From ab16477991a47e614e07422c4915c2d0367b9e70 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 11 Sep 2019 17:56:44 +0200 Subject: [PATCH] Add CVX optimization section --- matlab/figs/LMI_obtained_comp_filters.pdf | Bin 0 -> 71072 bytes matlab/figs/LMI_obtained_comp_filters.png | Bin 0 -> 116719 bytes matlab/figs/comp_cvx_h2i_hinf_norm.pdf | Bin 0 -> 67732 bytes matlab/figs/comp_cvx_h2i_hinf_norm.png | Bin 0 -> 103450 bytes .../figs/compare_cvx_h2hinf_comp_filters.pdf | Bin 0 -> 100169 bytes .../figs/compare_cvx_h2hinf_comp_filters.png | Bin 0 -> 127444 bytes matlab/figs/cps_compare_cvx_h2i.pdf | Bin 0 -> 80555 bytes matlab/figs/cps_compare_cvx_h2i.png | Bin 0 -> 133733 bytes matlab/figs/psd_compare_cvx_h2i.pdf | Bin 0 -> 72005 bytes matlab/figs/psd_compare_cvx_h2i.png | Bin 0 -> 176594 bytes ...per_sensor_uncertainty_compare_cvx_h2i.pdf | Bin 0 -> 133364 bytes ...per_sensor_uncertainty_compare_cvx_h2i.png | Bin 0 -> 143811 bytes matlab/index.html | 1139 +++++++++++------ matlab/index.org | 460 ++++++- tikz/index.org | 2 +- 15 files changed, 1179 insertions(+), 422 deletions(-) create mode 100644 matlab/figs/LMI_obtained_comp_filters.pdf create mode 100644 matlab/figs/LMI_obtained_comp_filters.png create mode 100644 matlab/figs/comp_cvx_h2i_hinf_norm.pdf create mode 100644 matlab/figs/comp_cvx_h2i_hinf_norm.png create mode 100644 matlab/figs/compare_cvx_h2hinf_comp_filters.pdf create mode 100644 matlab/figs/compare_cvx_h2hinf_comp_filters.png create mode 100644 matlab/figs/cps_compare_cvx_h2i.pdf create mode 100644 matlab/figs/cps_compare_cvx_h2i.png create mode 100644 matlab/figs/psd_compare_cvx_h2i.pdf create mode 100644 matlab/figs/psd_compare_cvx_h2i.png create mode 100644 matlab/figs/super_sensor_uncertainty_compare_cvx_h2i.pdf create mode 100644 matlab/figs/super_sensor_uncertainty_compare_cvx_h2i.png diff --git a/matlab/figs/LMI_obtained_comp_filters.pdf b/matlab/figs/LMI_obtained_comp_filters.pdf new file mode 100644 index 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Table of Contents

    -
  • 1. Optimal Sensor Fusion - Minimize the Super Sensor Noise +
  • 1. Optimal Sensor Fusion - Minimize the Super Sensor Noise
  • -
  • 2. Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty +
  • 2. Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty
  • -
  • 3. Optimal Sensor Fusion - Mixed Synthesis +
  • 3. Optimal Sensor Fusion - Mixed Synthesis
  • -
  • 4. H-Infinity synthesis to ensure both performance and robustness +
  • 4. Mixed Synthesis - LMI Optimization
  • -
  • 5. Equivalent Super Sensor +
  • 5. H-Infinity synthesis to ensure both performance and robustness
  • -
  • 6. Optimal And Robust Sensor Fusion in Practice +
  • 6. Equivalent Super Sensor
      -
    • 6.1. Measurement of the noise characteristics of the sensors -
    • -
    • 6.2. Estimate the dynamic uncertainty of the sensors
    • -
    • 6.3. Optimal and Robust synthesis of the complementary filters
    • +
    • 7. Optimal And Robust Sensor Fusion in Practice +
        +
      • 7.1. Measurement of the noise characteristics of the sensors +
      • -
      • 7. Methods of complementary filter synthesis - @@ -397,20 +410,20 @@ Two sensors are considered with both different noise characteristics and dynamic

          -
        • in section 1: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor's noise is minimized
        • -
        • in section 2: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor's uncertainty is bonded to acceptable values
        • -
        • in section 3: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor's uncertainty and to lower the RMS value of the super sensor's noise
        • -
        • in section 4: the \(\mathcal{H}_\infty\) synthesis is used for both limiting the noise and uncertainty of the super sensor
        • -
        • in section 5: we try to find the characteristics of the super sensor from the characteristics of the individual sensors and of the complementary filters
        • -
        • in section 6: a methodology is proposed to apply optimal and robust sensor fusion in practice
        • -
        • in section 7: methods of complementary filter synthesis are proposed
        • +
        • in section 1: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor's noise is minimized
        • +
        • in section 2: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor's uncertainty is bonded to acceptable values
        • +
        • in section 3: the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor's uncertainty and to lower the RMS value of the super sensor's noise
        • +
        • in section 5: the \(\mathcal{H}_\infty\) synthesis is used for both limiting the noise and uncertainty of the super sensor
        • +
        • in section 6: we try to find the characteristics of the super sensor from the characteristics of the individual sensors and of the complementary filters
        • +
        • in section 7: a methodology is proposed to apply optimal and robust sensor fusion in practice
        • +
        • in section 8: methods of complementary filter synthesis are proposed
        -
        -

        1 Optimal Sensor Fusion - Minimize the Super Sensor Noise

        +
        +

        1 Optimal Sensor Fusion - Minimize the Super Sensor Noise

        - +

        The idea is to combine sensors that works in different frequency range using complementary filters. @@ -431,23 +444,23 @@ The Matlab scripts is accessible here

        -
        -

        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{org51f2bdd} +\label{orgecfbe8a} \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

        @@ -457,16 +470,16 @@ 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

        @@ -476,7 +489,7 @@ We obtain the architecture of figure 2. \(H_1(s)\) and \(H_2(s)\) are complementary filters:

        \begin{equation} -\label{org7eb0521} +\label{orgc62aef9} H_1(s) + H_2(s) = 1 \end{equation} @@ -493,14 +506,14 @@ 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{org29e3384} +\label{orgb11ad80} \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 \(N_2\): @@ -520,7 +533,7 @@ N2 = ( +

        noise_characteristics_sensors.png

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

        @@ -528,8 +541,8 @@ 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: @@ -542,11 +555,11 @@ 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

        @@ -575,7 +588,7 @@ 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;
        @@ -600,15 +613,15 @@ 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.

        @@ -616,7 +629,7 @@ 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)

        @@ -630,7 +643,7 @@ 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)

        @@ -644,7 +657,7 @@ 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)

        @@ -652,15 +665,15 @@ CPS_H2 = 1 -

        1.4 Alternative H-Two Synthesis

        +
        +

        1.4 Alternative H-Two Synthesis

        -An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is shown in Fig. 8. +An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is shown in Fig. 8.

        -
        +

        h_infinity_optimal_comp_filters_bis.png

        Figure 8: Alternative formulation of the \(\mathcal{H}_2\) synthesis

        @@ -681,8 +694,8 @@ An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is sho
        -
        -

        1.5 H-Infinity Synthesis - method A

        +
        +

        1.5 H-Infinity Synthesis - method A

        Another objective that we may have is that the noise of the super sensor \(n_{SS}\) is following the minimum of the noise of the two sensors \(n_1\) and \(n_2\): @@ -706,12 +719,12 @@ 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. 9. +The weighting functions used as well as the obtained complementary filters are shown in Fig. 9.

        @@ -780,7 +793,7 @@ Test bounds: 0.1000 < gamma <= 10500.0000
        -
        +

        weights_comp_filters_Hinfa.png

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

        @@ -798,8 +811,8 @@ CPS_Ha = 1 -

        1.6 H-Infinity Synthesis - method B

        +
        +

        1.6 H-Infinity Synthesis - method B

        We have that: @@ -844,7 +857,7 @@ 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. 10. +The weighting function and the obtained complementary filters are shown in Fig. 10.

        @@ -904,7 +917,7 @@ Test bounds: 0.0000 < gamma <= 32.8125
        -
        +

        weights_comp_filters_Hinfb.png

        Figure 10: Weights and Complementary Fitlers obtained (png, pdf)

        @@ -918,8 +931,8 @@ CPS_Hb = 1 -

        1.7 H-Infinity Synthesis - method C

        +
        +

        1.7 H-Infinity Synthesis - method C

        Wp = 0.56*(inv(N1)+inv(N2))/(1 + s/2/pi/1000);
        @@ -976,7 +989,7 @@ Test bounds:      0.0000 <  gamma  <=     36.7543
         
        -
        +

        weights_comp_filters_Hinfc.png

        Figure 11: Weights and Complementary Fitlers obtained (png, pdf)

        @@ -990,26 +1003,26 @@ CPS_Hc = 1 -

        1.8 Comparison of the methods

        +
        +

        1.8 Comparison of the methods

        The three methods are now compared.

        -The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. 12. +The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. 12.

        -The Cumulative Power Spectrum for the same sensors are shown on Fig. 13. +The Cumulative Power Spectrum for the same sensors are shown on Fig. 13.

        -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.

        - +
        @@ -1058,14 +1071,14 @@ The RMS value of the obtained super sensors are shown on table + -
        +

        comparison_cps_noise.png

        Figure 13: Comparison of the obtained Cumulative Power Spectrum using the three methods (png, pdf)

        @@ -1073,8 +1086,8 @@ The RMS value of the obtained super sensors are shown on table -

        1.9 Obtained Super Sensor's noise uncertainty

        +
        +

        1.9 Obtained Super Sensor's noise uncertainty

        We would like to verify if the obtained sensor fusion architecture is robust to change in the sensor dynamics. @@ -1107,7 +1120,7 @@ G2 = 1

        The super sensor uncertain model is defined below using the complementary filters obtained with the \(\mathcal{H}_2\) synthesis. -The dynamical uncertainty bounds of the super sensor is shown in Fig. 14. +The dynamical uncertainty bounds of the super sensor is shown in Fig. 14. Right Half Plane zero might be introduced in the super sensor dynamics which will render the feedback system unstable.

        @@ -1117,7 +1130,7 @@ Right Half Plane zero might be introduced in the super sensor dynamics which wil
        -
        +

        uncertainty_super_sensor_H2_syn.png

        Figure 14: Uncertianty regions of both individual sensors and of the super sensor when using the \(\mathcal{H}_2\) synthesis (png, pdf)

        @@ -1125,8 +1138,8 @@ Right Half Plane zero might be introduced in the super sensor dynamics which wil
        -
        -

        1.10 Conclusion

        +
        +

        1.10 Conclusion

        From the above complementary filter design with the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) synthesis, it still seems that the \(\mathcal{H}_2\) synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the \(\mathcal{H}_2\) norm). @@ -1139,11 +1152,11 @@ However, the synthesis does not take into account the robustness of the sensor f

        -
        -

        2 Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty

        +
        +

        2 Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty

        - +

        We initially considered perfectly known sensor dynamics so that it can be perfectly inverted. @@ -1151,11 +1164,11 @@ We initially considered perfectly known sensor dynamics so that it can be perfec

        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. 15. +To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Fig. 15.

        -
        +

        sensor_fusion_dynamic_uncertainty.png

        Figure 15: Sensor fusion architecture with sensor dynamics uncertainty

        @@ -1172,16 +1185,16 @@ The Matlab scripts is accessible here<
        -
        -

        2.1 Super Sensor Dynamical Uncertainty

        +
        +

        2.1 Super Sensor Dynamical Uncertainty

        In practical systems, the sensor dynamics has always some level of uncertainty. -Let's represent that with multiplicative input uncertainty as shown on figure 15. +Let's represent that with multiplicative input uncertainty as shown on figure 15.

        -
        +

        sensor_fusion_dynamic_uncertainty.png

        Figure 16: Fusion of two sensors with input multiplicative uncertainty

        @@ -1203,7 +1216,7 @@ We see that as soon as we have some uncertainty in the sensor dynamics, we have

        -The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (figure 17). +The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (figure 17).

        @@ -1212,7 +1225,7 @@ We then have that the angle introduced by the super sensor is bounded by \(\arcs

        -
        +

        uncertainty_gain_phase_variation.png

        Figure 17: Maximum phase variation

        @@ -1220,8 +1233,8 @@ We then have that the angle introduced by the super sensor is bounded by \(\arcs
        -
        -

        2.2 Dynamical uncertainty of the individual sensors

        +
        +

        2.2 Dynamical uncertainty of the individual sensors

        Let say we want to merge two sensors: @@ -1247,7 +1260,7 @@ w2 = w2*18 with the upper and lower bounds on the magnitude and on the phase. +From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. 18 with the upper and lower bounds on the magnitude and on the phase.

        G1 = 1 + w1*ultidyn('Delta',[1 1]);
        @@ -1256,7 +1269,7 @@ G2 = 1 
         
        -
        +

        uncertainty_dynamics_sensors.png

        Figure 18: Dynamic uncertainty of the two sensors (png, pdf)

        @@ -1264,11 +1277,11 @@ G2 = 1
        -
        -

        2.3 Synthesis objective

        +
        +

        2.3 Synthesis objective

        -The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. 17. +The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. 17.

        @@ -1297,8 +1310,8 @@ The maximum phase shift due to dynamic uncertainty at frequency \(\omega\) will

        -
        -

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

        +
        +

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

        We know try to express this requirement in terms of an \(\mathcal{H}_\infty\) norm. @@ -1322,7 +1335,7 @@ Then: Which is approximately equivalent to (with an error of maximum \(\sqrt{2}\)):

        \begin{equation} -\label{org11aa148} +\label{org03fb6ef} \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} @@ -1333,12 +1346,12 @@ Thus, at these frequencies, \(|w_\phi|\) should be smaller than \(1\).
        -
        -

        2.5 Weighting Function used to bound the super sensor uncertainty

        +
        +

        2.5 Weighting Function used to bound the super sensor uncertainty

        Let's define \(w_\phi(s)\) in order to bound the maximum allowed phase uncertainty \(\Delta\phi_\text{max}\) of the super sensor dynamics. -The magnitude \(|w_\phi(j\omega)|\) is shown in Fig. 19 and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. 20. +The magnitude \(|w_\phi(j\omega)|\) is shown in Fig. 19 and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. 20.

        @@ -1353,25 +1366,25 @@ W2 = w2*wphi;
        -
        +

        magnitude_wphi.png

        Figure 19: Magnitude of the weght \(w_\phi(s)\) that is used to bound the uncertainty of the super sensor (png, pdf)

        -
        +

        maximum_wanted_phase_uncertainty.png

        Figure 20: 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. 21. +The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. 21.

        -
        +

        upper_bounds_comp_filter_max_phase_uncertainty.png

        Figure 21: 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)

        @@ -1379,15 +1392,15 @@ The obtained upper bounds on the complementary filters in order to limit the pha
        -
        -

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

        +
        +

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

        -The \(\mathcal{H}_\infty\) synthesis architecture used for the complementary filters is shown in Fig. 22. +The \(\mathcal{H}_\infty\) synthesis architecture used for the complementary filters is shown in Fig. 22.

        -
        +

        h_infinity_robust_fusion.png

        Figure 22: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

        @@ -1443,10 +1456,10 @@ And \(H_1(s)\) is defined as the complementary of \(H_2(s)\).

        -The obtained complementary filters are shown in Fig. 23. +The obtained complementary filters are shown in Fig. 23.

        -
        +

        comp_filter_hinf_uncertainty.png

        Figure 23: Obtained complementary filters (png, pdf)

        @@ -1454,11 +1467,11 @@ The obtained complementary filters are shown in Fig. 23
        -
        -

        2.7 Super sensor uncertainty

        +
        +

        2.7 Super sensor uncertainty

        -We can now compute the uncertainty of the super sensor. The result is shown in Fig. 24. +We can now compute the uncertainty of the super sensor. The result is shown in Fig. 24.

        @@ -1467,7 +1480,7 @@ We can now compute the uncertainty of the super sensor. The result is shown in F
        -
        +

        super_sensor_uncertainty_bode_plot.png

        Figure 24: Uncertainty on the dynamics of the super sensor (png, pdf)

        @@ -1484,8 +1497,8 @@ For instance, we could improve the dynamical uncertainty of the super sensor by
        -
        -

        2.8 Super sensor noise

        +
        +

        2.8 Super sensor noise

        We now compute the obtain Power Spectral Density of the super sensor's noise. @@ -1502,19 +1515,19 @@ N2 = (25. -The CPS of both sensor and of the super sensor is shown in Fig. 26. +The PSD of both sensor and of the super sensor is shown in Fig. 25. +The CPS of both sensor and of the super sensor is shown in Fig. 26.

        -
        +

        psd_sensors_hinf_synthesis.png

        Figure 25: Power Spectral Density of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (png, pdf)

        -
        +

        cps_sensors_hinf_synthesis.png

        Figure 26: Cumulative Power Spectrum of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (png, cps)

        @@ -1522,8 +1535,8 @@ The CPS of both sensor and of the super sensor is shown in Fig. -

        2.9 Conclusion

        +
        +

        2.9 Conclusion

        Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values. @@ -1536,11 +1549,11 @@ However, the RMS of the super sensor noise is not optimized as it was the case w

        -
        -

        3 Optimal Sensor Fusion - Mixed Synthesis

        +
        +

        3 Optimal Sensor Fusion - Mixed Synthesis

        - +

        @@ -1549,9 +1562,8 @@ The Matlab scripts is accessible -

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

        +
        +

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

        The goal is to design complementary filters such that: @@ -1571,8 +1583,8 @@ The Matlab function for that is h2hinfsyn ( -

        3.2 Noise characteristics and Uncertainty of the individual sensors

        +
        +

        3.2 Noise characteristics and Uncertainty of the individual sensors

        We define the weights that are used to characterize the dynamic uncertainty of the sensors. This will be used for the \(\mathcal{H}_\infty\) part of the synthesis. @@ -1601,11 +1613,11 @@ N2 = (27. +Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. 27.

        -
        +

        mixed_synthesis_noise_uncertainty_sensors.png

        Figure 27: Noise characteristsics and Dynamical uncertainty of the individual sensors (png, pdf)

        @@ -1613,12 +1625,12 @@ Both dynamical uncertainty and noise characteristics of the individual sensors a
        -
        -

        3.3 Weighting Functions on the uncertainty of the super sensor

        +
        +

        3.3 Weighting Functions on the uncertainty of the super sensor

        We design weights for the \(\mathcal{H}_\infty\) part of the synthesis in order to limit the dynamical uncertainty of the super sensor. -The maximum wanted multiplicative uncertainty is shown in Fig. 28. The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here. +The maximum wanted multiplicative uncertainty is shown in Fig. 28. The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.

        @@ -1629,18 +1641,18 @@ wphi = 0. +

        mixed_syn_hinf_weight.png

        Figure 28: Wanted maximum module uncertainty of the super sensor (png, pdf)

        -The equivalent Magnitude and Phase uncertainties are shown in Fig. 29. +The equivalent Magnitude and Phase uncertainties are shown in Fig. 29.

        -
        +

        mixed_syn_objective_hinf.png

        Figure 29: \(\mathcal{H}_\infty\) synthesis objective part of the mixed-synthesis (png, pdf)

        @@ -1648,11 +1660,11 @@ The equivalent Magnitude and Phase uncertainties are shown in Fig. -

        3.4 Mixed Synthesis Architecture

        +
        +

        3.4 Mixed Synthesis Architecture

        -The synthesis architecture that is used here is shown in Fig. 30. +The synthesis architecture that is used here is shown in Fig. 30.

        @@ -1665,7 +1677,7 @@ The controller \(K\) is synthesized such that it: -

        +

        mixed_h2_hinf_synthesis.png

        Figure 30: Mixed H2/H-Infinity Synthesis

        @@ -1713,8 +1725,8 @@ P = [W1u -

        3.5 Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis

        +
        +

        3.5 Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis

        The mixed $\mathcal{H}2$/\(\mathcal{H}_\infty\) synthesis is performed below. @@ -1729,11 +1741,11 @@ H1 = 1

        -The obtained complementary filters are shown in Fig. 31. +The obtained complementary filters are shown in Fig. 31.

        -
        +

        comp_filters_mixed_synthesis.png

        Figure 31: Obtained complementary filters after mixed $\mathcal{H}2$/\(\mathcal{H}_\infty\) synthesis (png, pdf)

        @@ -1741,15 +1753,15 @@ The obtained complementary filters are shown in Fig. 31
        -
        -

        3.6 Obtained Super Sensor's noise

        +
        +

        3.6 Obtained Super Sensor's noise

        -The PSD and CPS of the super sensor's noise are shown in Fig. 32 and Fig. 33 respectively. +The PSD and CPS of the super sensor's noise are shown in Fig. 32 and Fig. 33 respectively.

        -
        +

        psd_super_sensor_mixed_syn.png

        Figure 32: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}2$/\(\mathcal{H}_\infty\) synthesis (png, pdf)

        @@ -1757,7 +1769,7 @@ The PSD and CPS of the super sensor's noise are shown in Fig. +

        cps_super_sensor_mixed_syn.png

        Figure 33: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}2$/\(\mathcal{H}_\infty\) synthesis (png, pdf)

        @@ -1765,15 +1777,15 @@ The PSD and CPS of the super sensor's noise are shown in Fig. -

        3.7 Obtained Super Sensor's Uncertainty

        +
        +

        3.7 Obtained Super Sensor's Uncertainty

        -The uncertainty on the super sensor's dynamics is shown in Fig. [[]]. +The uncertainty on the super sensor's dynamics is shown in Fig. 34.

        -
        +

        super_sensor_dyn_uncertainty_mixed_syn.png

        Figure 34: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis (png, pdf)

        @@ -1781,8 +1793,8 @@ The uncertainty on the super sensor's dynamics is shown in Fig. [[]].
        -
        -

        3.8 Conclusion

        +
        +

        3.8 Conclusion

        This synthesis methods allows both to: @@ -1795,11 +1807,302 @@ This synthesis methods allows both to:

        -
        -

        4 H-Infinity synthesis to ensure both performance and robustness

        +
        +

        4 Mixed Synthesis - LMI Optimization

        +
        +
        +

        4.1 Introduction

        +

        - +The following matlab scripts was written by Mohit. +

        +
        +
        + +
        +

        4.2 Noise characteristics and Uncertainty of the individual sensors

        +
        +

        +We define the weights that are used to characterize the dynamic uncertainty of the sensors. This will be used for the \(\mathcal{H}_\infty\) part of the synthesis. +

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

        +We define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\). This will be used for the \(\mathcal{H}_2\) part of the synthesis. +

        +
        +
        omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
        +N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
        +
        +omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
        +N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
        +
        +
        +
        +
        + +
        +

        4.3 Weights

        +
        +

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

        + +
        +
        n = 4; w0 = 2*pi*900; G0 = 9; G1 = 1; Gc = 1.1;
        +H = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
        +wphi = 0.2*(s+3.142e04)/(s+628.3)/H;
        +
        +W1u = ss(w1*wphi); W2u = ss(w2*wphi); % Weight on the uncertainty
        +W1n = ss(N1); W2n = ss(N2); % Weight on the noise
        +
        +
        + +
        +
        P = [W1u -W1u;
        +     0    W2u;
        +     W1n -W1n;
        +     0    W2n;
        +     1    0];
        +
        +
        +
        +
        + +
        +

        4.4 LMI Optimization

        +
        +

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

        + +

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

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

        +We Define all the variables. +

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

        +We define the minimization objective. +

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

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

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

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

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

        +And we run the optimization. +

        +
        +
        cvx_end
        +cvx_status
        +
        +
        + +

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

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

        4.5 Result

        +
        +

        +The obtained complementary filters are compared with the required upper bounds on Fig. 35. +

        + + +
        +

        LMI_obtained_comp_filters.png +

        +

        Figure 35: Obtained complementary filters using the LMI optimization (png, pdf)

        +
        +
        +
        + +
        +

        4.6 Comparison with the matlab Mixed Synthesis

        +
        +

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

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

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

        + + +
        +

        compare_cvx_h2hinf_comp_filters.png +

        +

        Figure 36: Comparison between the complementary filters obtained with the CVX toolbox and with the h2hinfsyn command (png, pdf)

        +
        +
        +
        + +
        +

        4.7 H-Infinity Objective

        +
        +

        +In terms of the \(\mathcal{H}_\infty\) objective, both synthesis method are satisfying the requirements as shown in Fig. 37. +

        + + +
        +

        comp_cvx_h2i_hinf_norm.png +

        +

        Figure 37: H-Infinity norm requirement and results (png, pdf)

        +
        +
        +
        + +
        +

        4.8 Obtained Super Sensor's noise

        +
        +

        +The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and h2hinfsyn command are compared in Fig. 38 and 39. +

        + + +
        +

        psd_compare_cvx_h2i.png +

        +

        Figure 38: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}2$/\(\mathcal{H}_\infty\) synthesis (png, pdf)

        +
        + + + +
        +

        cps_compare_cvx_h2i.png +

        +

        Figure 39: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}2$/\(\mathcal{H}_\infty\) synthesis (png, pdf)

        +
        +
        +
        + +
        +

        4.9 Obtained Super Sensor's Uncertainty

        +
        +

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

        + + +
        +

        super_sensor_uncertainty_compare_cvx_h2i.png +

        +

        Figure 40: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis (png, pdf)

        +
        +
        +
        +
        + +
        +

        5 H-Infinity synthesis to ensure both performance and robustness

        +
        +

        +

        @@ -1809,9 +2112,9 @@ The Matlab scripts is accessible here

        -
        -

        4.1 Introduction

        -
        +
        +

        5.1 Introduction

        +

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

        @@ -1877,9 +2180,9 @@ We may not want to follow the minimum uncertainty.
        -
        -

        4.2 Dynamical uncertainty and Noise level of the individual sensors

        -
        +
        +

        5.2 Dynamical uncertainty and Noise level of the individual sensors

        +

        Uncertainty on the individual sensors:

        @@ -1907,17 +2210,17 @@ N2 = ( +

        noise_uncertainty_sensors_hinf.png

        -

        Figure 35: Noise and Uncertainty characteristics of the sensors (png, pdf)

        +

        Figure 41: Noise and Uncertainty characteristics of the sensors (png, pdf)

        -
        -

        4.3 Weights for uncertainty and performance

        -
        +
        +

        5.3 Weights for uncertainty and performance

        +

        We design weights that are used to describe the wanted upper bound on the super sensor's noise and super sensor's uncertainty.

        @@ -1949,31 +2252,31 @@ Wp = inv(G236. +The noise and uncertainty weights of the individual sensors and the asked noise/uncertainty of the super sensor are displayed in Fig. 42.

        -
        +

        charac_sensors_weights.png

        -

        Figure 36: Upper bounds on the super sensor's noise and super sensor's uncertainty (png, pdf)

        +

        Figure 42: Upper bounds on the super sensor's noise and super sensor's uncertainty (png, pdf)

        -The corresponding maximum norms of the filters to have the perf/robust asked are shown in Fig. 37. +The corresponding maximum norms of the filters to have the perf/robust asked are shown in Fig. 43.

        -
        +

        upper_bound_complementary_filters_perf_robust.png

        -

        Figure 37: Upper bounds on the complementary filters (png, pdf)

        +

        Figure 43: Upper bounds on the complementary filters (png, pdf)

        -
        -

        4.4 H-infinity synthesis with 4 outputs corresponding to the 4 weights

        -
        +
        +

        5.4 H-infinity synthesis with 4 outputs corresponding to the 4 weights

        +

        We do the \(\mathcal{H}_\infty\) synthesis with 4 weights and 4 outputs.

        @@ -2047,44 +2350,44 @@ Test bounds: 1.4139 < gamma <= 65.6899

        -The obtained complementary filters with the upper bounds are shown in Fig. 38. +The obtained complementary filters with the upper bounds are shown in Fig. 44.

        -
        +

        hinf_result_comp_filters_4_outputs.png

        -

        Figure 38: caption (png, pdf)

        +

        Figure 44: caption (png, pdf)

        -
        +

        upper_bounds_perf_robust_result_4_outputs.png

        -

        Figure 39: Obtained PSD and uncertainty with the corresponding upper bounds (png, pdf)

        +

        Figure 45: Obtained PSD and uncertainty with the corresponding upper bounds (png, pdf)

        -
        +

        4outputs_hinf_psd_cps2svg.png

        -

        Figure 40: PSD and CPS (png, pdf)

        +

        Figure 46: PSD and CPS (png, pdf)

        -
        +

        4outputs_uncertainty.png

        -

        Figure 41: Dynamical uncertainty (png, pdf)

        +

        Figure 47: Dynamical uncertainty (png, pdf)

        -
        -

        4.5 Conclusion

        -
        +
        +

        5.5 Conclusion

        +

        The \(\mathcal{H}_\infty\) synthesis has been used to design complementary filters that permits to robustly merge sensors while ensuring a maximum noise level. However, no guarantee is made that the RMS value of the super sensor's noise is minimized. @@ -2093,30 +2396,30 @@ However, no guarantee is made that the RMS value of the super sensor's noise is

        -
        -

        5 Equivalent Super Sensor

        -
        +
        +

        6 Equivalent Super Sensor

        +

        - +

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

        -
        -

        5.1 Sensor Fusion Architecture

        -
        +
        +

        6.1 Sensor Fusion Architecture

        +

        -Let consider figure 42 where two sensors are merged. +Let consider figure 48 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 42: Sensor fusion architecture (png, pdf).

        +

        Figure 48: Sensor fusion architecture (png, pdf).

        @@ -2143,28 +2446,28 @@ And the noise of the super sensor is:
        -
        -

        5.2 Equivalent Configuration

        -
        +
        +

        6.2 Equivalent Configuration

        +

        -We try to determine \(w_{ss}(s)\) and \(N_{ss}(s)\) such that the sensor on figure 43 is equivalent to the super sensor of figure 42. +We try to determine \(w_{ss}(s)\) and \(N_{ss}(s)\) such that the sensor on figure 49 is equivalent to the super sensor of figure 48.

        -
        +

        sensor_fusion_equivalent.png

        -

        Figure 43: Equivalent Super Sensor (png, pdf).

        +

        Figure 49: Equivalent Super Sensor (png, pdf).

        -
        -

        5.3 Model the uncertainty of the super sensor

        -
        +
        +

        6.3 Model the uncertainty of the super sensor

        +

        -At each frequency \(\omega\), the uncertainty set of the super sensor shown on figure 42 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 43 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 48 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 49 is a circle centered on \(1\) with a radius equal to \(|w_{ss}(j\omega)|\) on the complex plane.

        @@ -2179,11 +2482,11 @@ Ideally, we want to find a weight \(w_{ss}(s)\) so that:

        -
        -

        5.4 Model the noise of the super sensor

        -
        +
        +

        6.4 Model the noise of the super sensor

        +

        -The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 42 is: +The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 48 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} \\ @@ -2191,7 +2494,7 @@ 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 43 is: +The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 49 is:

        \begin{align*} \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\ @@ -2210,9 +2513,9 @@ Ideally, we want to find a weight \(N_{ss}(s)\) such that:
        -
        -

        5.5 First guess

        -
        +
        +

        6.5 First guess

        +

        We could choose

        @@ -2239,11 +2542,11 @@ and
        -
        -

        6 Optimal And Robust Sensor Fusion in Practice

        -
        +
        +

        7 Optimal And Robust Sensor Fusion in Practice

        +

        - +

        Here are the steps in order to apply optimal and robust sensor fusion: @@ -2255,26 +2558,26 @@ Here are the steps in order to apply optimal and robust sensor fusion:

      • Apply H2/H-infinity synthesis of the complementary filters
      • -
        -

        6.1 Measurement of the noise characteristics of the sensors

        -
        +
        +

        7.1 Measurement of the noise characteristics of the sensors

        +
        -
        -

        6.1.1 Huddle Test

        -
        +
        +

        7.1.1 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 44. +Let's consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity \(x\) as shown in figure 50.

        -
        +

        huddle_test.png

        -

        Figure 44: Huddle test block diagram

        +

        Figure 50: Huddle test block diagram

        @@ -2290,7 +2593,7 @@ We also assume that their dynamics is ideal: \(G_1(s) = G_2(s) = 1\). We then have:

        \begin{equation} -\label{orgee171d5} +\label{org5197e00} \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} @@ -2298,16 +2601,16 @@ We then have: Since the input signal \(x\) and the instrumental noise \(n\) are incoherent:

        \begin{equation} -\label{orgd8add68} +\label{org7504a36} |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)| \end{equation}

        -From equations \eqref{orgee171d5} and \eqref{orgd8add68}, we finally obtain +From equations \eqref{org5197e00} and \eqref{org7504a36}, we finally obtain

        \begin{equation} -\label{orga080a7f} +\label{orgb46a810} |\Phi_n(\omega)| = |\Phi_{\hat{x}}(\omega)| \left( 1 - \sqrt{\gamma_{\hat{x}_1\hat{x}_2}^2(\omega)} \right) \end{equation} @@ -2315,9 +2618,9 @@ From equations \eqref{orgee171d5} and \eqref{orgd8add68}, we finally obtain
        -
        -

        6.1.2 Weights that represents the noises' PSD

        -
        +
        +

        7.1.2 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.

        @@ -2335,21 +2638,21 @@ 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 45. +The sensor with a normalized noise input is shown in figure 51.

        -
        +

        one_sensor_normalized_noise.png

        -

        Figure 45: One sensor with normalized noise

        +

        Figure 51: One sensor with normalized noise

        -
        -

        6.1.3 Comparison of the noises' PSD

        -
        +
        +

        7.1.3 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.

        @@ -2364,9 +2667,9 @@ Ideally, the PSD of the noise are such that:
        -
        -

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

        -
        +
        +

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

        +

        The coherence between signals \(x\) and \(y\) is defined as follow \[ \gamma^2_{xy}(\omega) = \frac{|\Phi_{xy}(\omega)|^2}{|\Phi_{x}(\omega)| |\Phi_{y}(\omega)|} \] @@ -2393,11 +2696,11 @@ where:

        -
        -

        6.2 Estimate the dynamic uncertainty of the sensors

        -
        +
        +

        7.2 Estimate the dynamic uncertainty of the sensors

        +

        -Let's consider one sensor represented on figure 46. +Let's consider one sensor represented on figure 52.

        @@ -2409,17 +2712,17 @@ The goal is to accurately determine \(w(s)\) for the sensors that have to be mer

        -
        +

        one_sensor_dyn_uncertainty.png

        -

        Figure 46: Sensor with dynamic uncertainty

        +

        Figure 52: Sensor with dynamic uncertainty

        -
        -

        6.3 Optimal and Robust synthesis of the complementary filters

        -
        +
        +

        7.3 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:

        @@ -2429,7 +2732,7 @@ 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 42 such that: +The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in figure 48 such that:

        • the uncertainty on the super sensor dynamics is minimized
        • @@ -2437,27 +2740,27 @@ The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in f
        -
        +

        sensor_fusion_full.png

        -

        Figure 47: Sensor fusion architecture with sensor dynamics uncertainty

        +

        Figure 53: Sensor fusion architecture with sensor dynamics uncertainty

        -
        -

        7 Methods of complementary filter synthesis

        -
        +
        +

        8 Methods of complementary filter synthesis

        +

        - +

        -
        -

        7.1 Complementary filters using analytical formula

        -
        +
        +

        8.1 Complementary filters using analytical formula

        +

        - +

        @@ -2467,9 +2770,9 @@ All the files (data and Matlab scripts) are accessible -

        7.1.1 Analytical 1st order complementary filters

        -
        +
        +

        8.1.1 Analytical 1st order complementary filters

        +

        First order complementary filters are defined with following equations:

        @@ -2479,7 +2782,7 @@ First order complementary filters are defined with following equations: \end{align}

        -Their bode plot is shown figure 48. +Their bode plot is shown figure 54.

        @@ -2491,17 +2794,17 @@ Hl1 = 1
        -
        +

        comp_filter_1st_order.png

        -

        Figure 48: Bode plot of first order complementary filter (png, pdf)

        +

        Figure 54: Bode plot of first order complementary filter (png, pdf)

        -
        -

        7.1.2 Second Order Complementary Filters

        -
        +
        +

        8.1.2 Second Order Complementary Filters

        +

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

        @@ -2526,15 +2829,15 @@ where:

        -This is illustrated on figure 49. +This is illustrated on figure 55. 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 49: Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (png, pdf)

        +

        Figure 55: Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (png, pdf)

        @@ -2549,17 +2852,17 @@ xlabel('$ +

        param_alpha_hinf_norm.png

        -

        Figure 50: Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (png, pdf)

        +

        Figure 56: Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (png, pdf)

        -
        -

        7.1.3 Third Order Complementary Filters

        -
        +
        +

        8.1.3 Third Order Complementary Filters

        +

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

        @@ -2577,7 +2880,7 @@ The parameters are:

        -The filters are defined below and the result is shown on figure 51. +The filters are defined below and the result is shown on figure 57.

        @@ -2591,20 +2894,20 @@ Hl3_ana = ( +

        complementary_filters_third_order.png

        -

        Figure 51: Third order complementary filters using the analytical formula (png, pdf)

        +

        Figure 57: Third order complementary filters using the analytical formula (png, pdf)

        -
        -

        7.2 H-Infinity synthesis of complementary filters

        -
        +
        +

        8.2 H-Infinity synthesis of complementary filters

        +

        - +

        @@ -2614,27 +2917,27 @@ All the files (data and Matlab scripts) are accessible -

        7.2.1 Synthesis Architecture

        -
        +
        +

        8.2.1 Synthesis Architecture

        +

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

        -In order to do so, we use the generalized plant shown on figure 52 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 58 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 52: Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters

        +

        Figure 58: 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 53) 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 59) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one: \[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \]

        @@ -2653,17 +2956,17 @@ 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 53: \(\mathcal{H}_\infty\) synthesis of the complementary filters

        +

        Figure 59: \(\mathcal{H}_\infty\) synthesis of the complementary filters

        -
        -

        7.2.2 Weights

        -
        +
        +

        8.2.2 Weights

        +
        omegab = 2*pi*9;
         wH = (omegab)^2/(s + omegab*sqrt(1e-5))^2;
        @@ -2673,17 +2976,17 @@ wL = (s 
        +

        weights_wl_wh.png

        -

        Figure 54: Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (png, pdf)

        +

        Figure 60: Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (png, pdf)

        -
        -

        7.2.3 H-Infinity Synthesis

        -
        +
        +

        8.2.3 H-Infinity Synthesis

        +

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

        @@ -2724,7 +3027,7 @@ 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 55. +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 61.

        Hh_hinf = 1 - Hl_hinf;
        @@ -2733,28 +3036,28 @@ We then define the high pass filter \(H_H = 1 - H_L\). The bode plot of both \(H
         
        -
        -

        7.2.4 Obtained Complementary Filters

        -
        +
        +

        8.2.4 Obtained Complementary Filters

        +

        -The obtained complementary filters are shown on figure 55. +The obtained complementary filters are shown on figure 61.

        -
        +

        hinf_filters_results.png

        -

        Figure 55: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (png, pdf)

        +

        Figure 61: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (png, pdf)

        -
        -

        7.3 Feedback Control Architecture to generate Complementary Filters

        -
        +
        +

        8.3 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. @@ -2771,14 +3074,14 @@ All the files (data and Matlab scripts) are accessible -

        7.3.1 Architecture

        -
        +
        +

        8.3.1 Architecture

        +
        -
        +

        complementary_filters_feedback_architecture.png

        -

        Figure 56: Architecture used to generate the complementary filters

        +

        Figure 62: Architecture used to generate the complementary filters

        @@ -2802,9 +3105,9 @@ Which contains two integrator and a lead. \(\omega_c\) is used to tune the cross

        -
        -

        7.3.2 Loop Gain Design

        -
        + -
        -

        7.3.3 Complementary Filters Obtained

        -
        +
        +

        8.3.3 Complementary Filters Obtained

        +

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

        @@ -2838,26 +3141,26 @@ Hh = 1/
        -
        +

        low_pass_high_pass_filters.png

        -

        Figure 58: Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (png, pdf)

        +

        Figure 64: Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (png, pdf)

        -
        -

        7.4 Analytical Formula found in the literature

        -
        +
        +

        8.4 Analytical Formula found in the literature

        +

        - +

        -
        -

        7.4.1 Analytical Formula

        -
        +
        +

        8.4.1 Analytical Formula

        +

        min15_compl_filter_desig_angle_estim

        @@ -2905,9 +3208,9 @@ Hh = 1/
        -
        -

        7.4.2 Matlab

        -
        +
        +

        8.4.2 Matlab

        +
        omega0 = 1*2*pi; % [rad/s]
         tau = 1/omega0; % [s]
        @@ -2924,17 +3227,17 @@ HL3 = (
        +

        comp_filters_literature.png

        -

        Figure 59: Comparison of some complementary filters found in the literature (png, pdf)

        +

        Figure 65: Comparison of some complementary filters found in the literature (png, pdf)

        -
        -

        7.4.3 Discussion

        -
        +
        +

        8.4.3 Discussion

        +

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

        @@ -2942,11 +3245,11 @@ Analytical Formula found in the literature provides either no parameter for tuni
        -
        -

        7.5 Comparison of the different methods of synthesis

        -
        +
        +

        8.5 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:

          @@ -2972,7 +3275,7 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti

        Author: Thomas Dehaeze

        -

        Created: 2019-09-03 mar. 09:06

        +

        Created: 2019-09-11 mer. 17:56

        Validate

        diff --git a/matlab/index.org b/matlab/index.org index 52bea39..22c1489 100644 --- a/matlab/index.org +++ b/matlab/index.org @@ -1502,8 +1502,7 @@ We see that the blue complementary filters with a lower maximum norm permits to :header-args:matlab+: :comments org :mkdirp yes :END: <> - -** ZIP file containing the data and matlab files :ignore: +** ZIP file containing the data and matlab files :ignore: #+begin_note The Matlab scripts is accessible [[file:matlab/mixed_synthesis_sensor_fusion.m][here]]. #+end_note @@ -1859,7 +1858,7 @@ The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_se [[file:figs/cps_super_sensor_mixed_syn.png]] ** Obtained Super Sensor's Uncertainty -The uncertainty on the super sensor's dynamics is shown in Fig. [[]]. +The uncertainty on the super sensor's dynamics is shown in Fig. [[fig:super_sensor_dyn_uncertainty_mixed_syn]]. #+begin_src matlab :exports none G1 = 1 + w1*ultidyn('Delta',[1 1]); @@ -1943,6 +1942,461 @@ This synthesis methods allows both to: - limit the dynamical uncertainty of the super sensor - minimize the RMS value of the estimation +* Mixed Synthesis - LMI Optimization +** Introduction +The following matlab scripts was written by Mohit. + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + freqs = logspace(-1, 3, 1000); +#+end_src + +** Noise characteristics and Uncertainty of the individual sensors +We define the weights that are used to characterize the dynamic uncertainty of the sensors. This will be used for the $\mathcal{H}_\infty$ part of the synthesis. +#+begin_src matlab + omegac = 100*2*pi; G0 = 0.1; Ginf = 10; + w1 = (Ginf*s/omegac + G0)/(s/omegac + 1); + + omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1; + w2 = (Ginf*s/omegac + G0)/(s/omegac + 1); + omegac = 5000*2*pi; G0 = 1; Ginf = 50; + w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1); +#+end_src + +We define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$. This will be used for the $\mathcal{H}_2$ part of the synthesis. +#+begin_src matlab + omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; + N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100); + + omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; + N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; +#+end_src + +** Weights +The weights for the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ part are defined below. + +#+begin_src matlab + n = 4; w0 = 2*pi*900; G0 = 9; G1 = 1; Gc = 1.1; + H = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; + wphi = 0.2*(s+3.142e04)/(s+628.3)/H; + + W1u = ss(w1*wphi); W2u = ss(w2*wphi); % Weight on the uncertainty + W1n = ss(N1); W2n = ss(N2); % Weight on the noise +#+end_src + +#+begin_src matlab + P = [W1u -W1u; + 0 W2u; + W1n -W1n; + 0 W2n; + 1 0]; +#+end_src + +** LMI Optimization +We are using the [[http://cvxr.com/cvx/][CVX toolbox]] to solve the optimization problem. + +We first put the generalized plant in a State-space form. +#+begin_src matlab + A = P.A; + Bw = P.B(:,1); + Bu = P.B(:,2); + Cz1 = P.C(1:2,:); Dz1w = P.D(1:2,1); Dz1u = P.D(1:2,2); % Hinf + Cz2 = P.C(3:4,:); Dz2w = P.D(1:2,1); Dz2u = P.D(1:2,2); % H2 + Cy = P.C(5,:); Dyw = P.D(5,1); Dyu = P.D(5,2); + + n = size(P.A,1); + ny = 1; % number of measurements + nu = 1; % number of control inputs + nz = 2; + nw = 1; + + Wtinf = 0; + Wt2 = 1; +#+end_src + +We Define all the variables. +#+begin_src matlab + cvx_startup; + + cvx_begin sdp + cvx_quiet true + cvx_solver sedumi + variable X(n,n) symmetric; + variable Y(n,n) symmetric; + variable W(nz,nz) symmetric; + variable Ah(n,n); + variable Bh(n,ny); + variable Ch(nu,n); + variable Dh(nu,ny); + variable eta; + variable gam; +#+end_src + +We define the minimization objective. +#+begin_src matlab + minimize Wt2*eta+Wtinf*gam % mix objective + subject to: +#+end_src + +The $\mathcal{H}_\infty$ constraint. +#+begin_src matlab + gam<=1; % Keep the Hinf norm less than 1 + + [ X, eye(n,n) ; + eye(n,n), Y ] >= 0 ; + + [ A*X + Bu*Ch + X*A' + Ch'*Bu', A+Bu*Dh*Cy+Ah', Bw+Bu*Dh*Dyw, X*Cz1' + Ch'*Dz1u' ; + (A+Bu*Dh*Cy+Ah')', Y*A + A'*Y + Bh*Cy + Cy'*Bh', Y*Bw + Bh*Dyw, (Cz1+Dz1u*Dh*Cy)' ; + (Bw+Bu*Dh*Dyw)', Bw'*Y + Dyw'*Bh', -eye(nw,nw), (Dz1w+Dz1u*Dh*Dyw)' ; + Cz1*X + Dz1u*Ch, Cz1+Dz1u*Dh*Cy, Dz1w+Dz1u*Dh*Dyw, -gam*eye(nz,nz)] <= 0 ; +#+end_src + +The $\mathcal{H}_2$ constraint. +#+begin_src matlab + trace(W) <= eta ; + + [ W, Cz2*X+Dz2u*Ch, Cz2*X+Dz2u*Ch; + X*Cz2'+Ch'*Dz2u', X, eye(n,n) ; + (Cz2*X+Dz2u*Ch)', eye(n,n), Y ] >= 0 ; + + [ A*X + Bu*Ch + X*A' + Ch'*Bu', A+Bu*Dh*Cy+Ah', Bw+Bu*Dh*Dyw ; + (A+Bu*Dh*Cy+Ah')', Y*A + A'*Y + Bh*Cy + Cy'*Bh', Y*Bw + Bh*Dyw ; + (Bw+Bu*Dh*Dyw)', Bw'*Y + Dyw'*Bh', -eye(nw,nw)] <= 0 ; +#+end_src + +And we run the optimization. +#+begin_src matlab + cvx_end + cvx_status +#+end_src + +#+begin_src matlab :exports none + if(strcmp(cvx_status,'Inaccurate/Solved')) + display('The solver was unable to make a determination to within the default numerical tolerance.'); + display('However, it determined that the results obtained satisfied a “relaxed” tolerance leve'); + display('and therefore may still be suitable for further use.'); + end +#+end_src + +Finally, we can compute the obtained complementary filters. +#+begin_src matlab + M = eye(n); + N = inv(M)*(eye(n,n)-Y*X); + Dk = Dh; + Ck = (Ch-Dk*Cy*X)*inv(M'); + Bk = inv(N)*(Bh-Y*Bu*Dk); + Ak = inv(N)*(Ah-Y*(A+Bu*Dk*Cy)*X-N*Bk*Cy*X-Y*Bu*Ck*M')*inv(M'); + + H2 = tf(ss(Ak,Bk,Ck,Dk)); + H1 = 1 - H2; +#+end_src + +** Result +The obtained complementary filters are compared with the required upper bounds on Fig. [[fig:LMI_obtained_comp_filters]]. + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1u, freqs, 'Hz'))), '--', 'DisplayName', '$W_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2u, freqs, 'Hz'))), '--', 'DisplayName', '$W_2$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); + + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([1e-3, 2]); + legend('location', 'southwest'); + + ax2 = subplot(2,1,2); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); + hold off; + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + set(gca, 'XScale', 'log'); + yticks([-360:90:360]); + ylim([-180, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/LMI_obtained_comp_filters.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:LMI_obtained_comp_filters +#+CAPTION: Obtained complementary filters using the LMI optimization ([[./figs/LMI_obtained_comp_filters.png][png]], [[./figs/LMI_obtained_comp_filters.pdf][pdf]]) +[[file:figs/LMI_obtained_comp_filters.png]] + +** Comparison with the matlab Mixed Synthesis +The Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis is performed below. +#+begin_src matlab + Nmeas = 1; Ncon = 1; Nz2 = 2; + + [H2m,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on'); + + H1m = 1 - H2m; +#+end_src + +The obtained filters are compare with the one obtained using the CVX toolbox in Fig. [[]]. + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{1,\mathcal{H}_2/\mathcal{H}_\infty}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{2,\mathcal{H}_2/\mathcal{H}_\infty}$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_{1, CVX}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_{2, CVX}$'); + + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([1e-3, 2]); + legend('location', 'southwest'); + + ax2 = subplot(2,1,2); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 180/pi*phase(squeeze(freqresp(H1m, freqs, 'Hz'))), '--'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*phase(squeeze(freqresp(H2m, freqs, 'Hz'))), '--'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); + hold off; + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + set(gca, 'XScale', 'log'); + yticks([-360:90:360]); + ylim([-180, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/compare_cvx_h2hinf_comp_filters.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:compare_cvx_h2hinf_comp_filters +#+CAPTION: Comparison between the complementary filters obtained with the CVX toolbox and with the =h2hinfsyn= command ([[./figs/compare_cvx_h2hinf_comp_filters.png][png]], [[./figs/compare_cvx_h2hinf_comp_filters.pdf][pdf]]) +[[file:figs/compare_cvx_h2hinf_comp_filters.png]] + +** H-Infinity Objective +In terms of the $\mathcal{H}_\infty$ objective, both synthesis method are satisfying the requirements as shown in Fig. [[fig:comp_cvx_h2i_hinf_norm]]. + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1u, freqs, 'Hz'))), '-.', 'DisplayName', '$1/W_{1u}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2u, freqs, 'Hz'))), '-.', 'DisplayName', '$1/W_{2u}$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{1,\mathcal{H}_2/\mathcal{H}_\infty}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2m, freqs, 'Hz'))), '--', 'DisplayName', '$H_{2,\mathcal{H}_2/\mathcal{H}_\infty}$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_{1, CVX}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_{2, CVX}$'); + + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([1e-3, 2]); + legend('location', 'southwest'); + xlim([freqs(1), freqs(end)]); + xticks([0.1, 1, 10, 100, 1000]); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comp_cvx_h2i_hinf_norm.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comp_cvx_h2i_hinf_norm +#+CAPTION: H-Infinity norm requirement and results ([[./figs/comp_cvx_h2i_hinf_norm.png][png]], [[./figs/comp_cvx_h2i_hinf_norm.pdf][pdf]]) +[[file:figs/comp_cvx_h2i_hinf_norm.png]] + +** Obtained Super Sensor's noise +The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h2hinfsyn= command are compared in Fig. [[fig:psd_compare_cvx_h2i]] and [[fig:cps_compare_cvx_h2i]]. + +#+begin_src matlab :exports none + PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2; + PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2; + PSD_cvx = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; + PSD_h2i = abs(squeeze(freqresp(N1*H1m, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2m, freqs, 'Hz'))).^2; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$'); + plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$'); + plot(freqs, PSD_cvx, 'k-', 'DisplayName', '$\Phi_{\hat{x}, CVX}$'); + plot(freqs, PSD_h2i, 'k--', 'DisplayName', '$\Phi_{\hat{x}, \mathcal{H}_2/\mathcal{H}_\infty}$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Power Spectral Density'); + hold off; + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/psd_compare_cvx_h2i.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:psd_compare_cvx_h2i +#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/psd_compare_cvx_h2i.png][png]], [[./figs/psd_compare_cvx_h2i.pdf][pdf]]) +[[file:figs/psd_compare_cvx_h2i.png]] + + +#+begin_src matlab :exports none + CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1); + CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2); + CPS_cvx = 1/pi*cumtrapz(2*pi*freqs, PSD_cvx); + CPS_h2i = 1/pi*cumtrapz(2*pi*freqs, PSD_h2i); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end)))); + plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end)))); + plot(freqs, CPS_cvx, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{CVX}} = %.1e$', sqrt(CPS_cvx(end)))); + plot(freqs, CPS_h2i, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2/\\mathcal{H}_\\infty}} = %.1e$', sqrt(CPS_h2i(end)))); + set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum'); + hold off; + xlim([2e-1, freqs(end)]); + ylim([1e-10 1e-5]); + legend('location', 'southeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cps_compare_cvx_h2i.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cps_compare_cvx_h2i +#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/cps_compare_cvx_h2i.png][png]], [[./figs/cps_compare_cvx_h2i.pdf][pdf]]) +[[file:figs/cps_compare_cvx_h2i.png]] + +** Obtained Super Sensor's Uncertainty +The uncertainty on the super sensor's dynamics is shown in Fig. [[]]. + +#+begin_src matlab :exports none + G1 = 1 + w1*ultidyn('Delta',[1 1]); + G2 = 1 + w2*ultidyn('Delta',[1 1]); + + % We here compute the maximum and minimum phase of the super sensor + Dphiss_cvx = 180/pi*asin(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz')))); + Dphiss_cvx(abs(squeeze(freqresp(w1*H1, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2, freqs, 'Hz'))) > 1) = 190; + + Dphiss_h2i = 180/pi*asin(abs(squeeze(freqresp(w1*H1m, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2m, freqs, 'Hz')))); + Dphiss_h2i(abs(squeeze(freqresp(w1*H1m, freqs, 'Hz')))+abs(squeeze(freqresp(w2*H2m, freqs, 'Hz'))) > 1) = 190; + + % We here compute the maximum and minimum phase of both sensors + Dphi1 = 180/pi*asin(abs(squeeze(freqresp(w1, freqs, 'Hz')))); + Dphi2 = 180/pi*asin(abs(squeeze(freqresp(w2, freqs, 'Hz')))); + Dphi1(abs(squeeze(freqresp(w1, freqs, 'Hz'))) > 1) = 190; + Dphi2(abs(squeeze(freqresp(w2, freqs, 'Hz'))) > 1) = 190; +#+end_src + +#+begin_src matlab :exports none + figure; + % Magnitude + ax1 = subplot(2,1,1); + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, 1 + abs(squeeze(freqresp(w1, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S1'); + set(gca,'ColorOrderIndex',1); + plot(freqs, max(1 - abs(squeeze(freqresp(w1, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',2); + plot(freqs, 1 + abs(squeeze(freqresp(w2, freqs, 'Hz'))), '--', 'DisplayName', 'Bounds - S2'); + set(gca,'ColorOrderIndex',2); + plot(freqs, max(1 - abs(squeeze(freqresp(w2, freqs, 'Hz'))), 0), '--', 'HandleVisibility', 'off'); + plot(freqs, 1 + abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 'k--', 'DisplayName', 'Bounds - CVX'); + plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1+w2*H2, freqs, 'Hz'))), 0), 'k--', 'HandleVisibility', 'off'); + plot(freqs, 1 + abs(squeeze(freqresp(w1*H1m+w2*H2m, freqs, 'Hz'))), 'k-', 'DisplayName', 'Bounds - $\mathcal{H}_2/\mathcal{H}_\infty$'); + plot(freqs, max(1 - abs(squeeze(freqresp(w1*H1m+w2*H2m, freqs, 'Hz'))), 0), 'k-', 'HandleVisibility', 'off'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + set(gca, 'XTickLabel',[]); + legend('location', 'southwest'); + ylabel('Magnitude'); + ylim([5e-2, 10]); + hold off; + + % Phase + ax2 = subplot(2,1,2); + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, Dphi1, '--'); + set(gca,'ColorOrderIndex',1); + plot(freqs, -Dphi1, '--'); + set(gca,'ColorOrderIndex',2); + plot(freqs, Dphi2, '--'); + set(gca,'ColorOrderIndex',2); + plot(freqs, -Dphi2, '--'); + plot(freqs, Dphiss_cvx, 'k--'); + plot(freqs, -Dphiss_cvx, 'k--'); + plot(freqs, Dphiss_h2i, 'k-'); + plot(freqs, -Dphiss_h2i, 'k-'); + set(gca,'xscale','log'); + yticks(-180:90:180); + ylim([-180 180]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + linkaxes([ax1,ax2],'x'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/super_sensor_uncertainty_compare_cvx_h2i.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:super_sensor_uncertainty_compare_cvx_h2i +#+CAPTION: Super Sensor Dynamical Uncertainty obtained with the mixed synthesis ([[./figs/super_sensor_uncertainty_compare_cvx_h2i.png][png]], [[./figs/super_sensor_uncertainty_compare_cvx_h2i.pdf][pdf]]) +[[file:figs/super_sensor_uncertainty_compare_cvx_h2i.png]] + * H-Infinity synthesis to ensure both performance and robustness :PROPERTIES: :header-args:matlab+: :tangle matlab/hinf_syn_perf_robust.m diff --git a/tikz/index.org b/tikz/index.org index 0e31f60..2847e2a 100644 --- a/tikz/index.org +++ b/tikz/index.org @@ -9,7 +9,7 @@ #+HTML_HEAD: #+HTML_HEAD: #+HTML_HEAD: -#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/MEGA/These/Papers/dehaeze19_desig_compl_filte/tikz/}{config.tex}") +#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/Cloud/These/Papers/dehaeze19_desig_compl_filte/tikz/}{config.tex}") #+PROPERTY: header-args:latex+ :imagemagick t :fit yes #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100
        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