Update all the analysis about robustness / uncertainty reduction
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matlab/figs/comp_filter_hinf_uncertainty.pdf
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@ -44,7 +44,7 @@ Then, three design methods for generating two complementary filters are proposed
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- in section [[sec:feedback_generate_comp_filters]], the classical feedback architecture is used
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- in section [[sec:analytical_formula_literature]], analytical formulas found in the literature are listed
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* Optimal Sensor Fusion for noise characteristics
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* Optimal Sensor Fusion - Minimize the Super Sensor Noise
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/optimal_comp_filters.m
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:header-args:matlab+: :comments org :mkdirp yes
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@ -625,7 +625,7 @@ The RMS value of the obtained super sensors are shown on table [[tab:rms_results
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** Conclusion
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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).
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* Robustness to sensor dynamics uncertainty
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* Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/comp_filter_robustness.m
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:header-args:matlab+: :comments org :mkdirp yes
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@ -633,26 +633,16 @@ From the above complementary filter design with the $\mathcal{H}_2$ and $\mathca
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<<sec:comp_filter_robustness>>
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** Introduction :ignore:
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Let's first consider ideal sensors where $G_1 = 1$ and $G_2 = 1$ (figure [[fig:fusion_two_noisy_sensors_with_dyn_bis]]).
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We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
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#+name: fig:fusion_two_noisy_sensors_with_dyn_bis
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#+caption: Fusion of two sensors
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[[file:figs/fusion_two_noisy_sensors_with_dyn_bis.png]]
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We now take into account the fact that the sensor dynamics is only partially known.
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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]].
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# #+name: fig:fusion_two_signals
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# #+caption: Fusion of two noisy measurements of $x$
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# [[file:figs/fusion_two_signals.png]]
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#+name: fig:sensor_fusion_dynamic_uncertainty
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#+caption: Sensor fusion architecture with sensor dynamics uncertainty
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[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]]
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We then have:
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\begin{align*}
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\hat{x} &= (x + n_1) H_1 + (x + n_2) H_2 \\
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&= x + n_1 H_1 + n_2 H_2
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\end{align*}
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So the estimation error is
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\[ \delta_x = \hat{x} - x = n_1 H_1 + n_2 H_2 \]
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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.
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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.
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** ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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@ -677,38 +667,22 @@ And we see that the complementary filters are only shaping the noise and that th
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<<matlab-init>>
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#+end_src
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** Static Gain Mismatch :noexport:
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Even though we here still consider that the two sensors have perfect dynamics, we consider gain mismatch for the two sensors:
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\begin{align*}
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G_1(s) &= 1 + \delta_1(s) \\
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G_2(s) &= 1 + \delta_2(s)
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\end{align*}
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Thus, we have:
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\begin{align*}
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\hat{x} &= (x + n_1) (1 + \delta_1) H_1 + (x + n_2) (1 + \delta_2) H_2 \\
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&= x (1 + \delta_1 H_1 + \delta_2 H_2) + n_1 (1 + \delta_1) H_1 + n_2(1 + \delta_2) H_2
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\end{align*}
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So the transfer function from $x$ to $\hat{x}$ is:
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\begin{align*}
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\frac{\hat{x}}{x} &= 1 + \delta_1 H_1 + \delta_2 H_2 \\
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&= 1 + \delta_1 H_1 + \delta_2 (1 - H_1) \\
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&= 1 + (\delta_1 - \delta_2) H_1 + \delta_2 \\
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\end{align*}
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#+begin_src matlab
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freqs = logspace(-1, 3, 1000);
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#+end_src
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** Unknown sensor dynamics dynamics
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In practical systems, the sensor dynamics has always some level of uncertainty.
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Let's represent that with multiplicative input uncertainty as shown on figure [[fig:fusion_gain_mismatch]].
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Let's represent that with multiplicative input uncertainty as shown on figure [[fig:sensor_fusion_dynamic_uncertainty]].
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#+name: fig:fusion_gain_mismatch
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#+name: fig:sensor_fusion_dynamic_uncertainty
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#+caption: Fusion of two sensors with input multiplicative uncertainty
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[[file:figs-tikz/fusion_gain_mismatch.png]]
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[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]]
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We have:
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The dynamics of the super sensor is represented by
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\begin{align*}
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\frac{\hat{x}}{x} &= (1 + W_1 \Delta_1) H_1 + (1 + W_2 \Delta_2) H_2 \\
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&= 1 + W_1 H_1 \Delta_1 + W_2 H_2 \Delta_2
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\frac{\hat{x}}{x} &= (1 + w_1 \Delta_1) H_1 + (1 + w_2 \Delta_2) H_2 \\
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&= 1 + w_1 H_1 \Delta_1 + w_2 H_2 \Delta_2
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\end{align*}
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With $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$.
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@ -741,31 +715,35 @@ We then have that the angle introduced by the super sensor is bounded by $\arcsi
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Thus, we choose should choose $\epsilon$ so that the maximum phase uncertainty introduced by the sensors is of an acceptable value.
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** Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor
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Let's say the two sensors dynamics $H_1$ and $H_2$ have been identified with the associated uncertainty weights $W_1$ and $W_2$.
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Let's say the two sensors dynamics have been identified with the associated uncertainty weights $w_1(s)$ and $w_2(s)$.
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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:
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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:
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\begin{align*}
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& arcsin(|H_1 W_1| + |H_2 W_2|) < 20 \text{ deg} \\
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\Longleftrightarrow & |H_1 W_1| + |H_2 W_2| < 0.34
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& 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 \\
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\Longleftrightarrow & |H_1 w_1| + |H_2 w_2| < 0.34 \quad \forall\omega
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\end{align*}
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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.
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For simplicity, let's suppose $W_1(s) = W_2(s) = 0.1$ ($10\%$ uncertainty in the sensor gain).
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\[ |H_1 W_1| + |H_2 W_2| < 3.4 \]
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Basically, at frequencies where $|w_i(j\omega)|$ is large, $|H_i(j\omega)|$ has to be made small.
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Thus, by limiting the norm of the complementary filters, we can limit the maximum unwanted phase introduced by the uncertainty on the sensors dynamics.
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This is of primary importance in order to ensure the stability of the feedback loop using the super sensor signal.
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** First Basic Example with gain mismatch
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Let's consider two ideal sensors except one sensor has not an expected gain of one but a gain of $0.6$.
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Let's consider two ideal sensors except one sensor has not an expected unity gain but a gain equal to $0.6$:
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\begin{align*}
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G_1(s) &= 1 \\
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G_2(s) &= 0.6
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\end{align*}
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#+begin_src matlab
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G1 = 1;
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G2 = 0.6;
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#+end_src
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Let's design two complementary filters as shown on figure [[fig:comp_filters_robustness_test]].
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Two pairs of complementary filters are designed and shown on figure [[fig:comp_filters_robustness_test]].
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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.
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#+begin_src matlab :exports none
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@ -822,7 +800,7 @@ The complementary filters shown in blue does not present a bump as the red ones
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#+CAPTION: The two complementary filters designed for the robustness test ([[./figs/comp_filters_robustness_test.png][png]], [[./figs/comp_filters_robustness_test.pdf][pdf]])
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[[file:figs/comp_filters_robustness_test.png]]
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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]]).
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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]]).
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We see that the blue complementary filters with a lower maximum norm permits to limit the phase lag introduced by the gain mismatch.
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@ -863,7 +841,367 @@ We see that the blue complementary filters with a lower maximum norm permits to
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#+CAPTION: Comparison of the obtained super sensor transfer functions ([[./figs/tf_super_sensor_comp.png][png]], [[./figs/tf_super_sensor_comp.pdf][pdf]])
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[[file:figs/tf_super_sensor_comp.png]]
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** TODO More Complete example with model uncertainty
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** More Complete example with dynamical uncertainty
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We want to merge two sensors:
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- sensor 1 that has unknown dynamics above 10Hz: $|w_1(j\omega)| > 1$ for $\omega > 10\text{ Hz}$
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- sensor 2 that has unknown dynamics below 1Hz and above 1kHz $|w_2(j\omega)| > 1$ for $\omega < 1\text{ Hz}$ and $\omega > 1\text{ kHz}$
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*** Dynamical uncertainty of the individual sensors
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We define the weights that are used to characterize the dynamic uncertainty of the sensors.
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#+begin_src matlab
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omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
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w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
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w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 5000*2*pi; G0 = 1; Ginf = 50;
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w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
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#+end_src
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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.
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#+begin_src matlab
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G1 = 1 + w1*ultidyn('Delta',[1 1]);
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G2 = 1 + w2*ultidyn('Delta',[1 1]);
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#+end_src
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#+begin_src matlab :exports none
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% Few random samples of the sensor dynamics are computed
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G1s = usample(G1, 10);
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G2s = usample(G2, 10);
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#+end_src
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#+begin_src matlab :exports none
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% We here compute the maximum and minimum phase of both sensors
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Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz'))));
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Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz'))));
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Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190;
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Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190;
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#+end_src
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#+begin_src matlab :exports none
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figure;
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% Magnitude
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ax1 = subaxis(2,1,1);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--');
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for i = 1:length(G1s)
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plot(freqs, abs(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
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plot(freqs, abs(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude');
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ylim([1e-1, 10]);
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hold off;
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% Phase
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ax2 = subaxis(2,1,2);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, Dphi1, '--');
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set(gca,'ColorOrderIndex',1);
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plot(freqs, -Dphi1, '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, Dphi2, '--');
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set(gca,'ColorOrderIndex',2);
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plot(freqs, -Dphi2, '--');
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for i = 1:length(G1s)
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plot(freqs, 180/pi*angle(squeeze(freqresp(G1s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0 0.4470 0.7410 0.4]);
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plot(freqs, 180/pi*angle(squeeze(freqresp(G2s(:, :, i, 1), freqs, 'Hz'))), '-', 'color', [0.8500 0.3250 0.0980 0.4]);
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end
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set(gca,'xscale','log');
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yticks(-180:90:180);
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ylim([-180 180]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+HEADER: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/uncertainty_dynamics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+NAME: fig:uncertainty_dynamics_sensors
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#+CAPTION: Dynamic uncertainty of the two sensors ([[./figs/uncertainty_dynamics_sensors.png][png]], [[./figs/uncertainty_dynamics_sensors.pdf][pdf]])
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[[file:figs/uncertainty_dynamics_sensors.png]]
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*** Synthesis objective
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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]].
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At each frequency $\omega$, the radius of the circle is $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$.
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Thus, the phase shift $\Delta\phi(\omega)$ due to the super sensor uncertainty is bounded by:
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\[ |\Delta\phi(\omega)| \leq \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) \]
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Let's define some allowed frequency depend phase shift $\Delta\phi_\text{max}(\omega) > 0$ such that:
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\[ |\Delta\phi(\omega)| < \Delta\phi_\text{max}(\omega), \quad \forall\omega \]
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If $H_1(s)$ and $H_2(s)$ are designed such that
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\[ |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big) \]
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The maximum phase shift due to dynamic uncertainty at frequency $\omega$ will be $\Delta\phi_\text{max}(\omega)$.
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*** Requirements as an $\mathcal{H}_\infty$ norm
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We know try to express this requirement in terms of an $\mathcal{H}_\infty$ norm.
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Let define one weight $w_\phi(s)$ that represents the maximum wanted phase uncertainty:
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\[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \]
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Then:
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\begin{align*}
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& |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 \\
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\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 \\
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\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
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\end{align*}
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Which is approximately equivalent to (with an approximation of maximum $\sqrt{2}$):
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#+name: eq:hinf_conf_phase_uncertainty
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\begin{equation}
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\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
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\end{equation}
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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.
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Thus, at these frequencies, $|w_\phi|$ should be smaller than $1$.
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*** H-Infinity Synthesis
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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 $\omega_0 = 500\text{ Hz}$
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#+begin_src matlab
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Dphi = 20; % [deg]
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n = 4; w0 = 2*pi*900; G0 = 1/sin(Dphi*pi/180); Ginf = 1/100; Gc = 1;
|
||||
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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Before Width: | Height: | Size: 19 KiB After Width: | Height: | Size: 12 KiB |
415
tikz/index.org
415
tikz/index.org
@ -140,6 +140,61 @@ Configuration file is accessible [[file:config.org][here]].
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#+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]]).
|
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#+RESULTS:
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||||
[[file:figs/h_infinity_optimal_comp_filters.png]]
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* Generate Complementary Filters using Feedback Control Architecture
|
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#+begin_src latex :file complementary_filters_feedback_architecture.pdf :tangle figs/complementary_filters_feedback_architecture.tex :exports both
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\begin{tikzpicture}
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\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
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\node[block, right=1 of addfb] (L){$L$};
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\node[addb={+}{}{}{}{}, right=1 of L] (adddy){};
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\draw[<-] (addfb.west) -- ++(-1, 0) node[above right]{$y_1$};
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\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]]
|
||||
|
||||
Loading…
Reference in New Issue
Block a user