First analysis of mixed H2/Hinf synthesis
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@ -1048,8 +1048,13 @@ The obtained upper bounds on the complementary filters in order to limit the pha
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#+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]])
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[[file:figs/upper_bounds_comp_filter_max_phase_uncertainty.png]]
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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]].
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The $\mathcal{H}_\infty$ synthesis architecture used 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/h_infinity_robust_fusion.png]]
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The generalized plant is defined below.
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#+begin_src matlab
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P = [W1 -W1;
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0 W2;
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@ -1089,6 +1094,8 @@ Test bounds: 0.0447 < gamma <= 1.3318
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H1 = 1 - H2;
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#+end_src
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The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
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#+begin_src matlab :exports none
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figure;
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@ -1214,6 +1221,327 @@ We can now compute the uncertainty of the super sensor. The result is shown in F
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We here just used very wimple weights. We could shape the dynamical uncertainty of the super sensor by using more complex weights.
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We could for instance ask for less uncertainty at low frequency.
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* Optimal Sensor Fusion - Mixed Synthesis
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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freqs = logspace(-1, 3, 1000);
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#+end_src
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** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis - Introduction
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The goal is to design complementary filters such that:
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- the maximum uncertainty on the super sensor is bounded
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- the RMS value of the super sensor noise is minimized
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To do so, we can use the Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis.
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The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/robust/ref/h2hinfsyn.html][doc]]).
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** Definition of the weights
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We define the weights that are used to characterize the dynamic uncertainty of the sensors.
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#+begin_src matlab
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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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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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#+end_src
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We define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$.
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#+begin_src matlab
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omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
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omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
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N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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#+end_src
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We define the generalized plant that will be used for the mixed synthesis.
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#+begin_src matlab
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P = [W1 -W1;
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0 W2;
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N1 -N1;
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0 N2;
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1 0];
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P = ss(P);
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#+end_src
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** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis
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#+begin_src matlab :results output replace :exports both
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Nmeas = 1; Ncon = 1; Nz2 = 2;
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[K,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [1, 10], 'H2MAX', 2, 'HINFMAX', 2, 'DKMAX', 0, 'TOL', 0.001, 'DISPLAY', 'on');
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#+end_src
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#+RESULTS:
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#+begin_example
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Nmeas = 1; Ncon = 1; Nz2 = 2;
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[K,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [1, 10], 'H2MAX', 2, 'HINFMAX', 2, 'DKMAX', 0, 'TOL', 0.001, 'DISPLAY', 'on');
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Optimization of 1.000 * G^2 + 10.000 * H^2 :
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----------------------------------------------
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Solver for linear objective minimization under LMI constraints
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Iterations : Best objective value so far
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27 39.619680
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28 36.192540
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29 35.073373
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30 35.073373
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31 33.083063
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32 33.083063
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33 33.083063
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34 28.479232
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35 28.479232
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36 28.479232
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37 26.389647
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38 23.237872
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39 17.636707
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40 13.600053
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41 10.293603
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42 9.361705
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43 9.361705
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44 9.361705
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,* switching to QR
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45 9.361705
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46 8.891291
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47 8.891291
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48 8.891291
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49 6.892529
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50 6.892529
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51 5.425127
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52 5.425127
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53 3.499118
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54 3.499118
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55 3.249775
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56 3.249775
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57 2.893020
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58 2.893020
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,*** new lower bound: 1.803459
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59 2.583410
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,*** new lower bound: 1.976383
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60 2.513031
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,*** new lower bound: 2.027973
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61 2.441505
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,*** new lower bound: 2.067580
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62 2.377727
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,*** new lower bound: 2.099478
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63 2.342173
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,*** new lower bound: 2.125297
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64 2.315672
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,*** new lower bound: 2.146487
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65 2.295832
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,*** new lower bound: 2.163923
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66 2.280942
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,*** new lower bound: 2.239118
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67 2.265181
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68 2.261547
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69 2.259300
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,*** new lower bound: 2.240010
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70 2.258097
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,*** new lower bound: 2.241899
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71 2.257082
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,*** new lower bound: 2.243477
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72 2.256225
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,*** new lower bound: 2.244794
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73 2.255501
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,*** new lower bound: 2.245895
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74 2.254596
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,*** new lower bound: 2.246815
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75 2.254112
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,*** new lower bound: 2.247580
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76 2.253705
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,*** new lower bound: 2.248219
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77 2.253196
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,*** new lower bound: 2.251056
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Result: feasible solution of required accuracy
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best objective value: 2.253196
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guaranteed relative accuracy: 9.50e-04
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f-radius saturation: 30.962% of R = 1.00e+08
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Guaranteed Hinf performance: 1.01e+00
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Guaranteed H2 performance: 2.49e-01
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#+end_example
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#+begin_src matlab
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H2 = zpk(K);
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H1 = 1-H2;
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
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plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
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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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#+end_src
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** Obtained Super Sensor's noise
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#+begin_src matlab
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PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
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PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
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PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
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plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
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plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
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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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#+end_src
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#+begin_src matlab
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CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
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CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
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CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
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plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
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plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
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set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
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hold off;
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xlim([2e-1, freqs(end)]);
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ylim([1e-10 1e-5]);
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legend('location', 'southeast');
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#+end_src
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** Obtained Super Sensor's Uncertainty
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#+begin_src matlab
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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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#+end_src
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#+begin_src matlab
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Gss = G1*H1 + G2*H2;
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#+end_src
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#+begin_src matlab :exports none
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Gsss = usample(Gss, 20);
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#+end_src
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#+begin_src matlab :exports none
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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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#+end_src
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#+begin_src matlab :exports none
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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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#+end_src
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#+begin_src matlab :exports none
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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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plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--');
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plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--');
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for i = 1:length(Gsss)
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plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
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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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% plot(freqs, Dphimax, 'r-');
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% plot(freqs, -Dphimax, 'r-');
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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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plot(freqs, Dphiss, 'k--');
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plot(freqs, -Dphiss, 'k--');
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for i = 1:length(Gsss)
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
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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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#+end_src
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* Equivalent Super Sensor
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The goal here is to find the parameters of a single sensor that would best represent a super sensor.
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