From 8a901b0e87552324305a9bbb429f299c9a5c3d4b Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 29 Aug 2019 22:47:20 +0200 Subject: [PATCH] Update all the analysis about robustness / uncertainty reduction --- matlab/figs/comp_filter_hinf_uncertainty.pdf | Bin 0 -> 75181 bytes matlab/figs/comp_filter_hinf_uncertainty.png | Bin 0 -> 117676 bytes .../figs/maximum_wanted_phase_uncertainty.pdf | Bin 0 -> 39768 bytes .../figs/maximum_wanted_phase_uncertainty.png | Bin 0 -> 55520 bytes .../super_sensor_uncertainty_bode_plot.pdf | Bin 0 -> 303155 bytes .../super_sensor_uncertainty_bode_plot.png | Bin 0 -> 139344 bytes matlab/figs/uncertainty_dynamics_sensors.pdf | Bin 0 -> 299042 bytes matlab/figs/uncertainty_dynamics_sensors.png | Bin 0 -> 198036 bytes ...unds_comp_filter_max_phase_uncertainty.pdf | Bin 0 -> 42932 bytes ...unds_comp_filter_max_phase_uncertainty.png | Bin 0 -> 106548 bytes matlab/index.html | 800 +++++++++++------- matlab/index.org | 450 ++++++++-- ...ementary_filters_feedback_architecture.pdf | Bin 11408 -> 49030 bytes ...ementary_filters_feedback_architecture.png | Bin 6005 -> 6236 bytes ...ementary_filters_feedback_architecture.svg | 62 +- .../sensor_fusion_dynamic_uncertainty.pdf | Bin 70614 -> 69539 bytes .../sensor_fusion_dynamic_uncertainty.png | Bin 47322 -> 33306 bytes .../sensor_fusion_dynamic_uncertainty.svg | 407 ++++----- .../figs/uncertainty_gain_phase_variation.pdf | Bin 12577 -> 29335 bytes .../figs/uncertainty_gain_phase_variation.png | Bin 7076 -> 8470 bytes .../figs/uncertainty_gain_phase_variation.svg | 119 +-- tikz/index.org | 415 +++++---- 22 files changed, 1370 insertions(+), 883 deletions(-) create mode 100644 matlab/figs/comp_filter_hinf_uncertainty.pdf create mode 100644 matlab/figs/comp_filter_hinf_uncertainty.png create mode 100644 matlab/figs/maximum_wanted_phase_uncertainty.pdf create mode 100644 matlab/figs/maximum_wanted_phase_uncertainty.png create mode 100644 matlab/figs/super_sensor_uncertainty_bode_plot.pdf create mode 100644 matlab/figs/super_sensor_uncertainty_bode_plot.png create mode 100644 matlab/figs/uncertainty_dynamics_sensors.pdf create mode 100644 matlab/figs/uncertainty_dynamics_sensors.png create mode 100644 matlab/figs/upper_bounds_comp_filter_max_phase_uncertainty.pdf create mode 100644 matlab/figs/upper_bounds_comp_filter_max_phase_uncertainty.png diff --git a/matlab/figs/comp_filter_hinf_uncertainty.pdf b/matlab/figs/comp_filter_hinf_uncertainty.pdf new file mode 100644 index 0000000000000000000000000000000000000000..61d672cb46c19e1a8e4111b85900bfd7e50eabae GIT binary patch literal 75181 zcma(1V{|58v<41#I!V9j;Eio{Y}>YN+qToOZ9D0>W7|&0w)uvW-<|tmX8rH1HD7A) zv+?YGPSvXOoLWUHFDy#KK+6VAI(!~m0L?-`Phe+g3C+z-Cv9SD=4?*D`ejk16Sc5* zHgP1N6SX#QHW4;4vNJZ}<%M=~b~G`tfp*WT>Xb^jXsyp#-6{8#9o+}HtIk%jra@6+C2n{0aNKthAn^)|Mtjvv999VZ6YAp0EzR%Vg z)B7|*yVLVqygF;U1wC~89@Kt*oIT?`aGyTEbo;%>*53Qzh~d3;@&UbVlaRVMUvGSp 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Table of Contents

    -
  • 1. Optimal Sensor Fusion for noise characteristics +
  • 1. Optimal Sensor Fusion - Minimize the Super Sensor Noise
  • -
  • 2. Robustness to sensor dynamics uncertainty +
  • 2. Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty
      -
    • 2.1. Unknown sensor dynamics dynamics
    • -
    • 2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor
    • -
    • 2.3. First Basic Example with gain mismatch
    • -
    • 2.4. TODO More Complete example with model uncertainty
    • +
    • 2.1. Unknown sensor dynamics dynamics
    • +
    • 2.2. Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor
    • +
    • 2.3. First Basic Example with gain mismatch
    • +
    • 2.4. More Complete example with dynamical uncertainty +
    • -
    • 3. Equivalent Super Sensor -
    • -
    • 4. Optimal And Robust Sensor Fusion in Practice +
    • 3. Equivalent Super Sensor
        -
      • 4.1. Measurement of the noise characteristics of the sensors -
      • -
      • 4.2. Estimate the dynamic uncertainty of the sensors
      • -
      • 4.3. Optimal and Robust synthesis of the complementary filters
      • +
      • 4. Optimal And Robust Sensor Fusion in Practice +
          +
        • 4.1. Measurement of the noise characteristics of the sensors +
        • -
        • 5. Methods of complementary filter synthesis - @@ -374,31 +382,31 @@ To achieve this, the sensors included in the filter should complement one anothe
            -
          • in section 1, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied
          • +
          • in section 1, the optimal design of the complementary filters in order to obtain the lowest resulting "super sensor" noise is studied

          When blending two sensors using complementary filters with unknown dynamics, phase lag may be introduced that renders the close-loop system unstable.

            -
          • in section 2, the blending robustness to sensor dynamic uncertainty is studied.
          • +
          • in section 2, the blending robustness to sensor dynamic uncertainty is studied.

          Then, three design methods for generating two complementary filters are proposed:

            -
          • in section 5.1, analytical formulas are proposed
          • -
          • in section 5.2, the \(\mathcal{H}_\infty\) synthesis is used
          • -
          • in section 5.3, the classical feedback architecture is used
          • -
          • in section 5.4, analytical formulas found in the literature are listed
          • +
          • in section 5.1, analytical formulas are proposed
          • +
          • in section 5.2, the \(\mathcal{H}_\infty\) synthesis is used
          • +
          • in section 5.3, the classical feedback architecture is used
          • +
          • in section 5.4, analytical formulas found in the literature are listed
          -
          -

          1 Optimal Sensor Fusion for noise characteristics

          +
          +

          1 Optimal Sensor Fusion - Minimize the Super Sensor Noise

          - +

          The idea is to combine sensors that works in different frequency range using complementary filters. @@ -419,23 +427,23 @@ All the files (data and Matlab scripts) are accessible -

          1.1 Architecture

          +
          +

          1.1 Architecture

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

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

          \begin{equation} -\label{org31f32a3} +\label{orgfdd39ec} \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1 \end{equation} -
          +

          fusion_two_noisy_sensors_weights.png

          Figure 1: Fusion of two sensors

          @@ -445,16 +453,16 @@ Let's consider the sensor fusion architecture shown on figure 2. +We obtain the architecture of figure 2.

          -
          +

          sensor_fusion_noisy_perfect_dyn.png

          Figure 2: Fusion of two sensors with ideal dynamics

          @@ -464,7 +472,7 @@ We obtain the architecture of figure 2. \(H_1(s)\) and \(H_2(s)\) are complementary filters:

          \begin{equation} -\label{orgf0f9605} +\label{org60c56f0} H_1(s) + H_2(s) = 1 \end{equation} @@ -481,14 +489,14 @@ We have that the Power Spectral Density (PSD) of \(\hat{x}\) is: And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\):

          \begin{equation} -\label{orge1ffe08} +\label{org796f363} \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega} \end{equation}
          -
          -

          1.2 Noise of the sensors

          +
          +

          1.2 Noise of the sensors

          Let's define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\): @@ -508,7 +516,7 @@ N2 = ( +

          noise_characteristics_sensors.png

          Figure 3: Noise Characteristics of the two sensors (png, pdf)

          @@ -516,8 +524,8 @@ N2 = ( -

          1.3 H-Two Synthesis

          +
          +

          1.3 H-Two Synthesis

          As \(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise: \(\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1\) and we have: @@ -530,11 +538,11 @@ For that, we use the \(\mathcal{H}_2\) Synthesis.

          -We use the generalized plant architecture shown on figure 4. +We use the generalized plant architecture shown on figure 4.

          -
          +

          h_infinity_optimal_comp_filters.png

          Figure 4: \(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters

          @@ -552,7 +560,7 @@ Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we mi

          -We define the generalized plant \(P\) on matlab as shown on figure 4. +We define the generalized plant \(P\) on matlab as shown on figure 4.

          P = [0   N2  1;
          @@ -577,15 +585,15 @@ Finally, we define \(H_2(s) = 1 - H_1(s)\).
           

          -The complementary filters obtained are shown on figure 5. +The complementary filters obtained are shown on figure 5.

          -The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. 6. +The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. 6.

          -The Cumulative Power Spectrum (CPS) is shown on Fig. 7. +The Cumulative Power Spectrum (CPS) is shown on Fig. 7.

          @@ -593,7 +601,7 @@ The obtained RMS value of the super sensor is lower than the RMS value of the in

          -
          +

          htwo_comp_filters.png

          Figure 5: Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis (png, pdf)

          @@ -607,7 +615,7 @@ PSD_H2 = abs(squeeze -
          +

          psd_sensors_htwo_synthesis.png

          Figure 6: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (png, pdf)

          @@ -621,7 +629,7 @@ CPS_H2 = 1 +

          cps_h2_synthesis.png

          Figure 7: Cumulative Power Spectrum of individual sensors and super sensor using the \(\mathcal{H}_2\) synthesis (png, pdf)

          @@ -629,8 +637,8 @@ CPS_H2 = 1 -

          1.4 H-Infinity Synthesis - method A

          +
          +

          1.4 H-Infinity Synthesis - method A

          Another objective that we may have is that the noise of the super sensor \(n_{SS}\) is following the minimum of the noise of the two sensors \(n_1\) and \(n_2\): @@ -654,12 +662,12 @@ We could try to approach that with the \(\mathcal{H}_\infty\) synthesis by using

          -As shown on Fig. 3, the frequency where the two sensors have the same noise level is around 9Hz. +As shown on Fig. 3, the frequency where the two sensors have the same noise level is around 9Hz. We will thus choose weighting functions such that the merging frequency is around 9Hz.

          -The weighting functions used as well as the obtained complementary filters are shown in Fig. 8. +The weighting functions used as well as the obtained complementary filters are shown in Fig. 8.

          @@ -728,7 +736,7 @@ Test bounds: 0.1000 < gamma <= 10500.0000
          -
          +

          weights_comp_filters_Hinfa.png

          Figure 8: Weights and Complementary Fitlers obtained (png, pdf)

          @@ -746,8 +754,8 @@ CPS_Ha = 1 -

          1.5 H-Infinity Synthesis - method B

          +
          +

          1.5 H-Infinity Synthesis - method B

          We have that: @@ -792,7 +800,7 @@ We use this as the weighting functions for the \(\mathcal{H}_\infty\) synthesis

          -The weighting function and the obtained complementary filters are shown in Fig. 9. +The weighting function and the obtained complementary filters are shown in Fig. 9.

          @@ -852,7 +860,7 @@ Test bounds: 0.0000 < gamma <= 32.8125
          -
          +

          weights_comp_filters_Hinfb.png

          Figure 9: Weights and Complementary Fitlers obtained (png, pdf)

          @@ -866,26 +874,26 @@ CPS_Hb = 1 -

          1.6 Comparison of the methods

          +
          +

          1.6 Comparison of the methods

          The three methods are now compared.

          -The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. 10. +The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. 10.

          -The Cumulative Power Spectrum for the same sensors are shown on Fig. 11. +The Cumulative Power Spectrum for the same sensors are shown on Fig. 11.

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

          - +
          @@ -929,14 +937,14 @@ The RMS value of the obtained super sensors are shown on table + -
          +

          comparison_cps_noise.png

          Figure 11: Comparison of the obtained Cumulative Power Spectrum using the three methods (png, pdf)

          @@ -944,8 +952,8 @@ The RMS value of the obtained super sensors are shown on table -

          1.7 Conclusion

          +
          +

          1.7 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). @@ -954,38 +962,30 @@ From the above complementary filter design with the \(\mathcal{H}_2\) and \(\mat

          -
          -

          2 Robustness to sensor dynamics uncertainty

          +
          +

          2 Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty

          - +

          -Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure 12). +We initially considered perfectly known sensor dynamics so that it can be perfectly inverted. +

          + +

          +We now take into account the fact that the sensor dynamics is only partially known. +To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Fig. 12.

          -
          -

          fusion_two_noisy_sensors_with_dyn_bis.png +

          +

          sensor_fusion_dynamic_uncertainty.png

          -

          Figure 12: Fusion of two sensors

          +

          Figure 12: Sensor fusion architecture with sensor dynamics uncertainty

          -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*} - -

          -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. +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.

          @@ -995,27 +995,27 @@ All the files (data and Matlab scripts) are accessible -

          2.1 Unknown sensor dynamics dynamics

          +
          +

          2.1 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 13. +Let's represent that with multiplicative input uncertainty as shown on figure 12.

          -
          -

          fusion_gain_mismatch.png +

          +

          sensor_fusion_dynamic_uncertainty.png

          Figure 13: Fusion of two sensors with input multiplicative uncertainty

          -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 \(\| \Delta_i \|_\infty < 1\). @@ -1049,7 +1049,7 @@ Which is approximately the same as requiring

          -The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(\epsilon\) (figure 14). +The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(\epsilon\) (figure 14).

          @@ -1058,7 +1058,7 @@ We then have that the angle introduced by the super sensor is bounded by \(\arcs

          -
          +

          uncertainty_gain_phase_variation.png

          Figure 14: Maximum phase variation

          @@ -1070,19 +1070,19 @@ Thus, we choose should choose \(\epsilon\) so that the maximum phase uncertainty
          -
          -

          2.2 Design the complementary filters in order to limit the phase and gain uncertainty of the super sensor

          +
          +

          2.2 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*}

          @@ -1090,11 +1090,7 @@ We can do that with the \(\mathcal{H}_\infty\) synthesis by setting upper bounds

          -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.

          @@ -1104,12 +1100,18 @@ This is of primary importance in order to ensure the stability of the feedback l
          -
          -

          2.3 First Basic Example with gain mismatch

          + +
          +

          2.3 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*} +
          G1 = 1;
           G2 = 0.6;
          @@ -1117,19 +1119,19 @@ G2 = 0.15.
          +Two pairs of complementary filters are designed and shown on figure 15.
           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.
           

          -
          +

          comp_filters_robustness_test.png

          Figure 15: The two complementary filters designed for the robustness test (png, pdf)

          -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 16). +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 16).

          @@ -1137,7 +1139,7 @@ We see that the blue complementary filters with a lower maximum norm permits to

          -
          +

          tf_super_sensor_comp.png

          Figure 16: Comparison of the obtained super sensor transfer functions (png, pdf)

          @@ -1145,32 +1147,264 @@ We see that the blue complementary filters with a lower maximum norm permits to
          -
          -

          2.4 TODO More Complete example with model uncertainty

          +
          +

          2.4 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 \(\omega > 1\text{ kHz}\)
          • +
          +
          + +
          +

          2.4.1 Dynamical uncertainty of the individual sensors

          +
          +

          +We define the weights that are used to characterize the dynamic uncertainty of the sensors. +

          +
          +
          omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
          +w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
          +
          +omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
          +w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
          +omegac = 5000*2*pi; G0 = 1; Ginf = 50;
          +w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
          +
          +
          + +

          +From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. 17 with the upper and lower bounds on the magnitude and on the phase. +

          +
          +
          G1 = 1 + w1*ultidyn('Delta',[1 1]);
          +G2 = 1 + w2*ultidyn('Delta',[1 1]);
          +
          +
          + + +
          +

          uncertainty_dynamics_sensors.png +

          +

          Figure 17: Dynamic uncertainty of the two sensors (png, pdf)

          -
          -

          3 Equivalent Super Sensor

          +
          + +
          +

          2.4.2 Synthesis objective

          +
          +

          +The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. 14. +

          + +

          +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 \(\Delta\phi_\text{max}(\omega)\). +

          +
          +
          + +
          +

          2.4.3 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 \(\sqrt{2}\)): +

          +\begin{equation} +\label{org6f4cf45} + \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 \(1\). +

          +
          +
          + +
          +

          2.4.4 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 \(\omega_0 = 500\text{ Hz}\) +

          + +
          +
          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;
          +
          +
          + + +
          +

          maximum_wanted_phase_uncertainty.png +

          +

          Figure 18: Maximum wanted phase uncertainty using this weight (png, pdf)

          +
          + +

          +The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. 19. +

          + + +
          +

          upper_bounds_comp_filter_max_phase_uncertainty.png +

          +

          Figure 19: Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz (png, pdf)

          +
          + +

          +The \(\mathcal{H}_\infty\) synthesis is performed using the defined weights and the obtained complementary filters are shown in Fig. 20. +

          + +
          +
          P = [W1 -W1;
          +     0   W2;
          +     1   0];
          +
          +
          + +

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

          +
          +
          [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
          +
          +
          + +
          +[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
          +Resetting value of Gamma min based on D_11, D_12, D_21 terms
          +
          +Test bounds:      0.0447 <  gamma  <=      1.3318
          +
          +  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
          +    1.332   1.3e+01 -1.0e-14   1.3e+00   -2.6e-18    0.0000    p
          +    0.688   1.3e-11# ********   1.3e+00   -6.7e-15  ********    f
          +    1.010   1.1e+01 -1.5e-14   1.3e+00   -2.5e-14    0.0000    p
          +    0.849   6.9e-11# ********   1.3e+00   -2.3e-14  ********    f
          +    0.930   5.2e-12# ********   1.3e+00   -6.1e-18  ********    f
          +    0.970   5.6e-11# ********   1.3e+00   -2.3e-14  ********    f
          +    0.990   5.0e-11# ********   1.3e+00   -1.7e-17  ********    f
          +    1.000   2.1e-10# ********   1.3e+00    0.0e+00  ********    f
          +    1.005   1.9e-10# ********   1.3e+00   -3.7e-14  ********    f
          +    1.008   1.1e+01 -9.1e-15   1.3e+00    0.0e+00    0.0000    p
          +    1.006   1.2e-09# ********   1.3e+00   -6.9e-16  ********    f
          +    1.007   1.1e+01 -4.6e-15   1.3e+00   -1.8e-16    0.0000    p
          +
          + Gamma value achieved:     1.0069
          +
          + +
          +
          H1 = 1 - H2;
          +
          +
          + + +
          +

          comp_filter_hinf_uncertainty.png +

          +

          Figure 20: Obtained complementary filters (png, pdf)

          +
          +
          +
          + +
          +

          2.4.5 Super sensor uncertainty

          +
          +

          +We can now compute the uncertainty of the super sensor. The result is shown in Fig. 21. +

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

          super_sensor_uncertainty_bode_plot.png +

          +

          Figure 21: Uncertainty on the dynamics of the super sensor (png, pdf)

          +
          + +

          +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. +

          +
          +
          +
          +
          + +
          +

          3 Equivalent Super Sensor

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

          -
          -

          3.1 Sensor Fusion Architecture

          +
          +

          3.1 Sensor Fusion Architecture

          -Let consider figure 17 where two sensors are merged. +Let consider figure 22 where two sensors are merged. The dynamic uncertainty of each sensor is represented by a weight \(w_i(s)\), the frequency characteristics each of the sensor noise is represented by the weights \(N_i(s)\). The noise sources \(\tilde{n}_i\) are considered to be white noise: \(\Phi_{\tilde{n}_i}(\omega) = 1, \ \forall\omega\).

          -
          +

          sensor_fusion_full.png

          -

          Figure 17: Sensor fusion architecture (png, pdf).

          +

          Figure 22: Sensor fusion architecture (png, pdf).

          @@ -1197,28 +1431,28 @@ And the noise of the super sensor is:
          -
          -

          3.2 Equivalent Configuration

          +
          +

          3.2 Equivalent Configuration

          -We try to determine \(w_{ss}(s)\) and \(N_{ss}(s)\) such that the sensor on figure 18 is equivalent to the super sensor of figure 17. +We try to determine \(w_{ss}(s)\) and \(N_{ss}(s)\) such that the sensor on figure 23 is equivalent to the super sensor of figure 22.

          -
          +

          sensor_fusion_equivalent.png

          -

          Figure 18: Equivalent Super Sensor (png, pdf).

          +

          Figure 23: Equivalent Super Sensor (png, pdf).

          -
          -

          3.3 Model the uncertainty of the super sensor

          +
          +

          3.3 Model the uncertainty of the super sensor

          -At each frequency \(\omega\), the uncertainty set of the super sensor shown on figure 17 is a circle centered on \(1\) with a radius equal to \(|H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|\) on the complex plane. -The uncertainty set of the sensor shown on figure 18 is a circle centered on \(1\) with a radius equal to \(|w_{ss}(j\omega)|\) on the complex plane. +At each frequency \(\omega\), the uncertainty set of the super sensor shown on figure 22 is a circle centered on \(1\) with a radius equal to \(|H_1(j\omega) w_1(j\omega)| + |H_2(j\omega) w_2(j\omega)|\) on the complex plane. +The uncertainty set of the sensor shown on figure 23 is a circle centered on \(1\) with a radius equal to \(|w_{ss}(j\omega)|\) on the complex plane.

          @@ -1233,11 +1467,11 @@ Ideally, we want to find a weight \(w_{ss}(s)\) so that:

          -
          -

          3.4 Model the noise of the super sensor

          +
          +

          3.4 Model the noise of the super sensor

          -The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 17 is: +The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 22 is:

          \begin{align*} \Phi_{\hat{x}}(\omega) &= | H_1(j\omega) N_1(j\omega) |^2 \Phi_{\tilde{n}_1} + | H_2(j\omega) N_2(j\omega) |^2 \Phi_{\tilde{n}_2} \\ @@ -1245,7 +1479,7 @@ The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown \end{align*}

          -The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 18 is: +The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 23 is:

          \begin{align*} \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\ @@ -1264,8 +1498,8 @@ Ideally, we want to find a weight \(N_{ss}(s)\) such that:
          -
          -

          3.5 First guess

          +
          +

          3.5 First guess

          We could choose @@ -1293,8 +1527,8 @@ and

          -
          -

          4 Optimal And Robust Sensor Fusion in Practice

          +
          +

          4 Optimal And Robust Sensor Fusion in Practice

          Here are the steps in order to apply optimal and robust sensor fusion: @@ -1307,26 +1541,26 @@ Here are the steps in order to apply optimal and robust sensor fusion:

          -
          -

          4.1 Measurement of the noise characteristics of the sensors

          +
          +

          4.1 Measurement of the noise characteristics of the sensors

          -
          -

          4.1.1 Huddle Test

          +
          +

          4.1.1 Huddle Test

          The technique to estimate the sensor noise is taken from barzilai98_techn_measur_noise_sensor_presen.

          -Let's consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity \(x\) as shown in figure 19. +Let's consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity \(x\) as shown in figure 24.

          -
          +

          huddle_test.png

          -

          Figure 19: Huddle test block diagram

          +

          Figure 24: Huddle test block diagram

          @@ -1342,7 +1576,7 @@ We also assume that their dynamics is ideal: \(G_1(s) = G_2(s) = 1\). We then have:

          \begin{equation} -\label{org456a8db} +\label{org51dae3b} \gamma_{\hat{x}_1\hat{x}_2}^2(\omega) = \frac{1}{1 + 2 \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right) + \left( \frac{|\Phi_n(\omega)|}{|\Phi_{\hat{x}}(\omega)|} \right)^2} \end{equation} @@ -1350,16 +1584,16 @@ We then have: Since the input signal \(x\) and the instrumental noise \(n\) are incoherent:

          \begin{equation} -\label{org0b69a39} +\label{orgec4c52a} |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)| \end{equation}

          -From equations \eqref{org456a8db} and \eqref{org0b69a39}, we finally obtain +From equations \eqref{org51dae3b} and \eqref{orgec4c52a}, we finally obtain

          \begin{equation} -\label{org47f030a} +\label{orgdc4f127} |\Phi_n(\omega)| = |\Phi_{\hat{x}}(\omega)| \left( 1 - \sqrt{\gamma_{\hat{x}_1\hat{x}_2}^2(\omega)} \right) \end{equation} @@ -1367,8 +1601,8 @@ From equations \eqref{org456a8db} and \eqref{org0b69a39}, we finally obtain
          -
          -

          4.1.2 Weights that represents the noises' PSD

          +
          +

          4.1.2 Weights that represents the noises' PSD

          For further complementary filter synthesis, it is preferred to consider a normalized noise source \(\tilde{n}\) that has a PSD equal to one (\(\Phi_{\tilde{n}}(\omega) = 1\)) and to use a weighting filter \(N(s)\) in order to represent the frequency dependence of the noise. @@ -1387,20 +1621,20 @@ These weighting filters can then be used to compare the noise level of sensors f

          -The sensor with a normalized noise input is shown in figure 20. +The sensor with a normalized noise input is shown in figure 25.

          -
          +

          one_sensor_normalized_noise.png

          -

          Figure 20: One sensor with normalized noise

          +

          Figure 25: One sensor with normalized noise

          -
          -

          4.1.3 Comparison of the noises' PSD

          +
          +

          4.1.3 Comparison of the noises' PSD

          Once the noise of the sensors to be merged have been characterized, the power spectral density of both sensors have to be compared. @@ -1416,8 +1650,8 @@ Ideally, the PSD of the noise are such that:

          -
          -

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

          +
          +

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

          The coherence between signals \(x\) and \(y\) is defined as follow @@ -1445,11 +1679,11 @@ where:

          -
          -

          4.2 Estimate the dynamic uncertainty of the sensors

          +
          +

          4.2 Estimate the dynamic uncertainty of the sensors

          -Let's consider one sensor represented on figure 21. +Let's consider one sensor represented on figure 26.

          @@ -1461,16 +1695,16 @@ The goal is to accurately determine \(w(s)\) for the sensors that have to be mer

          -
          +

          one_sensor_dyn_uncertainty.png

          -

          Figure 21: Sensor with dynamic uncertainty

          +

          Figure 26: Sensor with dynamic uncertainty

          -
          -

          4.3 Optimal and Robust synthesis of the complementary filters

          +
          +

          4.3 Optimal and Robust synthesis of the complementary filters

          Once the noise characteristics and dynamic uncertainty of both sensors have been determined and we have determined the following weighting functions: @@ -1481,7 +1715,7 @@ Once the noise characteristics and dynamic uncertainty of both sensors have been

          -The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in figure 17 such that: +The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in figure 22 such that:

          • the uncertainty on the super sensor dynamics is minimized
          • @@ -1489,24 +1723,24 @@ The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in f
          -
          +

          sensor_fusion_full.png

          -

          Figure 22: Sensor fusion architecture with sensor dynamics uncertainty

          +

          Figure 27: Sensor fusion architecture with sensor dynamics uncertainty

          -
          -

          5 Methods of complementary filter synthesis

          +
          +

          5 Methods of complementary filter synthesis

          -
          -

          5.1 Complementary filters using analytical formula

          +
          +

          5.1 Complementary filters using analytical formula

          - +

          @@ -1516,8 +1750,8 @@ All the files (data and Matlab scripts) are accessible -

          5.1.1 Analytical 1st order complementary filters

          +
          +

          5.1.1 Analytical 1st order complementary filters

          First order complementary filters are defined with following equations: @@ -1528,7 +1762,7 @@ First order complementary filters are defined with following equations: \end{align}

          -Their bode plot is shown figure 23. +Their bode plot is shown figure 28.

          @@ -1540,16 +1774,16 @@ Hl1 = 1
          -
          +

          comp_filter_1st_order.png

          -

          Figure 23: Bode plot of first order complementary filter (png, pdf)

          +

          Figure 28: Bode plot of first order complementary filter (png, pdf)

          -
          -

          5.1.2 Second Order Complementary Filters

          +
          +

          5.1.2 Second Order Complementary Filters

          We here use analytical formula for the complementary filters \(H_L\) and \(H_H\). @@ -1575,15 +1809,15 @@ where:

          -This is illustrated on figure 24. +This is illustrated on figure 29. The slope of those filters at high and low frequencies is \(-2\) and \(2\) respectively for \(H_L\) and \(H_H\).

          -
          +

          comp_filters_param_alpha.png

          -

          Figure 24: Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (png, pdf)

          +

          Figure 29: Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (png, pdf)

          @@ -1598,16 +1832,16 @@ xlabel('$ +

          param_alpha_hinf_norm.png

          -

          Figure 25: Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (png, pdf)

          +

          Figure 30: Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (png, pdf)

          -
          -

          5.1.3 Third Order Complementary Filters

          +
          +

          5.1.3 Third Order Complementary Filters

          The following formula gives complementary filters with slopes of \(-3\) and \(3\): @@ -1626,7 +1860,7 @@ The parameters are:

          -The filters are defined below and the result is shown on figure 26. +The filters are defined below and the result is shown on figure 31.

          @@ -1640,20 +1874,20 @@ Hl3_ana = ( +

          complementary_filters_third_order.png

          -

          Figure 26: Third order complementary filters using the analytical formula (png, pdf)

          +

          Figure 31: Third order complementary filters using the analytical formula (png, pdf)

          -
          -

          5.2 H-Infinity synthesis of complementary filters

          +
          +

          5.2 H-Infinity synthesis of complementary filters

          - +

          @@ -1663,8 +1897,8 @@ All the files (data and Matlab scripts) are accessible -

          5.2.1 Synthesis Architecture

          +
          +

          5.2.1 Synthesis Architecture

          We here synthesize the complementary filters using the \(\mathcal{H}_\infty\) synthesis. @@ -1672,18 +1906,18 @@ The goal is to specify upper bounds on the norms of \(H_L\) and \(H_H\) while en

          -In order to do so, we use the generalized plant shown on figure 27 where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively. +In order to do so, we use the generalized plant shown on figure 32 where \(w_L\) and \(w_H\) weighting transfer functions that will be used to shape \(H_L\) and \(H_H\) respectively.

          -
          +

          sf_hinf_filters_plant_b.png

          -

          Figure 27: Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters

          +

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

          -The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure 28) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one: +The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_L\) (figure 33) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one: \[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \]

          @@ -1702,16 +1936,16 @@ We then see that \(w_L\) and \(w_H\) can be used to shape both \(H_L\) and \(H_H

          -
          +

          sf_hinf_filters_b.png

          -

          Figure 28: \(\mathcal{H}_\infty\) synthesis of the complementary filters

          +

          Figure 33: \(\mathcal{H}_\infty\) synthesis of the complementary filters

          -
          -

          5.2.2 Weights

          +
          +

          5.2.2 Weights

          omegab = 2*pi*9;
          @@ -1722,16 +1956,16 @@ wL = (s 
          +

          weights_wl_wh.png

          -

          Figure 29: Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (png, pdf)

          +

          Figure 34: Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (png, pdf)

          -
          -

          5.2.3 H-Infinity Synthesis

          +
          +

          5.2.3 H-Infinity Synthesis

          We define the generalized plant \(P\) on matlab. @@ -1773,7 +2007,7 @@ Test bounds: 0.0000 < gamma <= 1.7285

          -We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure 30. +We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H_L\) and \(H_H\) is shown on figure 35.

          Hh_hinf = 1 - Hl_hinf;
          @@ -1782,28 +2016,28 @@ We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H
           
          -
          -

          5.2.4 Obtained Complementary Filters

          +
          +

          5.2.4 Obtained Complementary Filters

          -The obtained complementary filters are shown on figure 30. +The obtained complementary filters are shown on figure 35.

          -
          +

          hinf_filters_results.png

          -

          Figure 30: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (png, pdf)

          +

          Figure 35: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (png, pdf)

          -
          -

          5.3 Feedback Control Architecture to generate Complementary Filters

          +
          +

          5.3 Feedback Control Architecture to generate Complementary Filters

          - +

          The idea is here to use the fact that in a classical feedback architecture, \(S + T = 1\), in order to design complementary filters. @@ -1820,14 +2054,14 @@ All the files (data and Matlab scripts) are accessible -

          5.3.1 Architecture

          +
          +

          5.3.1 Architecture

          -
          +

          complementary_filters_feedback_architecture.png

          -

          Figure 31: Architecture used to generate the complementary filters

          +

          Figure 36: Architecture used to generate the complementary filters

          @@ -1851,8 +2085,8 @@ Which contains two integrator and a lead. \(\omega_c\) is used to tune the cross

          -
          -

          5.3.2 Loop Gain Design

          +
          -
          -

          5.3.3 Complementary Filters Obtained

          +
          +

          5.3.3 Complementary Filters Obtained

          We then compute the resulting low pass and high pass filters. @@ -1887,25 +2121,25 @@ Hh = 1/

          -
          +

          low_pass_high_pass_filters.png

          -

          Figure 33: Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (png, pdf)

          +

          Figure 38: Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (png, pdf)

          -
          -

          5.4 Analytical Formula found in the literature

          +
          +

          5.4 Analytical Formula found in the literature

          - +

          -
          -

          5.4.1 Analytical Formula

          +
          +

          5.4.1 Analytical Formula

          min15_compl_filter_desig_angle_estim @@ -1954,8 +2188,8 @@ Hh = 1/

          -
          -

          5.4.2 Matlab

          +
          +

          5.4.2 Matlab

          omega0 = 1*2*pi; % [rad/s]
          @@ -1973,16 +2207,16 @@ HL3 = (
          +

          comp_filters_literature.png

          -

          Figure 34: Comparison of some complementary filters found in the literature (png, pdf)

          +

          Figure 39: Comparison of some complementary filters found in the literature (png, pdf)

          -
          -

          5.4.3 Discussion

          +
          +

          5.4.3 Discussion

          Analytical Formula found in the literature provides either no parameter for tuning the robustness / performance trade-off. @@ -1991,11 +2225,11 @@ Analytical Formula found in the literature provides either no parameter for tuni

          -
          -

          5.5 Comparison of the different methods of synthesis

          +
          +

          5.5 Comparison of the different methods of synthesis

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

            @@ -2021,7 +2255,7 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti

          Author: Thomas Dehaeze

          -

          Created: 2019-08-29 jeu. 14:53

          +

          Created: 2019-08-29 jeu. 22:47

          Validate

          diff --git a/matlab/index.org b/matlab/index.org index 3ec04ab..831ad4c 100644 --- a/matlab/index.org +++ b/matlab/index.org @@ -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 <> ** 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 <> #+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") + <> +#+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") + <> +#+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") + <> +#+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") + <> +#+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") + <> +#+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. 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zD&zvY@qb7H%J6aDt0eM5YH>Ok`6lEnPejqCH?@{@0rLQAobF4!M>m6;Z3tAz$f!l)&Qc?!((+w8;(lx$0ou zk>Sel0Q6n$@TqXqnxP?*1rim+&+pY1$jkgN6~q%LHs1!J0b1{3sZy3ZRoUV90PlDR zABx@AiEMpHZlDSxD?27*wQf6ehbUUD(BB-b)igo)*#vtcfcHH0{_$`|*iu zp~%x^TieQhp|0@1QJ*?o`*iBH$DiL^G-)Vb(M4 - + - + - + - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - + - + - - - - + - - + - - - - - - - + + + - - - - - - - - - - - - - - - - - + - + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - + diff --git a/tikz/index.org b/tikz/index.org index 47d703c..d342eaf 100644 --- a/tikz/index.org +++ b/tikz/index.org @@ -140,6 +140,61 @@ Configuration file is accessible [[file:config.org][here]]. #+caption: H-Infinity - Optimal Complementary Filters ([[./figs/h_infinity_optimal_comp_filters.png][png]], [[./figs/h_infinity_optimal_comp_filters.pdf][pdf]], [[./figs/h_infinity_optimal_comp_filters.tex][tex]]). #+RESULTS: [[file:figs/h_infinity_optimal_comp_filters.png]] +* Generate Complementary Filters using Feedback Control Architecture +#+begin_src latex :file complementary_filters_feedback_architecture.pdf :tangle figs/complementary_filters_feedback_architecture.tex :exports both + \begin{tikzpicture} + \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){}; + \node[block, right=1 of addfb] (L){$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$}; + \end{tikzpicture} +#+end_src + +#+name: fig:complementary_filters_feedback_architecture +#+caption: Generation of Complementary Filters using the feedback architecture ([[./figs/complementary_filters_feedback_architecture.png][png]], [[./figs/complementary_filters_feedback_architecture.pdf][pdf]], [[./figs/complementary_filters_feedback_architecture.tex][tex]]). +#+RESULTS: +[[file:figs/complementary_filters_feedback_architecture.png]] + +* H-Infinity Synthesis for Robust Sensor Fusion +#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both + <> + \begin{tikzpicture} + \node[block={4.0cm}{3.0cm}, dashed] (P) {}; + \node[above] at (P.north) {$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$}; + \end{tikzpicture} +#+end_src + +#+name: fig:h_infinity_robust_fusion +#+caption: H-Infinity Synthesis for Robust Sensor Fusion ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]). +#+RESULTS: +[[file:figs/h_infinity_robust_fusion.png]] * Fusion of two noisy sensors #+begin_src latex :file fusion_two_noisy_sensors.pdf :tangle figs/fusion_two_noisy_sensors.tex <> @@ -274,113 +329,26 @@ Configuration file is accessible [[file:config.org][here]]. #+RESULTS: [[file:figs/fusion_two_noisy_sensors_weights.png]] -* Fusion of two noisy signals -#+begin_src latex :file fusion_two_signals.pdf :tangle figs/fusion_two_signals.tex - \begin{tikzpicture} - \node[addb] (add) at (0, 0){}; - \node[block, above left=0.5 and 0.8 of add] (H1) {$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}$}; - \end{tikzpicture} -#+end_src - -#+name: fig:fusion_two_noisy_sensors_with_dyn_bis -#+caption: Fusion of two noisy sensors_with_dyn_bis ([[./figs/fusion_two_noisy_sensors_with_dyn_bis.png][png]], [[./figs/fusion_two_noisy_sensors_with_dyn_bis.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_with_dyn_bis.tex][tex]]). -#+RESULTS: -[[file:figs/fusion_two_noisy_sensors_with_dyn_bis.png]] - -* Fusion of two sensors with mismatch dynamics -#+begin_src latex :file fusion_gain_mismatch.pdf :tangle figs/fusion_gain_mismatch.tex +* Sensor fusion architecture with sensor dynamics uncertainty +#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/sensor_fusion_dynamic_uncertainty.tex <> \begin{tikzpicture} \node[branch] (x) at (0, 0); - \node[addb, above right=1 and 3.5 of x](add1){}; - \node[addb, below right=1 and 3.5 of x](add2){}; - \node[block, above left= of add1](delta1){$\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 - <> - \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 <> @@ -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 + <> + \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]]
          Table 1: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters

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