Reworked all the analysis about optimal sensor fusion

This commit is contained in:
Thomas Dehaeze 2019-08-29 14:55:03 +02:00
parent 148ede19ad
commit 28d394df34
33 changed files with 1752 additions and 833 deletions

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@ -86,61 +86,65 @@ The complementary filters have to be designed in order to minimize the effect no
#+end_src
** Architecture
Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_with_dyn]] where two sensors 1 and 2 are measuring the same quantity $x$ with different noise characteristics determined by $W_1$ and $W_2$.
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 $N_2(s)$.
$n_1$ and $n_2$ are white noise (constant power spectral density over all frequencies).
$\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise:
#+name: eq:normalized_noise
\begin{equation}
\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1
\end{equation}
#+name: fig:fusion_two_noisy_sensors_with_dyn
#+name: fig:fusion_two_noisy_sensors_weights
#+caption: Fusion of two sensors
[[file:figs-tikz/fusion_two_noisy_sensors_with_dyn.png]]
[[file:figs-tikz/fusion_two_noisy_sensors_weights.png]]
We consider that the two sensor dynamics $G_1$ and $G_2$ are ideal ($G_1 = G_2 = 1$). We obtain the architecture of figure [[fig:fusion_two_noisy_sensors]].
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}
#+name: fig:fusion_two_noisy_sensors
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/fusion_two_noisy_sensors.png]]
[[file:figs-tikz/sensor_fusion_noisy_perfect_dyn.png]]
$H_1$ and $H_2$ are complementary filters ($H_1 + H_2 = 1$). The goal is to design $H_1$ and $H_2$ such that the effect of the noise sources $n_1$ and $n_2$ has the smallest possible effect on the estimation $\hat{x}$.
$H_1(s)$ 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:
\[ \Gamma_{\hat{x}} = |H_1 W_1|^2 \Gamma_{n_1} + |H_2 W_2|^2 \Gamma_{n_2} \]
\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]
And the goal is the minimize the Root Mean Square (RMS) value of $\hat{x}$:
\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Gamma_{\hat{x}}(\omega) d\omega} \]
As $n_1$ and $n_2$ are white noise: $\Gamma_{n_1} = \Gamma_{n_2} = 1$ and we have:
\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 W_1|^2(\omega) + |H_2 W_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2 \]
Thus, the goal is to design $H_1$ and $H_2$ such that $H_1 + H_2 = 1$ and such that $\left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2$ is minimized.
For that, we will use the $\mathcal{H}_2$ Synthesis.
#+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 $W_1$ and $W_2$:
- Sensor 1 characterized by $W_1$ has low noise at low frequency (for instance a geophone)
- Sensor 2 characterized by $W_2$ has low noise at high frequency (for instance an accelerometer)
#+begin_src matlab :exports none
omegac = 2*pi; G0 = 1e-2; Ginf = 1e-6;
W1 = ((sqrt(G0))/(s/omegac + 1))^2;
omegac = 100*2*pi; G0 = 1e-6; Ginf = 1e-2;
W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
#+end_src
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;
W1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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(W1, freqs, 'Hz'))), '-', 'DisplayName', '$W_1$');
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '-', 'DisplayName', '$W_2$');
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;
@ -149,15 +153,21 @@ Let's define the noise characteristics of the two sensors by choosing $W_1$ and
#+end_src
#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nosie_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
#+begin_src matlab :var filepath="figs/noise_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+NAME: fig:nosie_characteristics_sensors
#+CAPTION: Noise Characteristics of the two sensors ([[./figs/nosie_characteristics_sensors.png][png]], [[./figs/nosie_characteristics_sensors.pdf][pdf]])
[[file:figs/nosie_characteristics_sensors.png]]
#+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
@ -165,16 +175,16 @@ We use the generalized plant architecture shown on figure [[fig:h_infinity_optim
[[file:figs-tikz/h_infinity_optimal_comp_filters.png]]
The transfer function from $[n_1, n_2]$ to $\hat{x}$ is:
\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \]
\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \]
If we define $H_2 = 1 - H_1$, we obtain:
\[ \begin{bmatrix} W_1 H_1 \\ W_2 H_2 \end{bmatrix} \]
\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]
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 W2 1;
W1 -W2 0];
P = [0 N2 1;
N1 -N2 0];
#+end_src
And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
@ -182,17 +192,18 @@ And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
[H1, ~, gamma] = h2syn(P, 1, 1);
#+end_src
What is minimized is =norm([W1*H1,W2*H2], 2)=.
Finally, we define $H_2 = 1 - H_1$.
Finally, we define $H_2(s) = 1 - H_1(s)$.
#+begin_src matlab
H2 = 1 - H1;
#+end_src
** Analysis
The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]. The PSD of the [[fig:psd_sensors_htwo_synthesis]].
Finally, the RMS value of $\hat{x}$ is shown on table [[tab:rms_results]].
The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor.
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;
@ -215,14 +226,20 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
#+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, abs(squeeze(freqresp(W1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|W_1|^2$');
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|W_2|^2$');
plot(freqs, abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|W_1H_1|^2+|W_2H_2|^2$');
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('Magnitude');
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
hold off;
xlim([freqs(1), freqs(end)]);
legend('location', 'northeast');
@ -237,77 +254,76 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
#+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 :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([norm([W1], 2);norm([W2], 2);norm([W1*H1 + W2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'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 | 1.1e-02 |
| Sensor 2 | 1.3e-03 |
| Optimal Sensor Fusion | 1.5e-04 |
** H-Infinity Synthesis
*** First idea
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 use filters with the highest possible order to approximate that.
However, we may have robustness and implementation problems.
#+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/4000);
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
n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
W1 = (((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;
W2 = (((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;
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, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$');
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
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([freqs(1), freqs(end)]);
legend('location', 'northeast');
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")
<<plt-matlab>>
#+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]]
** 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 = [W1 -W1;
0 W2;
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
[H2, ~, 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');
#+end_src
#+RESULTS:
#+begin_example
[H2, ~, 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
@ -343,7 +359,7 @@ Test bounds: 0.1000 < gamma <= 10500.0000
#+end_example
#+begin_src matlab
H1 = 1 - H2;
H1a = 1 - H2a;
#+end_src
#+begin_src matlab :exports none
@ -352,14 +368,14 @@ Test bounds: 0.1000 < gamma <= 10500.0000
ax1 = subplot(2,1,1);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
plot(freqs, 1./abs(squeeze(freqresp(W1a, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
set(gca,'ColorOrderIndex',2)
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
plot(freqs, 1./abs(squeeze(freqresp(W2a, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
plot(freqs, abs(squeeze(freqresp(H1a, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
plot(freqs, abs(squeeze(freqresp(H2a, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@ -371,9 +387,9 @@ Test bounds: 0.1000 < gamma <= 10500.0000
ax2 = subplot(2,1,2);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
plot(freqs, 180/pi*phase(squeeze(freqresp(H1a, freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
plot(freqs, 180/pi*phase(squeeze(freqresp(H2a, freqs, 'Hz'))), '-');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
@ -384,135 +400,100 @@ Test bounds: 0.1000 < gamma <= 10500.0000
xticks([0.1, 1, 10, 100, 1000]);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_1|^2$');
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_2|^2$');
plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
legend('location', 'northeast');
#+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")
<<plt-matlab>>
#+end_src
#+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)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
#+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
#+RESULTS:
| | rms value |
|-----------------------+-----------|
| Sensor 1 | 1.1e-02 |
| Sensor 2 | 1.3e-03 |
| Optimal Sensor Fusion | 3.2e-04 |
*** Second idea
** H-Infinity Synthesis - method B
We have that:
\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \]
\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \]
We model the Power Spectral Densities of the two sensors by weighting functions $N_1(s)$ and $N_2(s)$.
\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \]
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 $|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2$.
Then, at frequencies where $|H_1| < |H_2|$ we would like that $|N_1| = 1$ and $|N_2| = 0$ as we discussed before.
Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2$.
However, if we design $H_2$ such that when $|N_1| < |N_2|$, we have that:
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_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\
\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|H_1 N_1\right|^2
&\approx \left|N_1\right|^2
\end{align*}
We have then that:
\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \]
We suppose that $|H_1| = 1$:
\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \]
We can use that for the maximum magnitude of $|H_2|$ at frequencies where $|N_1| < |N_2|$
Similarly, we have that the maximum magnitude of $|H_1|$ at frequencies where $|N_1| > |N_2|$ is:
Similarly, we design $H_1(s)$ such that at frequencies where $|N_1| > |N_2|$:
\[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \]
We could take $\epsilon = 1$, then we increase the noise of the super sensor just by a factor $\sqrt{2}$ over the all bandwidth from the idea case.
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.
This could be used as weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
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
omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
epsilon = 2;
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;
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
W1 = N1/N2;
W2 = N2/N1;
W1 = W1/(1 + s/2/pi/1000); % this is added so that it is proper
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$');
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$');
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 = [W1 -W1;
0 W2;
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
[H2, ~, 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');
#+end_src
#+RESULTS:
#+begin_example
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
Test bounds: 0.0000 < gamma <= 2625.0000
[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
2.625e+03 1.8e+01 -2.3e-12 6.3e+00 -1.7e-14 0.0000 p
1.313e+03 1.8e+01 -3.3e-12 6.3e+00 -5.9e-13 0.0000 p
656.250 1.8e+01 -1.8e-12 6.3e+00 -1.2e-13 0.0000 p
328.125 1.8e+01 -8.3e-13 6.3e+00 -7.5e-14 0.0000 p
164.063 1.8e+01 -3.8e-13 6.3e+00 -6.1e-14 0.0000 p
82.031 1.8e+01 -3.3e-22 6.3e+00 -5.4e-14 0.0000 p
41.016 1.8e+01 -3.4e-12 6.3e+00 -1.4e-16 0.0000 p
20.508 1.8e+01 -6.4e-14 6.3e+00 -4.4e-13 0.0000 p
10.254 1.8e+01 -1.4e-13 6.3e+00 -4.9e-14 0.0000 p
5.127 1.7e+01 -1.1e-12 6.3e+00 -1.6e-13 0.0000 p
2.563 1.7e+01 -6.5e-13 6.3e+00 -7.3e-14 0.0000 p
1.282 1.4e+01 -8.9e+04# 6.3e+00 -1.6e-14 0.0000 f
1.923 1.6e+01 -4.6e+05# 6.3e+00 -2.1e-13 0.0000 f
2.243 1.7e+01 -1.9e+06# 6.3e+00 -9.9e-14 0.0000 f
2.403 1.7e+01 -3.7e-11 6.3e+00 -2.1e-12 0.0000 p
2.323 1.7e+01 -4.6e+06# 6.3e+00 -5.8e-14 0.0000 f
2.363 1.7e+01 -1.2e+07# 6.3e+00 -3.3e-14 0.0000 f
2.383 1.7e+01 -6.8e+07# 6.3e+00 -1.2e-13 0.0000 f
2.393 1.7e+01 -5.5e-11 6.3e+00 -1.8e-12 0.0000 p
2.388 1.7e+01 1.3e-22 6.3e+00 -2.5e-15 0.0000 p
2.386 1.7e+01 -1.5e+08# 6.3e+00 -1.4e-13 0.0000 f
2.387 1.7e+01 -4.0e+08# 6.3e+00 -1.0e-13 0.0000 f
2.388 1.7e+01 -2.1e+09# 6.3e+00 -4.2e-13 0.0000 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: 2.3882
Gamma value achieved: 1.1390
#+end_example
#+begin_src matlab
H1 = 1 - H2;
H1b = 1 - H2b;
#+end_src
#+begin_src matlab :exports none
@ -521,14 +502,14 @@ Test bounds: 0.0000 < gamma <= 2625.0000
ax1 = subplot(2,1,1);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
plot(freqs, 1./abs(squeeze(freqresp(W1b, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
set(gca,'ColorOrderIndex',2)
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
plot(freqs, 1./abs(squeeze(freqresp(W2b, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
plot(freqs, abs(squeeze(freqresp(H1b, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
plot(freqs, abs(squeeze(freqresp(H2b, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
@ -540,9 +521,9 @@ Test bounds: 0.0000 < gamma <= 2625.0000
ax2 = subplot(2,1,2);
hold on;
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
plot(freqs, 180/pi*phase(squeeze(freqresp(H1b, freqs, 'Hz'))), '-');
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
plot(freqs, 180/pi*phase(squeeze(freqresp(H2b, freqs, 'Hz'))), '-');
hold off;
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
set(gca, 'XScale', 'log');
@ -553,32 +534,96 @@ Test bounds: 0.0000 < gamma <= 2625.0000
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")
<<plt-matlab>>
#+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
** 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)], {'Sensor 1', 'Sensor 2', 'H2 Fusion', 'H-Infinity a', 'H-Infinity b'}, {'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 |
#+begin_src matlab :exports none
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_1|^2$');
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_2|^2$');
plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$');
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
plot(freqs, PSD_Ha, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},a}$');
plot(freqs, PSD_Hb, 'k-.', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},b}$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
hold off;
xlim([freqs(1), freqs(end)]);
legend('location', 'northeast');
#+end_src
#+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)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
#+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")
<<plt-matlab>>
#+end_src
#+RESULTS:
| | rms value |
|-----------------------+-----------|
| Sensor 1 | 1.1e-02 |
| Sensor 2 | 1.3e-03 |
| Optimal Sensor Fusion | 1.9e-04 |
#+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]]
*** Conclusion
From the above two experiments with $\mathcal{H}_\infty$ 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)
#+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))));
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))));
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")
<<plt-matlab>>
#+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]]
** 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).
* Robustness to sensor dynamics uncertainty
:PROPERTIES:

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@ -108,7 +108,7 @@ Configuration file is accessible [[file:config.org][here]].
#+begin_src latex :file h_infinity_optimal_comp_filters.pdf :tangle figs/h_infinity_optimal_comp_filters.tex
<<tikz_default>>
\begin{tikzpicture}
\node[block={5.0cm}{3.0cm}, dashed] (P) {};
\node[block={5.0cm}{3.0cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P$};
\coordinate[] (inputn1) at ($(P.south west)!0.8!(P.north west) + (-\cdist, 0)$);
@ -118,18 +118,18 @@ Configuration file is accessible [[file:config.org][here]].
\coordinate[] (outputx) at ($(P.south east)!0.5!(P.north east) + ( \cdist, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( \cdist, 0)$);
\node[block, right=1.5 of inputn1] (W1){$W_1$};
\node[block, right=1.5 of inputn2] (W2){$W_2$};
\node[addb={+}{}{}{}{-}, right=of W1] (sub) {};
\node[addb, right=2 of W2] (add) {};
\node[block, right=1.5 of inputn1] (N1){$N_1$};
\node[block, right=1.5 of inputn2] (N2){$N_2$};
\node[addb={+}{}{}{}{-}, right=of N1] (sub) {};
\node[addb, right=2 of N2] (add) {};
\node[block, below=of P] (H1) {$H_1$};
\draw[->] (inputn1)node[above right]{$n_1$} -- (W1.west);
\draw[->] (inputn2)node[above right]{$n_2$} -- (W2.west);
\draw[->] (W1) -- (sub.west);
\draw[->] (W2) -| (sub.south);
\draw[->] (W2-|sub.south) node[branch]{} -- (add.west);
\draw[->] (inputn1)node[above right]{$\tilde{n}_1$} -- (N1.west);
\draw[->] (inputn2)node[above right]{$\tilde{n}_2$} -- (N2.west);
\draw[->] (N1) -- (sub.west);
\draw[->] (N2) -| (sub.south);
\draw[->] (N2-|sub.south) node[branch]{} -- (add.west);
\draw[->] (sub.east) -- ++(0.5, 0) |- ($(outputv) + (-0.3, 0)$) |- (H1.east);
\draw[->] (H1.west) -| ($(inputu) + (0.3, 0)$) -| (add.south);
\draw[->] (add.east) -- (outputx) node[above left]{$\hat{x}$};
@ -142,28 +142,40 @@ Configuration file is accessible [[file:config.org][here]].
[[file:figs/h_infinity_optimal_comp_filters.png]]
* Fusion of two noisy sensors
#+begin_src latex :file fusion_two_noisy_sensors.pdf :tangle figs/fusion_two_noisy_sensors.tex
<<tikz_default>>
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=1.5 and 1 of x](add1){};
\node[addb, below right=1.5 and 1 of x](add2){};
\node[block, above=0.5 of add1](W1){$W_1$};
\node[block, above=0.5 of add2](W2){$W_2$};
\node[block, right=1 of add1](H1){$H_1$};
\node[block, right=1 of add2](H2){$H_2$};
\node[addb, right=4 of x](add){};
\node[block, above right=0.5 and 0.5 of x](G1){$G_1(s)$};
\node[block, below right=0.5 and 0.5 of x](G2){$G_2(s)$};
\node[addb, right=0.8 of G1](add1){};
\node[addb, right=0.8 of G2](add2){};
\node[block, right=0.8 of add1](H1){$H_1(s)$};
\node[block, right=0.8 of add2](H2){$H_2(s)$};
\node[addb, right=5 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (add1.west);
\draw[->] (x) |- (add2.west);
\draw[->] (W1.south) -- (add1.north);
\draw[->] (W2.south) -- (add2.north);
\draw[<-] (W1.north) -- ++(0, 0.8)node[below right]{$n_1$};
\draw[<-] (W2.north) -- ++(0, 0.8)node[below right]{$n_2$};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (G1.west);
\draw[->] (x.center) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (add2.west);
\draw[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){$n_1$};
\draw[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){$n_2$};
\draw[->] (add1.east) -- (H1.west);
\draw[->] (add2.east) -- (H2.west);
\draw[->] (H1) -| (add.north);
\draw[->] (H2) -| (add.south);
\draw[->] (add.east) -- ++(1, 0) node[above left]{$\hat{x}$};
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1};
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2};
\end{scope}
\end{tikzpicture}
#+end_src
@ -171,6 +183,97 @@ Configuration file is accessible [[file:config.org][here]].
#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_noisy_sensors.png][png]], [[./figs/fusion_two_noisy_sensors.pdf][pdf]], [[./figs/fusion_two_noisy_sensors.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_noisy_sensors.png]]
* Fusion of two noisy sensors with perfect dynamics
#+begin_src latex :file sensor_fusion_noisy_perfect_dyn.pdf :tangle figs/sensor_fusion_noisy_perfect_dyn.tex
<<tikz_default>>
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=1.2 and 1.8 of x](addn1){};
\node[addb, below right=1.2 and 1.8 of x](addn2){};
\node[block, above=0.5 of addn1](N1) {$N_1(s)$};
\node[block, above=0.5 of addn2](N2) {$N_2(s)$};
\node[block, right=1.2 of addn1](H1){$H_1(s)$};
\node[block, right=1.2 of addn2](H2){$H_2(s)$};
\node[addb, right=4.5 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (addn1.west);
\draw[->] (x.center) |- (addn2.west);
\draw[->] (addn1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
\draw[->] (addn2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
\draw[->] ($(N1.north)+(0,0.7)$) node[below right](n1){$\tilde{n}_1$} -- (N1.north);
\draw[->] ($(N2.north)+(0,0.7)$) node[below right](n2){$\tilde{n}_2$} -- (N2.north);
\draw[->] (N1.south) -- (addn1.north)node[above right]{$n_1$};
\draw[->] (N2.south) -- (addn2.north)node[above right]{$n_2$};
\draw[->] (H1.east) -| (add.north);
\draw[->] (H2.east) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(addn2.south-|x)+(-0.2, -0.4)$) ($(n1.north-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[block, fit={($(x|-addn1.south)+(0.2, -0.2)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1};
\node[block, fit={($(x|-addn2.south)+(0.2, -0.2)$) ($(n2.north-|N2.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:sensor_fusion_noisy_perfect_dyn
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_noisy_perfect_dyn.png][png]], [[./figs/sensor_fusion_noisy_perfect_dyn.pdf][pdf]], [[./figs/sensor_fusion_noisy_perfect_dyn.tex][tex]]).
#+RESULTS:
[[file:figs/sensor_fusion_noisy_perfect_dyn.png]]
* Fusion of two noisy sensors with sensors weights
#+begin_src latex :file fusion_two_noisy_sensors_weights.pdf :tangle figs/fusion_two_noisy_sensors_weights.tex
<<tikz_default>>
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=1 and 0.5 of x](G1){$G_1(s)$};
\node[block, below right=1 and 0.5 of x](G2){$G_2(s)$};
\node[addb, right=0.8 of G1](add1){};
\node[addb, right=0.8 of G2](add2){};
\node[block, above=0.5 of add1](N1){$N_1(s)$};
\node[block, above=0.5 of add2](N2){$N_2(s)$};
\node[block, right=0.9 of add1](H1){$H_1(s)$};
\node[block, right=0.9 of add2](H2){$H_2(s)$};
\node[addb, right=5 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (G1.west);
\draw[->] (x.center) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (add2.west);
\draw[<-] (N1.north) -- ++(0, 0.6)node[below right](n1){$\tilde{n}_1$};
\draw[<-] (N2.north) -- ++(0, 0.6)node[below right](n2){$\tilde{n}_2$};
\draw[->] (N1.south) -- (add1.north)node[above right]{$n_1$};
\draw[->] (N2.south) -- (add2.north)node[above right]{$n_2$};
\draw[->] (add1.east) -- (H1.west);
\draw[->] (add2.east) -- (H2.west);
\draw[->] (H1) -| (add.north);
\draw[->] (H2) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer]
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1};
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_noisy_sensors_weights
#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_noisy_sensors_weights.png][png]], [[./figs/fusion_two_noisy_sensors_weights.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_weights.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_noisy_sensors_weights.png]]
* Fusion of two noisy signals
#+begin_src latex :file fusion_two_signals.pdf :tangle figs/fusion_two_signals.tex
\begin{tikzpicture}