Update all the analysis about robustness / uncertainty reduction

This commit is contained in:
Thomas Dehaeze 2019-08-29 22:47:20 +02:00
parent 28d394df34
commit 8a901b0e87
22 changed files with 1370 additions and 883 deletions

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@ -44,7 +44,7 @@ Then, three design methods for generating two complementary filters are proposed
- in section [[sec:feedback_generate_comp_filters]], the classical feedback architecture is used
- in section [[sec:analytical_formula_literature]], analytical formulas found in the literature are listed
* Optimal Sensor Fusion for noise characteristics
* 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
@ -625,7 +625,7 @@ The RMS value of the obtained super sensors are shown on table [[tab:rms_results
** 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
* 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
@ -633,28 +633,18 @@ From the above complementary filter design with the $\mathcal{H}_2$ and $\mathca
<<sec:comp_filter_robustness>>
** Introduction :ignore:
Let's first consider ideal sensors where $G_1 = 1$ and $G_2 = 1$ (figure [[fig:fusion_two_noisy_sensors_with_dyn_bis]]).
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
#+name: fig:fusion_two_noisy_sensors_with_dyn_bis
#+caption: Fusion of two sensors
[[file:figs/fusion_two_noisy_sensors_with_dyn_bis.png]]
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:fusion_two_signals
# #+caption: Fusion of two noisy measurements of $x$
# [[file:figs/fusion_two_signals.png]]
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]]
We then have:
\begin{align*}
\hat{x} &= (x + n_1) H_1 + (x + n_2) H_2 \\
&= x + n_1 H_1 + n_2 H_2
\end{align*}
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.
So the estimation error is
\[ \delta_x = \hat{x} - x = n_1 H_1 + n_2 H_2 \]
And we see that the complementary filters are only shaping the noise and that they do not impact the transfer function from $x$ to $\hat{x}$ that is in the feedback path.
** ZIP file containing the data and matlab files :ignore:
** ZIP file containing the data and matlab files :ignore:
#+begin_src bash :exports none :results none
if [ matlab/comp_filter_robustness.m -nt data/comp_filter_robustness.zip ]; then
cp matlab/comp_filter_robustness.m comp_filter_robustness.m;
@ -677,38 +667,22 @@ And we see that the complementary filters are only shaping the noise and that th
<<matlab-init>>
#+end_src
** Static Gain Mismatch :noexport:
Even though we here still consider that the two sensors have perfect dynamics, we consider gain mismatch for the two sensors:
\begin{align*}
G_1(s) &= 1 + \delta_1(s) \\
G_2(s) &= 1 + \delta_2(s)
\end{align*}
Thus, we have:
\begin{align*}
\hat{x} &= (x + n_1) (1 + \delta_1) H_1 + (x + n_2) (1 + \delta_2) H_2 \\
&= x (1 + \delta_1 H_1 + \delta_2 H_2) + n_1 (1 + \delta_1) H_1 + n_2(1 + \delta_2) H_2
\end{align*}
So the transfer function from $x$ to $\hat{x}$ is:
\begin{align*}
\frac{\hat{x}}{x} &= 1 + \delta_1 H_1 + \delta_2 H_2 \\
&= 1 + \delta_1 H_1 + \delta_2 (1 - H_1) \\
&= 1 + (\delta_1 - \delta_2) H_1 + \delta_2 \\
\end{align*}
#+begin_src matlab
freqs = logspace(-1, 3, 1000);
#+end_src
** Unknown sensor dynamics dynamics
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:fusion_gain_mismatch]].
Let's represent that with multiplicative input uncertainty as shown on figure [[fig:sensor_fusion_dynamic_uncertainty]].
#+name: fig:fusion_gain_mismatch
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Fusion of two sensors with input multiplicative uncertainty
[[file:figs-tikz/fusion_gain_mismatch.png]]
[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]]
We have:
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
\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 $\| \Delta_i \|_\infty < 1$.
@ -741,31 +715,35 @@ We then have that the angle introduced by the super sensor is bounded by $\arcsi
Thus, we choose should choose $\epsilon$ so that the maximum phase uncertainty introduced by the sensors is of an acceptable value.
** Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor
Let's say the two sensors dynamics $H_1$ and $H_2$ have been identified with the associated uncertainty weights $W_1$ and $W_2$.
Let's say the two sensors dynamics have been identified with the associated uncertainty weights $w_1(s)$ and $w_2(s)$.
If we want to have a maximum phase introduced by the sensors of 20 degrees, we have to design $H_1$ and $H_2$ such that:
If we want to have a maximum phase introduced by the sensors of 20 degrees, we have to design $H_1(s)$ and $H_2(s)$ such that:
\begin{align*}
& arcsin(|H_1 W_1| + |H_2 W_2|) < 20 \text{ deg} \\
\Longleftrightarrow & |H_1 W_1| + |H_2 W_2| < 0.34
& arcsin\Big( |H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)| \Big) < 20 \text{ deg} \quad \forall\omega \\
\Longleftrightarrow & |H_1 w_1| + |H_2 w_2| < 0.34 \quad \forall\omega
\end{align*}
We can do that with the $\mathcal{H}_\infty$ synthesis by setting upper bounds on the complementary filters using weights that corresponds to the sensor dynamics uncertainty.
For simplicity, let's suppose $W_1(s) = W_2(s) = 0.1$ ($10\%$ uncertainty in the sensor gain).
\[ |H_1 W_1| + |H_2 W_2| < 3.4 \]
Basically, at frequencies where $|w_i(j\omega)|$ is large, $|H_i(j\omega)|$ has to be made small.
Thus, by limiting the norm of the complementary filters, we can limit the maximum unwanted phase introduced by the uncertainty on the sensors dynamics.
This is of primary importance in order to ensure the stability of the feedback loop using the super sensor signal.
** First Basic Example with gain mismatch
Let's consider two ideal sensors except one sensor has not an expected gain of one but a gain of $0.6$.
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
Let's design two complementary filters as shown on figure [[fig:comp_filters_robustness_test]].
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
@ -822,7 +800,7 @@ The complementary filters shown in blue does not present a bump as the red ones
#+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 $H_1*G_1 + H_2*G_2$ for both complementary filters pair (figure [[fig:tf_super_sensor_comp]]).
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.
@ -863,7 +841,367 @@ We see that the blue complementary filters with a lower maximum norm permits to
#+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]]
** TODO More Complete example with model uncertainty
** More Complete example with dynamical uncertainty
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 $\omega > 1\text{ kHz}$
*** Dynamical uncertainty of the individual sensors
We define the weights that are used to characterize the dynamic uncertainty of the sensors.
#+begin_src matlab
omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
#+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 = subaxis(2,1,1);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',1);
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--');
for i = 1:length(G1s)
plot(freqs, abs(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
plot(freqs, abs(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude');
ylim([1e-1, 10]);
hold off;
% Phase
ax2 = subaxis(2,1,2);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, Dphi1, '--');
set(gca,'ColorOrderIndex',1);
plot(freqs, -Dphi1, '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, Dphi2, '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, -Dphi2, '--');
for i = 1:length(G1s)
plot(freqs, 180/pi*angle(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
end
set(gca,'xscale','log');
yticks(-180:90:180);
ylim([-180 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
#+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")
<<plt-matlab>>
#+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 know try to express this requirement in terms of an $\mathcal{H}_\infty$ norm.
Let 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 approximation 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}
On 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 $1$.
*** H-Infinity Synthesis
Let's define $w_\phi(s)$ such that the maximum allowed phase uncertainty $\Delta\phi_\text{max}$ of the super sensor dynamics is $30\text{ deg}$ until frequency $\omega_0 = 500\text{ Hz}$
#+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
% 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")
<<plt-matlab>>
#+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-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+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]]
The $\mathcal{H}_\infty$ synthesis is performed using the defined weights and the obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
#+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
#+begin_src matlab
H1 = 1 - H2;
#+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(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")
<<plt-matlab>>
#+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 = subaxis(2,1,1);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',1);
plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--');
plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--');
plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--');
for i = 1:length(Gsss)
plot(freqs, abs(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]);
ylabel('Magnitude');
ylim([1e-1, 10]);
hold off;
% Phase
ax2 = subaxis(2,1,2);
hold on;
% plot(freqs, Dphimax, 'r-');
% plot(freqs, -Dphimax, 'r-');
set(gca,'ColorOrderIndex',1);
plot(freqs, Dphi1, '--');
set(gca,'ColorOrderIndex',1);
plot(freqs, -Dphi1, '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, Dphi2, '--');
set(gca,'ColorOrderIndex',2);
plot(freqs, -Dphi2, '--');
plot(freqs, Dphiss, 'k--');
plot(freqs, -Dphiss, 'k--');
for i = 1:length(Gsss)
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsss(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0 0 0.2]);
end
set(gca,'xscale','log');
yticks(-180:90:180);
ylim([-180 180]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
#+end_src
#+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")
<<plt-matlab>>
#+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]]
We here just used very wimple weights. We could shape the dynamical uncertainty of the super sensor by using more complex weights.
We could for instance ask for less uncertainty at low frequency.
* Equivalent Super Sensor
The goal here is to find the parameters of a single sensor that would best represent a super sensor.

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@ -140,6 +140,61 @@ Configuration file is accessible [[file:config.org][here]].
#+caption: H-Infinity - Optimal Complementary Filters ([[./figs/h_infinity_optimal_comp_filters.png][png]], [[./figs/h_infinity_optimal_comp_filters.pdf][pdf]], [[./figs/h_infinity_optimal_comp_filters.tex][tex]]).
#+RESULTS:
[[file:figs/h_infinity_optimal_comp_filters.png]]
* Generate Complementary Filters using Feedback Control Architecture
#+begin_src latex :file complementary_filters_feedback_architecture.pdf :tangle figs/complementary_filters_feedback_architecture.tex :exports both
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=1 of addfb] (L){$L$};
\node[addb={+}{}{}{}{}, right=1 of L] (adddy){};
\draw[<-] (addfb.west) -- ++(-1, 0) node[above right]{$y_1$};
\draw[->] (addfb.east) -- (L.west);
\draw[->] (L.east) -- (adddy.west);
\draw[->] (adddy.east) -- ++(1, 0) node[above left]{$y_s$};
\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -1) -| (addfb.south);
\draw[<-] (adddy.north) -- ++(0, 1) node[below right]{$y_2$};
\end{tikzpicture}
#+end_src
#+name: fig:complementary_filters_feedback_architecture
#+caption: Generation of Complementary Filters using the feedback architecture ([[./figs/complementary_filters_feedback_architecture.png][png]], [[./figs/complementary_filters_feedback_architecture.pdf][pdf]], [[./figs/complementary_filters_feedback_architecture.tex][tex]]).
#+RESULTS:
[[file:figs/complementary_filters_feedback_architecture.png]]
* H-Infinity Synthesis for Robust Sensor Fusion
#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
<<tikz_default>>
\begin{tikzpicture}
\node[block={4.0cm}{3.0cm}, dashed] (P) {};
\node[above] at (P.north) {$P$};
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-\cdist, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-\cdist, 0)$);
\coordinate[] (outputh) at ($(P.south east)!0.8!(P.north east) + ( \cdist, 0)$);
\coordinate[] (outputl) at ($(P.south east)!0.4!(P.north east) + ( \cdist, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( \cdist, 0)$);
\node[block, left=2*\cdist of outputl] (W1){$w_1$};
\node[block, left=2*\cdist of outputh] (W2){$w_2$};
\node[addb={+}{}{}{}{-}, left=of W2] (sub) {};
\node[block, below=\cdist of P] (H1) {$H_1$};
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
\draw[->] (H1.west) -| ($(inputu)+(0.5*\cdist, 0)$) -- (W1.west);
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
\draw[->] (sub.east) -- (W2.west);
\draw[->] ($(inputw)+(2*\cdist, 0)$) node[branch]{} |- ($(outputv)+(-0.5*\cdist, 0)$) |- (H1.east);
\draw[->] (W2.east) -- (outputh)node[above left]{$z_2$};
\draw[->] (W1.east) -- (outputl)node[above left]{$z_1$};
\end{tikzpicture}
#+end_src
#+name: fig:h_infinity_robust_fusion
#+caption: H-Infinity Synthesis for Robust Sensor Fusion ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]).
#+RESULTS:
[[file:figs/h_infinity_robust_fusion.png]]
* Fusion of two noisy sensors
#+begin_src latex :file fusion_two_noisy_sensors.pdf :tangle figs/fusion_two_noisy_sensors.tex
<<tikz_default>>
@ -274,113 +329,26 @@ Configuration file is accessible [[file:config.org][here]].
#+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}
\node[addb] (add) at (0, 0){};
\node[block, above left=0.5 and 0.8 of add] (H1) {$H_1$};
\node[block, below left=0.5 and 0.8 of add] (H2) {$H_2$};
\draw[->] ($(H1.west)+(-1.5, 0)$)node[above right]{$x + n_1$} -- (H1.west);
\draw[->] ($(H2.west)+(-1.5, 0)$)node[above right]{$x + n_2$} -- (H2.west);
\draw[->] (H1) -| (add.north);
\draw[->] (H2) -| (add.south);
\draw[->] (add.east) -- ++(1, 0) node[above left]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_signals
#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_signals.png][png]], [[./figs/fusion_two_signals.pdf][pdf]], [[./figs/fusion_two_signals.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_signals.png]]
* Fusion of two noisy sensors with Dynamics
#+begin_src latex :file fusion_two_noisy_sensors_with_dyn.pdf :tangle figs/fusion_two_noisy_sensors_with_dyn.tex
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=1.5 and 0.5 of x](G1){$G_1$};
\node[block, below right=1.5 and 0.5 of x](G2){$G_2$};
\node[addb, right=1 of G1](add1){};
\node[addb, right=1 of G2](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=6 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (G1.west);
\draw[->] (x) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (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[->] (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]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_noisy_sensors_with_dyn
#+caption: Fusion of two noisy sensors_with_dyn ([[./figs/fusion_two_noisy_sensors_with_dyn.png][png]], [[./figs/fusion_two_noisy_sensors_with_dyn.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_with_dyn.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_noisy_sensors_with_dyn.png]]
* Fusion of two noisy sensors with Dynamics - Bis
#+begin_src latex :file fusion_two_noisy_sensors_with_dyn_bis.pdf :tangle figs/fusion_two_noisy_sensors_with_dyn_bis.tex
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=0.5 and 0.5 of x](G1){$G_1$};
\node[block, below right=0.5 and 0.5 of x](G2){$G_2$};
\node[addb, right=1 of G1](add1){};
\node[addb, right=1 of G2](add2){};
\node[block, right=1 of add1](H1){$H_1$};
\node[block, right=1 of add2](H2){$H_2$};
\node[addb, right=6 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (G1.west);
\draw[->] (x) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (add2.west);
\draw[<-] (add1.north) -- ++(0, 0.8)node[below right]{$n_1$};
\draw[<-] (add2.north) -- ++(0, 0.8)node[below right]{$n_2$};
\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]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_noisy_sensors_with_dyn_bis
#+caption: Fusion of two noisy sensors_with_dyn_bis ([[./figs/fusion_two_noisy_sensors_with_dyn_bis.png][png]], [[./figs/fusion_two_noisy_sensors_with_dyn_bis.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_with_dyn_bis.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_noisy_sensors_with_dyn_bis.png]]
* Fusion of two sensors with mismatch dynamics
#+begin_src latex :file fusion_gain_mismatch.pdf :tangle figs/fusion_gain_mismatch.tex
* Sensor fusion architecture with sensor dynamics uncertainty
#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/sensor_fusion_dynamic_uncertainty.tex
<<tikz_default>>
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=1 and 3.5 of x](add1){};
\node[addb, below right=1 and 3.5 of x](add2){};
\node[block, above left= of add1](delta1){$\Delta_1$};
\node[block, above left= of add2](delta2){$\Delta_2$};
\node[block, left= of delta1](W1){$W_1$};
\node[block, left= of delta2](W2){$W_2$};
\node[block, right= of add1](H1){$H_1$};
\node[block, right= of add2](H2){$H_2$};
\node[addb, right=5.5 of x](add){};
\node[addb, above right=0.8 and 3.7 of x](add1){};
\node[addb, below right=0.8 and 3.7 of x](add2){};
\node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$};
\node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$};
\node[block, left=0.5 of delta1](W1){$w_1(s)$};
\node[block, left=0.5 of delta2](W2){$w_2(s)$};
\node[block, right=0.5 of add1](H1){$H_1(s)$};
\node[block, right=0.5 of add2](H2){$H_2(s)$};
\node[addb, right=5.8 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (add1.west);
\draw[->] (x) |- (add2.west);
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch]{} |- (W2.west);
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (add1.west);
\draw[->] (x.center) |- (add2.west);
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west);
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S1){} |- (W2.west);
\draw[->] (W1.east) -- (delta1.west);
\draw[->] (W2.east) -- (delta2.west);
\draw[->] (delta1.east) -| (add1.north);
@ -389,89 +357,24 @@ Configuration file is accessible [[file:config.org][here]].
\draw[->] (add2.east) -- (H2.west);
\draw[->] (H1.east) -| (add.north);
\draw[->] (H2.east) -| (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={($(add2.south-|x)+(-0.2, -0.4)$) ($(delta1.north-|add.east)+(0.2, 0.4)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor};
\node[block, fit={($(W1.north-|S1)+(-0.2, 0.2)$) ($(add1.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[above right] at (sensor1.south west) {Sensor 1};
\node[block, fit={($(W2.north-|S1)+(-0.2, 0.2)$) ($(add2.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
\node[above right] at (sensor2.south west) {Sensor 2};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:fusion_gain_mismatch
#+caption: Fusion of two noisy sensors_with_dyn ([[./figs/fusion_gain_mismatch.png][png]], [[./figs/fusion_gain_mismatch.pdf][pdf]], [[./figs/fusion_gain_mismatch.tex][tex]]).
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_dynamic_uncertainty.png][png]], [[./figs/sensor_fusion_dynamic_uncertainty.pdf][pdf]], [[./figs/sensor_fusion_dynamic_uncertainty.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_gain_mismatch.png]]
[[file:figs/sensor_fusion_dynamic_uncertainty.png]]
* Uncertainty to Phase and Gain variation
#+begin_src latex :file uncertainty_gain_phase_variation.pdf :tangle figs/uncertainty_gain_phase_variation.tex :exports both
\begin{tikzpicture}
\draw[->] (-0.5, 0) -- (7, 0) node[below left]{Real};
\draw[->] (0, -1) -- (0, 1) node[left]{Im};
\node[branch] (1) at (5, 0){};
\node[below] at (1){$1$};
\node[circle, draw] (c) at (1)[minimum size=2cm]{};
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2) -- node[midway, right]{$|H_i W_i|$} (1);
\draw[dashed] (2, 0) arc (0:23:1) node[midway, right]{$\Delta \phi$};
% \draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=1) coordinate(cbot);
\end{tikzpicture}
#+end_src
#+name: fig:uncertainty_gain_phase_variation
#+caption: Uncertainty to Phase and Gain variation ([[./figs/uncertainty_gain_phase_variation.png][png]], [[./figs/uncertainty_gain_phase_variation.pdf][pdf]], [[./figs/uncertainty_gain_phase_variation.tex][tex]]).
#+RESULTS:
[[file:figs/uncertainty_gain_phase_variation.png]]
* Generate Complementary Filters using Feedback Control Architecture
#+begin_src latex :file complementary_filters_feedback_architecture.pdf :tangle figs/complementary_filters_feedback_architecture.tex :exports both
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=1 of addfb] (L){$L$};
\node[addb={+}{}{}{}{}, right=1 of L] (adddy){};
\draw[<-] (addfb.west) -- ++(-1, 0) node[above right]{$y_1$};
\draw[->] (addfb.east) -- (L.west);
\draw[->] (L.east) -- (adddy.west);
\draw[->] (adddy.east) -- ++(1, 0) node[above left]{$y_s$};
\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -1) -| (addfb.south);
\draw[<-] (adddy.north) -- ++(0, 1) node[below right]{$y_2$};
\end{tikzpicture}
#+end_src
#+name: fig:complementary_filters_feedback_architecture
#+caption: Generation of Complementary Filters using the feedback architecture ([[./figs/complementary_filters_feedback_architecture.png][png]], [[./figs/complementary_filters_feedback_architecture.pdf][pdf]], [[./figs/complementary_filters_feedback_architecture.tex][tex]]).
#+RESULTS:
[[file:figs/complementary_filters_feedback_architecture.png]]
* H-Infinity Synthesis for Robust Sensor Fusion
#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
<<tikz_default>>
\begin{tikzpicture}
\node[block={4.0cm}{3.0cm}, dashed] (P) {};
\node[above] at (P.north) {$P$};
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-\cdist, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-\cdist, 0)$);
\coordinate[] (outputh) at ($(P.south east)!0.8!(P.north east) + ( \cdist, 0)$);
\coordinate[] (outputl) at ($(P.south east)!0.4!(P.north east) + ( \cdist, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( \cdist, 0)$);
\node[block, left=2*\cdist of outputl] (W1){$w_1$};
\node[block, left=2*\cdist of outputh] (W2){$w_2$};
\node[addb={+}{}{}{}{-}, left=of W2] (sub) {};
\node[block, below=\cdist of P] (H1) {$H_1$};
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
\draw[->] (H1.west) -| ($(inputu)+(0.5*\cdist, 0)$) -- (W1.west);
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
\draw[->] (sub.east) -- (W2.west);
\draw[->] ($(inputw)+(2*\cdist, 0)$) node[branch]{} |- ($(outputv)+(-0.5*\cdist, 0)$) |- (H1.east);
\draw[->] (W2.east) -- (outputh)node[above left]{$z_2$};
\draw[->] (W1.east) -- (outputl)node[above left]{$z_1$};
\end{tikzpicture}
#+end_src
#+name: fig:h_infinity_robust_fusion
#+caption: H-Infinity Synthesis for Robust Sensor Fusion ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]).
#+RESULTS:
[[file:figs/h_infinity_robust_fusion.png]]
* Sensor fusion architecture with sensor dynamics uncertainty and noise
#+begin_src latex :file sensor_fusion_full.pdf :tangle figs/sensor_fusion_full.tex
<<tikz_default>>
@ -708,3 +611,153 @@ Configuration file is accessible [[file:config.org][here]].
#+caption: Huddle Test ([[./figs/huddle_test.png][png]], [[./figs/huddle_test.pdf][pdf]], [[./figs/huddle_test.tex][tex]]).
#+RESULTS:
[[file:figs/huddle_test.png]]
* Uncertainty to Phase and Gain variation
#+begin_src latex :file uncertainty_gain_phase_variation.pdf :tangle figs/uncertainty_gain_phase_variation.tex :exports both
\begin{tikzpicture}
% Uncertainty Circle
\node[circle, draw, fill=black!20!white] (c) at (4.5, 0)[minimum size=2cm]{};
% Real and Imaginary Axis
\draw[->] (-0.5, 0) -- (6.2, 0) node[below left]{Re};
\draw[->] (0, -0.5) -- (0, 1) node[below left]{Im};
% X = 1
\node[branch] (1) at (4.5, 0){};
\node[below] at (1){$1$};
\draw[dashed] (0, 0)coordinate(A) -- (tangent cs:node=c,point={(0, 0)},solution=2)coordinate(B) -- node[midway, right]{$\epsilon$} (1);
\draw[dashed] (2, 0) arc (0:23:1) node[midway, right]{$\Delta \phi$};
\end{tikzpicture}
#+end_src
#+name: fig:uncertainty_gain_phase_variation
#+caption: Uncertainty to Phase and Gain variation ([[./figs/uncertainty_gain_phase_variation.png][png]], [[./figs/uncertainty_gain_phase_variation.pdf][pdf]], [[./figs/uncertainty_gain_phase_variation.tex][tex]]).
#+RESULTS:
[[file:figs/uncertainty_gain_phase_variation.png]]
* Fusion of two noisy signals
#+begin_src latex :file fusion_two_signals.pdf :tangle figs/fusion_two_signals.tex
\begin{tikzpicture}
\node[addb] (add) at (0, 0){};
\node[block, above left=0.5 and 0.8 of add] (H1) {$H_1$};
\node[block, below left=0.5 and 0.8 of add] (H2) {$H_2$};
\draw[->] ($(H1.west)+(-1.5, 0)$)node[above right]{$x + n_1$} -- (H1.west);
\draw[->] ($(H2.west)+(-1.5, 0)$)node[above right]{$x + n_2$} -- (H2.west);
\draw[->] (H1) -| (add.north);
\draw[->] (H2) -| (add.south);
\draw[->] (add.east) -- ++(1, 0) node[above left]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_signals
#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_signals.png][png]], [[./figs/fusion_two_signals.pdf][pdf]], [[./figs/fusion_two_signals.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_signals.png]]
* Fusion of two noisy sensors with Dynamics
#+begin_src latex :file fusion_two_noisy_sensors_with_dyn.pdf :tangle figs/fusion_two_noisy_sensors_with_dyn.tex
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=1.5 and 0.5 of x](G1){$G_1$};
\node[block, below right=1.5 and 0.5 of x](G2){$G_2$};
\node[addb, right=1 of G1](add1){};
\node[addb, right=1 of G2](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=6 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (G1.west);
\draw[->] (x) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (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[->] (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]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_noisy_sensors_with_dyn
#+caption: Fusion of two noisy sensors_with_dyn ([[./figs/fusion_two_noisy_sensors_with_dyn.png][png]], [[./figs/fusion_two_noisy_sensors_with_dyn.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_with_dyn.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_noisy_sensors_with_dyn.png]]
* Fusion of two noisy sensors with Dynamics - Bis
#+begin_src latex :file fusion_two_noisy_sensors_with_dyn_bis.pdf :tangle figs/fusion_two_noisy_sensors_with_dyn_bis.tex
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[block, above right=0.5 and 0.5 of x](G1){$G_1$};
\node[block, below right=0.5 and 0.5 of x](G2){$G_2$};
\node[addb, right=1 of G1](add1){};
\node[addb, right=1 of G2](add2){};
\node[block, right=1 of add1](H1){$H_1$};
\node[block, right=1 of add2](H2){$H_2$};
\node[addb, right=6 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (G1.west);
\draw[->] (x) |- (G2.west);
\draw[->] (G1.east) -- (add1.west);
\draw[->] (G2.east) -- (add2.west);
\draw[<-] (add1.north) -- ++(0, 0.8)node[below right]{$n_1$};
\draw[<-] (add2.north) -- ++(0, 0.8)node[below right]{$n_2$};
\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]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_two_noisy_sensors_with_dyn_bis
#+caption: Fusion of two noisy sensors_with_dyn_bis ([[./figs/fusion_two_noisy_sensors_with_dyn_bis.png][png]], [[./figs/fusion_two_noisy_sensors_with_dyn_bis.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_with_dyn_bis.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_two_noisy_sensors_with_dyn_bis.png]]
* Fusion of two sensors with mismatch dynamics
#+begin_src latex :file fusion_gain_mismatch.pdf :tangle figs/fusion_gain_mismatch.tex
<<tikz_default>>
\begin{tikzpicture}
\node[branch] (x) at (0, 0);
\node[addb, above right=1 and 3.5 of x](add1){};
\node[addb, below right=1 and 3.5 of x](add2){};
\node[block, above left= of add1](delta1){$\Delta_1$};
\node[block, above left= of add2](delta2){$\Delta_2$};
\node[block, left= of delta1](W1){$W_1$};
\node[block, left= of delta2](W2){$W_2$};
\node[block, right= of add1](H1){$H_1$};
\node[block, right= of add2](H2){$H_2$};
\node[addb, right=5.5 of x](add){};
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
\draw[->] (x) |- (add1.west);
\draw[->] (x) |- (add2.west);
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch]{} |- (W1.west);
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch]{} |- (W2.west);
\draw[->] (W1.east) -- (delta1.west);
\draw[->] (W2.east) -- (delta2.west);
\draw[->] (delta1.east) -| (add1.north);
\draw[->] (delta2.east) -| (add2.north);
\draw[->] (add1.east) -- (H1.west);
\draw[->] (add2.east) -- (H2.west);
\draw[->] (H1.east) -| (add.north);
\draw[->] (H2.east) -| (add.south);
\draw[->] (add.east) -- ++(1, 0) node[above left]{$\hat{x}$};
\end{tikzpicture}
#+end_src
#+name: fig:fusion_gain_mismatch
#+caption: Fusion of two noisy sensors_with_dyn ([[./figs/fusion_gain_mismatch.png][png]], [[./figs/fusion_gain_mismatch.pdf][pdf]], [[./figs/fusion_gain_mismatch.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_gain_mismatch.png]]