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First analysis of mixed H2/Hinf synthesis

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Thomas Dehaeze
2019-08-30 10:36:15 +02:00
parent d4a2695e7c
commit 34a68a0e36
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@@ -1048,8 +1048,13 @@ The obtained upper bounds on the complementary filters in order to limit the pha
#+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]]) #+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]] [[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]]. The $\mathcal{H}_\infty$ synthesis architecture used is shown in Fig. [[fig:h_infinity_robust_fusion]].
#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
[[file:figs/h_infinity_robust_fusion.png]]
The generalized plant is defined below.
#+begin_src matlab #+begin_src matlab
P = [W1 -W1; P = [W1 -W1;
0 W2; 0 W2;
@@ -1089,6 +1094,8 @@ Test bounds: 0.0447 < gamma <= 1.3318
H1 = 1 - H2; H1 = 1 - H2;
#+end_src #+end_src
The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
@@ -1214,6 +1221,327 @@ We can now compute the uncertainty of the super sensor. The result is shown in F
We here just used very wimple weights. We could shape the dynamical uncertainty of the super sensor by using more complex weights. We here just used very wimple weights. We could shape the dynamical uncertainty of the super sensor by using more complex weights.
We could for instance ask for less uncertainty at low frequency. We could for instance ask for less uncertainty at low frequency.
* Optimal Sensor Fusion - Mixed Synthesis
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab
freqs = logspace(-1, 3, 1000);
#+end_src
** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis - Introduction
The goal is to design complementary filters such that:
- the maximum uncertainty on the super sensor is bounded
- the RMS value of the super sensor noise is minimized
To do so, we can use the Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis.
The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/robust/ref/h2hinfsyn.html][doc]]).
** Definition of the weights
We define the weights that are used to characterize the dynamic uncertainty of the sensors.
#+begin_src matlab
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);
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;
#+end_src
We define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$.
#+begin_src matlab
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;
#+end_src
We define the generalized plant that will be used for the mixed synthesis.
#+begin_src matlab
P = [W1 -W1;
0 W2;
N1 -N1;
0 N2;
1 0];
P = ss(P);
#+end_src
** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis
#+begin_src matlab :results output replace :exports both
Nmeas = 1; Ncon = 1; Nz2 = 2;
[K,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [1, 10], 'H2MAX', 2, 'HINFMAX', 2, 'DKMAX', 0, 'TOL', 0.001, 'DISPLAY', 'on');
#+end_src
#+RESULTS:
#+begin_example
Nmeas = 1; Ncon = 1; Nz2 = 2;
[K,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [1, 10], 'H2MAX', 2, 'HINFMAX', 2, 'DKMAX', 0, 'TOL', 0.001, 'DISPLAY', 'on');
Optimization of 1.000 * G^2 + 10.000 * H^2 :
----------------------------------------------
Solver for linear objective minimization under LMI constraints
Iterations : Best objective value so far
1
2
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4
5
6
7
8
9
10
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27 39.619680
28 36.192540
29 35.073373
30 35.073373
31 33.083063
32 33.083063
33 33.083063
34 28.479232
35 28.479232
36 28.479232
37 26.389647
38 23.237872
39 17.636707
40 13.600053
41 10.293603
42 9.361705
43 9.361705
44 9.361705
,* switching to QR
45 9.361705
46 8.891291
47 8.891291
48 8.891291
49 6.892529
50 6.892529
51 5.425127
52 5.425127
53 3.499118
54 3.499118
55 3.249775
56 3.249775
57 2.893020
58 2.893020
,*** new lower bound: 1.803459
59 2.583410
,*** new lower bound: 1.976383
60 2.513031
,*** new lower bound: 2.027973
61 2.441505
,*** new lower bound: 2.067580
62 2.377727
,*** new lower bound: 2.099478
63 2.342173
,*** new lower bound: 2.125297
64 2.315672
,*** new lower bound: 2.146487
65 2.295832
,*** new lower bound: 2.163923
66 2.280942
,*** new lower bound: 2.239118
67 2.265181
68 2.261547
69 2.259300
,*** new lower bound: 2.240010
70 2.258097
,*** new lower bound: 2.241899
71 2.257082
,*** new lower bound: 2.243477
72 2.256225
,*** new lower bound: 2.244794
73 2.255501
,*** new lower bound: 2.245895
74 2.254596
,*** new lower bound: 2.246815
75 2.254112
,*** new lower bound: 2.247580
76 2.253705
,*** new lower bound: 2.248219
77 2.253196
,*** new lower bound: 2.251056
Result: feasible solution of required accuracy
best objective value: 2.253196
guaranteed relative accuracy: 9.50e-04
f-radius saturation: 30.962% of R = 1.00e+08
Guaranteed Hinf performance: 1.01e+00
Guaranteed H2 performance: 2.49e-01
#+end_example
#+begin_src matlab
H2 = zpk(K);
H1 = 1-H2;
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
legend('location', 'northeast');
#+end_src
** Obtained Super Sensor's noise
#+begin_src matlab
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;
#+end_src
#+begin_src matlab :exports none
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');
#+end_src
#+begin_src matlab
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);
#+end_src
#+begin_src matlab :exports none
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');
#+end_src
** Obtained Super Sensor's Uncertainty
#+begin_src matlab
G1 = 1 + w1*ultidyn('Delta',[1 1]);
G2 = 1 + w2*ultidyn('Delta',[1 1]);
#+end_src
#+begin_src matlab
Gss = G1*H1 + G2*H2;
#+end_src
#+begin_src matlab :exports none
Gsss = usample(Gss, 20);
#+end_src
#+begin_src matlab :exports none
% 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;
#+end_src
#+begin_src matlab :exports none
% 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;
#+end_src
#+begin_src matlab :exports none
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');
#+end_src
* Equivalent Super Sensor * Equivalent Super Sensor
The goal here is to find the parameters of a single sensor that would best represent a super sensor. The goal here is to find the parameters of a single sensor that would best represent a super sensor.