Add analysis about Mixed synthesis
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matlab/figs/4outputs_hinf_psd_cps2svg.pdf
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matlab/figs/charac_sensors_weights.pdf
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matlab/figs/mixed_syn_hinf_weight.pdf
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matlab/figs/mixed_synthesis_noise_uncertainty_sensors.pdf
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@ -4,7 +4,352 @@ clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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% First Basic Example with gain mismatch
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% Dynamical uncertainty of the individual sensors
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% Let say we want to merge two sensors:
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% - sensor 1 that has unknown dynamics above 10Hz: $|w_1(j\omega)| > 1$ for $\omega > 10\text{ Hz}$
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% - sensor 2 that has unknown dynamics below 1Hz and above 1kHz $|w_2(j\omega)| > 1$ for $\omega < 1\text{ Hz}$ and $\omega > 1\text{ kHz}$
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% We define the weights that are used to characterize the dynamic uncertainty of the sensors.
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freqs = logspace(-1, 3, 1000);
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omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
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w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
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w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 5000*2*pi; G0 = 1; Ginf = 50;
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w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
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% From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. [[fig:uncertainty_dynamics_sensors]] with the upper and lower bounds on the magnitude and on the phase.
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G1 = 1 + w1*ultidyn('Delta',[1 1]);
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G2 = 1 + w2*ultidyn('Delta',[1 1]);
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% Few random samples of the sensor dynamics are computed
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G1s = usample(G1, 10);
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G2s = usample(G2, 10);
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% We here compute the maximum and minimum phase of both sensors
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Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
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Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
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Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
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Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
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figure;
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% Magnitude
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ax1 = subaxis(2,1,1);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--');
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for i = 1:length(G1s)
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plot(freqs, abs(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
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plot(freqs, abs(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude');
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ylim([1e-1, 10]);
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hold off;
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% Phase
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ax2 = subaxis(2,1,2);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, Dphi1, '--');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, -Dphi1, '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, Dphi2, '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, -Dphi2, '--');
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for i = 1:length(G1s)
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plot(freqs, 180/pi*angle(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
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plot(freqs, 180/pi*angle(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
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end
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set(gca,'xscale','log');
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yticks(-180:90:180);
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ylim([-180 180]);
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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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% Weighting Function used to bound the super sensor uncertainty
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% Let's define $w_\phi(s)$ in order to bound the maximum allowed phase uncertainty $\Delta\phi_\text{max}$ of the super sensor dynamics.
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% The magnitude $|w_\phi(j\omega)|$ is shown in Fig. [[fig:magnitude_wphi]] and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. [[fig:maximum_wanted_phase_uncertainty]].
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Dphi = 20; % [deg]
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n = 4; w0 = 2*pi*900; G0 = 1/sin(Dphi*pi/180); Ginf = 1/100; Gc = 1;
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wphi = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (G0/Gc)^(1/n))/((1/Ginf)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (1/Gc)^(1/n)))^n;
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W1 = w1*wphi;
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W2 = w2*wphi;
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(wphi, freqs, 'Hz'))), '-', 'DisplayName', '$w_\phi(s)$');
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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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hold off;
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xlim([freqs(1), freqs(end)]);
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legend('location', 'northeast');
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% #+NAME: fig:magnitude_wphi
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% #+CAPTION: Magnitude of the weght $w_\phi(s)$ that is used to bound the uncertainty of the super sensor ([[./figs/magnitude_wphi.png][png]], [[./figs/magnitude_wphi.pdf][pdf]])
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% [[file:figs/magnitude_wphi.png]]
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% We here compute the wanted maximum and minimum phase of the super sensor
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Dphimax = 180/pi*asin(1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))));
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Dphimax(1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))) > 1) = 190;
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figure;
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hold on;
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plot(freqs, Dphimax, 'k--');
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plot(freqs, -Dphimax, 'k--');
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set(gca, 'XScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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hold off;
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xlim([freqs(1), freqs(end)]);
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ylim([-180 180]);
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yticks(-180:45:180);
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% #+NAME: fig:maximum_wanted_phase_uncertainty
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% #+CAPTION: Maximum wanted phase uncertainty using this weight ([[./figs/maximum_wanted_phase_uncertainty.png][png]], [[./figs/maximum_wanted_phase_uncertainty.pdf][pdf]])
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% [[file:figs/maximum_wanted_phase_uncertainty.png]]
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% The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. [[fig:upper_bounds_comp_filter_max_phase_uncertainty]].
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figure;
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hold on;
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '-', 'DisplayName', '$1/|w_1w_\phi|$');
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '-', 'DisplayName', '$1/|w_2w_\phi|$');
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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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hold off;
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xlim([freqs(1), freqs(end)]);
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legend('location', 'northeast');
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% $\mathcal{H}_\infty$ Synthesis
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% The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filters is shown in Fig. [[fig:h_infinity_robust_fusion]].
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% #+name: fig:h_infinity_robust_fusion
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% #+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
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% [[file:figs-tikz/h_infinity_robust_fusion.png]]
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% The generalized plant is defined below.
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P = [W1 -W1;
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0 W2;
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1 0];
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% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
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[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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% #+RESULTS:
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% #+begin_example
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% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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% Resetting value of Gamma min based on D_11, D_12, D_21 terms
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% Test bounds: 0.0447 < gamma <= 1.3318
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% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
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% 1.332 1.3e+01 -1.0e-14 1.3e+00 -2.6e-18 0.0000 p
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% 0.688 1.3e-11# ******** 1.3e+00 -6.7e-15 ******** f
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% 1.010 1.1e+01 -1.5e-14 1.3e+00 -2.5e-14 0.0000 p
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% 0.849 6.9e-11# ******** 1.3e+00 -2.3e-14 ******** f
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% 0.930 5.2e-12# ******** 1.3e+00 -6.1e-18 ******** f
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% 0.970 5.6e-11# ******** 1.3e+00 -2.3e-14 ******** f
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% 0.990 5.0e-11# ******** 1.3e+00 -1.7e-17 ******** f
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% 1.000 2.1e-10# ******** 1.3e+00 0.0e+00 ******** f
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% 1.005 1.9e-10# ******** 1.3e+00 -3.7e-14 ******** f
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% 1.008 1.1e+01 -9.1e-15 1.3e+00 0.0e+00 0.0000 p
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% 1.006 1.2e-09# ******** 1.3e+00 -6.9e-16 ******** f
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% 1.007 1.1e+01 -4.6e-15 1.3e+00 -1.8e-16 0.0000 p
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% Gamma value achieved: 1.0069
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% #+end_example
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% And $H_1(s)$ is defined as the complementary of $H_2(s)$.
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H1 = 1 - H2;
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% The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
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figure;
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ax1 = subplot(2,1,1);
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude');
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set(gca, 'XTickLabel',[]);
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legend('location', 'northeast');
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ax2 = subplot(2,1,2);
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hold on;
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
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hold off;
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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set(gca, 'XScale', 'log');
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yticks([-360:90:360]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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xticks([0.1, 1, 10, 100, 1000]);
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% Super sensor uncertainty
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% We can now compute the uncertainty of the super sensor. The result is shown in Fig. [[fig:super_sensor_uncertainty_bode_plot]].
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Gss = G1*H1 + G2*H2;
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Gsss = usample(Gss, 20);
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% We here compute the maximum and minimum phase of the super sensor
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Dphiss = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))));
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Dphiss(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190;
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figure;
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% Magnitude
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ax1 = subaxis(2,1,1);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S1');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S2');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
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plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--', 'DisplayName', 'Bounds - SS');
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plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(Gsss(1, 1, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'DisplayName', 'SS Dynamics');
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for i = 2:length(Gsss)
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plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'HandleVisibility', 'off');
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]);
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legend('location', 'southwest');
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ylabel('Magnitude');
|
||||
ylim([5e-2, 10]);
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
plot(freqs, Dphiss, 'k--');
|
||||
plot(freqs, -Dphiss, 'k--');
|
||||
for i = 1:length(Gsss)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
|
||||
end
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Super sensor noise
|
||||
% We now compute the obtain Power Spectral Density of the super sensor's noise.
|
||||
% The noise characteristics of both individual sensor are defined below.
|
||||
|
||||
|
||||
omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
|
||||
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
|
||||
|
||||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
|
||||
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
|
||||
|
||||
|
||||
|
||||
% The PSD of both sensor and of the super sensor is shown in Fig. [[fig:psd_sensors_hinf_synthesis]].
|
||||
% The CPS of both sensor and of the super sensor is shown in Fig. [[fig:cps_sensors_hinf_synthesis]].
|
||||
|
||||
|
||||
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
|
||||
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
|
||||
PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
|
||||
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
|
||||
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:psd_sensors_hinf_synthesis
|
||||
% #+CAPTION: Power Spectral Density of the obtained super sensor using the $\mathcal{H}_\infty$ synthesis ([[./figs/psd_sensors_hinf_synthesis.png][png]], [[./figs/psd_sensors_hinf_synthesis.pdf][pdf]])
|
||||
% [[file:figs/psd_sensors_hinf_synthesis.png]]
|
||||
|
||||
|
||||
CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
|
||||
CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
|
||||
CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
|
||||
plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
|
||||
hold off;
|
||||
xlim([2e-1, freqs(end)]);
|
||||
ylim([1e-10 1e-5]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
% First Basic Example with gain mismatch :noexport:
|
||||
% Let's consider two ideal sensors except one sensor has not an expected unity gain but a gain equal to $0.6$:
|
||||
% \begin{align*}
|
||||
% G_1(s) &= 1 \\
|
||||
@ -96,283 +441,3 @@ xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% Dynamical uncertainty of the individual sensors
|
||||
% We define the weights that are used to characterize the dynamic uncertainty of the sensors.
|
||||
|
||||
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
|
||||
|
||||
% From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. [[fig:uncertainty_dynamics_sensors]] with the upper and lower bounds on the magnitude and on the phase.
|
||||
|
||||
G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||||
|
||||
% Few random samples of the sensor dynamics are computed
|
||||
G1s = usample(G1, 10);
|
||||
G2s = usample(G2, 10);
|
||||
|
||||
% We here compute the maximum and minimum phase of both sensors
|
||||
Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
|
||||
Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
|
||||
Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
|
||||
Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--');
|
||||
for i = 1:length(G1s)
|
||||
plot(freqs, abs(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
|
||||
plot(freqs, abs(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Magnitude');
|
||||
ylim([1e-1, 10]);
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
for i = 1:length(G1s)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
|
||||
end
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% H-Infinity Synthesis
|
||||
% Let's define $w_\phi(s)$ in order to bound the maximum allowed phase uncertainty $\Delta\phi_\text{max}$ of the super sensor dynamics.
|
||||
|
||||
|
||||
Dphi = 20; % [deg]
|
||||
|
||||
n = 4; w0 = 2*pi*900; G0 = 1/sin(Dphi*pi/180); Ginf = 1/100; Gc = 1;
|
||||
wphi = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (G0/Gc)^(1/n))/((1/Ginf)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
W1 = w1*wphi;
|
||||
W2 = w2*wphi;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(wphi, freqs, 'Hz'))), '-', 'DisplayName', '$w_\phi(s)$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:magnitude_wphi
|
||||
% #+CAPTION: Magnitude of the weght $w_\phi(s)$ that is used to bound the uncertainty of the super sensor ([[./figs/magnitude_wphi.png][png]], [[./figs/magnitude_wphi.pdf][pdf]])
|
||||
% [[file:figs/magnitude_wphi.png]]
|
||||
|
||||
|
||||
% We here compute the wanted maximum and minimum phase of the super sensor
|
||||
Dphimax = 180/pi*asin(1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))));
|
||||
Dphimax(1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, Dphimax, 'k--');
|
||||
plot(freqs, -Dphimax, 'k--');
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([-180 180]);
|
||||
yticks(-180:45:180);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:maximum_wanted_phase_uncertainty
|
||||
% #+CAPTION: Maximum wanted phase uncertainty using this weight ([[./figs/maximum_wanted_phase_uncertainty.png][png]], [[./figs/maximum_wanted_phase_uncertainty.pdf][pdf]])
|
||||
% [[file:figs/maximum_wanted_phase_uncertainty.png]]
|
||||
|
||||
% The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. [[fig:upper_bounds_comp_filter_max_phase_uncertainty]].
|
||||
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '-', 'DisplayName', '$1/|w_1w_\phi|$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '-', 'DisplayName', '$1/|w_2w_\phi|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:upper_bounds_comp_filter_max_phase_uncertainty
|
||||
% #+CAPTION: Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz ([[./figs/upper_bounds_comp_filter_max_phase_uncertainty.png][png]], [[./figs/upper_bounds_comp_filter_max_phase_uncertainty.pdf][pdf]])
|
||||
% [[file:figs/upper_bounds_comp_filter_max_phase_uncertainty.png]]
|
||||
|
||||
% The $\mathcal{H}_\infty$ synthesis is performed using the defined weights and the obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
|
||||
|
||||
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
|
||||
|
||||
|
||||
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
% Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
% Test bounds: 0.0447 < gamma <= 1.3318
|
||||
|
||||
% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
|
||||
% 1.332 1.3e+01 -1.0e-14 1.3e+00 -2.6e-18 0.0000 p
|
||||
% 0.688 1.3e-11# ******** 1.3e+00 -6.7e-15 ******** f
|
||||
% 1.010 1.1e+01 -1.5e-14 1.3e+00 -2.5e-14 0.0000 p
|
||||
% 0.849 6.9e-11# ******** 1.3e+00 -2.3e-14 ******** f
|
||||
% 0.930 5.2e-12# ******** 1.3e+00 -6.1e-18 ******** f
|
||||
% 0.970 5.6e-11# ******** 1.3e+00 -2.3e-14 ******** f
|
||||
% 0.990 5.0e-11# ******** 1.3e+00 -1.7e-17 ******** f
|
||||
% 1.000 2.1e-10# ******** 1.3e+00 0.0e+00 ******** f
|
||||
% 1.005 1.9e-10# ******** 1.3e+00 -3.7e-14 ******** f
|
||||
% 1.008 1.1e+01 -9.1e-15 1.3e+00 0.0e+00 0.0000 p
|
||||
% 1.006 1.2e-09# ******** 1.3e+00 -6.9e-16 ******** f
|
||||
% 1.007 1.1e+01 -4.6e-15 1.3e+00 -1.8e-16 0.0000 p
|
||||
|
||||
% Gamma value achieved: 1.0069
|
||||
% #+end_example
|
||||
|
||||
|
||||
H1 = 1 - H2;
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
|
||||
% Super sensor uncertainty
|
||||
% We can now compute the uncertainty of the super sensor. The result is shown in Fig. [[fig:super_sensor_uncertainty_bode_plot]].
|
||||
|
||||
|
||||
Gss = G1*H1 + G2*H2;
|
||||
|
||||
Gsss = usample(Gss, 20);
|
||||
|
||||
% We here compute the maximum and minimum phase of the super sensor
|
||||
Dphiss = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))));
|
||||
Dphiss(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--');
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--');
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--');
|
||||
for i = 1:length(Gsss)
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Magnitude');
|
||||
ylim([1e-1, 10]);
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
% plot(freqs, Dphimax, 'r-');
|
||||
% plot(freqs, -Dphimax, 'r-');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
plot(freqs, Dphiss, 'k--');
|
||||
plot(freqs, -Dphiss, 'k--');
|
||||
for i = 1:length(Gsss)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
|
||||
end
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
@ -1,53 +0,0 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
% Matlab
|
||||
|
||||
omega0 = 1*2*pi; % [rad/s]
|
||||
tau = 1/omega0; % [s]
|
||||
|
||||
% From cite:corke04_inert_visual_sensin_system_small_auton_helic
|
||||
HL1 = 1/(s/omega0 + 1); HH1 = s/omega0/(s/omega0 + 1);
|
||||
|
||||
% From cite:jensen13_basic_uas
|
||||
HL2 = (2*omega0*s + omega0^2)/(s+omega0)^2; HH2 = s^2/(s+omega0)^2;
|
||||
|
||||
% From cite:shaw90_bandw_enhan_posit_measur_using_measur_accel
|
||||
HL3 = (3*tau*s + 1)/(tau*s + 1)^3; HH3 = (tau^3*s^3 + 3*tau^2*s^2)/(tau*s + 1)^3;
|
||||
|
||||
freqs = logspace(-1, 1, 1000);
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(HH1, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(HL1, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(HH2, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(HL2, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',3); plot(freqs, abs(squeeze(freqresp(HH3, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',3); plot(freqs, abs(squeeze(freqresp(HL3, freqs, 'Hz'))));
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Magnitude');
|
||||
hold off;
|
||||
ylim([1e-2 2]);
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(HH1, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(HL1, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(HH2, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(HL2, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',3); plot(freqs, 180/pi*angle(squeeze(freqresp(HH3, freqs, 'Hz'))));
|
||||
set(gca,'ColorOrderIndex',3); plot(freqs, 180/pi*angle(squeeze(freqresp(HL3, freqs, 'Hz'))));
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
359
matlab/matlab/hinf_syn_perf_robust.m
Normal file
359
matlab/matlab/hinf_syn_perf_robust.m
Normal file
@ -0,0 +1,359 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
% Dynamical uncertainty and Noise level of the individual sensors
|
||||
% Uncertainty on the individual sensors:
|
||||
|
||||
omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
|
||||
|
||||
% Noise level of the individual sensors:
|
||||
|
||||
omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
|
||||
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
|
||||
|
||||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
|
||||
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
|
||||
|
||||
figure;
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2(j\omega)|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(w1, freqs, 'Hz'))), '-', 'DisplayName', '$|w_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(w2, freqs, 'Hz'))), '-', 'DisplayName', '$|w_2(j\omega)|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% Weights for uncertainty and performance
|
||||
% We design weights that are used to describe the wanted upper bound on the super sensor's noise and super sensor's uncertainty.
|
||||
|
||||
% Weight on the uncertainty:
|
||||
|
||||
n = 4; w0 = 2*pi*500; G0 = 6; G1 = 1; Gc = 1.1;
|
||||
H = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
Wu = 0.2*(s+3.142e04)/(s+628.3)/H;
|
||||
|
||||
|
||||
|
||||
% Weight on the performance:
|
||||
|
||||
n = 1; w0 = 2*pi*9; A = 6;
|
||||
a = sqrt(2*A^(2/n) - 1 + 2*A^(1/n)*sqrt(A^(2/n) - 1));
|
||||
G = ((1 + s/(w0/a))*(1 + s/(w0*a))/(1 + s/w0)^2)^n;
|
||||
|
||||
n = 2; w0 = 2*pi*9; G0 = 1e-2; G1 = 1; Gc = 5e-1;
|
||||
G2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
Wp = inv(G2)*inv(G)*inv(N2);
|
||||
|
||||
|
||||
|
||||
% The noise and uncertainty weights of the individual sensors and the asked noise/uncertainty of the super sensor are displayed in Fig. [[fig:charac_sensors_weights]].
|
||||
|
||||
figure;
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2(j\omega)|$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(Wp, freqs, 'Hz'))), 'k--', 'DisplayName', '$|w_r(j\omega)|^{-1}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(w1, freqs, 'Hz'))), '-', 'DisplayName', '$|w_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(w2, freqs, 'Hz'))), '-', 'DisplayName', '$|w_2(j\omega)|$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(Wu, freqs, 'Hz'))), 'k--', 'DisplayName', '$|w_u(j\omega)|^{-1}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:charac_sensors_weights
|
||||
% #+CAPTION: Upper bounds on the super sensor's noise and super sensor's uncertainty ([[./figs/charac_sensors_weights.png][png]], [[./figs/charac_sensors_weights.pdf][pdf]])
|
||||
% [[file:figs/charac_sensors_weights.png]]
|
||||
|
||||
|
||||
% The corresponding maximum norms of the filters to have the perf/robust asked are shown in Fig. [[fig:upper_bound_complementary_filters_perf_robust]].
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(N1*Wp, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1| - perf$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(N2*Wp, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2| - perf$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(w1*Wu, freqs, 'Hz'))), '--', 'DisplayName', '$|N_1| - robu$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(w2*Wu, freqs, 'Hz'))), '--', 'DisplayName', '$|N_2| - robu$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
% H-infinity synthesis with 4 outputs corresponding to the 4 weights
|
||||
% We do the $\mathcal{H}_\infty$ synthesis with 4 weights and 4 outputs.
|
||||
|
||||
% \begin{equation*}
|
||||
% \left\| \begin{matrix}
|
||||
% W_{1p}(s) (1 - N_2(s)) \\
|
||||
% W_{2p}(s) N_2(s) \\
|
||||
% W_{1u}(s) (1 - N_2(s)) \\
|
||||
% W_{2u}(s) N_2(s)
|
||||
% \end{matrix} \right\|_\infty < 1
|
||||
% \end{equation*}
|
||||
|
||||
|
||||
|
||||
W1p = N1*Wp/(1+s/2/pi/1000); % Used to render W1p proper
|
||||
W2p = N2*Wp;
|
||||
W1u = w1*Wu;
|
||||
W2u = w2*Wu;
|
||||
|
||||
P = [W1p -W1p;
|
||||
0 W2p;
|
||||
W1u -W1u;
|
||||
0 W2u;
|
||||
1 0];
|
||||
|
||||
|
||||
|
||||
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
% Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
% Test bounds: 1.4139 < gamma <= 65.6899
|
||||
|
||||
% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
|
||||
% 65.690 1.3e+00 -6.7e-15 1.3e+00 -4.5e-13 0.0000 p
|
||||
% 33.552 1.3e+00 -9.4e-15 1.3e+00 -3.7e-14 0.0000 p
|
||||
% 17.483 1.3e+00 -5.6e-16 1.3e+00 -4.8e-13 0.0000 p
|
||||
% 9.448 1.3e+00 -3.2e-15 1.3e+00 -1.2e-13 0.0000 p
|
||||
% 5.431 1.3e+00 -2.3e-16 1.3e+00 -3.6e-13 0.0000 p
|
||||
% 3.422 1.3e+00 -7.3e-16 1.3e+00 -2.6e-15 0.0000 p
|
||||
% 2.418 1.3e+00 9.3e-17 1.3e+00 -3.0e-14 0.0000 p
|
||||
% 1.916 1.3e+00 2.4e-17 1.3e+00 -2.2e-14 0.0000 p
|
||||
% 1.665 1.3e+00 -2.5e-16 1.3e+00 -2.1e-14 0.0000 p
|
||||
% 1.539 1.3e+00 -6.9e-15 1.3e+00 -5.3e-14 0.0000 p
|
||||
% 1.477 1.3e+00 -2.1e-14 1.3e+00 -2.3e-13 0.0000 p
|
||||
% 1.445 1.3e+00 -1.3e-16 1.3e+00 -2.6e-15 0.0000 p
|
||||
% 1.430 1.3e+00 -4.9e-13 1.3e+00 -2.2e-13 0.0000 p
|
||||
% 1.422 1.3e+00 -1.2e+08# 1.3e+00 -2.6e-13 0.0000 f
|
||||
% 1.426 1.3e+00 -6.3e-13 1.3e+00 -3.3e-14 0.0000 p
|
||||
% 1.424 1.3e+00 -3.4e+08# 1.3e+00 -4.5e-14 0.0000 f
|
||||
% 1.425 1.3e+00 -1.7e+09# 1.3e+00 -5.2e-13 0.0000 f
|
||||
|
||||
% Gamma value achieved: 1.4256
|
||||
% #+end_example
|
||||
|
||||
|
||||
H1 = 1 - H2;
|
||||
|
||||
|
||||
|
||||
% The obtained complementary filters with the upper bounds are shown in Fig. [[fig:hinf_result_comp_filters_4_outputs]].
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1p, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1| - perf$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2p, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2| - perf$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1u, freqs, 'Hz'))), '--', 'DisplayName', '$|N_1| - robu$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2u, freqs, 'Hz'))), '--', 'DisplayName', '$|N_2| - robu$');
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|H_1|$');
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), 'k--', 'DisplayName', '$|H_2|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:hinf_result_comp_filters_4_outputs
|
||||
% #+CAPTION: caption ([[./figs/hinf_result_comp_filters_4_outputs.png][png]], [[./figs/hinf_result_comp_filters_4_outputs.pdf][pdf]])
|
||||
% [[file:figs/hinf_result_comp_filters_4_outputs.png]]
|
||||
|
||||
|
||||
|
||||
|
||||
figure;
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2(j\omega)|$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(Wp, freqs, 'Hz'))), 'k--', 'DisplayName', '$|w_r(j\omega)|^{-1}$');
|
||||
plot(freqs, sqrt(abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2), 'k-', 'DisplayName', '$|N_{ss}(j\omega)|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(w1, freqs, 'Hz'))), '-', 'DisplayName', '$|w_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(w2, freqs, 'Hz'))), '-', 'DisplayName', '$|w_2(j\omega)|$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(Wu, freqs, 'Hz'))), 'k--', 'DisplayName', '$|w_u(j\omega)|^{-1}$');
|
||||
plot(freqs, abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))), 'k-', 'DisplayName', '$|w_{ss}(j\omega)|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:upper_bounds_perf_robust_result_4_outputs
|
||||
% #+CAPTION: Obtained PSD and uncertainty with the corresponding upper bounds ([[./figs/upper_bounds_perf_robust_result_4_outputs.png][png]], [[./figs/upper_bounds_perf_robust_result_4_outputs.pdf][pdf]])
|
||||
% [[file:figs/upper_bounds_perf_robust_result_4_outputs.png]]
|
||||
|
||||
|
||||
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
|
||||
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
|
||||
PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
|
||||
|
||||
CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
|
||||
CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
|
||||
CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
|
||||
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
|
||||
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
|
||||
plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
|
||||
hold off;
|
||||
ylim([1e-10 1e-5]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:4outputs_hinf_psd_cps2svg
|
||||
% #+CAPTION: PSD and CPS ([[./figs/4outputs_hinf_psd_cps2svg.png][png]], [[./figs/4outputs_hinf_psd_cps2svg.pdf][pdf]])
|
||||
% [[file:figs/4outputs_hinf_psd_cps2svg.png]]
|
||||
|
||||
|
||||
|
||||
G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||||
|
||||
Gss = G1*H1 + G2*H2;
|
||||
Gsss = usample(Gss, 20);
|
||||
|
||||
% We here compute the maximum and minimum phase of the super sensor
|
||||
Dphiss = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))));
|
||||
Dphiss(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
% We here compute the maximum and minimum phase of both sensors
|
||||
Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
|
||||
Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
|
||||
Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
|
||||
Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S1');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S2');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--', 'DisplayName', 'Bounds - SS');
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(1, 1, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'DisplayName', 'SS Dynamics');
|
||||
for i = 2:length(Gsss)
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southwest');
|
||||
ylabel('Magnitude');
|
||||
ylim([5e-2, 10]);
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
plot(freqs, Dphiss, 'k--');
|
||||
plot(freqs, -Dphiss, 'k--');
|
||||
for i = 1:length(Gsss)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
|
||||
end
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
351
matlab/matlab/mixed_synthesis_sensor_fusion.m
Normal file
351
matlab/matlab/mixed_synthesis_sensor_fusion.m
Normal file
@ -0,0 +1,351 @@
|
||||
%% Clear Workspace and Close figures
|
||||
clear; close all; clc;
|
||||
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
freqs = logspace(-1, 3, 1000);
|
||||
|
||||
% Noise characteristics and Uncertainty of the individual sensors
|
||||
% We define the weights that are used to characterize the dynamic uncertainty of the sensors. This will be used for the $\mathcal{H}_\infty$ part of the synthesis.
|
||||
|
||||
omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
|
||||
|
||||
% We define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$. This will be used for the $\mathcal{H}_2$ part of the synthesis.
|
||||
|
||||
omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
|
||||
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
|
||||
|
||||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
|
||||
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
|
||||
|
||||
|
||||
|
||||
% Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. [[fig:mixed_synthesis_noise_uncertainty_sensors]].
|
||||
|
||||
|
||||
figure;
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2(j\omega)|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(w1, freqs, 'Hz'))), '-', 'DisplayName', '$|w_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(w2, freqs, 'Hz'))), '-', 'DisplayName', '$|w_2(j\omega)|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% Weighting Functions on the uncertainty of the super sensor
|
||||
% We design weights for the $\mathcal{H}_\infty$ part of the synthesis in order to limit the dynamical uncertainty of the super sensor.
|
||||
% The maximum wanted multiplicative uncertainty is shown in Fig. [[fig:mixed_syn_hinf_weight]]. The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
|
||||
|
||||
|
||||
n = 4; w0 = 2*pi*900; G0 = 9; G1 = 1; Gc = 1.1;
|
||||
H = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
wphi = 0.2*(s+3.142e04)/(s+628.3)/H;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(w1, freqs, 'Hz'))), '-', 'DisplayName', '$|w_1(j\omega)|$');
|
||||
plot(freqs, abs(squeeze(freqresp(w2, freqs, 'Hz'))), '-', 'DisplayName', '$|w_2(j\omega)|$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))), 'k--', 'DisplayName', '$|w_u(j\omega)|^{-1}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:mixed_syn_hinf_weight
|
||||
% #+CAPTION: Wanted maximum module uncertainty of the super sensor ([[./figs/mixed_syn_hinf_weight.png][png]], [[./figs/mixed_syn_hinf_weight.pdf][pdf]])
|
||||
% [[file:figs/mixed_syn_hinf_weight.png]]
|
||||
|
||||
% The equivalent Magnitude and Phase uncertainties are shown in Fig. [[fig:mixed_syn_objective_hinf]].
|
||||
|
||||
|
||||
G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||||
|
||||
% Few random samples of the sensor dynamics are computed
|
||||
G1s = usample(G1, 10);
|
||||
G2s = usample(G2, 10);
|
||||
|
||||
% We here compute the maximum and minimum phase of both sensors
|
||||
Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
|
||||
Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
|
||||
Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
|
||||
Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
% We here compute the wanted maximum and minimum phase of the super sensor
|
||||
Dphimax = 180/pi*asin(1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))));
|
||||
Dphimax(1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S1');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S2');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
plot(freqs, 1 + 1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))), 'k--', 'DisplayName', 'Synthesis Obj.');
|
||||
plot(freqs, max(1 - 1./abs(squeeze(freqresp(wphi, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off');
|
||||
for i = 1:length(G1s)
|
||||
plot(freqs, abs(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4], 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4], 'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Magnitude');
|
||||
ylim([1e-1, 10]);
|
||||
hold off;
|
||||
legend('location', 'southwest');
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
for i = 1:length(G1s)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
|
||||
end
|
||||
plot(freqs, Dphimax, 'k--');
|
||||
plot(freqs, -Dphimax, 'k--');
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Mixed Synthesis Architecture
|
||||
% The synthesis architecture that is used here is shown in Fig. [[fig:mixed_h2_hinf_synthesis]].
|
||||
|
||||
% The controller $K$ is synthesized such that it:
|
||||
% - Keeps the $\mathcal{H}_\infty$ norm $G$ of the transfer function from $w$ to $z_\infty$ bellow some specified value
|
||||
% - Keeps the $\mathcal{H}_2$ norm $H$ of the transfer function from $w$ to $z_2$ bellow some specified value
|
||||
% - Minimizes a trade-off criterion of the form $W_1 G^2 + W_2 H^2$ where $W_1$ and $W_2$ are specified values
|
||||
|
||||
% #+name: fig:mixed_h2_hinf_synthesis
|
||||
% #+caption: Mixed H2/H-Infinity Synthesis
|
||||
% [[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
|
||||
|
||||
% Here, we define $P$ such that:
|
||||
% \begin{align*}
|
||||
% \left\| \frac{z_\infty}{w} \right\|_\infty &= \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty \\
|
||||
% \left\| \frac{z_2}{w} \right\|_2 &= \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2
|
||||
% \end{align*}
|
||||
|
||||
% Then:
|
||||
% - we specify the maximum value for the $\mathcal{H}_\infty$ norm between $w$ and $z_\infty$ to be $1$
|
||||
% - we don't specify any maximum value for the $\mathcal{H}_2$ norm between $w$ and $z_2$
|
||||
% - we choose $W_1 = 0$ and $W_2 = 1$ such that the objective is to minimize the $\mathcal{H}_2$ norm between $w$ and $z_2$
|
||||
|
||||
% The synthesis objective is to have:
|
||||
% \[ \left\| \frac{z_\infty}{w} \right\|_\infty = \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty < 1 \]
|
||||
% and to minimize:
|
||||
% \[ \left\| \frac{z_2}{w} \right\|_2 = \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2 \]
|
||||
% which is what we wanted.
|
||||
|
||||
% We define the generalized plant that will be used for the mixed synthesis.
|
||||
|
||||
W1u = ss(w1*wphi); W2u = ss(w2*wphi); % Weight on the uncertainty
|
||||
W1n = ss(N1); W2n = ss(N2); % Weight on the noise
|
||||
|
||||
P = [W1u -W1u;
|
||||
0 W2u;
|
||||
W1n -W1n;
|
||||
0 W2n;
|
||||
1 0];
|
||||
|
||||
% Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis
|
||||
% The mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis is performed below.
|
||||
|
||||
Nmeas = 1; Ncon = 1; Nz2 = 2;
|
||||
|
||||
[H2,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
|
||||
|
||||
H1 = 1 - H2;
|
||||
|
||||
|
||||
|
||||
% The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_synthesis]].
|
||||
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1u, freqs, 'Hz'))), '--', 'DisplayName', '$W_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2u, freqs, 'Hz'))), '--', 'DisplayName', '$W_2$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([1e-3, 2]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
|
||||
% Obtained Super Sensor's noise
|
||||
% The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_sensor_mixed_syn]] and Fig. [[fig:cps_super_sensor_mixed_syn]] respectively.
|
||||
|
||||
|
||||
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
|
||||
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
|
||||
PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
|
||||
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
|
||||
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:psd_super_sensor_mixed_syn
|
||||
% #+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/psd_super_sensor_mixed_syn.png][png]], [[./figs/psd_super_sensor_mixed_syn.pdf][pdf]])
|
||||
% [[file:figs/psd_super_sensor_mixed_syn.png]]
|
||||
|
||||
|
||||
|
||||
CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
|
||||
CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
|
||||
CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
|
||||
plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
|
||||
hold off;
|
||||
xlim([2e-1, freqs(end)]);
|
||||
ylim([1e-10 1e-5]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
% Obtained Super Sensor's Uncertainty
|
||||
% The uncertainty on the super sensor's dynamics is shown in Fig. [[]].
|
||||
|
||||
|
||||
G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||||
|
||||
Gss = G1*H1 + G2*H2;
|
||||
Gsss = usample(Gss, 20);
|
||||
|
||||
% We here compute the maximum and minimum phase of the super sensor
|
||||
Dphiss = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))));
|
||||
Dphiss(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
% We here compute the maximum and minimum phase of both sensors
|
||||
Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
|
||||
Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
|
||||
Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
|
||||
Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S1');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S2');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--', 'DisplayName', 'Bounds - SS');
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(1, 1, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'DisplayName', 'SS Dynamics');
|
||||
for i = 2:length(Gsss)
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southwest');
|
||||
ylabel('Magnitude');
|
||||
ylim([5e-2, 10]);
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
plot(freqs, Dphiss, 'k--');
|
||||
plot(freqs, -Dphiss, 'k--');
|
||||
for i = 1:length(Gsss)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
|
||||
end
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
@ -41,6 +41,17 @@ legend('location', 'northeast');
|
||||
% #+caption: $\mathcal{H}_2$ Synthesis - Generalized plant used for the optimal generation of complementary filters
|
||||
% [[file:figs-tikz/h_infinity_optimal_comp_filters.png]]
|
||||
|
||||
% \begin{equation*}
|
||||
% \begin{pmatrix}
|
||||
% z \\ v
|
||||
% \end{pmatrix} = \begin{pmatrix}
|
||||
% 0 & N_2 & 1 \\
|
||||
% N_1 & -N_2 & 0
|
||||
% \end{pmatrix} \begin{pmatrix}
|
||||
% w_1 \\ w_2 \\ u
|
||||
% \end{pmatrix}
|
||||
% \end{equation*}
|
||||
|
||||
% The transfer function from $[n_1, n_2]$ to $\hat{x}$ is:
|
||||
% \[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \]
|
||||
% If we define $H_2 = 1 - H_1$, we obtain:
|
||||
@ -382,6 +393,101 @@ xticks([0.1, 1, 10, 100, 1000]);
|
||||
PSD_Hb = abs(squeeze(freqresp(N1*H1b, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2b, freqs, 'Hz'))).^2;
|
||||
CPS_Hb = 1/pi*cumtrapz(2*pi*freqs, PSD_Hb);
|
||||
|
||||
% H-Infinity Synthesis - method C
|
||||
|
||||
Wp = 0.56*(inv(N1)+inv(N2))/(1 + s/2/pi/1000);
|
||||
|
||||
W1c = N1*Wp;
|
||||
W2c = N2*Wp;
|
||||
|
||||
P = [W1c -W1c;
|
||||
0 W2c;
|
||||
1 0];
|
||||
|
||||
|
||||
|
||||
% And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
|
||||
[H2c, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
|
||||
|
||||
% #+RESULTS:
|
||||
% #+begin_example
|
||||
% [H2c, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
% Test bounds: 0.0000 < gamma <= 36.7543
|
||||
|
||||
% gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
|
||||
% 36.754 5.7e+00 -1.0e-13 6.3e+00 -6.2e-25 0.0000 p
|
||||
% 18.377 5.7e+00 -1.4e-12 6.3e+00 -1.8e-13 0.0000 p
|
||||
% 9.189 5.7e+00 -4.3e-13 6.3e+00 -4.7e-15 0.0000 p
|
||||
% 4.594 5.7e+00 -9.4e-13 6.3e+00 -4.7e-15 0.0000 p
|
||||
% 2.297 5.7e+00 -1.3e-16 6.3e+00 -6.8e-14 0.0000 p
|
||||
% 1.149 5.7e+00 -1.6e-17 6.3e+00 -1.5e-15 0.0000 p
|
||||
% 0.574 5.7e+00 -5.2e+02# 6.3e+00 -5.9e-14 0.0000 f
|
||||
% 0.861 5.7e+00 -3.1e+04# 6.3e+00 -3.8e-14 0.0000 f
|
||||
% 1.005 5.7e+00 -1.6e-12 6.3e+00 -1.1e-14 0.0000 p
|
||||
% 0.933 5.7e+00 -1.1e+05# 6.3e+00 -7.2e-14 0.0000 f
|
||||
% 0.969 5.7e+00 -3.3e+05# 6.3e+00 -5.6e-14 0.0000 f
|
||||
% 0.987 5.7e+00 -1.2e+06# 6.3e+00 -4.5e-15 0.0000 f
|
||||
% 0.996 5.7e+00 -6.5e-16 6.3e+00 -1.7e-15 0.0000 p
|
||||
% 0.992 5.7e+00 -2.9e+06# 6.3e+00 -6.1e-14 0.0000 f
|
||||
% 0.994 5.7e+00 -9.7e+06# 6.3e+00 -3.0e-16 0.0000 f
|
||||
% 0.995 5.7e+00 -8.0e-10 6.3e+00 -1.9e-13 0.0000 p
|
||||
% 0.994 5.7e+00 -2.3e+07# 6.3e+00 -4.3e-14 0.0000 f
|
||||
|
||||
% Gamma value achieved: 0.9949
|
||||
% #+end_example
|
||||
|
||||
|
||||
H1c = 1 - H2c;
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1c, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2c, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1c, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2c, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([5e-4, 20]);
|
||||
legend('location', 'northeast');
|
||||
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1c, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2c, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
yticks([-360:90:360]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
|
||||
|
||||
|
||||
% #+NAME: fig:weights_comp_filters_Hinfc
|
||||
% #+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfc.png][png]], [[./figs/weights_comp_filters_Hinfc.pdf][pdf]])
|
||||
% [[file:figs/weights_comp_filters_Hinfc.png]]
|
||||
|
||||
|
||||
PSD_Hc = abs(squeeze(freqresp(N1*H1c, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2c, freqs, 'Hz'))).^2;
|
||||
CPS_Hc = 1/pi*cumtrapz(2*pi*freqs, PSD_Hc);
|
||||
|
||||
|
||||
|
||||
% #+name: tab:rms_results
|
||||
@ -394,6 +500,7 @@ CPS_Hb = 1/pi*cumtrapz(2*pi*freqs, PSD_Hb);
|
||||
% | H2 Fusion | 1.2e-04 |
|
||||
% | H-Infinity a | 2.4e-04 |
|
||||
% | H-Infinity b | 1.4e-04 |
|
||||
% | H-Infinity c | 2.2e-04 |
|
||||
|
||||
|
||||
|
||||
@ -401,9 +508,10 @@ figure;
|
||||
hold on;
|
||||
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
|
||||
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
|
||||
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
|
||||
plot(freqs, PSD_Ha, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},a}$');
|
||||
plot(freqs, PSD_Hb, 'k-.', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},b}$');
|
||||
plot(freqs, PSD_H2, 'r-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
|
||||
plot(freqs, PSD_Ha, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},a}$');
|
||||
plot(freqs, PSD_Hb, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},b}$');
|
||||
plot(freqs, PSD_Hc, 'k-.', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},c}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
|
||||
hold off;
|
||||
@ -421,12 +529,104 @@ figure;
|
||||
hold on;
|
||||
plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
|
||||
plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
plot(freqs, CPS_Ha, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, a}} = %.1e$', sqrt(CPS_Ha(end))));
|
||||
plot(freqs, CPS_Hb, 'k-.', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, b}} = %.1e$', sqrt(CPS_Hb(end))));
|
||||
plot(freqs, CPS_H2, 'r-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
plot(freqs, CPS_Ha, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, a}} = %.1e$', sqrt(CPS_Ha(end))));
|
||||
plot(freqs, CPS_Hb, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, b}} = %.1e$', sqrt(CPS_Hb(end))));
|
||||
plot(freqs, CPS_Hc, 'k-.', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, c}} = %.1e$', sqrt(CPS_Hc(end))));
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
|
||||
hold off;
|
||||
xlim([2e-1, freqs(end)]);
|
||||
ylim([1e-10 1e-5]);
|
||||
legend('location', 'southeast');
|
||||
|
||||
% Obtained Super Sensor's noise uncertainty
|
||||
% We would like to verify if the obtained sensor fusion architecture is robust to change in the sensor dynamics.
|
||||
|
||||
% To study the dynamical uncertainty on the super sensor, we defined some multiplicative uncertainty on both sensor dynamics.
|
||||
% Two weights $w_1(s)$ and $w_2(s)$ are used to described the amplitude of the dynamical uncertainty.
|
||||
|
||||
|
||||
omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
|
||||
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
|
||||
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
|
||||
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
|
||||
|
||||
|
||||
|
||||
% The sensor uncertain models are defined below.
|
||||
|
||||
G1 = 1 + w1*ultidyn('Delta',[1 1]);
|
||||
G2 = 1 + w2*ultidyn('Delta',[1 1]);
|
||||
|
||||
% We here compute the maximum and minimum phase of both sensors
|
||||
Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
|
||||
Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
|
||||
Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
|
||||
Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
|
||||
|
||||
% The super sensor uncertain model is defined below using the complementary filters obtained with the $\mathcal{H}_2$ synthesis.
|
||||
% The dynamical uncertainty bounds of the super sensor is shown in Fig. [[fig:uncertainty_super_sensor_H2_syn]].
|
||||
% Right Half Plane zero might be introduced in the super sensor dynamics which will render the feedback system unstable.
|
||||
|
||||
|
||||
Gss = G1*H1 + G2*H2;
|
||||
|
||||
Gsss = usample(Gss, 20);
|
||||
|
||||
% We here compute the maximum and minimum phase of the super sensor
|
||||
Dphiss = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))));
|
||||
Dphiss(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subaxis(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S1');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S2');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off');
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(w1*H1, freqs, 'Hz'))) + abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))), 'k--', 'DisplayName', 'Bounds - SS');
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1, freqs, 'Hz'))) - abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off');
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(1, 1, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'DisplayName', 'SS Dynamics');
|
||||
for i = 2:length(Gsss)
|
||||
plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2], 'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
legend('location', 'southwest');
|
||||
ylabel('Magnitude');
|
||||
ylim([5e-2, 10]);
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subaxis(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, -Dphi1, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, Dphi2, '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, -Dphi2, '--');
|
||||
plot(freqs, Dphiss, 'k--');
|
||||
plot(freqs, -Dphiss, 'k--');
|
||||
for i = 1:length(Gsss)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
|
||||
end
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
Loading…
Reference in New Issue
Block a user