown in Figure [[fig:ex_inputs_signals]].
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% Time Vector
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t = 0:1e-4:0.5;
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% Reference Input
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r = zeros(size(t));
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r(t>0.05 & t<=0.25) = 1e-4*(t(t>0.05 & t<=0.25)-0.05);
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r(t>0.25) = 2e-5;
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% Measurement Noise
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Fs = 1e3; % Sampling Frequency [Hz]
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Ts = 1/Fs; % Sampling Time [s]
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n = sqrt(Fs/2)*randn(1, length(t)); % Signal with an ASD equal to one
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n = lsim(1e-6*(s + 2*pi*1e2)^2/(s + 2*pi*1e3)^2/(1+s/2/pi/500), n, t)'; % Shaped noise
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% Disturbance
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d = zeros(size(t));
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d(t>0.3) = 5e-6;
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
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nexttile;
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plot(t, 1e6*r, 'k-')
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xlabel('Time [s]');
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ylabel('Reference Signal $[\mu m]$');
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nexttile;
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plot(t, 1e6*n, 'k-')
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xlabel('Time [s]');
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ylabel('Measurement Noise $[\mu m]$');
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nexttile;
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plot(t, 1e6*d, 'k-')
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xlabel('Time [s]');
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ylabel('Disturbance Motion $[\mu m]$');
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% #+name: fig:ex_general_plant_sim
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% #+caption: Generalized plant that will be used for simulations
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% #+RESULTS:
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% [[file:figs/ex_general_plant_sim.png]]
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% The Generalized plant of Figure [[fig:ex_general_plant_sim]] is defined on Matlab as follows:
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Psim = [0 0 Gd G
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0 0 0 1
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1 -1 -Gd -G];
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Psim.InputName = {'r', 'n', 'd', 'u'};
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Psim.OutputName = {'y', 'u', 'e'};
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% Modern Interpretation of control specifications
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% <<sec:ex_specification_interpretation>>
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% #+begin_exercice
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% 1. Translate the control specifications into wanted shape of closed-loop transfer functions
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% 2. Conclude and the closed-loop transfer functions to be shaped
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% 3. Chose a general configuration architecture that allows to shape these transfer function
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% 4. Using Matlab, define the generalized plant
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% #+HTML: <details><summary>Hint</summary>
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% #+LATEX: \tcbsubtitle{Hint}
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% 1. Make use of Table [[tab:usual_shaping_gang_four]]
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% 2. Make use of Table [[tab:usual_shaping_gang_four]]
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% 3. See Section [[sec:shaping_multiple_tf]]
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% 4. See Section [[sec:shaping_multiple_tf]]
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% #+HTML: </details>
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% #+end_exercice
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% After converting the control specifications into wanted shape of closed-loop transfer functions, we might come up with the Table [[tab:ex_specification_shapes]].
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% In such case, we want to shape $S$, $GS$ and $T$.
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% #+name: tab:ex_specification_shapes
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% #+caption: Control Specifications and associated wanted shape of the closed-loop transfer functions
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% #+attr_latex: :environment tabularx :width 0.7\linewidth :align llX
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% | <l> | <c> | <l> |
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% | *Specification* | *TF* | *Wanted Shape* |
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% |--------------------------+-----------+------------------------------------|
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% | Follow Step Reference | $S$ | +40dB of slope at low frequency |
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% | Reject Disturbances | $S$, $GS$ | Small gain |
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% | Reject measurement noise | $T$ | Small high frequency (>100Hz) gain |
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% | Robust System | $S$ | Small $\Vert S \Vert_\infty$ |
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% To do so, we use to generalized plant shown in Figure [[fig:ex_general_plant]] for the synthesis where the three closed-loop tranfert functions from $w$ to $[z_1\,,z_2\,,z_3]$ are respectively $S$, $GS$ and $T$.
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% This generalized plant is defined on Matlab as follows:
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P = [1 -1
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G -G
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0 1
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G -G];
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% Step 1 - Shaping of $S$
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% <<sec:ex_shaping_S>>
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% Let's first shape the Sensitivity function as it is usually the most important of the /Gang of four/ closed-loop transfer functions.
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% To do so, we have to create a weight $W_1(s)$ that defines the wanted upper bound on $|S(j\omega)|$:
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% - small low frequency gain: $|S(j\cdot 0)| = 10^{-3}$
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% - minimum crossover frequency of $\approx 10Hz$: $|S(j2\pi 10)| < \frac{1}{\sqrt{2}}$
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% - small maximum peak magnitude for robustness properties: $\|S\|_\infty < 2$
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% The weighting function is design using the =generateWeigh= function and its inverse shape can be seen in Figure
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W1 = generateWeight('G0', 1e3, ...
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'G1', 1/2, ...
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'Gc', sqrt(2), 'wc', 2*pi*10, ...
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'n', 1);
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% To not constrain $GS$ and $T$ for the shaping of $S$, $W_2$ and $W_3$ are first taken as very small gains:
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W2 = tf(1e-8);
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W3 = tf(1e-8);
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% The $\mathcal{H}_\infty$ synthesis is performed and the obtained closed-loop transfer functions $S$, $GS$, and $T$ and compared with the upper bounds set by the weighting functions in Figure [[fig:ex_results_1]].
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Pw = blkdiag(W1, W2, W3, 1)*P;
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K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
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% #+RESULTS:
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% #+begin_example
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% Test bounds: 0.5 <= gamma <= 0.51
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% gamma X>=0 Y>=0 rho(XY)<1 p/f
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% 5.05e-01 0.0e+00 0.0e+00 5.511e-14 p
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% Limiting gains...
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% 5.05e-01 0.0e+00 0.0e+00 1.867e-14 p
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% Best performance (actual): 0.502
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% #+end_example
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% Closed-loop tranfer functions are computed
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Z1 = lft(P, K1);
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freqs = logspace(-1, 3, 1000);
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
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ax1 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W|^{-1}$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('S');
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legend('location', 'southeast');
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ax2 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('GS');
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ax3 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('T');
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ax4 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('K');
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linkaxes([ax1, ax2, ax3, ax4], 'x')
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:ex_results_1
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% #+caption: Obtained Shape Closed-Loop transfer functions (dashed black lines indicate inverse magnitude of the weighting functions)
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% #+RESULTS:
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% [[file:figs/ex_results_1.png]]
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% Time domain simulation is then performed and the obtained output displacement and control inputs are shown in Figure [[fig:ex_time_domain_1]].
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% We can see:
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% - we are not able to follow the ramp input. This have to be solved by modifying the weighting function $W_1(s)$
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% - we have poor rejection of disturbances. This we be solve by shaping $GS$ in Section [[sec:ex_shaping_GS]]
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% - we have quite large effect of the measurement noise. This will be solved by shaping $T$ in Section [[sec:ex_shaping_T]]
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% Closed-loop simulation
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Z1sim = lft(Psim, K1);
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z = lsim(Z1sim, [r;n;d], t);
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y1 = z(:,1);
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u1 = z(:,2);
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figure;
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tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
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nexttile;
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hold on;
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plot(t, 1e6*y1, 'DisplayName', 'Step 1');
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plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
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hold off;
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xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
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legend('location', 'southeast');
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% #+name: fig:ex_time_domain_1
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% #+caption: Time domain simulation results
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% #+RESULTS:
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% [[file:figs/ex_time_domain_1.png]]
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% Remember that in order to follow ramp inputs, the sensitivity function should have a slope of +40dB/dec at low frequency (Table [[tab:usual_shaping_gang_four]]).
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% To do so, let's modify $W_1$ to impose a slope of +40dB/dec at low frequency.
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% This can simple be done by using a second order weight:
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W1 = generateWeight('G0', 1e3, ...
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'G1', 1/2, ...
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'Gc', sqrt(2), 'wc', 2*pi*15, ...
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'n', 2);
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% The $\mathcal{H}_\infty$ synthesis is performed using the new weights and the obtained closed-loop shaped are shown in figure [[fig:ex_results_1b]].
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% The time domain signals are shown in Figure [[fig:ex_time_domain_1b]] and it is confirmed that the ramps are now follows without static errors.
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Pw = blkdiag(W1, W2, W3, 1)*P;
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K1b = hinfsyn(Pw, 1, 1, 'Display', 'on');
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% Compute the closed-loop transfer function
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Z1b = lft(P, K1b);
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freqs = logspace(-1, 3, 1000);
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
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ax1 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
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plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1b');
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W^{-1}|$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('S');
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legend('location', 'southeast');
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ax2 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('GS');
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ax3 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('T');
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ax4 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('K');
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linkaxes([ax1, ax2, ax3, ax4], 'x')
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:ex_results_1b
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% #+caption: Obtained Shape Closed-Loop transfer functions
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% #+RESULTS:
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% [[file:figs/ex_results_1b.png]]
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% Closed-loop simulations
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Z1bsim = lft(Psim, K1b);
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z = lsim(Z1bsim, [r;n;d], t);
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y1b = z(:,1);
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u1b = z(:,2);
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figure;
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tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
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hold on;
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plot(t, 1e6*y1, 'DisplayName', 'Step 1');
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plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
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plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
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hold off;
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xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
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legend('location', 'southeast');
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% Step 2 - Shaping of $GS$
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% <<sec:ex_shaping_GS>>
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% Looking at Figure [[fig:ex_results_2]], it is clear that the rejection of disturbances is not satisfactory.
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% This can also be seen by the large peak of $GS$ in Figure [[fig:ex_results_1b]].
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% This poor rejection of disturbances is actually due to the fact that the obtain controller has at notch at the resonance frequency of the plant.
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% To overcome this issue, we can simply increase the magnitude of $W_2$ to limit the peak magnitude of $GS$
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% Let's take $W_2$ as a simple constant gain:
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W2 = tf(4e5);
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% The $\mathcal{H}_\infty$ Synthesis is performed and the obtained closed-loop transfer functions are shown in Figure [[fig:ex_results_2]].
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Pw = blkdiag(W1, W2, W3, 1)*P;
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K2 = hinfsyn(Pw, 1, 1, 'Display', 'on');
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Z2 = lft(P, K2);
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freqs = logspace(-1, 3, 1000);
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
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ax1 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
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plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1b');
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plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 2');
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W^{-1}|$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('S');
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legend('location', 'southeast');
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ax2 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('GS');
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ax3 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('T');
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ax4 = nexttile;
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hold on;
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plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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title('K');
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linkaxes([ax1, ax2, ax3, ax4], 'x')
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:ex_results_2
|
||||
% #+caption: Obtained Shape Closed-Loop transfer functions
|
||||
% #+RESULTS:
|
||||
% [[file:figs/ex_results_2.png]]
|
||||
|
||||
% Time domain simulation results are shown in Figure [[fig:ex_time_domain_2]].
|
||||
% If is shown that indeed, the disturbance rejection performance are much better and only very small oscillation is obtained.
|
||||
|
||||
|
||||
Z2sim = lft(Psim, K2);
|
||||
|
||||
z = lsim(Z2sim, [r;n;d], t);
|
||||
y2 = z(:,1);
|
||||
u2 = z(:,2);
|
||||
|
||||
figure;
|
||||
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
hold on;
|
||||
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
|
||||
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
|
||||
plot(t, 1e6*y2, 'DisplayName', 'Step 2');
|
||||
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
|
||||
legend('location', 'southeast');
|
||||
|
||||
% Step 3 - Shaping of $T$
|
||||
% <<sec:ex_shaping_T>>
|
||||
|
||||
% Finally, we want to limit the effect of the noise on the displacement output.
|
||||
|
||||
% To do so, $T$ is shaped such that its high frequency gain is reduced.
|
||||
|
||||
% This is done by increasing the high frequency gain of the weighting function $W_3$ until the $\mathcal{H}_\infty$ synthesis gives $\gamma \approx 1$.
|
||||
|
||||
% The final weighting function $W_3$ is defined as follows:
|
||||
|
||||
W3 = generateWeight('G0', 1e-1, ...
|
||||
'G1', 1e4, ...
|
||||
'Gc', 1, 'wc', 2*pi*70, ...
|
||||
'n', 2);
|
||||
|
||||
|
||||
|
||||
% The $\mathcal{H}_\infty$ synthesis is performed and $\gamma$ is closed to one.
|
||||
|
||||
Pw = blkdiag(W1, W2, W3, 1)*P;
|
||||
|
||||
K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% Test bounds: 0.622 <= gamma <= 1.62
|
||||
|
||||
% gamma X>=0 Y>=0 rho(XY)<1 p/f
|
||||
% 1.01e+00 -1.3e+07 # -6.4e-18 3.459e-04 f
|
||||
% 1.28e+00 0.0e+00 0.0e+00 5.369e-02 p
|
||||
% 1.13e+00 3.8e-15 0.0e+00 1.695e-01 p
|
||||
% 1.07e+00 -5.0e-08 -1.2e-19 1.622e+00 # f
|
||||
% 1.10e+00 0.0e+00 0.0e+00 3.147e-01 p
|
||||
% 1.08e+00 0.0e+00 0.0e+00 5.327e-01 p
|
||||
% 1.08e+00 3.4e-15 0.0e+00 8.052e-01 p
|
||||
|
||||
% Best performance (actual): 1.08
|
||||
% #+end_example
|
||||
|
||||
% The obtained closed-loop transfer functions are shown in Figure [[fig:ex_results_3]] and we can obverse that:
|
||||
% - The high frequency gain of $T$ is indeed reduced
|
||||
% - This comes as the expense of a large magnitude both $GS$ and $S$.
|
||||
% This means we will probably have slightly less good disturbance rejection and tracking performances.
|
||||
|
||||
|
||||
Z3 = lft(P, K3);
|
||||
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
|
||||
plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1b');
|
||||
plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 2');
|
||||
plot(freqs, abs(squeeze(freqresp(Z3(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 3');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W^{-1}|$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
title('S');
|
||||
legend('location', 'southeast');
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Z3(2,1), freqs, 'Hz'))));
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
title('GS');
|
||||
|
||||
ax3 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Z3(3,1), freqs, 'Hz'))));
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k--');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
title('T');
|
||||
|
||||
ax4 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(K3, freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
title('K');
|
||||
|
||||
linkaxes([ax1, ax2, ax3, ax4], 'x')
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
% #+name: fig:ex_results_3
|
||||
% #+caption: Obtained Shape Closed-Loop transfer functions
|
||||
% #+RESULTS:
|
||||
% [[file:figs/ex_results_3.png]]
|
||||
|
||||
% The time domain simulation signals are shown in Figure [[fig:ex_time_domain_3]].
|
||||
% We can indeed see a slightly less good disturbance rejection.
|
||||
% However, the vibrations induced by the sensor noise is well reduced.
|
||||
% This can be seen when zooming on the output signal in Figure [[fig:ex_time_domaim_3_zoom]].
|
||||
|
||||
|
||||
Z3sim = lft(Psim, K3);
|
||||
|
||||
z = lsim(Z3sim, [r;n;d], t);
|
||||
y3 = z(:,1);
|
||||
u3 = z(:,2);
|
||||
|
||||
figure;
|
||||
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
hold on;
|
||||
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
|
||||
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
|
||||
plot(t, 1e6*y2, 'DisplayName', 'Step 2');
|
||||
plot(t, 1e6*y3, 'DisplayName', 'Step 3');
|
||||
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
|
||||
legend('location', 'southeast');
|
||||
|
||||
|
||||
|
||||
% #+name: fig:ex_time_domain_3
|
||||
% #+caption: Time domain simulation results
|
||||
% #+RESULTS:
|
||||
% [[file:figs/ex_time_domain_3.png]]
|
||||
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
|
||||
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
|
||||
plot(t, 1e6*y2, 'DisplayName', 'Step 2');
|
||||
plot(t, 1e6*y3, 'DisplayName', 'Step 3');
|
||||
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
|
||||
legend('location', 'southeast');
|
||||
xlim([0.4, 0.5])
|
||||
Reference in new issue
Block a user