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b/matlab/index_old.html @@ -0,0 +1,3686 @@ + + + + + + +Robust and Optimal Sensor Fusion - Matlab Computation + + + + + + + + + + + + +
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Robust and Optimal Sensor Fusion - Matlab Computation

+
+

Table of Contents

+
+ +
+
+ +

+In this document, the optimal and robust design of complementary filters is studied. +

+ +

+Two sensors are considered with both different noise characteristics and dynamical uncertainties represented by multiplicative input uncertainty. +

+ +
    +
  • in section 2: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’s noise is minimized
    • +
  • in section 3: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’s uncertainty is bonded to acceptable values
    • +
  • in section 4: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor’s uncertainty and to lower the RMS value of the super sensor’s noise
    • +
  • in section 6: the \(\mathcal{H}_\infty\) synthesis is used for both limiting the noise and uncertainty of the super sensor
    • +
  • in section 7: we try to find the characteristics of the super sensor from the characteristics of the individual sensors and of the complementary filters
    • +
  • in section 8: a methodology is proposed to apply optimal and robust sensor fusion in practice
    • +
  • in section 9: methods of complementary filter synthesis are proposed
    • +
+ +
+

1 Comparison with Bibliographic example

+
+
+
+

1.1 Bendat, J., Optimum filters for independent measurements of two related perturbed messages (1957)

+
+

+(Bendat 1957) +

+ +
+
freqs = logspace(-1, 2, 1000);
+
+
+ +

+Weights to shape the noise of both sensors: +

+
+
K1 = 100;
+K2 = 1;
+b = 10;
+b1 = b;
+b2 = b;
+
+N1 = sqrt(K1)*b1/(b1+s)/(s + 1e-2);
+N2 = sqrt(K2)*b2/(b2+s);
+
+
+ + +
+

bendat57_noise_weights.png +

+

Figure 1: Weights

+
+ +

+\(\mathcal{H}_2\) synthesis: +

+
+
P = [0   N2  1;
+     N1 -N2  0];
+[H1, ~, gamma] = h2syn(P, 1, 1);
+H2 = 1 - H1;
+
+
+ +

+The optimal obtained filter (from the paper) is: +

+
+
a = sqrt(K2/K1);
+G = (a*s + 1 + a*b)/(1 + a*b)/(a*s + 1);
+
+
+ + +
+

bendat57_optimal_filters.png +

+

Figure 2: Obtain Filters

+
+ + +
+

bendat57_psd_estimation.png +

+

Figure 3: PSD of the individual sensors + super sensor

+
+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 RMS
Sensor 122.93
Sensor 22.37
H2 Synthesis1.74
Paper1.74
+
+
+ +
+

1.2 Plummer, A. R., Optimal complementary filters and their application in motion measurement (2006)

+
+

+(Plummer 2006) +

+ +

+Weights +

+
+
N1 = 24.3e-6*(s + 2*pi*0.1)*(s + 1220)/1220*(1/(1 + s/2/pi/1e4)/(1 + s/2/pi/1e4));
+N2 = 0.363/(s + 0.01)*(s + 12.2)/(s + 0.01);
+
+
+ + +
+

plummer06_noise_weights.png +

+

Figure 4: Weights

+
+ +

+\(\mathcal{H}_2\) synthesis: +

+
+
P = [0   N2  1;
+     N1 -N2  0];
+
+[H1, ~, gamma] = h2syn(P, 1, 1);
+
+H2 = 1 - H1;
+
+
+ +

+The optimal obtained filter (from the paper) is: +

+
+
G = (0.0908*s + 1)/(5.51e-7*s^3 + 7.47e-4*s^2 + 0.0908*s + 1);
+
+
+ + +
+

plummer06_optimal_filters.png +

+

Figure 5: Obtain Filters

+
+ + +
+

plummer06_psd_estimation.png +

+

Figure 6: PSD of the individual sensors + super sensor

+
+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 RMS
Sensor 1130.0
Sensor 20.00753
H2 Synthesis0.00091
Paper0.0107
+ +

+Parameters of the time domain simulation. +

+
+
Fs = 2.5e3; % Sampling Frequency [Hz]
+Ts = 1/Fs; % Sampling Time [s]
+
+t = 0:Ts:2; % Time Vector [s]
+
+
+ +

+Generate noises in velocity corresponding to sensor 1 and 2: +

+
+
n1 = lsim(N1, sqrt(Fs/2)*randn(length(t), 1), t);
+n2 = lsim(N2, sqrt(Fs/2)*randn(length(t), 1), t);
+
+
+ + +
+

plummer06_time_domain_signals.png +

+

Figure 7: Time domain signals

+
+
+
+ +
+

1.3 Robert Grover Brown, P. Y. C. H., Introduction to random signals and applied kalman filtering with matlab exercises (2012)

+
+

+(Robert Grover Brown 2012) Section 8.6 +

+ +
+
w0 = 1; % [rad/s]
+wc = 20*w0; % [rad/s]
+
+k1 = sqrt(200*sqrt(2)*w0^3); % [m]
+k2 = sqrt(100*pi/wc); % [m]
+
+N1 = k1/(s^2 + sqrt(2)*s + 1);
+N2 = k2/(1 + s/(wc*(2/pi)));
+
+
+ + +
+

robert12_noise_weights.png +

+

Figure 8: Weights

+
+ +

+And we do the \(\mathcal{H}_2\) synthesis using the h2syn command. +

+
+
P = [0   N2  1;
+     N1 -N2  0];
+
+[H1, ~, gamma] = h2syn(P, 1, 1);
+
+H2 = 1 - H1;
+
+
+ + +
+

robert12_optimal_filters.png +

+

Figure 9: Obtain Filters

+
+ + +
+

robert12_psd_estimation.png +

+

Figure 10: PSD of the individual sensors + super sensor

+
+ +

+We can see that the optimal \(\mathcal{H}_2\) control gives similar results as Kalman filtering. +

+ + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + +
MethodMean Square Error
Kalman Filter21.47
Euristic35.32
Optimal H220.96
+ + + + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + +
MethodMean Square Error
Kalman Filter21.47
Euristic35.32
Optimal H221.40
+
+
+
+ +
+

2 Optimal Sensor Fusion - Minimize the Super Sensor Noise

+
+

+ +

+

+The idea is to combine sensors that works in different frequency range using complementary filters. +

+ +

+Doing so, one “super sensor” is obtained that can have better noise characteristics than the individual sensors over a large frequency range. +

+ +

+The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise. +

+
+

+The Matlab scripts is accessible here. +

+ +
+
+ +
+

2.1 Architecture

+
+

+Let’s consider the sensor fusion architecture shown on figure 11 where two sensors (sensor 1 and sensor 2) are measuring the same quantity \(x\) with different noise characteristics determined by \(N_1(s)\) and \(N_2(s)\). +

+ +

+\(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise: +

+\begin{equation} +\label{org800b631} + \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1 +\end{equation} + + +
+

fusion_two_noisy_sensors_weights.png +

+

Figure 11: Fusion of two sensors

+
+ +

+We consider that the two sensor dynamics \(G_1(s)\) and \(G_2(s)\) are ideal: +

+\begin{equation} +\label{orgc324337} + G_1(s) = G_2(s) = 1 +\end{equation} + +

+We obtain the architecture of figure 12. +

+ + +
+

sensor_fusion_noisy_perfect_dyn.png +

+

Figure 12: Fusion of two sensors with ideal dynamics

+
+ +

+\(H_1(s)\) and \(H_2(s)\) are complementary filters: +

+\begin{equation} +\label{org8e22dff} + H_1(s) + H_2(s) = 1 +\end{equation} + +

+The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the estimation \(\hat{x}\). +

+ +

+We have that the Power Spectral Density (PSD) of \(\hat{x}\) is: +\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \] +

+ +

+And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\): +

+\begin{equation} +\label{orgeef7d07} + \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega} +\end{equation} +
+
+ +
+

2.2 Noise of the sensors

+
+

+Let’s define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\): +

+
    +
  • Sensor 1 characterized by \(N_1(s)\) has low noise at low frequency (for instance a geophone)
    • +
  • Sensor 2 characterized by \(N_2(s)\) has low noise at high frequency (for instance an accelerometer)
    • +
+ +
+
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;
+
+
+ + +
+

noise_characteristics_sensors.png +

+

Figure 13: Noise Characteristics of the two sensors (png, pdf)

+
+
+
+ +
+

2.3 H-Two Synthesis

+
+

+As \(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise: \(\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1\) and we have: +\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \] +Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized. +

+ +

+For that, we use the \(\mathcal{H}_2\) Synthesis. +

+ +

+We use the generalized plant architecture shown on figure 14. +

+ + +
+

h_infinity_optimal_comp_filters.png +

+

Figure 14: \(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters

+
+ +\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: +\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \] +

+ +

+Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we minimize the RMS value of \(\hat{x}\). +

+ +

+We define the generalized plant \(P\) on matlab as shown on figure 14. +

+
+
P = [0   N2  1;
+     N1 -N2  0];
+
+
+ +

+And we do the \(\mathcal{H}_2\) synthesis using the h2syn command. +

+
+
[H1, ~, gamma] = h2syn(P, 1, 1);
+
+
+ +

+Finally, we define \(H_2(s) = 1 - H_1(s)\). +

+
+
H2 = 1 - H1;
+
+
+ +

+The complementary filters obtained are shown on figure 15. +

+ +

+The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. 16. +

+ +

+The Cumulative Power Spectrum (CPS) is shown on Fig. 17. +

+ +

+The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors. +

+ + +
+

htwo_comp_filters.png +

+

Figure 15: Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis (png, pdf)

+
+ +
+
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;
+
+
+ + +
+

psd_sensors_htwo_synthesis.png +

+

Figure 16: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (png, pdf)

+
+ +
+
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);
+
+
+ + +
+

cps_h2_synthesis.png +

+

Figure 17: Cumulative Power Spectrum of individual sensors and super sensor using the \(\mathcal{H}_2\) synthesis (png, pdf)

+
+
+
+ +
+

2.4 Alternative H-Two Synthesis

+
+

+An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is shown in Fig. 18. +

+ + +
+

h_infinity_optimal_comp_filters_bis.png +

+

Figure 18: Alternative formulation of the \(\mathcal{H}_2\) synthesis

+
+ +\begin{equation*} +\begin{pmatrix} + z_1 \\ z_2 \\ v +\end{pmatrix} = \begin{pmatrix} + N_1 & -N_1 \\ + 0 & N_2 \\ + 1 & 0 +\end{pmatrix} \begin{pmatrix} + w \\ u +\end{pmatrix} +\end{equation*} +
+
+ + +
+

2.5 H-Infinity Synthesis - method A

+
+

+Another objective that we may have is that the noise of the super sensor \(n_{SS}\) is following the minimum of the noise of the two sensors \(n_1\) and \(n_2\): +\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \] +

+ +

+In order to obtain that ideal case, we need that the complementary filters be designed such that: +

+\begin{align*} + & |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ + & |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) +\end{align*} + +

+Which is indeed impossible in practice. +

+ +

+We could try to approach that with the \(\mathcal{H}_\infty\) synthesis by using high order filters. +

+ +

+As shown on Fig. 13, the frequency where the two sensors have the same noise level is around 9Hz. +We will thus choose weighting functions such that the merging frequency is around 9Hz. +

+ +

+The weighting functions used as well as the obtained complementary filters are shown in Fig. 19. +

+ +
+
n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
+W1a = (((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;
+
+n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2;
+W2a = (((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;
+
+
+ +
+
P = [W1a -W1a;
+     0    W2a;
+     1    0];
+
+
+ +

+And we do the \(\mathcal{H}_\infty\) synthesis using the hinfsyn command. +

+
+
[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+
+ +
+[H2a, ~, 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:      0.1000 <  gamma  <=  10500.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+04   2.1e+01 -3.0e-07   7.8e+00   -1.3e-15    0.0000    p
+5.250e+03   2.1e+01 -1.5e-08   7.8e+00   -5.8e-14    0.0000    p
+2.625e+03   2.1e+01   2.5e-10   7.8e+00   -3.7e-12    0.0000    p
+1.313e+03   2.1e+01 -3.2e-11   7.8e+00   -7.3e-14    0.0000    p
+  656.344   2.1e+01 -2.2e-10   7.8e+00   -1.1e-15    0.0000    p
+  328.222   2.1e+01 -1.1e-10   7.8e+00   -1.2e-15    0.0000    p
+  164.161   2.1e+01 -2.4e-08   7.8e+00   -8.9e-16    0.0000    p
+   82.130   2.1e+01   2.0e-10   7.8e+00   -9.1e-31    0.0000    p
+   41.115   2.1e+01 -6.8e-09   7.8e+00   -4.1e-13    0.0000    p
+   20.608   2.1e+01   3.3e-10   7.8e+00   -1.4e-12    0.0000    p
+   10.354   2.1e+01 -9.8e-09   7.8e+00   -1.8e-15    0.0000    p
+    5.227   2.1e+01 -4.1e-09   7.8e+00   -2.5e-12    0.0000    p
+    2.663   2.1e+01   2.7e-10   7.8e+00   -4.0e-14    0.0000    p
+    1.382   2.1e+01 -3.2e+05#  7.8e+00   -3.5e-14    0.0000    f
+    2.023   2.1e+01 -5.0e-10   7.8e+00    0.0e+00    0.0000    p
+    1.702   2.1e+01 -2.4e+07#  7.8e+00   -1.6e-13    0.0000    f
+    1.862   2.1e+01 -6.0e+08#  7.8e+00   -1.0e-12    0.0000    f
+    1.942   2.1e+01 -2.8e-09   7.8e+00   -8.1e-14    0.0000    p
+    1.902   2.1e+01 -2.5e-09   7.8e+00   -1.1e-13    0.0000    p
+    1.882   2.1e+01 -9.3e-09   7.8e+00   -2.0e-15    0.0001    p
+    1.872   2.1e+01 -1.3e+09#  7.8e+00   -3.6e-22    0.0000    f
+    1.877   2.1e+01 -2.6e+09#  7.8e+00   -1.2e-13    0.0000    f
+    1.880   2.1e+01 -5.6e+09#  7.8e+00   -1.4e-13    0.0000    f
+    1.881   2.1e+01 -1.2e+10#  7.8e+00   -3.3e-12    0.0000    f
+    1.882   2.1e+01 -3.2e+10#  7.8e+00   -8.5e-14    0.0001    f
+
+ Gamma value achieved:     1.8824
+
+ +
+
H1a = 1 - H2a;
+
+
+ + +
+

weights_comp_filters_Hinfa.png +

+

Figure 19: Weights and Complementary Fitlers obtained (png, pdf)

+
+ +

+We then compute the Power Spectral Density as well as the Cumulative Power Spectrum. +

+ +
+
PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2;
+CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha);
+
+
+
+
+ +
+

2.6 H-Infinity Synthesis - method B

+
+

+We have that: +\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \] +

+ +

+Then, at frequencies where \(|H_1(j\omega)| < |H_2(j\omega)|\) we would like that \(|N_1(j\omega)| = 1\) and \(|N_2(j\omega)| = 0\) as we discussed before. +Then \(|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2\). +

+ +

+We know that this is impossible in practice. A more realistic choice is to design \(H_2(s)\) such that when \(|N_2(j\omega)| > |N_1(j\omega)|\), we have that: +\[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \] +

+ +

+Which is equivalent to have (by supposing \(|H_1| \approx 1\)): +\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] +

+ +

+And we have: +

+\begin{align*} + \Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ + &= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\ + &\approx \left|N_1\right|^2 +\end{align*} + +

+Similarly, we design \(H_1(s)\) such that at frequencies where \(|N_1| > |N_2|\): +\[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \] +

+ +

+For instance, is we take \(\epsilon = 1\), then the PSD of \(\hat{x}\) is increased by just by a factor \(\sqrt{2}\) over the all frequencies from the idea case. +

+ +

+We use this as the weighting functions for the \(\mathcal{H}_\infty\) synthesis of the complementary filters. +

+ +

+The weighting function and the obtained complementary filters are shown in Fig. 20. +

+ +
+
epsilon = 2;
+
+W1b = 1/epsilon*N1/N2;
+W2b = 1/epsilon*N2/N1;
+
+W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper
+
+
+ +
+
P = [W1b -W1b;
+     0    W2b;
+     1    0];
+
+
+ +

+And we do the \(\mathcal{H}_\infty\) synthesis using the hinfsyn command. +

+
+
[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+
+ +
+[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Test bounds:      0.0000 <  gamma  <=     32.8125
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+   32.812   1.8e+01   3.4e-10   6.3e+00   -2.9e-13    0.0000    p
+   16.406   1.8e+01   3.4e-10   6.3e+00   -1.2e-15    0.0000    p
+    8.203   1.8e+01   3.3e-10   6.3e+00   -2.6e-13    0.0000    p
+    4.102   1.8e+01   3.3e-10   6.3e+00   -2.1e-13    0.0000    p
+    2.051   1.7e+01   3.4e-10   6.3e+00   -7.2e-16    0.0000    p
+    1.025   1.6e+01 -1.3e+06#  6.3e+00   -8.3e-14    0.0000    f
+    1.538   1.7e+01   3.4e-10   6.3e+00   -2.0e-13    0.0000    p
+    1.282   1.7e+01   3.4e-10   6.3e+00   -7.9e-17    0.0000    p
+    1.154   1.7e+01   3.6e-10   6.3e+00   -1.8e-13    0.0000    p
+    1.089   1.7e+01 -3.4e+06#  6.3e+00   -1.7e-13    0.0000    f
+    1.122   1.7e+01 -1.0e+07#  6.3e+00   -3.2e-13    0.0000    f
+    1.138   1.7e+01 -1.3e+08#  6.3e+00   -1.8e-13    0.0000    f
+    1.146   1.7e+01   3.2e-10   6.3e+00   -3.0e-13    0.0000    p
+    1.142   1.7e+01   5.5e-10   6.3e+00   -2.8e-13    0.0000    p
+    1.140   1.7e+01 -1.5e-10   6.3e+00   -2.3e-13    0.0000    p
+    1.139   1.7e+01 -4.8e+08#  6.3e+00   -6.2e-14    0.0000    f
+    1.139   1.7e+01   1.3e-09   6.3e+00   -8.9e-17    0.0000    p
+
+ Gamma value achieved:     1.1390
+
+ +
+
H1b = 1 - H2b;
+
+
+ + +
+

weights_comp_filters_Hinfb.png +

+

Figure 20: Weights and Complementary Fitlers obtained (png, pdf)

+
+ +
+
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);
+
+
+
+
+ +
+

2.7 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');
+
+
+ +
+[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
+
+ +
+
H1c = 1 - H2c;
+
+
+ + +
+

weights_comp_filters_Hinfc.png +

+

Figure 21: Weights and Complementary Fitlers obtained (png, pdf)

+
+ +
+
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);
+
+
+
+
+ +
+

2.8 Comparison of the methods

+
+

+The three methods are now compared. +

+ +

+The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. 22. +

+ +

+The Cumulative Power Spectrum for the same sensors are shown on Fig. 23. +

+ +

+The RMS value of the obtained super sensors are shown on table 1. +

+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters
 rms value
Sensor 11.3e-03
Sensor 21.3e-03
H2 Fusion1.2e-04
H-Infinity a2.4e-04
H-Infinity b1.4e-04
H-Infinity c2.2e-04
+ + + +
+

comparison_psd_noise.png +

+

Figure 22: Comparison of the obtained Power Spectral Density using the three methods (png, pdf)

+
+ + +
+

comparison_cps_noise.png +

+

Figure 23: Comparison of the obtained Cumulative Power Spectrum using the three methods (png, pdf)

+
+
+
+ +
+

2.9 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]);
+
+
+ +

+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. 24. +Right Half Plane zero might be introduced in the super sensor dynamics which will render the feedback system unstable. +

+ +
+
Gss = G1*H1 + G2*H2;
+
+
+ + +
+

uncertainty_super_sensor_H2_syn.png +

+

Figure 24: Uncertianty regions of both individual sensors and of the super sensor when using the \(\mathcal{H}_2\) synthesis (png, pdf)

+
+
+
+ +
+

2.10 Conclusion

+
+

+From the above complementary filter design with the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) synthesis, it still seems that the \(\mathcal{H}_2\) synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the \(\mathcal{H}_2\) norm). +

+ +

+However, the synthesis does not take into account the robustness of the sensor fusion. +

+
+
+
+ +
+

3 Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty

+
+

+ +

+

+We initially considered perfectly known sensor dynamics so that it can be perfectly inverted. +

+ +

+We now take into account the fact that the sensor dynamics is only partially known. +To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Fig. 25. +

+ + +
+

sensor_fusion_dynamic_uncertainty.png +

+

Figure 25: Sensor fusion architecture with sensor dynamics uncertainty

+
+ +

+The objective here is to design complementary filters \(H_1(s)\) and \(H_2(s)\) in order to minimize the dynamical uncertainty of the super sensor. +

+
+

+The Matlab scripts is accessible here. +

+ +
+
+ +
+

3.1 Super Sensor Dynamical Uncertainty

+
+

+In practical systems, the sensor dynamics has always some level of uncertainty. +Let’s represent that with multiplicative input uncertainty as shown on figure 25. +

+ + +
+

sensor_fusion_dynamic_uncertainty.png +

+

Figure 26: Fusion of two sensors with input multiplicative uncertainty

+
+ +

+The dynamics of the super sensor is represented by +

+\begin{align*} + \frac{\hat{x}}{x} &= (1 + w_1 \Delta_1) H_1 + (1 + w_2 \Delta_2) H_2 \\ + &= 1 + w_1 H_1 \Delta_1 + w_2 H_2 \Delta_2 +\end{align*} +

+with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\). +

+ +

+We see that as soon as we have some uncertainty in the sensor dynamics, we have that the complementary filters have some effect on the transfer function from \(x\) to \(\hat{x}\). +

+ +

+The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (figure 27). +

+ +

+We then have that the angle introduced by the super sensor is bounded by \(\arcsin(\epsilon)\): +\[ \angle \frac{\hat{x}}{x}(j\omega) \le \arcsin \Big(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\Big) \] +

+ + +
+

uncertainty_gain_phase_variation.png +

+

Figure 27: Maximum phase variation

+
+
+
+ +
+

3.2 Dynamical uncertainty of the individual sensors

+
+

+Let say we want to merge two sensors: +

+
    +
  • sensor 1 that has unknown dynamics above 10Hz: \(|w_1(j\omega)| > 1\) for \(\omega > 10\text{ Hz}\)
    • +
  • sensor 2 that has unknown dynamics below 1Hz and above 1kHz \(|w_2(j\omega)| > 1\) for \(\omega < 1\text{ Hz}\) and \(\omega > 1\text{ kHz}\)
    • +
+ +

+We define the weights that are used to characterize the dynamic uncertainty of the 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);
+
+
+ +

+From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. 28 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]);
+
+
+ + +
+

uncertainty_dynamics_sensors.png +

+

Figure 28: Dynamic uncertainty of the two sensors (png, pdf)

+
+
+
+ +
+

3.3 Synthesis objective

+
+

+The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. 27. +

+ +

+At each frequency \(\omega\), the radius of the circle is \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\). +

+ +

+Thus, the phase shift \(\Delta\phi(\omega)\) due to the super sensor uncertainty is bounded by: +\[ |\Delta\phi(\omega)| \leq \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) \] +

+ +

+Let’s define some allowed frequency depend phase shift \(\Delta\phi_\text{max}(\omega) > 0\) such that: +\[ |\Delta\phi(\omega)| < \Delta\phi_\text{max}(\omega), \quad \forall\omega \] +

+ + +

+If \(H_1(s)\) and \(H_2(s)\) are designed such that +\[ |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big) \] +

+ +

+The maximum phase shift due to dynamic uncertainty at frequency \(\omega\) will be \(\Delta\phi_\text{max}(\omega)\). +

+
+
+ +
+

3.4 Requirements as an \(\mathcal{H}_\infty\) norm

+
+

+We now try to express this requirement in terms of an \(\mathcal{H}_\infty\) norm. +

+ +

+Let’s define one weight \(w_\phi(s)\) that represents the maximum wanted phase uncertainty: +\[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \] +

+ +

+Then: +

+\begin{align*} + & |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big), \quad \forall\omega \\ + \Longleftrightarrow & |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < |w_\phi(j\omega)|^{-1}, \quad \forall\omega \\ + \Longleftrightarrow & \left| w_1(j\omega) H_1(j\omega) w_\phi(j\omega) \right| + \left| w_2(j\omega) H_2(j\omega) w_\phi(j\omega) \right| < 1, \quad \forall\omega +\end{align*} + +

+Which is approximately equivalent to (with an error of maximum \(\sqrt{2}\)): +

+\begin{equation} +\label{org6b95a9d} + \left\| \begin{matrix} w_1(s) w_\phi(s) H_1(s) \\ w_2(s) w_\phi(s) H_2(s) \end{matrix} \right\|_\infty < 1 +\end{equation} + +

+One should not forget that at frequency where both sensors has unknown dynamics (\(|w_1(j\omega)| > 1\) and \(|w_2(j\omega)| > 1\)), the super sensor dynamics will also be unknown and the phase uncertainty cannot be bounded. +Thus, at these frequencies, \(|w_\phi|\) should be smaller than \(1\). +

+
+
+ +
+

3.5 Weighting Function used to bound the super sensor uncertainty

+
+

+Let’s define \(w_\phi(s)\) in order to bound the maximum allowed phase uncertainty \(\Delta\phi_\text{max}\) of the super sensor dynamics. +The magnitude \(|w_\phi(j\omega)|\) is shown in Fig. 29 and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. 30. +

+ +
+
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;
+
+
+ + +
+

magnitude_wphi.png +

+

Figure 29: Magnitude of the weght \(w_\phi(s)\) that is used to bound the uncertainty of the super sensor (png, pdf)

+
+ + +
+

maximum_wanted_phase_uncertainty.png +

+

Figure 30: Maximum wanted phase uncertainty using this weight (png, pdf)

+
+ +

+The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. 31. +

+ + +
+

upper_bounds_comp_filter_max_phase_uncertainty.png +

+

Figure 31: Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz (png, pdf)

+
+
+
+ +
+

3.6 \(\mathcal{H}_\infty\) Synthesis

+
+

+The \(\mathcal{H}_\infty\) synthesis architecture used for the complementary filters is shown in Fig. 32. +

+ + +
+

h_infinity_robust_fusion.png +

+

Figure 32: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

+
+ +

+The generalized plant is defined below. +

+
+
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');
+
+
+ +
+[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
+
+ +

+And \(H_1(s)\) is defined as the complementary of \(H_2(s)\). +

+
+
H1 = 1 - H2;
+
+
+ +

+The obtained complementary filters are shown in Fig. 33. +

+ +
+

comp_filter_hinf_uncertainty.png +

+

Figure 33: Obtained complementary filters (png, pdf)

+
+
+
+ +
+

3.7 Super sensor uncertainty

+
+

+We can now compute the uncertainty of the super sensor. The result is shown in Fig. 34. +

+ +
+
Gss = G1*H1 + G2*H2;
+
+
+ + +
+

super_sensor_uncertainty_bode_plot.png +

+

Figure 34: Uncertainty on the dynamics of the super sensor (png, pdf)

+
+ +

+The uncertainty of the super sensor cannot be made smaller than both the individual sensor. Ideally, it would follow the minimum uncertainty of both sensors. +

+ +

+We here just used very wimple weights. +For instance, we could improve the dynamical uncertainty of the super sensor by making \(|w_\phi(j\omega)|\) smaller bellow 2Hz where the dynamical uncertainty of the sensor 1 is small. +

+
+
+ +
+

3.8 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. 35. +The CPS of both sensor and of the super sensor is shown in Fig. 36. +

+ + +
+

psd_sensors_hinf_synthesis.png +

+

Figure 35: Power Spectral Density of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (png, pdf)

+
+ + +
+

cps_sensors_hinf_synthesis.png +

+

Figure 36: Cumulative Power Spectrum of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (png, cps)

+
+
+
+ +
+

3.9 Conclusion

+
+

+Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values. +

+ +

+However, the RMS of the super sensor noise is not optimized as it was the case with the \(\mathcal{H}_2\) synthesis +

+
+
+
+ +
+

4 Optimal Sensor Fusion - Mixed Synthesis

+
+

+ +

+
+

+The Matlab scripts is accessible here. +

+ +
+
+
+

4.1 Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction

+
+

+The goal is to design complementary filters such that: +

+
    +
  • the maximum uncertainty of 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 (doc). +

+
+
+ +
+

4.2 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. 37. +

+ + +
+

mixed_synthesis_noise_uncertainty_sensors.png +

+

Figure 37: Noise characteristsics and Dynamical uncertainty of the individual sensors (png, pdf)

+
+
+
+ +
+

4.3 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. 38. 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;
+
+
+ + +
+

mixed_syn_hinf_weight.png +

+

Figure 38: Wanted maximum module uncertainty of the super sensor (png, pdf)

+
+ +

+The equivalent Magnitude and Phase uncertainties are shown in Fig. 39. +

+ + +
+

mixed_syn_objective_hinf.png +

+

Figure 39: \(\mathcal{H}_\infty\) synthesis objective part of the mixed-synthesis (png, pdf)

+
+
+
+ +
+

4.4 Mixed Synthesis Architecture

+
+

+The synthesis architecture that is used here is shown in Fig. 40. +

+ +

+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
    • +
+ + +
+

mixed_h2_hinf_synthesis.png +

+

Figure 40: Mixed H2/H-Infinity Synthesis

+
+ +

+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];
+
+
+
+
+ +
+

4.5 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. 41. +

+ + +
+

comp_filters_mixed_synthesis.png +

+

Figure 41: Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (png, pdf)

+
+
+
+ +
+

4.6 Obtained Super Sensor’s noise

+
+

+The PSD and CPS of the super sensor’s noise are shown in Fig. 42 and Fig. 43 respectively. +

+ + +
+

psd_super_sensor_mixed_syn.png +

+

Figure 42: Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (png, pdf)

+
+ + + +
+

cps_super_sensor_mixed_syn.png +

+

Figure 43: Cumulative Power Spectrum of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (png, pdf)

+
+
+
+ +
+

4.7 Obtained Super Sensor’s Uncertainty

+
+

+The uncertainty on the super sensor’s dynamics is shown in Fig. 44. +

+ + +
+

super_sensor_dyn_uncertainty_mixed_syn.png +

+

Figure 44: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis (png, pdf)

+
+
+
+ +
+

4.8 Conclusion

+
+

+This synthesis methods allows both to: +

+
    +
  • limit the dynamical uncertainty of the super sensor
    • +
  • minimize the RMS value of the estimation
    • +
+
+
+
+ +
+

5 Mixed Synthesis - LMI Optimization

+
+
+
+

5.1 Introduction

+
+

+The following matlab scripts was written by Mohit. +

+
+
+ +
+

5.2 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;
+
+
+
+
+ +
+

5.3 Weights

+
+

+The weights for the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) part are defined below. +

+ +
+
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;
+
+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];
+
+
+
+
+ +
+

5.4 LMI Optimization

+
+

+We are using the CVX toolbox to solve the optimization problem. +

+ +

+We first put the generalized plant in a State-space form. +

+
+
A = P.A;
+Bw = P.B(:,1);
+Bu = P.B(:,2);
+Cz1 = P.C(1:2,:); Dz1w = P.D(1:2,1); Dz1u = P.D(1:2,2); % Hinf
+Cz2 = P.C(3:4,:); Dz2w = P.D(1:2,1); Dz2u = P.D(1:2,2); % H2
+Cy = P.C(5,:); Dyw = P.D(5,1); Dyu = P.D(5,2);
+
+n = size(P.A,1);
+ny = 1; % number of measurements
+nu = 1; % number of control inputs
+nz = 2;
+nw = 1;
+
+Wtinf = 0;
+Wt2 = 1;
+
+
+ +

+We Define all the variables. +

+
+
cvx_startup;
+
+cvx_begin sdp
+cvx_quiet true
+cvx_solver sedumi
+variable X(n,n) symmetric;
+variable Y(n,n) symmetric;
+variable W(nz,nz) symmetric;
+variable Ah(n,n);
+variable Bh(n,ny);
+variable Ch(nu,n);
+variable Dh(nu,ny);
+variable eta;
+variable gam;
+
+
+ +

+We define the minimization objective. +

+
+
minimize Wt2*eta+Wtinf*gam % mix objective
+subject to:
+
+
+ +

+The \(\mathcal{H}_\infty\) constraint. +

+
+
gam<=1; % Keep the Hinf norm less than 1
+
+[ X, eye(n,n) ;
+  eye(n,n), Y ] >= 0 ;
+
+[ A*X + Bu*Ch + X*A' + Ch'*Bu', A+Bu*Dh*Cy+Ah', Bw+Bu*Dh*Dyw, X*Cz1' + Ch'*Dz1u' ;
+  (A+Bu*Dh*Cy+Ah')', Y*A + A'*Y + Bh*Cy + Cy'*Bh', Y*Bw + Bh*Dyw, (Cz1+Dz1u*Dh*Cy)' ;
+  (Bw+Bu*Dh*Dyw)', Bw'*Y + Dyw'*Bh', -eye(nw,nw), (Dz1w+Dz1u*Dh*Dyw)' ;
+  Cz1*X + Dz1u*Ch, Cz1+Dz1u*Dh*Cy, Dz1w+Dz1u*Dh*Dyw, -gam*eye(nz,nz)] <= 0 ;
+
+
+ +

+The \(\mathcal{H}_2\) constraint. +

+
+
trace(W) <= eta ;
+
+[ W, Cz2*X+Dz2u*Ch, Cz2*X+Dz2u*Ch;
+  X*Cz2'+Ch'*Dz2u', X, eye(n,n) ;
+  (Cz2*X+Dz2u*Ch)', eye(n,n), Y ] >= 0 ;
+
+[ A*X + Bu*Ch + X*A' + Ch'*Bu', A+Bu*Dh*Cy+Ah', Bw+Bu*Dh*Dyw ;
+  (A+Bu*Dh*Cy+Ah')', Y*A + A'*Y + Bh*Cy + Cy'*Bh', Y*Bw + Bh*Dyw ;
+  (Bw+Bu*Dh*Dyw)', Bw'*Y + Dyw'*Bh', -eye(nw,nw)] <= 0 ;
+
+
+ +

+And we run the optimization. +

+
+
cvx_end
+cvx_status
+
+
+ +

+Finally, we can compute the obtained complementary filters. +

+
+
M = eye(n);
+N = inv(M)*(eye(n,n)-Y*X);
+Dk = Dh;
+Ck = (Ch-Dk*Cy*X)*inv(M');
+Bk = inv(N)*(Bh-Y*Bu*Dk);
+Ak = inv(N)*(Ah-Y*(A+Bu*Dk*Cy)*X-N*Bk*Cy*X-Y*Bu*Ck*M')*inv(M');
+
+H2 = tf(ss(Ak,Bk,Ck,Dk));
+H1 = 1 - H2;
+
+
+
+
+ +
+

5.5 Result

+
+

+The obtained complementary filters are compared with the required upper bounds on Fig. 45. +

+ + +
+

LMI_obtained_comp_filters.png +

+

Figure 45: Obtained complementary filters using the LMI optimization (png, pdf)

+
+
+
+ +
+

5.6 Comparison with the matlab Mixed Synthesis

+
+

+The Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis is performed below. +

+
+
Nmeas = 1; Ncon = 1; Nz2 = 2;
+
+[H2m,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
+
+H1m = 1 - H2m;
+
+
+ +

+The obtained filters are compare with the one obtained using the CVX toolbox in Fig. [[]]. +

+ + +
+

compare_cvx_h2hinf_comp_filters.png +

+

Figure 46: Comparison between the complementary filters obtained with the CVX toolbox and with the h2hinfsyn command (png, pdf)

+
+
+
+ +
+

5.7 H-Infinity Objective

+
+

+In terms of the \(\mathcal{H}_\infty\) objective, both synthesis method are satisfying the requirements as shown in Fig. 47. +

+ + +
+

comp_cvx_h2i_hinf_norm.png +

+

Figure 47: H-Infinity norm requirement and results (png, pdf)

+
+
+
+ +
+

5.8 Obtained Super Sensor’s noise

+
+

+The PSD and CPS of the super sensor’s noise obtained with the CVX toolbox and h2hinfsyn command are compared in Fig. 48 and 49. +

+ + +
+

psd_compare_cvx_h2i.png +

+

Figure 48: Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (png, pdf)

+
+ + + +
+

cps_compare_cvx_h2i.png +

+

Figure 49: Cumulative Power Spectrum of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (png, pdf)

+
+
+
+ +
+

5.9 Obtained Super Sensor’s Uncertainty

+
+

+The uncertainty on the super sensor’s dynamics is shown in Fig. [[]]. +

+ + +
+

super_sensor_uncertainty_compare_cvx_h2i.png +

+

Figure 50: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis (png, pdf)

+
+
+
+
+ +
+

6 H-Infinity synthesis to ensure both performance and robustness

+
+

+ +

+
+

+The Matlab scripts is accessible here. +

+ +
+
+ +
+

6.1 Introduction

+
+

+The idea is to use only the \(\mathcal{H}_\infty\) norm to express both the maximum wanted super sensor uncertainty and the fact that we want to minimize the super sensor’s noise. +

+ +

+For performance, we may want to obtain a super sensor’s noise that is close to the minimum of the individual sensor noises. +

+ +

+The noise of the super sensor is: +\[ |N_{ss}(j\omega)|^2 = | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \] +

+ +

+The minimum noise that we can obtain follows the minimum noise of the individual sensor: +

+\begin{align*} + & |N_{ss}(j\omega)| \approx |N_1(j\omega)| \quad \text{when} \quad |N_1(j\omega)| < |N_2(j\omega)| \\ + & |N_{ss}(j\omega)| \approx |N_2(j\omega)| \quad \text{when} \quad |N_2(j\omega)| < |N_1(j\omega)| +\end{align*} + +

+To do so, we want to design the complementary filters such that: +

+\begin{align*} + & |H_2(j\omega)| \ll 1 \quad \text{when} \quad |N_1(j\omega)| < |N_2(j\omega)| \\ + & |H_1(j\omega)| \ll 1 \quad \text{when} \quad |N_2(j\omega)| < |N_1(j\omega)| +\end{align*} + + + + +

+For the uncertainty of the super sensor. +The equivalent super sensor uncertainty is: +\[ |w_{ss}(j\omega)| = |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \] +

+ +

+The minimum uncertainty that we can obtain follows the minimum uncertainty of the individual sensor: +

+\begin{align*} + & |w_{ss}(j\omega)| \approx |w_1(j\omega)| \quad \text{when} \quad |w_1(j\omega)| < |w_2(j\omega)| \\ + & |w_{ss}(j\omega)| \approx |w_2(j\omega)| \quad \text{when} \quad |w_2(j\omega)| < |w_1(j\omega)| +\end{align*} + +

+To do so, we want to design the complementary filters such that: +

+\begin{align*} + & |H_2(j\omega)| \ll 1 \quad \text{when} \quad |w_1(j\omega)| < |w_2(j\omega)| \\ + & |H_1(j\omega)| \ll 1 \quad \text{when} \quad |w_2(j\omega)| < |w_1(j\omega)| +\end{align*} + + +

+Of course, the conditions for performance and uncertainty may not be compatible. +

+ +

+We may not want to follow the minimum uncertainty. +

+
+
+ +
+

6.2 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;
+
+
+ + +
+

noise_uncertainty_sensors_hinf.png +

+

Figure 51: Noise and Uncertainty characteristics of the sensors (png, pdf)

+
+
+
+ +
+

6.3 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. 52. +

+ +
+

charac_sensors_weights.png +

+

Figure 52: Upper bounds on the super sensor’s noise and super sensor’s uncertainty (png, pdf)

+
+ + +

+The corresponding maximum norms of the filters to have the perf/robust asked are shown in Fig. 53. +

+ +
+

upper_bound_complementary_filters_perf_robust.png +

+

Figure 53: Upper bounds on the complementary filters (png, pdf)

+
+
+
+ +
+

6.4 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');
+
+
+ +
+[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
+
+ +
+
H1 = 1 - H2;
+
+
+ +

+The obtained complementary filters with the upper bounds are shown in Fig. 54. +

+ +
+

hinf_result_comp_filters_4_outputs.png +

+

Figure 54: caption (png, pdf)

+
+ + + + +
+

upper_bounds_perf_robust_result_4_outputs.png +

+

Figure 55: Obtained PSD and uncertainty with the corresponding upper bounds (png, pdf)

+
+ + +
+

4outputs_hinf_psd_cps2svg.png +

+

Figure 56: PSD and CPS (png, pdf)

+
+ + + +
+

4outputs_uncertainty.png +

+

Figure 57: Dynamical uncertainty (png, pdf)

+
+
+
+ +
+

6.5 Conclusion

+
+

+The \(\mathcal{H}_\infty\) synthesis has been used to design complementary filters that permits to robustly merge sensors while ensuring a maximum noise level. +However, no guarantee is made that the RMS value of the super sensor’s noise is minimized. +

+
+
+
+ +
+

7 Equivalent Super Sensor

+
+

+ +

+

+The goal here is to find the parameters of a single sensor that would best represent a super sensor. +

+
+
+

7.1 Sensor Fusion Architecture

+
+

+Let consider figure 58 where two sensors are merged. +The dynamic uncertainty of each sensor is represented by a weight \(w_i(s)\), the frequency characteristics each of the sensor noise is represented by the weights \(N_i(s)\). +The noise sources \(\tilde{n}_i\) are considered to be white noise: \(\Phi_{\tilde{n}_i}(\omega) = 1, \ \forall\omega\). +

+ + +
+

sensor_fusion_full.png +

+

Figure 58: Sensor fusion architecture (png, pdf).

+
+ + +\begin{align*} + \hat{x} &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\ + &\quad \quad + \Big(H_1(s) \big(1 + w_1(s) \Delta_1(s)\big) + H_2(s) \big(1 + w_2(s) \Delta_2(s)\big)\Big) x \\ + &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\ + &\quad \quad + \big(1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)\big) x +\end{align*} + +

+To the dynamics of the super sensor is: +

+\begin{equation} + \frac{\hat{x}}{x} = 1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s) +\end{equation} + +

+And the noise of the super sensor is: +

+\begin{equation} + n_{ss} = H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 +\end{equation} +
+
+ +
+

7.2 Equivalent Configuration

+
+

+We try to determine \(w_{ss}(s)\) and \(N_{ss}(s)\) such that the sensor on figure 59 is equivalent to the super sensor of figure 58. +

+ + +
+

sensor_fusion_equivalent.png +

+

Figure 59: Equivalent Super Sensor (png, pdf).

+
+
+
+ +
+

7.3 Model the uncertainty of the super sensor

+
+

+At each frequency \(\omega\), the uncertainty set of the super sensor shown on figure 58 is a circle centered on \(1\) with a radius equal to \(|H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|\) on the complex plane. +The uncertainty set of the sensor shown on figure 59 is a circle centered on \(1\) with a radius equal to \(|w_{ss}(j\omega)|\) on the complex plane. +

+ +

+Ideally, we want to find a weight \(w_{ss}(s)\) so that: +

+
+

+\[ |w_{ss}(j\omega)| = |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \] +

+ +
+
+
+ +
+

7.4 Model the noise of the super sensor

+
+

+The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 58 is: +

+\begin{align*} + \Phi_{\hat{x}}(\omega) &= | H_1(j\omega) N_1(j\omega) |^2 \Phi_{\tilde{n}_1} + | H_2(j\omega) N_2(j\omega) |^2 \Phi_{\tilde{n}_2} \\ + &= | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 +\end{align*} + +

+The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 59 is: +

+\begin{align*} + \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\ + &= | N_{ss}(j\omega) |^2 +\end{align*} + +

+Ideally, we want to find a weight \(N_{ss}(s)\) such that: +

+
+

+\[ |N_{ss}(j\omega)|^2 = | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \] +

+ +
+
+
+ +
+

7.5 First guess

+
+

+We could choose +

+\begin{align*} + w_{ss}(s) &= H_1(s) w_1(s) + H_2(s) w_2(s) \\ + N_{ss}(s) &= H_1(s) N_1(s) + H_2(s) N_2(s) +\end{align*} + +

+But we would have: +

+\begin{align*} + |w_{ss}(j\omega)| &= |H_1(j\omega) w_1(j\omega) + H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \\ + &\neq |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega +\end{align*} +

+and +

+\begin{align*} + |N_{ss}(j\omega)|^2 &= | H_1(j\omega) N_1(j\omega) + H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\ + &\neq | H_1(j\omega) N_1(j\omega)|^2 + |H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\ +\end{align*} +
+
+
+ +
+

8 Optimal And Robust Sensor Fusion in Practice

+
+

+ +

+

+Here are the steps in order to apply optimal and robust sensor fusion: +

+ +
    +
  • Measure the noise characteristics of the sensors to be merged (necessary for “optimal” part of the fusion)
    • +
  • Measure/Estimate the dynamic uncertainty of the sensors (necessary for “robust” part of the fusion)
    • +
  • Apply H2/H-infinity synthesis of the complementary filters
    • +
+
+
+

8.1 Measurement of the noise characteristics of the sensors

+
+
+
+

8.1.1 Huddle Test

+
+

+The technique to estimate the sensor noise is taken from (Barzilai, VanZandt, and Kenny 1998). +

+ +

+Let’s consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity \(x\) as shown in figure 60. +

+ + +
+

huddle_test.png +

+

Figure 60: Huddle test block diagram

+
+ +

+Each sensor has uncorrelated noise \(n_1\) and \(n_2\) and internal dynamics \(G_1(s)\) and \(G_2(s)\) respectively. +

+ +

+We here suppose that each sensor has the same magnitude of instrumental noise: \(n_1 = n_2 = n\). +We also assume that their dynamics is ideal: \(G_1(s) = G_2(s) = 1\). +

+ +

+We then have: +

+\begin{equation} +\label{orgdad26a1} + \gamma_{\hat{x}_1\hat{x}_2}^2(\omega) = \frac{1}{1 + 2 \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right) + \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right)^2} +\end{equation} + +

+Since the input signal \(x\) and the instrumental noise \(n\) are incoherent: +

+\begin{equation} +\label{org320aa52} + |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)| +\end{equation} + +

+From equations \eqref{eq:coh_bis} and \eqref{eq:incoherent_noise}, we finally obtain +

+
+\begin{equation} +\label{org57c4428} + |\Phi_n(\omega)| = |\Phi_{\hat{x}}(\omega)| \left( 1 - \sqrt{\gamma_{\hat{x}_1\hat{x}_2}^2(\omega)} \right) +\end{equation} + +
+
+
+ +
+

8.1.2 Weights that represents the noises’ PSD

+
+

+For further complementary filter synthesis, it is preferred to consider a normalized noise source \(\tilde{n}\) that has a PSD equal to one (\(\Phi_{\tilde{n}}(\omega) = 1\)) and to use a weighting filter \(N(s)\) in order to represent the frequency dependence of the noise. +

+ +

+The weighting filter \(N(s)\) should be designed such that: +

+\begin{align*} + & \Phi_n(\omega) \approx |N(j\omega)|^2 \Phi_{\tilde{n}}(\omega) \quad \forall \omega \\ + \Longleftrightarrow & |N(j\omega)| \approx \sqrt{\Phi_n(\omega)} \quad \forall \omega +\end{align*} + +

+These weighting filters can then be used to compare the noise level of sensors for the synthesis of complementary filters. +

+ +

+The sensor with a normalized noise input is shown in figure 61. +

+ + +
+

one_sensor_normalized_noise.png +

+

Figure 61: One sensor with normalized noise

+
+
+
+ +
+

8.1.3 Comparison of the noises’ PSD

+
+

+Once the noise of the sensors to be merged have been characterized, the power spectral density of both sensors have to be compared. +

+ +

+Ideally, the PSD of the noise are such that: +

+\begin{align*} + \Phi_{n_1}(\omega) &< \Phi_{n_2}(\omega) \text{ for } \omega < \omega_m \\ + \Phi_{n_1}(\omega) &> \Phi_{n_2}(\omega) \text{ for } \omega > \omega_m +\end{align*} +
+
+ +
+

8.1.4 Computation of the coherence, power spectral density and cross spectral density of signals

+
+

+The coherence between signals \(x\) and \(y\) is defined as follow +\[ \gamma^2_{xy}(\omega) = \frac{|\Phi_{xy}(\omega)|^2}{|\Phi_{x}(\omega)| |\Phi_{y}(\omega)|} \] +where \(|\Phi_x(\omega)|\) is the output Power Spectral Density (PSD) of signal \(x\) and \(|\Phi_{xy}(\omega)|\) is the Cross Spectral Density (CSD) of signal \(x\) and \(y\). +

+ +

+The PSD and CSD are defined as follow: +

+\begin{align} + |\Phi_x(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} \left| X_k(\omega, T) \right|^2 \\ + |\Phi_{xy}(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} [ X_k^*(\omega, T) ] [ Y_k(\omega, T) ] +\end{align} +

+where: +

+
    +
  • \(n_d\) is the number for records averaged
    • +
  • \(T\) is the length of each record
    • +
  • \(X_k(\omega, T)\) is the finite Fourier transform of the \(k^{\text{th}}\) record
    • +
  • \(X_k^*(\omega, T)\) is its complex conjugate
    • +
+
+
+
+ +
+

8.2 Estimate the dynamic uncertainty of the sensors

+
+

+Let’s consider one sensor represented on figure 62. +

+ +

+The dynamic uncertainty is represented by an input multiplicative uncertainty where \(w(s)\) is a weight that represents the level of the uncertainty. +

+ +

+The goal is to accurately determine \(w(s)\) for the sensors that have to be merged. +

+ + +
+

one_sensor_dyn_uncertainty.png +

+

Figure 62: Sensor with dynamic uncertainty

+
+
+
+ +
+

8.3 Optimal and Robust synthesis of the complementary filters

+
+

+Once the noise characteristics and dynamic uncertainty of both sensors have been determined and we have determined the following weighting functions: +

+
    +
  • \(w_1(s)\) and \(w_2(s)\) representing the dynamic uncertainty of both sensors
    • +
  • \(N_1(s)\) and \(N_2(s)\) representing the noise characteristics of both sensors
    • +
+ +

+The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in figure 58 such that: +

+
    +
  • the uncertainty on the super sensor dynamics is minimized
    • +
  • the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the lowest possible effect on the estimation \(\hat{x}\)
    • +
+ + +
+

sensor_fusion_full.png +

+

Figure 63: Sensor fusion architecture with sensor dynamics uncertainty

+
+
+
+
+ +
+

9 Methods of complementary filter synthesis

+
+

+ +

+
+
+

9.1 Complementary filters using analytical formula

+
+

+ +

+
+

+All the files (data and Matlab scripts) are accessible here. +

+ +
+
+ +
+

9.1.1 Analytical 1st order complementary filters

+
+

+First order complementary filters are defined with following equations: +

+\begin{align} + H_L(s) = \frac{1}{1 + \frac{s}{\omega_0}}\\ + H_H(s) = \frac{\frac{s}{\omega_0}}{1 + \frac{s}{\omega_0}} +\end{align} + +

+Their bode plot is shown figure 64. +

+ +
+
w0 = 2*pi; % [rad/s]
+
+Hh1 = (s/w0)/((s/w0)+1);
+Hl1 = 1/((s/w0)+1);
+
+
+ + +
+

comp_filter_1st_order.png +

+

Figure 64: Bode plot of first order complementary filter (png, pdf)

+
+
+
+ +
+

9.1.2 Second Order Complementary Filters

+
+

+We here use analytical formula for the complementary filters \(H_L\) and \(H_H\). +

+ +

+The first two formulas that are used to generate complementary filters are: +

+\begin{align*} + H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\ + H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)} +\end{align*} +

+where: +

+
    +
  • \(\omega_0\) is the blending frequency in rad/s.
    • +
  • \(\alpha\) is used to change the shape of the filters: +
      +
    • Small values for \(\alpha\) will produce high magnitude of the filters \(|H_L(j\omega)|\) and \(|H_H(j\omega)|\) near \(\omega_0\) but smaller value for \(|H_L(j\omega)|\) above \(\approx 1.5 \omega_0\) and for \(|H_H(j\omega)|\) below \(\approx 0.7 \omega_0\)
      • +
    • A large \(\alpha\) will do the opposite
      • +
    • +
+ +

+This is illustrated on figure 65. +The slope of those filters at high and low frequencies is \(-2\) and \(2\) respectively for \(H_L\) and \(H_H\). +

+ + +
+

comp_filters_param_alpha.png +

+

Figure 65: Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (png, pdf)

+
+ +

+We now study the maximum norm of the filters function of the parameter \(\alpha\). As we saw that the maximum norm of the filters is important for the robust merging of filters. +

+
+
figure;
+plot(alphas, infnorms)
+set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
+xlabel('$\alpha$'); ylabel('$\|H_1\|_\infty$');
+
+
+ + +
+

param_alpha_hinf_norm.png +

+

Figure 66: Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (png, pdf)

+
+
+
+ +
+

9.1.3 Third Order Complementary Filters

+
+

+The following formula gives complementary filters with slopes of \(-3\) and \(3\): +

+\begin{align*} + H_L(s) &= \frac{\left(1+(\alpha+1)(\beta+1)\right) (\frac{s}{\omega_0})^2 + (1+\alpha+\beta)(\frac{s}{\omega_0}) + 1}{\left(\frac{s}{\omega_0} + 1\right) \left( (\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1 \right) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)}\\ + H_H(s) &= \frac{(\frac{s}{\omega_0})^3 \left( (\frac{s}{\omega_0})^2 + (1+\alpha+\beta) (\frac{s}{\omega_0}) + (1+(\alpha+1)(\beta+1)) \right)}{\left(\frac{s}{\omega_0} + 1\right) \left( (\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1 \right) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)} +\end{align*} + +

+The parameters are: +

+
    +
  • \(\omega_0\) is the blending frequency in rad/s
    • +
  • \(\alpha\) and \(\beta\) that are used to change the shape of the filters similarly to the parameter \(\alpha\) for the second order complementary filters
    • +
+ +

+The filters are defined below and the result is shown on figure 67. +

+ +
+
alpha = 1;
+beta = 10;
+w0 = 2*pi*14;
+
+Hh3_ana = (s/w0)^3 * ((s/w0)^2 + (1+alpha+beta)*(s/w0) + (1+(alpha+1)*(beta+1)))/((s/w0 + 1)*((s/w0)^2+alpha*(s/w0)+1)*((s/w0)^2+beta*(s/w0)+1));
+Hl3_ana = ((1+(alpha+1)*(beta+1))*(s/w0)^2 + (1+alpha+beta)*(s/w0) + 1)/((s/w0 + 1)*((s/w0)^2+alpha*(s/w0)+1)*((s/w0)^2+beta*(s/w0)+1));
+
+
+ + +
+

complementary_filters_third_order.png +

+

Figure 67: Third order complementary filters using the analytical formula (png, pdf)

+
+
+
+
+ +
+

9.2 H-Infinity synthesis of complementary filters

+
+

+ +

+
+

+All the files (data and Matlab scripts) are accessible here. +

+ +
+
+ +
+

9.2.1 Synthesis Architecture

+
+

+We here synthesize the complementary filters using the \(\mathcal{H}_\infty\) synthesis. +The goal is to specify upper bounds on the norms of \(H_L\) and \(H_H\) while ensuring their complementary property (\(H_L + H_H = 1\)). +

+ +

+In order to do so, we use the generalized plant shown on figure 68 where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively. +

+ + +
+

sf_hinf_filters_plant_b.png +

+

Figure 68: Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters

+
+ +

+The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure 69) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one: +\[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \] +

+ +

+Thus, if the above condition is verified, we can define \(H_H = 1 - H_L\) and we have that: +\[ \left\| \begin{array}{c} H_L w_L \\ H_H w_H \end{array} \right\|_\infty < 1 \] +Which is almost (with an maximum error of \(\sqrt{2}\)) equivalent to: +

+\begin{align*} + |H_L| &< \frac{1}{|w_L|}, \quad \forall \omega \\ + |H_H| &< \frac{1}{|w_H|}, \quad \forall \omega +\end{align*} + +

+We then see that \(w_L\) and \(w_H\) can be used to shape both \(H_L\) and \(H_H\) while ensuring (by definition of \(H_H = 1 - H_L\)) their complementary property. +

+ + +
+

sf_hinf_filters_b.png +

+

Figure 69: \(\mathcal{H}_\infty\) synthesis of the complementary filters

+
+
+
+ +
+

9.2.2 Weights

+
+
+
omegab = 2*pi*9;
+wH = (omegab)^2/(s + omegab*sqrt(1e-5))^2;
+omegab = 2*pi*28;
+wL = (s + omegab/(4.5)^(1/3))^3/(s*(1e-4)^(1/3) + omegab)^3;
+
+
+ + +
+

weights_wl_wh.png +

+

Figure 70: Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (png, pdf)

+
+
+
+ +
+

9.2.3 H-Infinity Synthesis

+
+

+We define the generalized plant \(P\) on matlab. +

+
+
P = [0   wL;
+     wH -wH;
+     1   0];
+
+
+ +

+And we do the \(\mathcal{H}_\infty\) synthesis using the hinfsyn command. +

+
+
[Hl_hinf, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+
+ +
+[Hl_hinf, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Test bounds:      0.0000 <  gamma  <=      1.7285
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+    1.729   4.1e+01   8.4e-12   1.8e-01    0.0e+00    0.0000    p
+    0.864   3.9e+01 -5.8e-02#  1.8e-01    0.0e+00    0.0000    f
+    1.296   4.0e+01   8.4e-12   1.8e-01    0.0e+00    0.0000    p
+    1.080   4.0e+01   8.5e-12   1.8e-01    0.0e+00    0.0000    p
+    0.972   3.9e+01 -4.2e-01#  1.8e-01    0.0e+00    0.0000    f
+    1.026   4.0e+01   8.5e-12   1.8e-01    0.0e+00    0.0000    p
+    0.999   3.9e+01   8.5e-12   1.8e-01    0.0e+00    0.0000    p
+    0.986   3.9e+01 -1.2e+00#  1.8e-01    0.0e+00    0.0000    f
+    0.993   3.9e+01 -8.2e+00#  1.8e-01    0.0e+00    0.0000    f
+    0.996   3.9e+01   8.5e-12   1.8e-01    0.0e+00    0.0000    p
+    0.994   3.9e+01   8.5e-12   1.8e-01    0.0e+00    0.0000    p
+    0.993   3.9e+01 -3.2e+01#  1.8e-01    0.0e+00    0.0000    f
+
+ Gamma value achieved:     0.9942
+
+ +

+We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure 71. +

+
+
Hh_hinf = 1 - Hl_hinf;
+
+
+
+
+ +
+

9.2.4 Obtained Complementary Filters

+
+

+The obtained complementary filters are shown on figure 71. +

+ + +
+

hinf_filters_results.png +

+

Figure 71: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (png, pdf)

+
+
+
+
+ +
+

9.3 Feedback Control Architecture to generate Complementary Filters

+
+

+ +

+

+The idea is here to use the fact that in a classical feedback architecture, \(S + T = 1\), in order to design complementary filters. +

+ +

+Thus, all the tools that has been developed for classical feedback control can be used for complementary filter design. +

+
+

+All the files (data and Matlab scripts) are accessible here. +

+ +
+
+ +
+

9.3.1 Architecture

+
+ +
+

complementary_filters_feedback_architecture.png +

+

Figure 72: Architecture used to generate the complementary filters

+
+ +

+We have: +\[ y = \underbrace{\frac{L}{L + 1}}_{H_L} y_1 + \underbrace{\frac{1}{L + 1}}_{H_H} y_2 \] +with \(H_L + H_H = 1\). +

+ +

+The only thing to design is \(L\) such that the complementary filters are stable with the wanted shape. +

+ +

+A simple choice is: +\[ L = \left(\frac{\omega_c}{s}\right)^2 \frac{\frac{s}{\omega_c / \alpha} + 1}{\frac{s}{\omega_c} + \alpha} \] +

+ +

+Which contains two integrator and a lead. \(\omega_c\) is used to tune the crossover frequency and \(\alpha\) the trade-off “bump” around blending frequency and filtering away from blending frequency. +

+
+
+ +
+

9.3.2 Loop Gain Design

+
+

+Let’s first define the loop gain \(L\). +

+
+
wc = 2*pi*1;
+alpha = 2;
+
+L = (wc/s)^2 * (s/(wc/alpha) + 1)/(s/wc + alpha);
+
+
+ + +
+

loop_gain_bode_plot.png +

+

Figure 73: Bode plot of the loop gain \(L\) (png, pdf)

+
+
+
+ +
+

9.3.3 Complementary Filters Obtained

+
+

+We then compute the resulting low pass and high pass filters. +

+
+
Hl = L/(L + 1);
+Hh = 1/(L + 1);
+
+
+ + +
+

low_pass_high_pass_filters.png +

+

Figure 74: Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (png, pdf)

+
+
+
+
+ +
+

9.4 Analytical Formula found in the literature

+
+

+ +

+
+ +
+

9.4.1 Analytical Formula

+
+

+(Min and Jeung 2015) +

+\begin{align*} + H_L(s) = \frac{K_p s + K_i}{s^2 + K_p s + K_i} \\ + H_H(s) = \frac{s^2}{s^2 + K_p s + K_i} +\end{align*} + +

+(Corke 2004) +

+\begin{align*} + H_L(s) = \frac{1}{s/p + 1} \\ + H_H(s) = \frac{s/p}{s/p + 1} +\end{align*} + +

+(Jensen, Coopmans, and Chen 2013) +

+\begin{align*} + H_L(s) = \frac{2 \omega_0 s + \omega_0^2}{(s + \omega_0)^2} \\ + H_H(s) = \frac{s^2}{(s + \omega_0)^2} +\end{align*} + +\begin{align*} + H_L(s) = \frac{C(s)}{C(s) + s} \\ + H_H(s) = \frac{s}{C(s) + s} +\end{align*} + +

+(Shaw and Srinivasan 1990) +

+\begin{align*} + H_L(s) = \frac{3 \tau s + 1}{(\tau s + 1)^3} \\ + H_H(s) = \frac{\tau^3 s^3 + 3 \tau^2 s^2}{(\tau s + 1)^3} +\end{align*} + +

+(Baerveldt and Klang 1997) +

+\begin{align*} + H_L(s) = \frac{2 \tau s + 1}{(\tau s + 1)^2} \\ + H_H(s) = \frac{\tau^2 s^2}{(\tau s + 1)^2} +\end{align*} +
+
+ +
+

9.4.2 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;
+
+
+ + +
+

comp_filters_literature.png +

+

Figure 75: Comparison of some complementary filters found in the literature (png, pdf)

+
+
+
+ +
+

9.4.3 Discussion

+
+

+Analytical Formula found in the literature provides either no parameter for tuning the robustness / performance trade-off. +

+
+
+
+ +
+

9.5 Comparison of the different methods of synthesis

+
+

+ +The generated complementary filters using \(\mathcal{H}_\infty\) and the analytical formulas are very close to each other. However there is some difference to note here: +

+
    +
  • the analytical formula provides a very simple way to generate the complementary filters (and thus the controller), they could even be used to tune the controller online using the parameters \(\alpha\) and \(\omega_0\). However, these formula have the property that \(|H_H|\) and \(|H_L|\) are symmetrical with the frequency \(\omega_0\) which may not be desirable.
    • +
  • while the \(\mathcal{H}_\infty\) synthesis of the complementary filters is not as straightforward as using the analytical formula, it provides a more optimized procedure to obtain the complementary filters
    • +
+
+
+
+ +
+

10 Real World Example of optimal sensor fusion

+
+

+(Moore, Fleming, and Yong 2019) +

+
+ + +
+

10.1 Matlab Code

+
+

+Take an Accelerometer and a Geophone both measuring the absolute motion of a structure. +

+ +

+Parameters of the inertial sensors. +

+
+
m_acc = 0.01;
+k_acc = 1e6;
+c_acc = 20;
+
+m_geo = 1;
+k_geo = 1e3;
+c_geo = 10;
+
+
+ +

+Transfer function from motion to measurement +

+ +

+For the accelerometer. +The measurement is the relative motion structure/inertial mass: +\[ \frac{d}{\ddot{w}} = \frac{-m}{ms^2 + cs + k} \] +

+ +

+For the geophone. +The measurement is the relative velocity structure/inertial mass: +\[ \frac{\dot{d}}{\dot{w}} = \frac{-ms^2}{ms^2 + cs + k} \] +

+ +
+
G_acc = -m_acc/(m_acc*s^2 + c_acc*s + k_acc); % [m/(m/s^2)]
+G_geo = -m_geo*s^2/(m_geo*s^2 + c_geo*s + k_geo); % [m/s/m/s]
+
+
+ +

+Suppose the measure of the relative motion for the accelerometer (capacitive sensor for instance) has a white noise characteristic: +Suppose the measure of the relative velocity (current flowing through the coil) has a white noise characteristic: +

+ +

+Define the noise characteristics +

+
+
n = 1; w0 = 2*pi*5e3; G0 = 5e-12; G1 = 1e-15; Gc = G0/2;
+L_acc = (((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;
+
+n = 1; w0 = 2*pi*5e3; G0 = 1e-6; G1 = 1e-8; Gc = G0/2;
+L_geo = (((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;
+
+
+ +

+Transfer function of the conversion to obtain the velocity: +

+
+
C_acc = (-k_acc/m_acc/(2*pi + s));
+C_geo = tf(-1);
+
+
+ +

+Let’s plot the noise of both sensors: +Dynamics of both sensors +

+
+
+
+

10.2 Time domain signals

+
+
+
Fs = 1e4; % Sampling Frequency [Hz]
+Ts = 1/Fs; % Sampling Time [s]
+
+t = 0:Ts:10; % Time Vector [s]
+
+
+ +
+
n_acc = lsim(L_acc*C_acc, sqrt(Fs/2)*randn(length(t), 1), t); % [m/s]
+n_geo = lsim(L_geo*C_geo, sqrt(Fs/2)*randn(length(t), 1), t); % [m/s]
+
+
+ +
+
figure;
+hold on;
+plot(t, n_geo)
+plot(t, n_acc)
+hold off;
+
+
+
+
+ +
+

10.3 H2 Synthesis

+
+
+
N1 = L_acc*C_acc;
+N2 = L_geo*C_geo;
+
+
+ +
+
bodeFig({N1, N2}, logspace(-1, 5, 1000))
+
+
+ +
+
P = [0   N2  1;
+     N1 -N2  0];
+
+
+ +

+And we do the \(\mathcal{H}_2\) synthesis using the h2syn command. +

+
+
[H1, ~, gamma] = h2syn(P, 1, 1);
+
+
+ +

+Finally, we define \(H_2(s) = 1 - H_1(s)\). +

+
+
H2 = 1 - H1;
+
+
+ +
+
bodeFig({H1, H2}, struct('phase', true))
+
+
+ +
+
n_acc_filt = lsim(H1, n_acc, t);
+n_geo_filt = lsim(H2, n_geo, t);
+
+
+ + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + +
 RMS
Accelerometer9.7e-05
Geophone5.9e-05
Super Sensor1.5e-05
+ +
+
figure;
+hold on;
+plot(t, n_geo)
+plot(t, n_acc)
+plot(t, n_acc_filt + n_geo_filt)
+hold off;
+
+
+
+
+ +
+

10.4 Signal and Noise

+
+

+Velocity Signal: +

+
+
v = lsim(1/(1 + s/2/pi/2), 1e-4*sqrt(Fs/2)*randn(length(t), 1), t);
+v = 1e-4 * sin(2*pi*100*t);
+
+
+ +
+
v_acc = lsim(s*G_acc*C_acc, v, t) + n_acc;
+v_geo = lsim(G_geo*C_geo,   v, t) + n_geo;
+
+
+ +
+
v_ss = lsim(H1, v_acc, t) + lsim(H2, v_geo, t);
+
+
+ +
+
figure;
+hold on;
+plot(t, v_geo)
+plot(t, v_acc)
+plot(t, v_ss)
+plot(t, v, 'k--')
+hold off;
+xlim([1, 1+0.1])
+
+
+
+
+ +
+

10.5 PSD and CPS

+
+
+
nx = length(n_acc);
+na = 16;
+win = hanning(floor(nx/na));
+
+[p_acc, f] = pwelch(n_acc, win, 0, [], Fs);
+[p_geo, ~] = pwelch(n_geo, win, 0, [], Fs);
+[p_ss,  ~] = pwelch(n_acc_filt + n_geo_filt, win, 0, [], Fs);
+
+
+
+
+ +
+

10.6 Transfer function of the super sensor

+
+
+
bodeFig({s*C_acc*G_acc, C_geo*G_geo, s*C_acc*G_acc*H1+C_geo*G_geo*H2}, struct('phase', true))
+
+
+
+
+
+ +

+ +

+ +

Bibliography

+
+
Baerveldt, A.-J., and R. Klang. 1997. “A Low-Cost and Low-Weight Attitude Estimation System for an Autonomous Helicopter.” In Proceedings of IEEE International Conference on Intelligent Engineering Systems, nil. https://doi.org/10.1109/ines.1997.632450.
+
Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” Review of Scientific Instruments 69 (7):2767–72. https://doi.org/10.1063/1.1149013.
+
Bendat, J. 1957. “Optimum Filters for Independent Measurements of Two Related Perturbed Messages.” IRE Transactions on Circuit Theory, --. https://doi.org/10.1109/tct.1957.1086345.
+
Corke, Peter. 2004. “An Inertial and Visual Sensing System for a Small Autonomous Helicopter.” Journal of Robotic Systems 21 (2):43–51. https://doi.org/10.1002/rob.10127.
+
Jensen, Austin, Cal Coopmans, and YangQuan Chen. 2013. “Basics and Guidelines of Complementary Filters for Small UAS Navigation.” In 2013 International Conference on Unmanned Aircraft Systems (ICUAS), nil. https://doi.org/10.1109/icuas.2013.6564726.
+
Min, Hyung Gi, and Eun Tae Jeung. 2015. “Complementary Filter Design for Angle Estimation Using Mems Accelerometer and Gyroscope.” Department of Control and Instrumentation, Changwon National University, Changwon, Korea, 641–773.
+
Moore, Steven Ian, Andrew J. Fleming, and Yuen Kuan Yong. 2019. “Capacitive Instrumentation and Sensor Fusion for High-Bandwidth Nanopositioning.” IEEE Sensors Letters 3 (8):1–3. https://doi.org/10.1109/lsens.2019.2933065.
+
Plummer, A. R. 2006. “Optimal Complementary Filters and Their Application in Motion Measurement.” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 220 (6):489–507. https://doi.org/10.1243/09596518JSCE229.
+
Robert Grover Brown, Patrick Y. C. Hwang. 2012. Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises. 4th ed. Wiley.
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Shaw, F.R., and K. Srinivasan. 1990. “Bandwidth Enhancement of Position Measurements Using Measured Acceleration.” Mechanical Systems and Signal Processing 4 (1):23–38. https://doi.org/10.1016/0888-3270(90)90038-m.
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Author: Thomas Dehaeze

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Created: 2020-09-28 lun. 17:27

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+ + diff --git a/matlab/index_old.org b/matlab/index_old.org new file mode 100644 index 0000000..9a5704b --- /dev/null +++ b/matlab/index_old.org @@ -0,0 +1,4914 @@ +#+TITLE: Robust and Optimal Sensor Fusion - Matlab Computation +:DRAWER: +#+HTML_LINK_HOME: ../index.html +#+HTML_LINK_UP: ../index.html + +#+LATEX_CLASS: cleanreport +#+LATEX_CLASS_OPTIONS: [tocnp, secbreak, minted] + +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :tangle no +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :exports both +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :noweb yes +#+PROPERTY: header-args:matlab+ :mkdirp yes +#+PROPERTY: header-args:matlab+ :output-dir figs +:END: + +* Introduction :ignore: +In this document, the optimal and robust design of complementary filters is studied. + +Two sensors are considered with both different noise characteristics and dynamical uncertainties represented by multiplicative input uncertainty. + +- in section [[sec:optimal_comp_filters]]: the $\mathcal{H}_2$ synthesis is used to design complementary filters such that the RMS value of the super sensor's noise is minimized +- in section [[sec:comp_filter_robustness]]: the $\mathcal{H}_\infty$ synthesis is used to design complementary filters such that the super sensor's uncertainty is bonded to acceptable values +- in section [[sec:mixed_synthesis_sensor_fusion]]: the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is used to both limit the super sensor's uncertainty and to lower the RMS value of the super sensor's noise +- in section [[sec:hinf_syn_perf_robust]]: the $\mathcal{H}_\infty$ synthesis is used for both limiting the noise and uncertainty of the super sensor +- in section [[sec:equi_super_sensor]]: we try to find the characteristics of the super sensor from the characteristics of the individual sensors and of the complementary filters +- in section [[sec:opti_robust_practice]]: a methodology is proposed to apply optimal and robust sensor fusion in practice +- in section [[sec:comp_filter_synthesis]]: methods of complementary filter synthesis are proposed + +* Comparison with Bibliographic example +** Bendat, J., Optimum filters for independent measurements of two related perturbed messages (1957) +cite:bendat57_optim_filter_indep_measur_two + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 2, 1000); +#+end_src + +Weights to shape the noise of both sensors: +#+begin_src matlab + K1 = 100; + K2 = 1; + b = 10; + b1 = b; + b2 = b; + + N1 = sqrt(K1)*b1/(b1+s)/(s + 1e-2); + N2 = sqrt(K2)*b2/(b2+s); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$N_1$'); + plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$N_2$'); + 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 + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/bendat57_noise_weights.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bendat57_noise_weights +#+caption: Weights +#+RESULTS: +[[file:figs/bendat57_noise_weights.png]] + +$\mathcal{H}_2$ synthesis: +#+begin_src matlab + P = [0 N2 1; + N1 -N2 0]; + [H1, ~, gamma] = h2syn(P, 1, 1); + H2 = 1 - H1; +#+end_src + +The optimal obtained filter (from the paper) is: +#+begin_src matlab + a = sqrt(K2/K1); + G = (a*s + 1 + a*b)/(1 + a*b)/(a*s + 1); +#+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', '$H_2$'); + plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k--', 'DisplayName', '$G_2$'); + plot(freqs, abs(squeeze(freqresp(1-G, freqs, 'Hz'))), 'k--', 'DisplayName', '$1-G_2$'); + 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 + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/bendat57_optimal_filters.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bendat57_optimal_filters +#+caption: Obtain Filters +#+RESULTS: +[[file:figs/bendat57_optimal_filters.png]] + +#+begin_src matlab :exports none + 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; + PSD_G = abs(squeeze(freqresp(N1*(1-G), freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*G, freqs, 'Hz'))).^2; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$'); + plot(freqs, abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$'); + plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$'); + plot(freqs, PSD_G, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty}}$'); + 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 :tangle no :exports results :results file replace + exportFig('figs/bendat57_psd_estimation.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bendat57_psd_estimation +#+caption: PSD of the individual sensors + super sensor +#+RESULTS: +[[file:figs/bendat57_psd_estimation.png]] + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([trapz(freqs, PSD_S1), trapz(freqs, PSD_S2), trapz(freqs, PSD_H2), trapz(freqs, PSD_G)]', {'Sensor 1', 'Sensor 2', 'H2 Synthesis', 'Paper'}, {'RMS'}, ' %.2f '); +#+end_src + +#+RESULTS: +| | RMS | +|--------------+-------| +| Sensor 1 | 22.93 | +| Sensor 2 | 2.37 | +| H2 Synthesis | 1.74 | +| Paper | 1.74 | + +** Plummer, A. R., Optimal complementary filters and their application in motion measurement (2006) +cite:plummer06_optim_compl_filter_their_applic_motion_measur + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); +#+end_src + +Weights +#+begin_src matlab + N1 = 24.3e-6*(s + 2*pi*0.1)*(s + 1220)/1220*(1/(1 + s/2/pi/1e4)/(1 + s/2/pi/1e4)); + N2 = 0.363/(s + 0.01)*(s + 12.2)/(s + 0.01); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$N_1$'); + plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$N_2$'); + 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 + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/plummer06_noise_weights.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:plummer06_noise_weights +#+caption: Weights +#+RESULTS: +[[file:figs/plummer06_noise_weights.png]] + +$\mathcal{H}_2$ synthesis: +#+begin_src matlab + P = [0 N2 1; + N1 -N2 0]; + + [H1, ~, gamma] = h2syn(P, 1, 1); + + H2 = 1 - H1; +#+end_src + +The optimal obtained filter (from the paper) is: +#+begin_src matlab + G = (0.0908*s + 1)/(5.51e-7*s^3 + 7.47e-4*s^2 + 0.0908*s + 1); +#+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,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), '--', 'DisplayName', '$G$'); + plot(freqs, abs(squeeze(freqresp(1-G, freqs, 'Hz'))), '--', 'DisplayName', '$1-G$'); + 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 + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/plummer06_optimal_filters.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:plummer06_optimal_filters +#+caption: Obtain Filters +#+RESULTS: +[[file:figs/plummer06_optimal_filters.png]] + +#+begin_src matlab :exports none + 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; + PSD_G = abs(squeeze(freqresp(N1*G, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*(1-G), freqs, 'Hz'))).^2; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$'); + plot(freqs, abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$'); + plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$'); + plot(freqs, PSD_G, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty}}$'); + 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 :tangle no :exports results :results file replace + exportFig('figs/plummer06_psd_estimation.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:plummer06_psd_estimation +#+caption: PSD of the individual sensors + super sensor +#+RESULTS: +[[file:figs/plummer06_psd_estimation.png]] + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([trapz(freqs, PSD_S1), trapz(freqs, PSD_S2), trapz(freqs, PSD_H2), trapz(freqs, PSD_G)]', {'Sensor 1', 'Sensor 2', 'H2 Synthesis', 'Paper'}, {'RMS'}, ' %.2e '); +#+end_src + +#+RESULTS: +| | RMS | +|--------------+---------| +| Sensor 1 | 130.0 | +| Sensor 2 | 0.00753 | +| H2 Synthesis | 0.00091 | +| Paper | 0.0107 | + +Parameters of the time domain simulation. +#+begin_src matlab + Fs = 2.5e3; % Sampling Frequency [Hz] + Ts = 1/Fs; % Sampling Time [s] + + t = 0:Ts:2; % Time Vector [s] +#+end_src + +Generate noises in velocity corresponding to sensor 1 and 2: +#+begin_src matlab + n1 = lsim(N1, sqrt(Fs/2)*randn(length(t), 1), t); + n2 = lsim(N2, sqrt(Fs/2)*randn(length(t), 1), t); +#+end_src + +#+begin_src matlab :exports none + figure; + ax1 = subplot(3,1,1); + plot(t, n1, 'DisplayName', 'Differentiated Position'); + ylabel('Velocity [m/s]'); + legend(); + ax2 = subplot(3,1,2); + plot(t, n2, 'DisplayName', 'Integrated Acceleration'); + ylabel('Velocity [m/s]'); + legend(); + ax3 = subplot(3,1,3); + hold on; + plot(t, lsim(G, n1, t)+lsim((1-G), n2, t), 'DisplayName', 'Hinf'); + plot(t, lsim(H1, n1, t)+lsim(H2, n2, t), 'DisplayName', 'H2'); + ylabel('Velocity [m/s]'); + legend(); + hold off; + + linkaxes([ax1,ax2,ax3],'x'); + xlim([0.4,0.6]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/plummer06_time_domain_signals.pdf', 'width', 'wide', 'height', 'full'); +#+end_src + +#+name: fig:plummer06_time_domain_signals +#+caption: Time domain signals +#+RESULTS: +[[file:figs/plummer06_time_domain_signals.png]] + +** Robert Grover Brown, P. Y. C. H., Introduction to random signals and applied kalman filtering with matlab exercises (2012) +cite:robert12_introd_random_signal_applied_kalman Section 8.6 + +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-2, 2, 5000); % [rad/s] +#+end_src + +#+begin_src matlab + w0 = 1; % [rad/s] + wc = 20*w0; % [rad/s] + + k1 = sqrt(200*sqrt(2)*w0^3); % [m] + k2 = sqrt(100*pi/wc); % [m] + + N1 = k1/(s^2 + sqrt(2)*s + 1); + N2 = k2/(1 + s/(wc*(2/pi))); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(N1, freqs))), '-', 'DisplayName', '$N_1$'); + plot(freqs, abs(squeeze(freqresp(N2, freqs))), '-', 'DisplayName', '$N_2$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [rad/s]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/robert12_noise_weights.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:robert12_noise_weights +#+caption: Weights +#+RESULTS: +[[file:figs/robert12_noise_weights.png]] + +And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command. +#+begin_src matlab + P = [0 N2 1; + N1 -N2 0]; + + [H1, ~, gamma] = h2syn(P, 1, 1); + + H2 = 1 - H1; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(H1, freqs))), '-', 'DisplayName', '$H_1$'); + plot(freqs, abs(squeeze(freqresp(H2, freqs))), '-', 'DisplayName', '$H_2$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [rad/s]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/robert12_optimal_filters.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:robert12_optimal_filters +#+caption: Obtain Filters +#+RESULTS: +[[file:figs/robert12_optimal_filters.png]] + +#+begin_src matlab :exports none + 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 :tangle no :exports results :results file replace + exportFig('figs/robert12_psd_estimation.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:robert12_psd_estimation +#+caption: PSD of the individual sensors + super sensor +#+RESULTS: +[[file:figs/robert12_psd_estimation.png]] + +We can see that the optimal $\mathcal{H}_2$ control gives similar results as Kalman filtering. +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([21.47, 35.32, 2*trapz(freqs, PSD_H2)]', {'Kalman Filter', 'Euristic', 'Optimal H2'}, {'Method', 'Mean Square Error'}, ' %.2f '); +#+end_src + +#+RESULTS: +| Method | Mean Square Error | +|---------------+-------------------| +| Kalman Filter | 21.47 | +| Euristic | 35.32 | +| Optimal H2 | 20.96 | + + + +| Method | Mean Square Error | +|---------------+-------------------| +| Kalman Filter | 21.47 | +| Euristic | 35.32 | +|---------------+-------------------| +| Optimal H2 | 21.40 | + +* Optimal Sensor Fusion - Minimize the Super Sensor Noise +:PROPERTIES: +:header-args:matlab+: :tangle matlab/optimal_comp_filters.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** Introduction :ignore: +The idea is to combine sensors that works in different frequency range using complementary filters. + +Doing so, one "super sensor" is obtained that can have better noise characteristics than the individual sensors over a large frequency range. + +The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise. + +** ZIP file containing the data and matlab files :ignore: +#+begin_note + The Matlab scripts is accessible [[file:matlab/optimal_comp_filters.m][here]]. +#+end_note + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +** Architecture +Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_weights]] where two sensors (sensor 1 and sensor 2) are measuring the same quantity $x$ with different noise characteristics determined by $N_1(s)$ and $N_2(s)$. + +$\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: +#+name: eq:normalized_noise +\begin{equation} + \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1 +\end{equation} + +#+name: fig:fusion_two_noisy_sensors_weights +#+caption: Fusion of two sensors +[[file:figs-tikz/fusion_two_noisy_sensors_weights.png]] + +We consider that the two sensor dynamics $G_1(s)$ and $G_2(s)$ are ideal: +#+name: eq:idea_dynamics +\begin{equation} + G_1(s) = G_2(s) = 1 +\end{equation} + +We obtain the architecture of figure [[fig:sensor_fusion_noisy_perfect_dyn]]. + +#+name: fig:sensor_fusion_noisy_perfect_dyn +#+caption: Fusion of two sensors with ideal dynamics +[[file:figs-tikz/sensor_fusion_noisy_perfect_dyn.png]] + +$H_1(s)$ and $H_2(s)$ are complementary filters: +#+name: eq:comp_filters_property +\begin{equation} + H_1(s) + H_2(s) = 1 +\end{equation} + +The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise sources $\tilde{n}_1$ and $\tilde{n}_2$ has the smallest possible effect on the estimation $\hat{x}$. + +We have that the Power Spectral Density (PSD) of $\hat{x}$ is: +\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \] + +And the goal is the minimize the Root Mean Square (RMS) value of $\hat{x}$: +#+name: eq:rms_value_estimation +\begin{equation} + \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega} +\end{equation} + +** Noise of the sensors +Let's define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$: +- Sensor 1 characterized by $N_1(s)$ has low noise at low frequency (for instance a geophone) +- Sensor 2 characterized by $N_2(s)$ has low noise at high frequency (for instance an accelerometer) + +#+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 + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$N_1$'); + plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$N_2$'); + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/noise_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:noise_characteristics_sensors +#+CAPTION: Noise Characteristics of the two sensors ([[./figs/noise_characteristics_sensors.png][png]], [[./figs/noise_characteristics_sensors.pdf][pdf]]) +[[file:figs/noise_characteristics_sensors.png]] + +** H-Two Synthesis +As $\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: $\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1$ and we have: +\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \] +Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized. + +For that, we use the $\mathcal{H}_2$ Synthesis. + +We use the generalized plant architecture shown on figure [[fig:h_infinity_optimal_comp_filters]]. + +#+name: fig:h_infinity_optimal_comp_filters +#+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: +\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \] + +Thus, if we minimize the $\mathcal{H}_2$ norm of this transfer function, we minimize the RMS value of $\hat{x}$. + +We define the generalized plant $P$ on matlab as shown on figure [[fig:h_infinity_optimal_comp_filters]]. +#+begin_src matlab + P = [0 N2 1; + N1 -N2 0]; +#+end_src + +And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command. +#+begin_src matlab + [H1, ~, gamma] = h2syn(P, 1, 1); +#+end_src + +Finally, we define $H_2(s) = 1 - H_1(s)$. +#+begin_src matlab + H2 = 1 - H1; +#+end_src + +The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]. + +The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. [[fig:psd_sensors_htwo_synthesis]]. + +The Cumulative Power Spectrum (CPS) is shown on Fig. [[fig:cps_h2_synthesis]]. + +The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors. + +#+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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/htwo_comp_filters.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:htwo_comp_filters +#+CAPTION: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis ([[./figs/htwo_comp_filters.png][png]], [[./figs/htwo_comp_filters.pdf][pdf]]) +[[file:figs/htwo_comp_filters.png]] + +#+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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/psd_sensors_htwo_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:psd_sensors_htwo_synthesis +#+CAPTION: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal ([[./figs/psd_sensors_htwo_synthesis.png][png]], [[./figs/psd_sensors_htwo_synthesis.pdf][pdf]]) +[[file:figs/psd_sensors_htwo_synthesis.png]] + +#+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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cps_h2_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cps_h2_synthesis +#+CAPTION: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis ([[./figs/cps_h2_synthesis.png][png]], [[./figs/cps_h2_synthesis.pdf][pdf]]) +[[file:figs/cps_h2_synthesis.png]] + +** Alternative H-Two Synthesis +An alternative Alternative formulation of the $\mathcal{H}_2$ synthesis is shown in Fig. [[fig:h_infinity_optimal_comp_filters_bis]]. + +#+name: fig:h_infinity_optimal_comp_filters_bis +#+caption: Alternative formulation of the $\mathcal{H}_2$ synthesis +[[file:figs-tikz/h_infinity_optimal_comp_filters_bis.png]] + +\begin{equation*} +\begin{pmatrix} + z_1 \\ z_2 \\ v +\end{pmatrix} = \begin{pmatrix} + N_1 & -N_1 \\ + 0 & N_2 \\ + 1 & 0 +\end{pmatrix} \begin{pmatrix} + w \\ u +\end{pmatrix} +\end{equation*} + + +** H-Infinity Synthesis - method A +Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$: +\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \] + +In order to obtain that ideal case, we need that the complementary filters be designed such that: +\begin{align*} + & |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ + & |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) +\end{align*} + +Which is indeed impossible in practice. + +We could try to approach that with the $\mathcal{H}_\infty$ synthesis by using high order filters. + +As shown on Fig. [[fig:noise_characteristics_sensors]], the frequency where the two sensors have the same noise level is around 9Hz. +We will thus choose weighting functions such that the merging frequency is around 9Hz. + +The weighting functions used as well as the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfa]]. + +#+begin_src matlab + n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2; + W1a = (((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; + + n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2; + W2a = (((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; +#+end_src + +#+begin_src matlab + P = [W1a -W1a; + 0 W2a; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+RESULTS: +#+begin_example +[H2a, ~, 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: 0.1000 < gamma <= 10500.0000 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f +1.050e+04 2.1e+01 -3.0e-07 7.8e+00 -1.3e-15 0.0000 p +5.250e+03 2.1e+01 -1.5e-08 7.8e+00 -5.8e-14 0.0000 p +2.625e+03 2.1e+01 2.5e-10 7.8e+00 -3.7e-12 0.0000 p +1.313e+03 2.1e+01 -3.2e-11 7.8e+00 -7.3e-14 0.0000 p + 656.344 2.1e+01 -2.2e-10 7.8e+00 -1.1e-15 0.0000 p + 328.222 2.1e+01 -1.1e-10 7.8e+00 -1.2e-15 0.0000 p + 164.161 2.1e+01 -2.4e-08 7.8e+00 -8.9e-16 0.0000 p + 82.130 2.1e+01 2.0e-10 7.8e+00 -9.1e-31 0.0000 p + 41.115 2.1e+01 -6.8e-09 7.8e+00 -4.1e-13 0.0000 p + 20.608 2.1e+01 3.3e-10 7.8e+00 -1.4e-12 0.0000 p + 10.354 2.1e+01 -9.8e-09 7.8e+00 -1.8e-15 0.0000 p + 5.227 2.1e+01 -4.1e-09 7.8e+00 -2.5e-12 0.0000 p + 2.663 2.1e+01 2.7e-10 7.8e+00 -4.0e-14 0.0000 p + 1.382 2.1e+01 -3.2e+05# 7.8e+00 -3.5e-14 0.0000 f + 2.023 2.1e+01 -5.0e-10 7.8e+00 0.0e+00 0.0000 p + 1.702 2.1e+01 -2.4e+07# 7.8e+00 -1.6e-13 0.0000 f + 1.862 2.1e+01 -6.0e+08# 7.8e+00 -1.0e-12 0.0000 f + 1.942 2.1e+01 -2.8e-09 7.8e+00 -8.1e-14 0.0000 p + 1.902 2.1e+01 -2.5e-09 7.8e+00 -1.1e-13 0.0000 p + 1.882 2.1e+01 -9.3e-09 7.8e+00 -2.0e-15 0.0001 p + 1.872 2.1e+01 -1.3e+09# 7.8e+00 -3.6e-22 0.0000 f + 1.877 2.1e+01 -2.6e+09# 7.8e+00 -1.2e-13 0.0000 f + 1.880 2.1e+01 -5.6e+09# 7.8e+00 -1.4e-13 0.0000 f + 1.881 2.1e+01 -1.2e+10# 7.8e+00 -3.3e-12 0.0000 f + 1.882 2.1e+01 -3.2e+10# 7.8e+00 -8.5e-14 0.0001 f + + Gamma value achieved: 1.8824 +#+end_example + +#+begin_src matlab + H1a = 1 - H2a; +#+end_src + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1a, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2a, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1a, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2a, 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(H1a, freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*phase(squeeze(freqresp(H2a, 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]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfa.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:weights_comp_filters_Hinfa +#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfa.png][png]], [[./figs/weights_comp_filters_Hinfa.pdf][pdf]]) +[[file:figs/weights_comp_filters_Hinfa.png]] + +We then compute the Power Spectral Density as well as the Cumulative Power Spectrum. + +#+begin_src matlab + PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2; + CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha); +#+end_src + +** H-Infinity Synthesis - method B +We have that: +\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \] + +Then, at frequencies where $|H_1(j\omega)| < |H_2(j\omega)|$ we would like that $|N_1(j\omega)| = 1$ and $|N_2(j\omega)| = 0$ as we discussed before. +Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2$. + +We know that this is impossible in practice. A more realistic choice is to design $H_2(s)$ such that when $|N_2(j\omega)| > |N_1(j\omega)|$, we have that: +\[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \] + +Which is equivalent to have (by supposing $|H_1| \approx 1$): +\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] + +And we have: +\begin{align*} + \Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ + &= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\ + &\approx \left|N_1\right|^2 +\end{align*} + +Similarly, we design $H_1(s)$ such that at frequencies where $|N_1| > |N_2|$: +\[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \] + +For instance, is we take $\epsilon = 1$, then the PSD of $\hat{x}$ is increased by just by a factor $\sqrt{2}$ over the all frequencies from the idea case. + +We use this as the weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters. + +The weighting function and the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfb]]. + +#+begin_src matlab + epsilon = 2; + + W1b = 1/epsilon*N1/N2; + W2b = 1/epsilon*N2/N1; + + W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper +#+end_src + +#+begin_src matlab + P = [W1b -W1b; + 0 W2b; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+RESULTS: +#+begin_example +[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +Test bounds: 0.0000 < gamma <= 32.8125 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f + 32.812 1.8e+01 3.4e-10 6.3e+00 -2.9e-13 0.0000 p + 16.406 1.8e+01 3.4e-10 6.3e+00 -1.2e-15 0.0000 p + 8.203 1.8e+01 3.3e-10 6.3e+00 -2.6e-13 0.0000 p + 4.102 1.8e+01 3.3e-10 6.3e+00 -2.1e-13 0.0000 p + 2.051 1.7e+01 3.4e-10 6.3e+00 -7.2e-16 0.0000 p + 1.025 1.6e+01 -1.3e+06# 6.3e+00 -8.3e-14 0.0000 f + 1.538 1.7e+01 3.4e-10 6.3e+00 -2.0e-13 0.0000 p + 1.282 1.7e+01 3.4e-10 6.3e+00 -7.9e-17 0.0000 p + 1.154 1.7e+01 3.6e-10 6.3e+00 -1.8e-13 0.0000 p + 1.089 1.7e+01 -3.4e+06# 6.3e+00 -1.7e-13 0.0000 f + 1.122 1.7e+01 -1.0e+07# 6.3e+00 -3.2e-13 0.0000 f + 1.138 1.7e+01 -1.3e+08# 6.3e+00 -1.8e-13 0.0000 f + 1.146 1.7e+01 3.2e-10 6.3e+00 -3.0e-13 0.0000 p + 1.142 1.7e+01 5.5e-10 6.3e+00 -2.8e-13 0.0000 p + 1.140 1.7e+01 -1.5e-10 6.3e+00 -2.3e-13 0.0000 p + 1.139 1.7e+01 -4.8e+08# 6.3e+00 -6.2e-14 0.0000 f + 1.139 1.7e+01 1.3e-09 6.3e+00 -8.9e-17 0.0000 p + + Gamma value achieved: 1.1390 +#+end_example + +#+begin_src matlab + H1b = 1 - H2b; +#+end_src + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1b, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2b, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1b, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2b, 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(H1b, freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*phase(squeeze(freqresp(H2b, 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]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfb.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:weights_comp_filters_Hinfb +#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfb.png][png]], [[./figs/weights_comp_filters_Hinfb.pdf][pdf]]) +[[file:figs/weights_comp_filters_Hinfb.png]] + +#+begin_src matlab + 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); +#+end_src + +** H-Infinity Synthesis - method C +#+begin_src matlab + Wp = 0.56*(inv(N1)+inv(N2))/(1 + s/2/pi/1000); + + W1c = N1*Wp; + W2c = N2*Wp; +#+end_src + +#+begin_src matlab + P = [W1c -W1c; + 0 W2c; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2c, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+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 + +#+begin_src matlab + H1c = 1 - H2c; +#+end_src + +#+begin_src matlab :exports none + 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]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + +#+begin_src matlab + 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); +#+end_src + +** Comparison of the methods +The three methods are now compared. + +The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. [[fig:comparison_psd_noise]]. + +The Cumulative Power Spectrum for the same sensors are shown on Fig. [[fig:comparison_cps_noise]]. + +The RMS value of the obtained super sensors are shown on table [[tab:rms_results]]. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([norm([N1], 2) ; norm([N2], 2) ; norm([N1*H1, N2*H2], 2) ; norm([N1*H1a, N2*H2a], 2) ; norm([N1*H1b, N2*H2b], 2) ; norm([N1*H1c, N2*H2c], 2)], {'Sensor 1', 'Sensor 2', 'H2 Fusion', 'H-Infinity a', 'H-Infinity b', 'H-Infinity c'}, {'rms value'}, ' %.1e'); +#+end_src + +#+name: tab:rms_results +#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters +#+RESULTS: +| | rms value | +|--------------+-----------| +| Sensor 1 | 1.3e-03 | +| Sensor 2 | 1.3e-03 | +| H2 Fusion | 1.2e-04 | +| H-Infinity a | 2.4e-04 | +| H-Infinity b | 1.4e-04 | +| H-Infinity c | 2.2e-04 | + + +#+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, '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; + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comparison_psd_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comparison_psd_noise +#+CAPTION: Comparison of the obtained Power Spectral Density using the three methods ([[./figs/comparison_psd_noise.png][png]], [[./figs/comparison_psd_noise.pdf][pdf]]) +[[file:figs/comparison_psd_noise.png]] + +#+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, '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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comparison_cps_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comparison_cps_noise +#+CAPTION: Comparison of the obtained Cumulative Power Spectrum using the three methods ([[./figs/comparison_cps_noise.png][png]], [[./figs/comparison_cps_noise.pdf][pdf]]) +[[file:figs/comparison_cps_noise.png]] + +** 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. + +#+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); +#+end_src + +The sensor uncertain models are defined below. +#+begin_src matlab + G1 = 1 + w1*ultidyn('Delta',[1 1]); + G2 = 1 + w2*ultidyn('Delta',[1 1]); +#+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 + +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. + +#+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 + figure; + % Magnitude + ax1 = subplot(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 = subplot(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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/uncertainty_super_sensor_H2_syn.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:uncertainty_super_sensor_H2_syn +#+CAPTION: Uncertianty regions of both individual sensors and of the super sensor when using the $\mathcal{H}_2$ synthesis ([[./figs/uncertainty_super_sensor_H2_syn.png][png]], [[./figs/uncertainty_super_sensor_H2_syn.pdf][pdf]]) +[[file:figs/uncertainty_super_sensor_H2_syn.png]] + +** Conclusion +From the above complementary filter design with the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ synthesis, it still seems that the $\mathcal{H}_2$ synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the $\mathcal{H}_2$ norm). + +However, the synthesis does not take into account the robustness of the sensor fusion. + +* Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty +:PROPERTIES: +:header-args:matlab+: :tangle matlab/comp_filter_robustness.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** Introduction :ignore: +We initially considered perfectly known sensor dynamics so that it can be perfectly inverted. + +We now take into account the fact that the sensor dynamics is only partially known. +To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Fig. [[fig:sensor_fusion_dynamic_uncertainty]]. + +#+name: fig:sensor_fusion_dynamic_uncertainty +#+caption: Sensor fusion architecture with sensor dynamics uncertainty +[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]] + +The objective here is to design complementary filters $H_1(s)$ and $H_2(s)$ in order to minimize the dynamical uncertainty of the super sensor. + +** ZIP file containing the data and matlab files :ignore: +#+begin_note + The Matlab scripts is accessible [[file:matlab/comp_filter_robustness.m][here]]. +#+end_note + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +** Super Sensor Dynamical Uncertainty +In practical systems, the sensor dynamics has always some level of uncertainty. +Let's represent that with multiplicative input uncertainty as shown on figure [[fig:sensor_fusion_dynamic_uncertainty]]. + +#+name: fig:sensor_fusion_dynamic_uncertainty +#+caption: Fusion of two sensors with input multiplicative uncertainty +[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]] + +The dynamics of the super sensor is represented by +\begin{align*} + \frac{\hat{x}}{x} &= (1 + w_1 \Delta_1) H_1 + (1 + w_2 \Delta_2) H_2 \\ + &= 1 + w_1 H_1 \Delta_1 + w_2 H_2 \Delta_2 +\end{align*} +with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$. + +We see that as soon as we have some uncertainty in the sensor dynamics, we have that the complementary filters have some effect on the transfer function from $x$ to $\hat{x}$. + +The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (figure [[fig:uncertainty_gain_phase_variation]]). + +We then have that the angle introduced by the super sensor is bounded by $\arcsin(\epsilon)$: +\[ \angle \frac{\hat{x}}{x}(j\omega) \le \arcsin \Big(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\Big) \] + +#+name: fig:uncertainty_gain_phase_variation +#+caption: Maximum phase variation +[[file:figs-tikz/uncertainty_gain_phase_variation.png]] + +** Dynamical uncertainty of the individual sensors +Let say we want to merge two sensors: +- sensor 1 that has unknown dynamics above 10Hz: $|w_1(j\omega)| > 1$ for $\omega > 10\text{ Hz}$ +- sensor 2 that has unknown dynamics below 1Hz and above 1kHz $|w_2(j\omega)| > 1$ for $\omega < 1\text{ Hz}$ and $\omega > 1\text{ kHz}$ + +We define the weights that are used to characterize the dynamic uncertainty of the sensors. + +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); +#+end_src + +#+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); +#+end_src + +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. +#+begin_src matlab + G1 = 1 + w1*ultidyn('Delta',[1 1]); + G2 = 1 + w2*ultidyn('Delta',[1 1]); +#+end_src + +#+begin_src matlab :exports none + % Few random samples of the sensor dynamics are computed + G1s = usample(G1, 10); + G2s = usample(G2, 10); +#+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 = subplot(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 = subplot(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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/uncertainty_dynamics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:uncertainty_dynamics_sensors +#+CAPTION: Dynamic uncertainty of the two sensors ([[./figs/uncertainty_dynamics_sensors.png][png]], [[./figs/uncertainty_dynamics_sensors.pdf][pdf]]) +[[file:figs/uncertainty_dynamics_sensors.png]] + +** Synthesis objective +The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. [[fig:uncertainty_gain_phase_variation]]. + +At each frequency $\omega$, the radius of the circle is $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$. + +Thus, the phase shift $\Delta\phi(\omega)$ due to the super sensor uncertainty is bounded by: +\[ |\Delta\phi(\omega)| \leq \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) \] + +Let's define some allowed frequency depend phase shift $\Delta\phi_\text{max}(\omega) > 0$ such that: +\[ |\Delta\phi(\omega)| < \Delta\phi_\text{max}(\omega), \quad \forall\omega \] + + +If $H_1(s)$ and $H_2(s)$ are designed such that +\[ |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big) \] + +The maximum phase shift due to dynamic uncertainty at frequency $\omega$ will be $\Delta\phi_\text{max}(\omega)$. + +** Requirements as an $\mathcal{H}_\infty$ norm +We now try to express this requirement in terms of an $\mathcal{H}_\infty$ norm. + +Let's define one weight $w_\phi(s)$ that represents the maximum wanted phase uncertainty: +\[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \] + +Then: +\begin{align*} + & |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big), \quad \forall\omega \\ + \Longleftrightarrow & |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < |w_\phi(j\omega)|^{-1}, \quad \forall\omega \\ + \Longleftrightarrow & \left| w_1(j\omega) H_1(j\omega) w_\phi(j\omega) \right| + \left| w_2(j\omega) H_2(j\omega) w_\phi(j\omega) \right| < 1, \quad \forall\omega +\end{align*} + +Which is approximately equivalent to (with an error of maximum $\sqrt{2}$): +#+name: eq:hinf_conf_phase_uncertainty +\begin{equation} + \left\| \begin{matrix} w_1(s) w_\phi(s) H_1(s) \\ w_2(s) w_\phi(s) H_2(s) \end{matrix} \right\|_\infty < 1 +\end{equation} + +One should not forget that at frequency where both sensors has unknown dynamics ($|w_1(j\omega)| > 1$ and $|w_2(j\omega)| > 1$), the super sensor dynamics will also be unknown and the phase uncertainty cannot be bounded. +Thus, at these frequencies, $|w_\phi|$ should be smaller than $1$. + +** Weighting Function used to bound the super sensor uncertainty +Let's define $w_\phi(s)$ in order to bound the maximum allowed phase uncertainty $\Delta\phi_\text{max}$ of the super sensor dynamics. +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]]. + +#+begin_src matlab + 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 + +#+begin_src matlab :exports none + 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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/magnitude_wphi.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + +#+begin_src matlab :exports none + % 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; +#+end_src + +#+begin_src matlab :exports none + 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); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/maximum_wanted_phase_uncertainty.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]]. + +#+begin_src matlab :exports none + 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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/upper_bounds_comp_filter_max_phase_uncertainty.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + +** $\mathcal{H}_\infty$ Synthesis +The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filters 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-tikz/h_infinity_robust_fusion.png]] + +The generalized plant is defined below. +#+begin_src matlab + P = [W1 -W1; + 0 W2; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+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 + +And $H_1(s)$ is defined as the complementary of $H_2(s)$. +#+begin_src matlab + H1 = 1 - H2; +#+end_src + +The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]]. +#+begin_src matlab :exports none + 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]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_filter_hinf_uncertainty.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_filter_hinf_uncertainty +#+CAPTION: Obtained complementary filters ([[./figs/comp_filter_hinf_uncertainty.png][png]], [[./figs/comp_filter_hinf_uncertainty.pdf][pdf]]) +[[file:figs/comp_filter_hinf_uncertainty.png]] + +** 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]]. + +#+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 + figure; + % Magnitude + ax1 = subplot(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 = subplot(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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/super_sensor_uncertainty_bode_plot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:super_sensor_uncertainty_bode_plot +#+CAPTION: Uncertainty on the dynamics of the super sensor ([[./figs/super_sensor_uncertainty_bode_plot.png][png]], [[./figs/super_sensor_uncertainty_bode_plot.pdf][pdf]]) +[[file:figs/super_sensor_uncertainty_bode_plot.png]] + +The uncertainty of the super sensor cannot be made smaller than both the individual sensor. Ideally, it would follow the minimum uncertainty of both sensors. + +We here just used very wimple weights. +For instance, we could improve the dynamical uncertainty of the super sensor by making $|w_\phi(j\omega)|$ smaller bellow 2Hz where the dynamical uncertainty of the sensor 1 is small. + +** 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. + +#+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 + +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]]. + +#+begin_src matlab :exports none + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/psd_sensors_hinf_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + +#+begin_src matlab :exports none + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cps_sensors_hinf_synthesis.cps" :var figsize="full-tall" :post cps2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cps_sensors_hinf_synthesis +#+CAPTION: Cumulative Power Spectrum of the obtained super sensor using the $\mathcal{H}_\infty$ synthesis ([[./figs/cps_sensors_hinf_synthesis.png][png]], [[./figs/cps_sensors_hinf_synthesis.cps][cps]]) +[[file:figs/cps_sensors_hinf_synthesis.png]] + +** Conclusion +Using the $\mathcal{H}_\infty$ synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values. + +However, the RMS of the super sensor noise is not optimized as it was the case with the $\mathcal{H}_2$ synthesis + +** 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 \\ + G_2(s) &= 0.6 +\end{align*} + +#+begin_src matlab + G1 = 1; + G2 = 0.6; +#+end_src + +Two pairs of complementary filters are designed and shown on figure [[fig:comp_filters_robustness_test]]. +The complementary filters shown in blue does not present a bump as the red ones but provides less sensor separation at high and low frequencies. + +#+begin_src matlab :exports none + freqs = logspace(-1, 1, 1000); +#+end_src + +#+begin_src matlab :exports none + w0 = 2*pi; + alpha = 2; + + H1a = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + H2a = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + + w0 = 2*pi; + alpha = 0.1; + + H1b = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + H2b = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(H1a, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(H2a, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(H1b, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(H2b, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude'); + hold off; + % Phase + ax2 = subplot(2,1,2); + hold on; + set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(H1a, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(H2a, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(H1b, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(H2b, 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_filters_robustness_test.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_filters_robustness_test +#+CAPTION: The two complementary filters designed for the robustness test ([[./figs/comp_filters_robustness_test.png][png]], [[./figs/comp_filters_robustness_test.pdf][pdf]]) +[[file:figs/comp_filters_robustness_test.png]] + +We then compute the bode plot of the super sensor transfer function $H_1(s)G_1(s) + H_2(s)G_2(s)$ for both complementary filters pair (figure [[fig:tf_super_sensor_comp]]). + +We see that the blue complementary filters with a lower maximum norm permits to limit the phase lag introduced by the gain mismatch. + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(H1a*G1 + H2a*G2, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(H1b*G1 + H2b*G2, freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude'); + ylim([1e-1, 1e1]); + hold off; + % Phase + ax2 = subplot(2,1,2); + hold on; + set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(H1a*G1 + H2a*G2, freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(H1b*G1 + H2b*G2, 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/tf_super_sensor_comp.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:tf_super_sensor_comp +#+CAPTION: Comparison of the obtained super sensor transfer functions ([[./figs/tf_super_sensor_comp.png][png]], [[./figs/tf_super_sensor_comp.pdf][pdf]]) +[[file:figs/tf_super_sensor_comp.png]] + +* Optimal Sensor Fusion - Mixed Synthesis +:PROPERTIES: +:header-args:matlab+: :tangle matlab/mixed_synthesis_sensor_fusion.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> +** ZIP file containing the data and matlab files :ignore: +#+begin_note + The Matlab scripts is accessible [[file:matlab/mixed_synthesis_sensor_fusion.m][here]]. +#+end_note + +** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis - Introduction +The goal is to design complementary filters such that: +- the maximum uncertainty of 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]]). + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +** 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. +#+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); +#+end_src + +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. +#+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 + +Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. [[fig:mixed_synthesis_noise_uncertainty_sensors]]. + +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/mixed_synthesis_noise_uncertainty_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:mixed_synthesis_noise_uncertainty_sensors +#+CAPTION: Noise characteristsics and Dynamical uncertainty of the individual sensors ([[./figs/mixed_synthesis_noise_uncertainty_sensors.png][png]], [[./figs/mixed_synthesis_noise_uncertainty_sensors.pdf][pdf]]) +[[file:figs/mixed_synthesis_noise_uncertainty_sensors.png]] + +** 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. + +#+begin_src matlab + 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; +#+end_src + +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/mixed_syn_hinf_weight.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]]. + +#+begin_src matlab :exports none + 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 = subplot(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 = subplot(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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/mixed_syn_objective_hinf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:mixed_syn_objective_hinf +#+CAPTION: $\mathcal{H}_\infty$ synthesis objective part of the mixed-synthesis ([[./figs/mixed_syn_objective_hinf.png][png]], [[./figs/mixed_syn_objective_hinf.pdf][pdf]]) +[[file:figs/mixed_syn_objective_hinf.png]] + +** 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. +#+begin_src matlab + 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]; +#+end_src + +** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis +The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed below. +#+begin_src matlab + 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; +#+end_src + +The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_synthesis]]. + +#+begin_src matlab :exports none + 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]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_filters_mixed_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_filters_mixed_synthesis +#+CAPTION: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/comp_filters_mixed_synthesis.png][png]], [[./figs/comp_filters_mixed_synthesis.pdf][pdf]]) +[[file:figs/comp_filters_mixed_synthesis.png]] + +** 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. + +#+begin_src matlab :exports none + 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', '$\Phi_{\hat{x}}$'); + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/psd_super_sensor_mixed_syn.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + + +#+begin_src matlab :exports none + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cps_super_sensor_mixed_syn.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cps_super_sensor_mixed_syn +#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/cps_super_sensor_mixed_syn.png][png]], [[./figs/cps_super_sensor_mixed_syn.pdf][pdf]]) +[[file:figs/cps_super_sensor_mixed_syn.png]] + +** Obtained Super Sensor's Uncertainty +The uncertainty on the super sensor's dynamics is shown in Fig. [[fig:super_sensor_dyn_uncertainty_mixed_syn]]. + +#+begin_src matlab :exports none + 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; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(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 = subplot(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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/super_sensor_dyn_uncertainty_mixed_syn.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:super_sensor_dyn_uncertainty_mixed_syn +#+CAPTION: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis ([[./figs/super_sensor_dyn_uncertainty_mixed_syn.png][png]], [[./figs/super_sensor_dyn_uncertainty_mixed_syn.pdf][pdf]]) +[[file:figs/super_sensor_dyn_uncertainty_mixed_syn.png]] + +** Conclusion +This synthesis methods allows both to: +- limit the dynamical uncertainty of the super sensor +- minimize the RMS value of the estimation + +* Mixed Synthesis - LMI Optimization +** Introduction +The following matlab scripts was written by Mohit. + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +** 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. +#+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); +#+end_src + +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. +#+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 + +** Weights +The weights for the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ part are defined below. + +#+begin_src matlab + 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; + + W1u = ss(w1*wphi); W2u = ss(w2*wphi); % Weight on the uncertainty + W1n = ss(N1); W2n = ss(N2); % Weight on the noise +#+end_src + +#+begin_src matlab + P = [W1u -W1u; + 0 W2u; + W1n -W1n; + 0 W2n; + 1 0]; +#+end_src + +** LMI Optimization +We are using the [[http://cvxr.com/cvx/][CVX toolbox]] to solve the optimization problem. + +We first put the generalized plant in a State-space form. +#+begin_src matlab + A = P.A; + Bw = P.B(:,1); + Bu = P.B(:,2); + Cz1 = P.C(1:2,:); Dz1w = P.D(1:2,1); Dz1u = P.D(1:2,2); % Hinf + Cz2 = P.C(3:4,:); Dz2w = P.D(1:2,1); Dz2u = P.D(1:2,2); % H2 + Cy = P.C(5,:); Dyw = P.D(5,1); Dyu = P.D(5,2); + + n = size(P.A,1); + ny = 1; % number of measurements + nu = 1; % number of control inputs + nz = 2; + nw = 1; + + Wtinf = 0; + Wt2 = 1; +#+end_src + +We Define all the variables. +#+begin_src matlab + cvx_startup; + + cvx_begin sdp + cvx_quiet true + cvx_solver sedumi + variable X(n,n) symmetric; + variable Y(n,n) symmetric; + variable W(nz,nz) symmetric; + variable Ah(n,n); + variable Bh(n,ny); + variable Ch(nu,n); + variable Dh(nu,ny); + variable eta; + variable gam; +#+end_src + +We define the minimization objective. +#+begin_src matlab + minimize Wt2*eta+Wtinf*gam % mix objective + subject to: +#+end_src + +The $\mathcal{H}_\infty$ constraint. +#+begin_src matlab + gam<=1; % Keep the Hinf norm less than 1 + + [ X, eye(n,n) ; + eye(n,n), Y ] >= 0 ; + + [ A*X + Bu*Ch + X*A' + Ch'*Bu', A+Bu*Dh*Cy+Ah', Bw+Bu*Dh*Dyw, X*Cz1' + Ch'*Dz1u' ; + (A+Bu*Dh*Cy+Ah')', Y*A + A'*Y + Bh*Cy + Cy'*Bh', Y*Bw + Bh*Dyw, (Cz1+Dz1u*Dh*Cy)' ; + (Bw+Bu*Dh*Dyw)', Bw'*Y + Dyw'*Bh', -eye(nw,nw), (Dz1w+Dz1u*Dh*Dyw)' ; + Cz1*X + Dz1u*Ch, Cz1+Dz1u*Dh*Cy, Dz1w+Dz1u*Dh*Dyw, -gam*eye(nz,nz)] <= 0 ; +#+end_src + +The $\mathcal{H}_2$ constraint. +#+begin_src matlab + trace(W) <= eta ; + + [ W, Cz2*X+Dz2u*Ch, Cz2*X+Dz2u*Ch; + X*Cz2'+Ch'*Dz2u', X, eye(n,n) ; + (Cz2*X+Dz2u*Ch)', eye(n,n), Y ] >= 0 ; + + [ A*X + Bu*Ch + X*A' + Ch'*Bu', A+Bu*Dh*Cy+Ah', Bw+Bu*Dh*Dyw ; + (A+Bu*Dh*Cy+Ah')', Y*A + A'*Y + Bh*Cy + Cy'*Bh', Y*Bw + Bh*Dyw ; + (Bw+Bu*Dh*Dyw)', Bw'*Y + Dyw'*Bh', -eye(nw,nw)] <= 0 ; +#+end_src + +And we run the optimization. +#+begin_src matlab + cvx_end + cvx_status +#+end_src + +#+begin_src matlab :exports none + if(strcmp(cvx_status,'Inaccurate/Solved')) + display('The solver was unable to make a determination to within the default numerical tolerance.'); + display('However, it determined that the results obtained satisfied a “relaxed” tolerance leve'); + display('and therefore may still be suitable for further use.'); + end +#+end_src + +Finally, we can compute the obtained complementary filters. +#+begin_src matlab + M = eye(n); + N = inv(M)*(eye(n,n)-Y*X); + Dk = Dh; + Ck = (Ch-Dk*Cy*X)*inv(M'); + Bk = inv(N)*(Bh-Y*Bu*Dk); + Ak = inv(N)*(Ah-Y*(A+Bu*Dk*Cy)*X-N*Bk*Cy*X-Y*Bu*Ck*M')*inv(M'); + + H2 = tf(ss(Ak,Bk,Ck,Dk)); + H1 = 1 - H2; +#+end_src + +** Result +The obtained complementary filters are compared with the required upper bounds on Fig. [[fig:LMI_obtained_comp_filters]]. + +#+begin_src matlab :exports none + 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]); + ylim([-180, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/LMI_obtained_comp_filters.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:LMI_obtained_comp_filters +#+CAPTION: Obtained complementary filters using the LMI optimization ([[./figs/LMI_obtained_comp_filters.png][png]], [[./figs/LMI_obtained_comp_filters.pdf][pdf]]) +[[file:figs/LMI_obtained_comp_filters.png]] + +** Comparison with the matlab Mixed Synthesis +The Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis is performed below. +#+begin_src matlab + Nmeas = 1; Ncon = 1; Nz2 = 2; + + [H2m,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on'); + + H1m = 1 - H2m; +#+end_src + +The obtained filters are compare with the one obtained using the CVX toolbox in Fig. [[]]. + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{1,\mathcal{H}_2/\mathcal{H}_\infty}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{2,\mathcal{H}_2/\mathcal{H}_\infty}$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_{1, CVX}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_{2, CVX}$'); + + 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(H1m, freqs, 'Hz'))), '--'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*phase(squeeze(freqresp(H2m, freqs, 'Hz'))), '--'); + + 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]); + ylim([-180, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/compare_cvx_h2hinf_comp_filters.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:compare_cvx_h2hinf_comp_filters +#+CAPTION: Comparison between the complementary filters obtained with the CVX toolbox and with the =h2hinfsyn= command ([[./figs/compare_cvx_h2hinf_comp_filters.png][png]], [[./figs/compare_cvx_h2hinf_comp_filters.pdf][pdf]]) +[[file:figs/compare_cvx_h2hinf_comp_filters.png]] + +** H-Infinity Objective +In terms of the $\mathcal{H}_\infty$ objective, both synthesis method are satisfying the requirements as shown in Fig. [[fig:comp_cvx_h2i_hinf_norm]]. + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1u, freqs, 'Hz'))), '-.', 'DisplayName', '$1/W_{1u}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2u, freqs, 'Hz'))), '-.', 'DisplayName', '$1/W_{2u}$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{1,\mathcal{H}_2/\mathcal{H}_\infty}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{2,\mathcal{H}_2/\mathcal{H}_\infty}$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_{1, CVX}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_{2, CVX}$'); + + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([1e-3, 2]); + legend('location', 'southwest'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_cvx_h2i_hinf_norm.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_cvx_h2i_hinf_norm +#+CAPTION: H-Infinity norm requirement and results ([[./figs/comp_cvx_h2i_hinf_norm.png][png]], [[./figs/comp_cvx_h2i_hinf_norm.pdf][pdf]]) +[[file:figs/comp_cvx_h2i_hinf_norm.png]] + +** Obtained Super Sensor's noise +The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h2hinfsyn= command are compared in Fig. [[fig:psd_compare_cvx_h2i]] and [[fig:cps_compare_cvx_h2i]]. + +#+begin_src matlab :exports none + PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2; + PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2; + PSD_cvx = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; + PSD_h2i = abs(squeeze(freqresp(N1*H1m, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2m, 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_cvx, 'k-', 'DisplayName', '$\Phi_{\hat{x}, CVX}$'); + plot(freqs, PSD_h2i, 'k--', 'DisplayName', '$\Phi_{\hat{x}, \mathcal{H}_2/\mathcal{H}_\infty}$'); + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/psd_compare_cvx_h2i.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:psd_compare_cvx_h2i +#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/psd_compare_cvx_h2i.png][png]], [[./figs/psd_compare_cvx_h2i.pdf][pdf]]) +[[file:figs/psd_compare_cvx_h2i.png]] + + +#+begin_src matlab :exports none + CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1); + CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2); + CPS_cvx = 1/pi*cumtrapz(2*pi*freqs, PSD_cvx); + CPS_h2i = 1/pi*cumtrapz(2*pi*freqs, PSD_h2i); +#+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_cvx, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{CVX}} = %.1e$', sqrt(CPS_cvx(end)))); + plot(freqs, CPS_h2i, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2/\\mathcal{H}_\\infty}} = %.1e$', sqrt(CPS_h2i(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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cps_compare_cvx_h2i.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cps_compare_cvx_h2i +#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/cps_compare_cvx_h2i.png][png]], [[./figs/cps_compare_cvx_h2i.pdf][pdf]]) +[[file:figs/cps_compare_cvx_h2i.png]] + +** Obtained Super Sensor's Uncertainty +The uncertainty on the super sensor's dynamics is shown in Fig. [[]]. + +#+begin_src matlab :exports none + G1 = 1 + w1*ultidyn('Delta',[1 1]); + G2 = 1 + w2*ultidyn('Delta',[1 1]); + + % We here compute the maximum and minimum phase of the super sensor + Dphiss_cvx = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz')))); + Dphiss_cvx(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190; + + Dphiss_h2i = 180/pi*asin(abs(squeeze(freqresp(w1*H1m, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2m, freqs, 'Hz')))); + Dphiss_h2i(abs(squeeze(freqresp(w1*H1m, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2m, 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; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(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 - CVX'); + plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off'); + plot(freqs, 1 + abs(squeeze(freqresp(w1*H1m+w2*H2m, freqs, 'Hz'))), 'k-', 'DisplayName', 'Bounds - $\mathcal{H}_2/\mathcal{H}_\infty$'); + plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1m+w2*H2m, freqs, 'Hz'))), 0), 'k-', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + legend('location', 'southwest'); + ylabel('Magnitude'); + ylim([5e-2, 10]); + hold off; + + % Phase + ax2 = subplot(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_cvx, 'k--'); + plot(freqs, -Dphiss_cvx, 'k--'); + plot(freqs, Dphiss_h2i, 'k-'); + plot(freqs, -Dphiss_h2i, 'k-'); + 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 + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/super_sensor_uncertainty_compare_cvx_h2i.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:super_sensor_uncertainty_compare_cvx_h2i +#+CAPTION: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis ([[./figs/super_sensor_uncertainty_compare_cvx_h2i.png][png]], [[./figs/super_sensor_uncertainty_compare_cvx_h2i.pdf][pdf]]) +[[file:figs/super_sensor_uncertainty_compare_cvx_h2i.png]] + +* H-Infinity synthesis to ensure both performance and robustness +:PROPERTIES: +:header-args:matlab+: :tangle matlab/hinf_syn_perf_robust.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** ZIP file containing the data and matlab files :ignore: +#+begin_note + The Matlab scripts is accessible [[file:matlab/hinf_syn_perf_robust.m][here]]. +#+end_note + +** Introduction +The idea is to use only the $\mathcal{H}_\infty$ norm to express both the maximum wanted super sensor uncertainty and the fact that we want to minimize the super sensor's noise. + +For *performance*, we may want to obtain a super sensor's noise that is close to the minimum of the individual sensor noises. + +The noise of the super sensor is: +\[ |N_{ss}(j\omega)|^2 = | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \] + +The minimum noise that we can obtain follows the minimum noise of the individual sensor: +\begin{align*} + & |N_{ss}(j\omega)| \approx |N_1(j\omega)| \quad \text{when} \quad |N_1(j\omega)| < |N_2(j\omega)| \\ + & |N_{ss}(j\omega)| \approx |N_2(j\omega)| \quad \text{when} \quad |N_2(j\omega)| < |N_1(j\omega)| +\end{align*} + +To do so, we want to design the complementary filters such that: +\begin{align*} + & |H_2(j\omega)| \ll 1 \quad \text{when} \quad |N_1(j\omega)| < |N_2(j\omega)| \\ + & |H_1(j\omega)| \ll 1 \quad \text{when} \quad |N_2(j\omega)| < |N_1(j\omega)| +\end{align*} + + + + +For the *uncertainty* of the super sensor. +The equivalent super sensor uncertainty is: +\[ |w_{ss}(j\omega)| = |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \] + +The minimum uncertainty that we can obtain follows the minimum uncertainty of the individual sensor: +\begin{align*} + & |w_{ss}(j\omega)| \approx |w_1(j\omega)| \quad \text{when} \quad |w_1(j\omega)| < |w_2(j\omega)| \\ + & |w_{ss}(j\omega)| \approx |w_2(j\omega)| \quad \text{when} \quad |w_2(j\omega)| < |w_1(j\omega)| +\end{align*} + +To do so, we want to design the complementary filters such that: +\begin{align*} + & |H_2(j\omega)| \ll 1 \quad \text{when} \quad |w_1(j\omega)| < |w_2(j\omega)| \\ + & |H_1(j\omega)| \ll 1 \quad \text{when} \quad |w_2(j\omega)| < |w_1(j\omega)| +\end{align*} + + +Of course, the conditions for performance and uncertainty may not be compatible. + +We may not want to follow the minimum uncertainty. + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +** Dynamical uncertainty and Noise level of the individual sensors +Uncertainty on the individual 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); +#+end_src + +Noise level of the individual sensors: +#+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 + +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/noise_uncertainty_sensors_hinf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:noise_uncertainty_sensors_hinf +#+CAPTION: Noise and Uncertainty characteristics of the sensors ([[./figs/noise_uncertainty_sensors_hinf.png][png]], [[./figs/noise_uncertainty_sensors_hinf.pdf][pdf]]) +[[file:figs/noise_uncertainty_sensors_hinf.png]] + +** 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: +#+begin_src matlab + 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; +#+end_src + +Weight on the performance: +#+begin_src matlab + 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); +#+end_src + +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]]. +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/charac_sensors_weights.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]]. +#+begin_src matlab :exports none + 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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/upper_bound_complementary_filters_perf_robust.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:upper_bound_complementary_filters_perf_robust +#+CAPTION: Upper bounds on the complementary filters ([[./figs/upper_bound_complementary_filters_perf_robust.png][png]], [[./figs/upper_bound_complementary_filters_perf_robust.pdf][pdf]]) +[[file:figs/upper_bound_complementary_filters_perf_robust.png]] + +** 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*} + + +#+begin_src matlab + W1p = N1*Wp/(1+s/2/pi/1000); % Used to render W1p proper + W2p = N2*Wp; + W1u = w1*Wu; + W2u = w2*Wu; +#+end_src + +#+begin_src matlab + P = [W1p -W1p; + 0 W2p; + W1u -W1u; + 0 W2u; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+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 + +#+begin_src matlab + H1 = 1 - H2; +#+end_src + +The obtained complementary filters with the upper bounds are shown in Fig. [[fig:hinf_result_comp_filters_4_outputs]]. +#+begin_src matlab :exports none + 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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hinf_result_comp_filters_4_outputs.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + + + +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/upper_bounds_perf_robust_result_4_outputs.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + +#+begin_src matlab :exports none + 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); +#+end_src + +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/4outputs_hinf_psd_cps2svg.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+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]] + + +#+begin_src matlab :exports none + 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; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(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 = subplot(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'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/4outputs_uncertainty.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:4outputs_uncertainty +#+CAPTION: Dynamical uncertainty ([[./figs/4outputs_uncertainty.png][png]], [[./figs/4outputs_uncertainty.pdf][pdf]]) +[[file:figs/4outputs_uncertainty.png]] + +** TODO Weight for both :noexport: +:PROPERTIES: +:header-args:matlab+: :tangle no +:END: +We may want to weights that capture both requirements. +We then have one weight for H1 and one weight for H2 (2 weights in total instead of 1). + +#+begin_src matlab + W1 = w1*Wu*(1+s/2/pi/40)^2/(1 + s/2/pi/1000)^2; + W2 = N2*Wp; +#+end_src + +#+begin_src matlab :exports none + 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', '$|N_2| - robu$'); + plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W_1| - robu$'); + plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W_2| - robu$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + legend('location', 'northeast'); +#+end_src + +The generalized plant $P$ is then: +#+begin_src matlab + P = [W1 -W1; + 0 W2; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+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: 0.8000 < gamma <= 1312.5112 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f +1.313e+03 1.3e+01 -1.7e-16 6.3e+00 -1.4e-19 0.0000 p + 656.656 1.3e+01 -3.4e-17 6.3e+00 -1.9e-13 0.0000 p + 328.728 1.3e+01 7.7e-17 6.3e+00 -1.3e-24 0.0000 p + 164.764 1.3e+01 2.6e-17 6.3e+00 -1.1e-13 0.0000 p + 82.782 1.3e+01 -2.0e-16 6.3e+00 -1.1e-13 0.0000 p + 41.791 1.3e+01 1.0e-16 6.3e+00 -8.9e-16 0.0000 p + 21.295 1.3e+01 -8.4e-17 6.3e+00 -6.3e-15 0.0000 p + 11.048 1.3e+01 8.5e-17 6.3e+00 -8.6e-14 0.0000 p + 5.924 1.3e+01 -2.5e-16 6.3e+00 -7.5e-14 0.0000 p + 3.362 1.3e+01 -1.7e-17 6.3e+00 -1.2e-13 0.0000 p + 2.081 1.2e+01 -5.1e-17 6.3e+00 -1.3e-13 0.0000 p + 1.440 1.1e+01 -2.4e+09# 6.3e+00 -3.4e-13 0.0000 f + 1.761 1.2e+01 -7.9e-17 6.3e+00 -3.3e-13 0.0000 p + 1.601 1.1e+01 -1.0e+10# 6.3e+00 -1.4e-13 0.0000 f + 1.681 1.2e+01 -3.1e+10# 6.3e+00 -1.5e-13 0.0000 f + 1.721 1.2e+01 -1.5e+11# 6.3e+00 -3.2e-13 0.0000 f + 1.741 1.2e+01 -4.6e-17 6.3e+00 -1.3e-13 0.0000 p + 1.731 1.2e+01 -1.3e+12# 6.3e+00 -1.6e-13 0.0000 f + 1.736 1.2e+01 1.4e-16 6.3e+00 -1.0e-13 0.0000 p + 1.733 1.2e+01 -1.7e-09 6.3e+00 -1.3e-13 0.0000 p + 1.732 1.2e+01 -1.3e+13# 6.3e+00 -1.4e-13 0.0000 f + 1.733 1.2e+01 5.3e-18 6.3e+00 -1.3e-13 0.0000 p + + Gamma value achieved: 1.7326 +#+end_example + +#+begin_src matlab + H1 = 1 - H2; +#+end_src + +#+begin_src matlab :exports none + 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', '$|N_2| - robu$'); + 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'); +#+end_src + +#+begin_src matlab :exports none + 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); +#+end_src + +#+begin_src matlab :exports none + 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)]); +#+end_src + +#+begin_src matlab :exports none + 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; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(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 = subplot(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'); +#+end_src + +** TODO Try to obtain better weight for the dynamical uncertainty :noexport: +:PROPERTIES: +:header-args:matlab+: :tangle no +:END: +Maybe we are asking too much for the limiting of the uncertainty. In reality, we should only limit the uncertainty around the merging frequency so that no RHP zero is introduced, and around the wanted crossover frequency. + +Weights about the 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); +#+end_src + +We make one guess about a nice weight that is just above the minimum of both uncertainty weights +#+begin_src matlab + bodeFig({w1, w2, 0.5*inv(inv(w1)+inv(w2))}) +#+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 + +Weight that is used to bound the uncertainty of the super sensor. +#+begin_src matlab + wu = inv(inv(w1)+inv(w2)); + + W1 = w1/wu; + W2 = w2/wu; +#+end_src + +#+begin_src matlab + bodeFig({1/W1, 1/W2}) +#+end_src + +The wanted shape of complementary filters: +#+begin_src matlab + H1w = 1/W1; + H2w = 1/W2; +#+end_src + +The maximum wanted uncertainty. +#+begin_src matlab + Dphiss = 180/pi*asin(abs(squeeze(freqresp(w1*H1w, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2w, freqs, 'Hz')))); + Dphiss(abs(squeeze(freqresp(w1*H1w, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2w, freqs, 'Hz'))) > 1) = 190; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(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*H1w, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2w, freqs, 'Hz')))), 'k--'); + plot(freqs, max(1 - (abs(squeeze(freqresp(w1*H1w, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2w, freqs, 'Hz')))), 0), 'k--'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude'); + ylim([1e-1, 10]); + hold off; + + % Phase + ax2 = subplot(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--'); + 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 + + + +** TODO New idea about weighting function for robustness :noexport: +:PROPERTIES: +:header-args:matlab+: :tangle no +:END: +Trying to limit the phase is too complicated, it is much easier to limit the radius of the uncertainty circle. +This is of course linked to the gain and phase uncertainty, but it is easier to work with. + +Ideally, we want to have: +\begin{align*} + & |w_{ss}(j\omega)| \approx |w_1(j\omega)| \quad \text{when} \quad |w_1(j\omega)| < |w_2(j\omega)| \\ + & |w_{ss}(j\omega)| \approx |w_2(j\omega)| \quad \text{when} \quad |w_2(j\omega)| < |w_1(j\omega)| +\end{align*} + +It is thus very similar to what is done for limiting the super sensor noise. + +#+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); +#+end_src + +Weights on the Robustness: +#+begin_src matlab + epsilon = 1; + + W1r = 1/epsilon*w1/w2; + W2r = 1/epsilon*w2/w1; +#+end_src + +#+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 + +Weights on the Noise: +#+begin_src matlab + epsilon = 1; + + W1n = 1/epsilon*N1/N2; + W2n = 1/epsilon*N2/N1; + + W1n = W1n/(1 + s/2/pi/1000); % this is added so that it is proper +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1r, freqs, 'Hz'))), '-', 'DisplayName', 'W1 - Robust.'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2r, freqs, 'Hz'))), '-', 'DisplayName', 'W2 - Robust.'); + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1n, freqs, 'Hz'))), '--', 'DisplayName', 'W1 - Noise'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2n, freqs, 'Hz'))), '--', 'DisplayName', 'W2 - Noise'); + 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 + +#+begin_src matlab + P = [W1n -W1n; + 0 W2r; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+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: 0.5000 < gamma <= 65.6270 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f + 65.627 1.4e+01 -4.7e-13 1.3e+00 -2.7e-12 0.0000 p + 33.063 1.4e+01 3.7e-13 1.3e+00 -1.1e-17 0.0000 p + 16.782 1.4e+01 -9.5e-13 1.3e+00 -6.9e-15 0.0000 p + 8.641 1.4e+01 5.6e-13 1.3e+00 -2.0e-13 0.0000 p + 4.570 1.4e+01 2.6e-13 1.3e+00 -4.3e-14 0.0000 p + 2.535 1.4e+01 4.6e-13 1.3e+00 -1.7e-13 0.0000 p + 1.518 1.4e+01 -5.7e-13 1.3e+00 -8.2e-14 0.0000 p + 1.009 1.3e+01 7.9e-14 1.3e+00 -2.5e-14 0.0000 p + 0.754 1.3e+01 -2.1e-12 1.3e+00 -4.9e-15 0.0000 p + 0.627 1.2e+01 -2.1e+04# 1.3e+00 -1.8e-14 0.0000 f + 0.691 1.3e+01 -1.1e+05# 1.3e+00 -3.5e-16 0.0000 f + 0.723 1.3e+01 -5.5e+05# 1.3e+00 -2.0e-14 0.0000 f + 0.738 1.3e+01 -8.4e-12 1.3e+00 -2.7e-14 0.0000 p + 0.731 1.3e+01 -2.3e+06# 1.3e+00 -3.3e-13 0.0000 f + 0.735 1.3e+01 -9.9e-11 1.3e+00 -2.1e-14 0.0000 p + 0.733 1.3e+01 -8.9e+06# 1.3e+00 -5.0e-13 0.0000 f + 0.734 1.3e+01 -2.2e-10 1.3e+00 -1.9e-14 0.0000 p + + Gamma value achieved: 0.7335 +#+end_example + +#+begin_src matlab + H1 = 1 - H2; +#+end_src + +#+begin_src matlab + 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; + 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; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(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 = subplot(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'); +#+end_src + +#+begin_src matlab :exports none + 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'); +#+end_src + +#+begin_src matlab :exports none + 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'); +#+end_src + +** Conclusion +The $\mathcal{H}_\infty$ synthesis has been used to design complementary filters that permits to robustly merge sensors while ensuring a maximum noise level. +However, no guarantee is made that the RMS value of the super sensor's noise is minimized. + +* Equivalent Super Sensor +<> +** Introduction :ignore: +The goal here is to find the parameters of a single sensor that would best represent a super sensor. + +** Sensor Fusion Architecture +Let consider figure [[fig:sensor_fusion_full]] where two sensors are merged. +The dynamic uncertainty of each sensor is represented by a weight $w_i(s)$, the frequency characteristics each of the sensor noise is represented by the weights $N_i(s)$. +The noise sources $\tilde{n}_i$ are considered to be white noise: $\Phi_{\tilde{n}_i}(\omega) = 1, \ \forall\omega$. + +#+name: fig:sensor_fusion_full +#+caption: Sensor fusion architecture ([[./figs/sensor_fusion_full.png][png]], [[./figs/sensor_fusion_full.pdf][pdf]]). +#+RESULTS: +[[file:figs-tikz/sensor_fusion_full.png]] + + +\begin{align*} + \hat{x} &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\ + &\quad \quad + \Big(H_1(s) \big(1 + w_1(s) \Delta_1(s)\big) + H_2(s) \big(1 + w_2(s) \Delta_2(s)\big)\Big) x \\ + &= H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 \\ + &\quad \quad + \big(1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s)\big) x +\end{align*} + +To the dynamics of the super sensor is: +\begin{equation} + \frac{\hat{x}}{x} = 1 + H_1(s) w_1(s) \Delta_1(s) + H_2(s) w_2(s) \Delta_2(s) +\end{equation} + +And the noise of the super sensor is: +\begin{equation} + n_{ss} = H_1(s) N_1(s) \tilde{n}_1 + H_2(s) N_2(s) \tilde{n}_2 +\end{equation} + +** Equivalent Configuration +We try to determine $w_{ss}(s)$ and $N_{ss}(s)$ such that the sensor on figure [[fig:sensor_fusion_equivalent]] is equivalent to the super sensor of figure [[fig:sensor_fusion_full]]. + +#+name: fig:sensor_fusion_equivalent +#+caption: Equivalent Super Sensor ([[./figs/sensor_fusion_equivalent.png][png]], [[./figs/sensor_fusion_equivalent.pdf][pdf]]). +#+RESULTS: +[[file:figs-tikz/sensor_fusion_equivalent.png]] + +** Model the uncertainty of the super sensor +At each frequency $\omega$, the uncertainty set of the super sensor shown on figure [[fig:sensor_fusion_full]] is a circle centered on $1$ with a radius equal to $|H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|$ on the complex plane. +The uncertainty set of the sensor shown on figure [[fig:sensor_fusion_equivalent]] is a circle centered on $1$ with a radius equal to $|w_{ss}(j\omega)|$ on the complex plane. + +Ideally, we want to find a weight $w_{ss}(s)$ so that: +#+begin_important +\[ |w_{ss}(j\omega)| = |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \] +#+end_important + +** Model the noise of the super sensor +The PSD of the estimation $\hat{x}$ when $x = 0$ of the configuration shown on figure [[fig:sensor_fusion_full]] is: +\begin{align*} + \Phi_{\hat{x}}(\omega) &= | H_1(j\omega) N_1(j\omega) |^2 \Phi_{\tilde{n}_1} + | H_2(j\omega) N_2(j\omega) |^2 \Phi_{\tilde{n}_2} \\ + &= | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 +\end{align*} + +The PSD of the estimation $\hat{x}$ when $x = 0$ of the configuration shown on figure [[fig:sensor_fusion_equivalent]] is: +\begin{align*} + \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\ + &= | N_{ss}(j\omega) |^2 +\end{align*} + +Ideally, we want to find a weight $N_{ss}(s)$ such that: +#+begin_important +\[ |N_{ss}(j\omega)|^2 = | H_1(j\omega) N_1(j\omega) |^2 + | H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \] +#+end_important + +** First guess +We could choose +\begin{align*} + w_{ss}(s) &= H_1(s) w_1(s) + H_2(s) w_2(s) \\ + N_{ss}(s) &= H_1(s) N_1(s) + H_2(s) N_2(s) +\end{align*} + +But we would have: +\begin{align*} + |w_{ss}(j\omega)| &= |H_1(j\omega) w_1(j\omega) + H_2(j\omega) w_2(j\omega)|, \quad \forall\omega \\ + &\neq |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|, \quad \forall\omega +\end{align*} +and +\begin{align*} + |N_{ss}(j\omega)|^2 &= | H_1(j\omega) N_1(j\omega) + H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\ + &\neq | H_1(j\omega) N_1(j\omega)|^2 + |H_2(j\omega) N_2(j\omega) |^2 \quad \forall\omega \\ +\end{align*} + +* Optimal And Robust Sensor Fusion in Practice +<> +** Introduction :ignore: +Here are the steps in order to apply optimal and robust sensor fusion: + +- Measure the noise characteristics of the sensors to be merged (necessary for "optimal" part of the fusion) +- Measure/Estimate the dynamic uncertainty of the sensors (necessary for "robust" part of the fusion) +- Apply H2/H-infinity synthesis of the complementary filters + +** Measurement of the noise characteristics of the sensors +*** Huddle Test +The technique to estimate the sensor noise is taken from cite:barzilai98_techn_measur_noise_sensor_presen. + +Let's consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity $x$ as shown in figure [[fig:huddle_test]]. + +#+NAME: fig:huddle_test +#+CAPTION: Huddle test block diagram +[[file:figs-tikz/huddle_test.png]] + +Each sensor has uncorrelated noise $n_1$ and $n_2$ and internal dynamics $G_1(s)$ and $G_2(s)$ respectively. + +We here suppose that each sensor has the same magnitude of instrumental noise: $n_1 = n_2 = n$. +We also assume that their dynamics is ideal: $G_1(s) = G_2(s) = 1$. + +We then have: +#+NAME: eq:coh_bis +\begin{equation} + \gamma_{\hat{x}_1\hat{x}_2}^2(\omega) = \frac{1}{1 + 2 \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right) + \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right)^2} +\end{equation} + +Since the input signal $x$ and the instrumental noise $n$ are incoherent: +#+NAME: eq:incoherent_noise +\begin{equation} + |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)| +\end{equation} + +From equations eqref:eq:coh_bis and eqref:eq:incoherent_noise, we finally obtain +#+begin_important +#+NAME: eq:noise_psd +\begin{equation} + |\Phi_n(\omega)| = |\Phi_{\hat{x}}(\omega)| \left( 1 - \sqrt{\gamma_{\hat{x}_1\hat{x}_2}^2(\omega)} \right) +\end{equation} +#+end_important + +*** Weights that represents the noises' PSD + +For further complementary filter synthesis, it is preferred to consider a normalized noise source $\tilde{n}$ that has a PSD equal to one ($\Phi_{\tilde{n}}(\omega) = 1$) and to use a weighting filter $N(s)$ in order to represent the frequency dependence of the noise. + +The weighting filter $N(s)$ should be designed such that: +\begin{align*} + & \Phi_n(\omega) \approx |N(j\omega)|^2 \Phi_{\tilde{n}}(\omega) \quad \forall \omega \\ + \Longleftrightarrow & |N(j\omega)| \approx \sqrt{\Phi_n(\omega)} \quad \forall \omega +\end{align*} + +These weighting filters can then be used to compare the noise level of sensors for the synthesis of complementary filters. + +The sensor with a normalized noise input is shown in figure [[fig:one_sensor_normalized_noise]]. + +#+name: fig:one_sensor_normalized_noise +#+caption: One sensor with normalized noise +[[file:figs-tikz/one_sensor_normalized_noise.png]] + +*** Comparison of the noises' PSD +Once the noise of the sensors to be merged have been characterized, the power spectral density of both sensors have to be compared. + +Ideally, the PSD of the noise are such that: +\begin{align*} + \Phi_{n_1}(\omega) &< \Phi_{n_2}(\omega) \text{ for } \omega < \omega_m \\ + \Phi_{n_1}(\omega) &> \Phi_{n_2}(\omega) \text{ for } \omega > \omega_m +\end{align*} + +*** Computation of the coherence, power spectral density and cross spectral density of signals +The coherence between signals $x$ and $y$ is defined as follow +\[ \gamma^2_{xy}(\omega) = \frac{|\Phi_{xy}(\omega)|^2}{|\Phi_{x}(\omega)| |\Phi_{y}(\omega)|} \] +where $|\Phi_x(\omega)|$ is the output Power Spectral Density (PSD) of signal $x$ and $|\Phi_{xy}(\omega)|$ is the Cross Spectral Density (CSD) of signal $x$ and $y$. + +The PSD and CSD are defined as follow: +\begin{align} + |\Phi_x(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} \left| X_k(\omega, T) \right|^2 \\ + |\Phi_{xy}(\omega)| &= \frac{2}{n_d T} \sum^{n_d}_{n=1} [ X_k^*(\omega, T) ] [ Y_k(\omega, T) ] +\end{align} +where: +- $n_d$ is the number for records averaged +- $T$ is the length of each record +- $X_k(\omega, T)$ is the finite Fourier transform of the $k^{\text{th}}$ record +- $X_k^*(\omega, T)$ is its complex conjugate + +** Estimate the dynamic uncertainty of the sensors + +Let's consider one sensor represented on figure [[fig:one_sensor_dyn_uncertainty]]. + +The dynamic uncertainty is represented by an input multiplicative uncertainty where $w(s)$ is a weight that represents the level of the uncertainty. + +The goal is to accurately determine $w(s)$ for the sensors that have to be merged. + +#+name: fig:one_sensor_dyn_uncertainty +#+caption: Sensor with dynamic uncertainty +[[file:figs-tikz/one_sensor_dyn_uncertainty.png]] + +** Optimal and Robust synthesis of the complementary filters +Once the noise characteristics and dynamic uncertainty of both sensors have been determined and we have determined the following weighting functions: +- $w_1(s)$ and $w_2(s)$ representing the dynamic uncertainty of both sensors +- $N_1(s)$ and $N_2(s)$ representing the noise characteristics of both sensors + +The goal is to design complementary filters $H_1(s)$ and $H_2(s)$ shown in figure [[fig:sensor_fusion_full]] such that: +- the uncertainty on the super sensor dynamics is minimized +- the noise sources $\tilde{n}_1$ and $\tilde{n}_2$ has the lowest possible effect on the estimation $\hat{x}$ + +#+name: fig:sensor_fusion_full +#+caption: Sensor fusion architecture with sensor dynamics uncertainty +[[file:figs-tikz/sensor_fusion_full.png]] + +* Methods of complementary filter synthesis +<> +** Complementary filters using analytical formula + :PROPERTIES: + :header-args:matlab+: :tangle matlab/comp_filters_analytical.m + :header-args:matlab+: :comments org :mkdirp yes + :END: + <> + +*** Introduction :ignore: +*** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/comp_filters_analytical.m -nt data/comp_filters_analytical.zip ]; then + cp matlab/comp_filters_analytical.m comp_filters_analytical.m; + zip data/comp_filters_analytical \ + comp_filters_analytical.m + rm comp_filters_analytical.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/comp_filters_analytical.zip][here]]. +#+end_note + +*** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +*** Analytical 1st order complementary filters +First order complementary filters are defined with following equations: +\begin{align} + H_L(s) = \frac{1}{1 + \frac{s}{\omega_0}}\\ + H_H(s) = \frac{\frac{s}{\omega_0}}{1 + \frac{s}{\omega_0}} +\end{align} + +Their bode plot is shown figure [[fig:comp_filter_1st_order]]. + +#+begin_src matlab + w0 = 2*pi; % [rad/s] + + Hh1 = (s/w0)/((s/w0)+1); + Hl1 = 1/((s/w0)+1); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-2, 2, 1000); + + figure; + % Magnitude + ax1 = subplot(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, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude'); + hold off; + % Phase + ax2 = subplot(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,'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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_filter_1st_order.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_filter_1st_order +#+CAPTION: Bode plot of first order complementary filter ([[./figs/comp_filter_1st_order.png][png]], [[./figs/comp_filter_1st_order.pdf][pdf]]) +[[file:figs/comp_filter_1st_order.png]] + +*** Second Order Complementary Filters +We here use analytical formula for the complementary filters $H_L$ and $H_H$. + +The first two formulas that are used to generate complementary filters are: +\begin{align*} + H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\ + H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)} +\end{align*} +where: +- $\omega_0$ is the blending frequency in rad/s. +- $\alpha$ is used to change the shape of the filters: + - Small values for $\alpha$ will produce high magnitude of the filters $|H_L(j\omega)|$ and $|H_H(j\omega)|$ near $\omega_0$ but smaller value for $|H_L(j\omega)|$ above $\approx 1.5 \omega_0$ and for $|H_H(j\omega)|$ below $\approx 0.7 \omega_0$ + - A large $\alpha$ will do the opposite + +This is illustrated on figure [[fig:comp_filters_param_alpha]]. +The slope of those filters at high and low frequencies is $-2$ and $2$ respectively for $H_L$ and $H_H$. + +#+begin_src matlab :exports none + freqs_study = logspace(-2, 2, 10000); + alphas = [0.1, 1, 10]; + w0 = 2*pi*1; + + figure; + ax1 = subplot(2,1,1); + hold on; + for i = 1:length(alphas) + alpha = alphas(i); + Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + set(gca,'ColorOrderIndex',i); + plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz')))); + end + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + ylabel('Magnitude'); + hold off; + ylim([1e-3, 20]); + % Phase + ax2 = subplot(2,1,2); + hold on; + for i = 1:length(alphas) + alpha = alphas(i); + Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + set(gca,'ColorOrderIndex',i); + plot(freqs_study, 180/pi*angle(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'DisplayName', sprintf('$\\alpha = %g$', alpha)); + set(gca,'ColorOrderIndex',i); + plot(freqs_study, 180/pi*angle(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'HandleVisibility', 'off'); + end + set(gca,'xscale','log'); + yticks(-180:90:180); + ylim([-180 180]); + xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Phase [deg]'); + legend('Location', 'northeast'); + hold off; + linkaxes([ax1,ax2],'x'); + xlim([freqs_study(1), freqs_study(end)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_filters_param_alpha.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_filters_param_alpha +#+CAPTION: Effect of the parameter $\alpha$ on the shape of the generated second order complementary filters ([[./figs/comp_filters_param_alpha.png][png]], [[./figs/comp_filters_param_alpha.pdf][pdf]]) +[[file:figs/comp_filters_param_alpha.png]] + +We now study the maximum norm of the filters function of the parameter $\alpha$. As we saw that the maximum norm of the filters is important for the robust merging of filters. +#+begin_src matlab :exports none + alphas = logspace(-2, 2, 100); + w0 = 2*pi*1; + infnorms = zeros(size(alphas)); + + for i = 1:length(alphas) + alpha = alphas(i); + Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1)); + infnorms(i) = norm(Hh2, 'inf'); + end +#+end_src + +#+begin_src matlab + figure; + plot(alphas, infnorms) + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlabel('$\alpha$'); ylabel('$\|H_1\|_\infty$'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/param_alpha_hinf_norm.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:param_alpha_hinf_norm +#+CAPTION: Evolution of the H-Infinity norm of the complementary filters with the parameter $\alpha$ ([[./figs/param_alpha_hinf_norm.png][png]], [[./figs/param_alpha_hinf_norm.pdf][pdf]]) +[[file:figs/param_alpha_hinf_norm.png]] + +*** Third Order Complementary Filters +The following formula gives complementary filters with slopes of $-3$ and $3$: +\begin{align*} + H_L(s) &= \frac{\left(1+(\alpha+1)(\beta+1)\right) (\frac{s}{\omega_0})^2 + (1+\alpha+\beta)(\frac{s}{\omega_0}) + 1}{\left(\frac{s}{\omega_0} + 1\right) \left( (\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1 \right) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)}\\ + H_H(s) &= \frac{(\frac{s}{\omega_0})^3 \left( (\frac{s}{\omega_0})^2 + (1+\alpha+\beta) (\frac{s}{\omega_0}) + (1+(\alpha+1)(\beta+1)) \right)}{\left(\frac{s}{\omega_0} + 1\right) \left( (\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1 \right) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)} +\end{align*} + +The parameters are: +- $\omega_0$ is the blending frequency in rad/s +- $\alpha$ and $\beta$ that are used to change the shape of the filters similarly to the parameter $\alpha$ for the second order complementary filters + +The filters are defined below and the result is shown on figure [[fig:complementary_filters_third_order]]. + +#+begin_src matlab + alpha = 1; + beta = 10; + w0 = 2*pi*14; + + Hh3_ana = (s/w0)^3 * ((s/w0)^2 + (1+alpha+beta)*(s/w0) + (1+(alpha+1)*(beta+1)))/((s/w0 + 1)*((s/w0)^2+alpha*(s/w0)+1)*((s/w0)^2+beta*(s/w0)+1)); + Hl3_ana = ((1+(alpha+1)*(beta+1))*(s/w0)^2 + (1+alpha+beta)*(s/w0) + 1)/((s/w0 + 1)*((s/w0)^2+alpha*(s/w0)+1)*((s/w0)^2+beta*(s/w0)+1)); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(Hl3_ana, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$ - Analytical'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(Hh3_ana, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$ - Analytical'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([1e-3, 10]); + xticks([0.1, 1, 10, 100, 1000]); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/complementary_filters_third_order.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:complementary_filters_third_order +#+CAPTION: Third order complementary filters using the analytical formula ([[./figs/complementary_filters_third_order.png][png]], [[./figs/complementary_filters_third_order.pdf][pdf]]) +[[file:figs/complementary_filters_third_order.png]] + +** H-Infinity synthesis of complementary filters + :PROPERTIES: + :header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m + :header-args:matlab+: :comments org :mkdirp yes + :END: + <> + +*** Introduction :ignore: +*** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/h_inf_synthesis_complementary_filters.m -nt data/h_inf_synthesis_complementary_filters.zip ]; then + cp matlab/h_inf_synthesis_complementary_filters.m h_inf_synthesis_complementary_filters.m; + zip data/h_inf_synthesis_complementary_filters \ + h_inf_synthesis_complementary_filters.m + rm h_inf_synthesis_complementary_filters.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/h_inf_synthesis_complementary_filters.zip][here]]. +#+end_note + +*** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +*** Synthesis Architecture +We here synthesize the complementary filters using the $\mathcal{H}_\infty$ synthesis. +The goal is to specify upper bounds on the norms of $H_L$ and $H_H$ while ensuring their complementary property ($H_L + H_H = 1$). + +In order to do so, we use the generalized plant shown on figure [[fig:sf_hinf_filters_plant_b]] where $w_L$ and $w_H$ weighting transfer functions that will be used to shape $H_L$ and $H_H$ respectively. + +#+name: fig:sf_hinf_filters_plant_b +#+caption: Generalized plant used for the $\mathcal{H}_\infty$ synthesis of the complementary filters +[[file:figs-tikz/sf_hinf_filters_plant_b.png]] + +The $\mathcal{H}_\infty$ synthesis applied on this generalized plant will give a transfer function $H_L$ (figure [[fig:sf_hinf_filters_b]]) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_H,\ z_L]$ is less than one: +\[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \] + +Thus, if the above condition is verified, we can define $H_H = 1 - H_L$ and we have that: +\[ \left\| \begin{array}{c} H_L w_L \\ H_H w_H \end{array} \right\|_\infty < 1 \] +Which is almost (with an maximum error of $\sqrt{2}$) equivalent to: +\begin{align*} + |H_L| &< \frac{1}{|w_L|}, \quad \forall \omega \\ + |H_H| &< \frac{1}{|w_H|}, \quad \forall \omega +\end{align*} + +We then see that $w_L$ and $w_H$ can be used to shape both $H_L$ and $H_H$ while ensuring (by definition of $H_H = 1 - H_L$) their complementary property. + +#+name: fig:sf_hinf_filters_b +#+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters +[[file:figs-tikz/sf_hinf_filters_b.png]] + +*** Weights + +#+begin_src matlab + omegab = 2*pi*9; + wH = (omegab)^2/(s + omegab*sqrt(1e-5))^2; + omegab = 2*pi*28; + wL = (s + omegab/(4.5)^(1/3))^3/(s*(1e-4)^(1/3) + omegab)^3; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(wL, freqs, 'Hz'))), '-', 'DisplayName', '$w_L$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(wH, freqs, 'Hz'))), '-', 'DisplayName', '$w_H$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([1e-3, 10]); + xticks([0.1, 1, 10, 100, 1000]); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_wl_wh.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:weights_wl_wh +#+CAPTION: Weights on the complementary filters $w_L$ and $w_H$ and the associated performance weights ([[./figs/weights_wl_wh.png][png]], [[./figs/weights_wl_wh.pdf][pdf]]) +[[file:figs/weights_wl_wh.png]] + +*** H-Infinity Synthesis +We define the generalized plant $P$ on matlab. +#+begin_src matlab + P = [0 wL; + wH -wH; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [Hl_hinf, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+RESULTS: +#+begin_example +[Hl_hinf, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +Test bounds: 0.0000 < gamma <= 1.7285 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f + 1.729 4.1e+01 8.4e-12 1.8e-01 0.0e+00 0.0000 p + 0.864 3.9e+01 -5.8e-02# 1.8e-01 0.0e+00 0.0000 f + 1.296 4.0e+01 8.4e-12 1.8e-01 0.0e+00 0.0000 p + 1.080 4.0e+01 8.5e-12 1.8e-01 0.0e+00 0.0000 p + 0.972 3.9e+01 -4.2e-01# 1.8e-01 0.0e+00 0.0000 f + 1.026 4.0e+01 8.5e-12 1.8e-01 0.0e+00 0.0000 p + 0.999 3.9e+01 8.5e-12 1.8e-01 0.0e+00 0.0000 p + 0.986 3.9e+01 -1.2e+00# 1.8e-01 0.0e+00 0.0000 f + 0.993 3.9e+01 -8.2e+00# 1.8e-01 0.0e+00 0.0000 f + 0.996 3.9e+01 8.5e-12 1.8e-01 0.0e+00 0.0000 p + 0.994 3.9e+01 8.5e-12 1.8e-01 0.0e+00 0.0000 p + 0.993 3.9e+01 -3.2e+01# 1.8e-01 0.0e+00 0.0000 f + + Gamma value achieved: 0.9942 +#+end_example + +We then define the high pass filter $H_H = 1 - H_L$. The bode plot of both $H_L$ and $H_H$ is shown on figure [[fig:hinf_filters_results]]. +#+begin_src matlab + Hh_hinf = 1 - Hl_hinf; +#+end_src + +*** Obtained Complementary Filters + +The obtained complementary filters are shown on figure [[fig:hinf_filters_results]]. + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(wL, freqs, 'Hz'))), '--', 'DisplayName', '$w_L$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(wH, freqs, 'Hz'))), '--', 'DisplayName', '$w_H$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(Hl_hinf, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$ - $\mathcal{H}_\infty$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(Hh_hinf, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$ - $\mathcal{H}_\infty$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([1e-3, 10]); + xticks([0.1, 1, 10, 100, 1000]); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hinf_filters_results.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:hinf_filters_results +#+CAPTION: Obtained complementary filters using $\mathcal{H}_\infty$ synthesis ([[./figs/hinf_filters_results.png][png]], [[./figs/hinf_filters_results.pdf][pdf]]) +[[file:figs/hinf_filters_results.png]] + +** Feedback Control Architecture to generate Complementary Filters + :PROPERTIES: + :header-args:matlab+: :tangle matlab/feedback_generate_comp_filters.m + :header-args:matlab+: :comments org :mkdirp yes + :END: + <> + +*** Introduction :ignore: +The idea is here to use the fact that in a classical feedback architecture, $S + T = 1$, in order to design complementary filters. + +Thus, all the tools that has been developed for classical feedback control can be used for complementary filter design. + +*** ZIP file containing the data and matlab files :ignore: +#+begin_src bash :exports none :results none + if [ matlab/feedback_generate_comp_filters.m -nt data/feedback_generate_comp_filters.zip ]; then + cp matlab/feedback_generate_comp_filters.m feedback_generate_comp_filters.m; + zip data/feedback_generate_comp_filters \ + feedback_generate_comp_filters.m + rm feedback_generate_comp_filters.m; + fi +#+end_src + +#+begin_note + All the files (data and Matlab scripts) are accessible [[file:data/feedback_generate_comp_filters.zip][here]]. +#+end_note + +*** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-2, 2, 1000); +#+end_src + +*** Architecture +#+name: fig:complementary_filters_feedback_architecture +#+caption: Architecture used to generate the complementary filters +[[file:figs-tikz/complementary_filters_feedback_architecture.png]] + +We have: +\[ y = \underbrace{\frac{L}{L + 1}}_{H_L} y_1 + \underbrace{\frac{1}{L + 1}}_{H_H} y_2 \] +with $H_L + H_H = 1$. + +The only thing to design is $L$ such that the complementary filters are stable with the wanted shape. + +A simple choice is: +\[ L = \left(\frac{\omega_c}{s}\right)^2 \frac{\frac{s}{\omega_c / \alpha} + 1}{\frac{s}{\omega_c} + \alpha} \] + +Which contains two integrator and a lead. $\omega_c$ is used to tune the crossover frequency and $\alpha$ the trade-off "bump" around blending frequency and filtering away from blending frequency. + +*** Loop Gain Design +Let's first define the loop gain $L$. +#+begin_src matlab + wc = 2*pi*1; + alpha = 2; + + L = (wc/s)^2 * (s/(wc/alpha) + 1)/(s/wc + alpha); +#+end_src + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(2,1,1); + plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), '-'); + ylabel('Magnitude'); + set(gca, 'XScale', 'log'); + set(gca, 'YScale', 'log'); + + ax2 = subplot(2,1,2); + plot(freqs, 180/pi*phase(squeeze(freqresp(L, freqs, 'Hz'))), '--'); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + set(gca, 'XScale', 'log'); + ylim([-180, 0]); + yticks([-360:90:360]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/loop_gain_bode_plot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:loop_gain_bode_plot +#+CAPTION: Bode plot of the loop gain $L$ ([[./figs/loop_gain_bode_plot.png][png]], [[./figs/loop_gain_bode_plot.pdf][pdf]]) +[[file:figs/loop_gain_bode_plot.png]] + +*** Complementary Filters Obtained +We then compute the resulting low pass and high pass filters. +#+begin_src matlab + Hl = L/(L + 1); + Hh = 1/(L + 1); +#+end_src + +#+begin_src matlab :exports none + alphas = [1, 2, 10]; + + figure; + hold on; + for i = 1:length(alphas) + alpha = alphas(i); + L = (wc/s)^2 * (s/(wc/alpha) + 1)/(s/wc + alpha); + Hl = L/(L + 1); + Hh = 1/(L + 1); + set(gca,'ColorOrderIndex',i) + plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), 'DisplayName', sprintf('$\\alpha = %.0f$', alpha)); + set(gca,'ColorOrderIndex',i) + plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), 'HandleVisibility', 'off'); + end + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Amplitude') + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/low_pass_high_pass_filters.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:low_pass_high_pass_filters +#+CAPTION: Low pass and High pass filters $H_L$ and $H_H$ for different values of $\alpha$ ([[./figs/low_pass_high_pass_filters.png][png]], [[./figs/low_pass_high_pass_filters.pdf][pdf]]) +[[file:figs/low_pass_high_pass_filters.png]] + +** Analytical Formula found in the literature + <> + +*** Analytical Formula +cite:min15_compl_filter_desig_angle_estim +\begin{align*} + H_L(s) = \frac{K_p s + K_i}{s^2 + K_p s + K_i} \\ + H_H(s) = \frac{s^2}{s^2 + K_p s + K_i} +\end{align*} + +cite:corke04_inert_visual_sensin_system_small_auton_helic +\begin{align*} + H_L(s) = \frac{1}{s/p + 1} \\ + H_H(s) = \frac{s/p}{s/p + 1} +\end{align*} + +cite:jensen13_basic_uas +\begin{align*} + H_L(s) = \frac{2 \omega_0 s + \omega_0^2}{(s + \omega_0)^2} \\ + H_H(s) = \frac{s^2}{(s + \omega_0)^2} +\end{align*} + +\begin{align*} + H_L(s) = \frac{C(s)}{C(s) + s} \\ + H_H(s) = \frac{s}{C(s) + s} +\end{align*} + +cite:shaw90_bandw_enhan_posit_measur_using_measur_accel +\begin{align*} + H_L(s) = \frac{3 \tau s + 1}{(\tau s + 1)^3} \\ + H_H(s) = \frac{\tau^3 s^3 + 3 \tau^2 s^2}{(\tau s + 1)^3} +\end{align*} + +cite:baerveldt97_low_cost_low_weigh_attit +\begin{align*} + H_L(s) = \frac{2 \tau s + 1}{(\tau s + 1)^2} \\ + H_H(s) = \frac{\tau^2 s^2}{(\tau s + 1)^2} +\end{align*} + +*** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +*** Matlab +#+begin_src 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; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-1, 1, 1000); + + figure; + % Magnitude + ax1 = subplot(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 = subplot(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)]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_filters_literature.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_filters_literature +#+CAPTION: Comparison of some complementary filters found in the literature ([[./figs/comp_filters_literature.png][png]], [[./figs/comp_filters_literature.pdf][pdf]]) +[[file:figs/comp_filters_literature.png]] + +*** Discussion +Analytical Formula found in the literature provides either no parameter for tuning the robustness / performance trade-off. + +** Comparison of the different methods of synthesis + <> +The generated complementary filters using $\mathcal{H}_\infty$ and the analytical formulas are very close to each other. However there is some difference to note here: +- the analytical formula provides a very simple way to generate the complementary filters (and thus the controller), they could even be used to tune the controller online using the parameters $\alpha$ and $\omega_0$. However, these formula have the property that $|H_H|$ and $|H_L|$ are symmetrical with the frequency $\omega_0$ which may not be desirable. +- while the $\mathcal{H}_\infty$ synthesis of the complementary filters is not as straightforward as using the analytical formula, it provides a more optimized procedure to obtain the complementary filters + +* Real World Example of optimal sensor fusion +** Introduction :ignore: +cite:moore19_capac_instr_sensor_fusion_high_bandw_nanop + + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +** Sensor Noise :noexport: +#+begin_src matlab + A1 = 19.13; % [uV2/Hz] + A2 = 0.1632; % [uV2/Hz] + A3 = 6.847; % [uV2/Hz] + wnc = 3057; % [rad] + wx = 7929; % [rad/s] + + Fx = 1/(1 - s/wx)/(1 - s/wx); + [A B C D] = butter(2, 0.5, 'low'); + Fx = ss(A, B, C, D); + + Sq = A3*wnc/s + A3; + Sx = A1*Fx + A2; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 5, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(Sq, freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Sx, freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); +#+end_src + +** Matlab Code +Take an Accelerometer and a Geophone both measuring the absolute motion of a structure. + +Parameters of the inertial sensors. +#+begin_src matlab + m_acc = 0.01; + k_acc = 1e6; + c_acc = 20; + + m_geo = 1; + k_geo = 1e3; + c_geo = 10; +#+end_src + +Transfer function from motion to measurement + +For the accelerometer. +The measurement is the relative motion structure/inertial mass: +\[ \frac{d}{\ddot{w}} = \frac{-m}{ms^2 + cs + k} \] + +For the geophone. +The measurement is the relative velocity structure/inertial mass: +\[ \frac{\dot{d}}{\dot{w}} = \frac{-ms^2}{ms^2 + cs + k} \] + +#+begin_src matlab + G_acc = -m_acc/(m_acc*s^2 + c_acc*s + k_acc); % [m/(m/s^2)] + G_geo = -m_geo*s^2/(m_geo*s^2 + c_geo*s + k_geo); % [m/s/m/s] +#+end_src + +Suppose the measure of the relative motion for the accelerometer (capacitive sensor for instance) has a white noise characteristic: +Suppose the measure of the relative velocity (current flowing through the coil) has a white noise characteristic: + +Define the noise characteristics +#+begin_src matlab + n = 1; w0 = 2*pi*5e3; G0 = 5e-12; G1 = 1e-15; Gc = G0/2; + L_acc = (((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; + + n = 1; w0 = 2*pi*5e3; G0 = 1e-6; G1 = 1e-8; Gc = G0/2; + L_geo = (((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; +#+end_src + +Transfer function of the conversion to obtain the velocity: +#+begin_src matlab + C_acc = (-k_acc/m_acc/(2*pi + s)); + C_geo = tf(-1); +#+end_src + +Let's plot the noise of both sensors: +#+begin_src matlab :exports none + freqs = logspace(-1, 4, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(L_acc*C_acc, freqs, 'Hz'))), 'DisplayName', 'Acc'); + plot(freqs, abs(squeeze(freqresp(L_geo*C_geo, freqs, 'Hz'))), 'DisplayName', 'Geo'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Noise ASD [$m/s/\sqrt{Hz}$]'); + legend('location', 'northeast') +#+end_src + +Dynamics of both sensors +#+begin_src matlab :exports none + freqs = logspace(-1, 4, 1000); + + figure; + hold on; + plot(freqs, abs(squeeze(freqresp(s*G_acc*C_acc, freqs, 'Hz'))), 'DisplayName', 'Acc'); + plot(freqs, abs(squeeze(freqresp(G_geo*C_geo, freqs, 'Hz'))), 'DisplayName', 'Geo'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + legend('location', 'northeast') +#+end_src + +** Time domain signals +#+begin_src matlab + Fs = 1e4; % Sampling Frequency [Hz] + Ts = 1/Fs; % Sampling Time [s] + + t = 0:Ts:10; % Time Vector [s] +#+end_src + +#+begin_src matlab + n_acc = lsim(L_acc*C_acc, sqrt(Fs/2)*randn(length(t), 1), t); % [m/s] + n_geo = lsim(L_geo*C_geo, sqrt(Fs/2)*randn(length(t), 1), t); % [m/s] +#+end_src + +#+begin_src matlab + figure; + hold on; + plot(t, n_geo) + plot(t, n_acc) + hold off; +#+end_src + +** H2 Synthesis +#+begin_src matlab + N1 = L_acc*C_acc; + N2 = L_geo*C_geo; +#+end_src + +#+begin_src matlab + bodeFig({N1, N2}, logspace(-1, 5, 1000)) +#+end_src + +#+begin_src matlab + P = [0 N2 1; + N1 -N2 0]; +#+end_src + +And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command. +#+begin_src matlab + [H1, ~, gamma] = h2syn(P, 1, 1); +#+end_src + +Finally, we define $H_2(s) = 1 - H_1(s)$. +#+begin_src matlab + H2 = 1 - H1; +#+end_src + +#+begin_src matlab + bodeFig({H1, H2}, struct('phase', true)) +#+end_src + +#+begin_src matlab + n_acc_filt = lsim(H1, n_acc, t); + n_geo_filt = lsim(H2, n_geo, t); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([rms(n_acc), rms(n_geo), rms(n_acc_filt + n_geo_filt)]', {'Accelerometer', 'Geophone', 'Super Sensor'}, {'RMS'}, ' %.1e '); +#+end_src + +#+RESULTS: +| | RMS | +|---------------+---------| +| Accelerometer | 9.7e-05 | +| Geophone | 5.9e-05 | +| Super Sensor | 1.5e-05 | + +#+begin_src matlab + figure; + hold on; + plot(t, n_geo) + plot(t, n_acc) + plot(t, n_acc_filt + n_geo_filt) + hold off; +#+end_src + +** Signal and Noise +Velocity Signal: +#+begin_src matlab + v = lsim(1/(1 + s/2/pi/2), 1e-4*sqrt(Fs/2)*randn(length(t), 1), t); + v = 1e-4 * sin(2*pi*100*t); +#+end_src + +#+begin_src matlab + v_acc = lsim(s*G_acc*C_acc, v, t) + n_acc; + v_geo = lsim(G_geo*C_geo, v, t) + n_geo; +#+end_src + +#+begin_src matlab + v_ss = lsim(H1, v_acc, t) + lsim(H2, v_geo, t); +#+end_src + +#+begin_src matlab + figure; + hold on; + plot(t, v_geo) + plot(t, v_acc) + plot(t, v_ss) + plot(t, v, 'k--') + hold off; + xlim([1, 1+0.1]) +#+end_src + +** PSD and CPS +#+begin_src matlab + nx = length(n_acc); + na = 16; + win = hanning(floor(nx/na)); + + [p_acc, f] = pwelch(n_acc, win, 0, [], Fs); + [p_geo, ~] = pwelch(n_geo, win, 0, [], Fs); + [p_ss, ~] = pwelch(n_acc_filt + n_geo_filt, win, 0, [], Fs); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(f, p_acc, 'DisplayName', 'Accelerometer'); + plot(f, p_geo, 'DisplayName', 'Geophone'); + plot(f, p_ss, 'DisplayName', 'Super Sensor'); + hold off; + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Power Spectral Density $\left[\frac{(m/s)^2}{Hz}\right]$'); + legend('location', 'southwest'); +#+end_src + +** Transfer function of the super sensor +#+begin_src matlab + bodeFig({s*C_acc*G_acc, C_geo*G_geo, s*C_acc*G_acc*H1+C_geo*G_geo*H2}, struct('phase', true)) +#+end_src + +* Bibliography :ignore: +bibliographystyle:unsrt +bibliography:ref.bib