From 28d394df347e5720ec5836749a5899a845a8b012 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 29 Aug 2019 14:55:03 +0200 Subject: [PATCH] Reworked all the analysis about optimal sensor fusion --- matlab/figs/comparison_cps_noise.pdf | Bin 0 -> 89491 bytes matlab/figs/comparison_cps_noise.png | Bin 0 -> 144328 bytes matlab/figs/comparison_psd_noise.pdf | Bin 0 -> 79895 bytes matlab/figs/comparison_psd_noise.png | Bin 0 -> 183412 bytes matlab/figs/cps_h2_synthesis.pdf | Bin 0 -> 66643 bytes matlab/figs/cps_h2_synthesis.png | Bin 0 -> 128433 bytes matlab/figs/htwo_comp_filters.pdf | Bin 40226 -> 38410 bytes matlab/figs/htwo_comp_filters.png | Bin 105064 -> 107179 bytes matlab/figs/noise_characteristics_sensors.pdf | Bin 0 -> 42889 bytes matlab/figs/noise_characteristics_sensors.png | Bin 0 -> 124504 bytes matlab/figs/nosie_characteristics_sensors.pdf | Bin 43343 -> 42889 bytes matlab/figs/nosie_characteristics_sensors.png | Bin 117543 -> 124504 bytes matlab/figs/psd_sensors_htwo_synthesis.pdf | Bin 58759 -> 60072 bytes matlab/figs/psd_sensors_htwo_synthesis.png | Bin 164228 -> 169450 bytes matlab/figs/weights_comp_filters_Hinfa.pdf | Bin 0 -> 73530 bytes matlab/figs/weights_comp_filters_Hinfa.png | Bin 0 -> 139956 bytes matlab/figs/weights_comp_filters_Hinfb.pdf | Bin 0 -> 71242 bytes matlab/figs/weights_comp_filters_Hinfb.png | Bin 0 -> 131588 bytes matlab/index.html | 914 +++++++++--------- matlab/index.org | 495 +++++----- tikz/figs/fusion_two_noisy_sensors.pdf | Bin 11197 -> 69315 bytes tikz/figs/fusion_two_noisy_sensors.png | Bin 13140 -> 28519 bytes tikz/figs/fusion_two_noisy_sensors.svg | 285 ++++-- .../figs/fusion_two_noisy_sensors_weights.pdf | Bin 0 -> 69905 bytes .../figs/fusion_two_noisy_sensors_weights.png | Bin 0 -> 36952 bytes .../figs/fusion_two_noisy_sensors_weights.svg | 330 +++++++ tikz/figs/h_infinity_optimal_comp_filters.pdf | Bin 12149 -> 66595 bytes tikz/figs/h_infinity_optimal_comp_filters.png | Bin 13821 -> 13974 bytes tikz/figs/h_infinity_optimal_comp_filters.svg | 105 +- tikz/figs/sensor_fusion_noisy_perfect_dyn.pdf | Bin 0 -> 69614 bytes tikz/figs/sensor_fusion_noisy_perfect_dyn.png | Bin 0 -> 34609 bytes tikz/figs/sensor_fusion_noisy_perfect_dyn.svg | 303 ++++++ tikz/index.org | 153 ++- 33 files changed, 1752 insertions(+), 833 deletions(-) create mode 100644 matlab/figs/comparison_cps_noise.pdf create mode 100644 matlab/figs/comparison_cps_noise.png create mode 100644 matlab/figs/comparison_psd_noise.pdf create mode 100644 matlab/figs/comparison_psd_noise.png create mode 100644 matlab/figs/cps_h2_synthesis.pdf create mode 100644 matlab/figs/cps_h2_synthesis.png create mode 100644 matlab/figs/noise_characteristics_sensors.pdf create mode 100644 matlab/figs/noise_characteristics_sensors.png create mode 100644 matlab/figs/weights_comp_filters_Hinfa.pdf create mode 100644 matlab/figs/weights_comp_filters_Hinfa.png create mode 100644 matlab/figs/weights_comp_filters_Hinfb.pdf create mode 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    Table of Contents

      -
    • 1. Optimal Sensor Fusion for noise characteristics +
    • 1. Optimal Sensor Fusion for noise characteristics
        -
      • 1.1. Architecture
      • -
      • 1.2. Noise of the sensors
      • -
      • 1.3. H-Two Synthesis
      • -
      • 1.4. Analysis
      • -
      • 1.5. H-Infinity Synthesis -
      • +
      • 2. Robustness to sensor dynamics uncertainty +
      • -
      • 2. Robustness to sensor dynamics uncertainty +
      • 3. Equivalent Super Sensor
      • -
      • 3. Equivalent Super Sensor +
      • 4. Optimal And Robust Sensor Fusion in Practice
          -
        • 3.1. Sensor Fusion Architecture
        • -
        • 3.2. Equivalent Configuration
        • -
        • 3.3. Model the uncertainty of the super sensor
        • -
        • 3.4. Model the noise of the super sensor
        • -
        • 3.5. First guess
        • +
        • 4.1. Measurement of the noise characteristics of the sensors +
        • -
        • 4. Optimal And Robust Sensor Fusion in Practice -
            -
          • 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
          • +
          • 5. Methods of complementary filter synthesis + @@ -378,31 +374,31 @@ 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 for noise characteristics

            - +

            The idea is to combine sensors that works in different frequency range using complementary filters. @@ -423,102 +419,122 @@ 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 1 and 2 are measuring the same quantity \(x\) with different noise characteristics determined by \(W_1\) and \(W_2\). +Let's consider the sensor fusion architecture shown on figure 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 \(N_2(s)\).

            -\(n_1\) and \(n_2\) are white noise (constant power spectral density over all frequencies). +\(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise:

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

            fusion_two_noisy_sensors_with_dyn.png +

            +

            fusion_two_noisy_sensors_weights.png

            Figure 1: Fusion of two sensors

            -We consider that the two sensor dynamics \(G_1\) and \(G_2\) are ideal (\(G_1 = G_2 = 1\)). We obtain the architecture of figure 2. +We consider that the two sensor dynamics \(G_1(s)\) and \(G_2(s)\) are ideal: +

            +\begin{equation} +\label{org10f3718} + G_1(s) = G_2(s) = 1 +\end{equation} + +

            +We obtain the architecture of figure 2.

            -
            -

            fusion_two_noisy_sensors.png +

            +

            sensor_fusion_noisy_perfect_dyn.png

            Figure 2: Fusion of two sensors with ideal dynamics

            -\(H_1\) and \(H_2\) are complementary filters (\(H_1 + H_2 = 1\)). The goal is to design \(H_1\) and \(H_2\) such that the effect of the noise sources \(n_1\) and \(n_2\) has the smallest possible effect on the estimation \(\hat{x}\). +\(H_1(s)\) and \(H_2(s)\) are complementary filters: +

            +\begin{equation} +\label{orgf0f9605} + H_1(s) + H_2(s) = 1 +\end{equation} + +

            +The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the estimation \(\hat{x}\).

            We have that the Power Spectral Density (PSD) of \(\hat{x}\) is: -\[ \Gamma_{\hat{x}} = |H_1 W_1|^2 \Gamma_{n_1} + |H_2 W_2|^2 \Gamma_{n_2} \] +\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]

            And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\): -\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Gamma_{\hat{x}}(\omega) d\omega} \] -

            - -

            -As \(n_1\) and \(n_2\) are white noise: \(\Gamma_{n_1} = \Gamma_{n_2} = 1\) and we have: -\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 W_1|^2(\omega) + |H_2 W_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2 \] -

            - -

            -Thus, the goal is to design \(H_1\) and \(H_2\) such that \(H_1 + H_2 = 1\) and such that \(\left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2\) is minimized. -

            - -

            -For that, we will use the \(\mathcal{H}_2\) Synthesis.

            +\begin{equation} +\label{orge1ffe08} + \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 \(W_1\) and \(W_2\): +Let's define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\):

              -
            • Sensor 1 characterized by \(W_1\) has low noise at low frequency (for instance a geophone)
            • -
            • Sensor 2 characterized by \(W_2\) has low noise at high frequency (for instance an accelerometer)
            • +
            • Sensor 1 characterized by \(N_1(s)\) has low noise at low frequency (for instance a geophone)
            • +
            • Sensor 2 characterized by \(N_2(s)\) has low noise at high frequency (for instance an accelerometer)
            omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
            -W1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
            +N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
             
             omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
            -W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
            +N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
             
            -
            -

            nosie_characteristics_sensors.png +

            +

            noise_characteristics_sensors.png

            -

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

            +

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

            -
            -

            1.3 H-Two Synthesis

            +
            +

            1.3 H-Two Synthesis

            -We use the generalized plant architecture shown on figure 4. +As \(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise: \(\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1\) and we have: +\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \] +Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized. +

            + +

            +For that, we use the \(\mathcal{H}_2\) Synthesis. +

            + +

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

            -
            +

            h_infinity_optimal_comp_filters.png

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

            @@ -526,9 +542,9 @@ We use the generalized plant architecture shown on figure

            The transfer function from \([n_1, n_2]\) to \(\hat{x}\) is: -\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \] +\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \] If we define \(H_2 = 1 - H_1\), we obtain: -\[ \begin{bmatrix} W_1 H_1 \\ W_2 H_2 \end{bmatrix} \] +\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]

            @@ -536,11 +552,11 @@ 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   W2  1;
            -     W1 -W2  0];
            +
            P = [0   N2  1;
            +     N1 -N2  0];
             
            @@ -553,83 +569,69 @@ And we do the \(\mathcal{H}_2\) synthesis using the h2syn command.

            -What is minimized is norm([W1*H1,W2*H2], 2). -

            - -

            -Finally, we define \(H_2 = 1 - H_1\). +Finally, we define \(H_2(s) = 1 - H_1(s)\).

            H2 = 1 - H1;
             
            -
            -
            -
            -

            1.4 Analysis

            -

            -The complementary filters obtained are shown on figure 5. The PSD of the 6. -Finally, the RMS value of \(\hat{x}\) is shown on table 1. -The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor. +The complementary filters obtained are shown on figure 5. +

            + +

            +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 obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.

            -
            +

            htwo_comp_filters.png

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

            +
            +
            PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
            +PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
            +PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
            +
            +
            -
            + +

            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)

            - - +
            +
            CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
            +CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
            +CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
            +
            +
            ---- - - - - - - - - - - - - - - - - - - - - - - -
            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
             rms value
            Sensor 11.1e-02
            Sensor 21.3e-03
            Optimal Sensor Fusion1.5e-04
            +
            +

            cps_h2_synthesis.png +

            +

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

            +
            -
            -

            1.5 H-Infinity Synthesis

            -
            -
            -
            -

            1.5.1 First idea

            -
            +
            +

            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 \(n_2\): \[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \] @@ -639,41 +641,40 @@ Another objective that we may have is that the noise of the super sensor \(n_{SS In order to obtain that ideal case, we need that the complementary filters be designed such that:

            \begin{align*} - & H_1(j\omega) = 1 \text{ and } H_2(j\omega) = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ - & H_1(j\omega) = 0 \text{ and } H_2(j\omega) = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) + & |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ + & |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) \end{align*}

            -Which is indeed impossible in practice. +Which is indeed impossible in practice.

            -We could try to use filters with the highest possible order to approximate that. -However, we may have robustness and implementation problems. +We could try to approach that with the \(\mathcal{H}_\infty\) synthesis by using high order filters. +

            + +

            +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.

            -
            -
            omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
            -N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
            -
            -omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
            -N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
            -
            -
            -
            n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
            -W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
            +W1a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
             
             n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2;
            -W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
            +W2a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
             
            -
            P = [W1 -W1;
            -     0   W2;
            -     1   0];
            +
            P = [W1a -W1a;
            +     0    W2a;
            +     1    0];
             
            @@ -681,12 +682,12 @@ W2 = ( -
            [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
            +
            [H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
             
            -[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
            +[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
             Resetting value of Gamma min based on D_11, D_12, D_21 terms
             
             Test bounds:      0.1000 <  gamma  <=  10500.0000
            @@ -722,115 +723,92 @@ Test bounds:      0.1000 <  gamma  <=  10500.0000
             
            -
            H1 = 1 - H2;
            +
            H1a = 1 - H2a;
             
            - +
            +

            weights_comp_filters_Hinfa.png +

            +

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

            +
            --+

            +We then compute the Power Spectral Density as well as the Cumulative Power Spectrum. +

            -- - - - - - - - - - - - - - - - - - - - - - - -
             rms value
            Sensor 11.1e-02
            Sensor 21.3e-03
            Optimal Sensor Fusion3.2e-04
            +
            +
            PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2;
            +CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha);
            +
            +
            -
            -

            1.5.2 Second idea

            -
            +
            +

            1.5 H-Infinity Synthesis - method B

            +

            We have that: -\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \] +\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \]

            -We model the Power Spectral Densities of the two sensors by weighting functions \(N_1(s)\) and \(N_2(s)\). -\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \] +Then, at frequencies where \(|H_1(j\omega)| < |H_2(j\omega)|\) we would like that \(|N_1(j\omega)| = 1\) and \(|N_2(j\omega)| = 0\) as we discussed before. +Then \(|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2\).

            -Then, at frequencies where \(|H_1| < |H_2|\) we would like that \(|N_1| = 1\) and \(|N_2| = 0\) as we discussed before. -Then \(|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2\). -

            - -

            -However, if we design \(H_2\) such that when \(|N_1| < |N_2|\), we have that: +We know that this is impossible in practice. A more realistic choice is to design \(H_2(s)\) such that when \(|N_2(j\omega)| > |N_1(j\omega)|\), we have that: \[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \] +

            + +

            +Which is equivalent to have (by supposing \(|H_1| \approx 1\)): +\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] +

            + +

            And we have:

            \begin{align*} - \Phi_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ - &= (1 + \epsilon)\left|H_1 N_1\right|^2 \\ - &\approx \left|H_1 N_1\right|^2 + \Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ + &= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\ + &\approx \left|N_1\right|^2 \end{align*}

            -We have then that: -\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \] -We suppose that \(|H_1| = 1\): -\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] -We can use that for the maximum magnitude of \(|H_2|\) at frequencies where \(|N_1| < |N_2|\) -

            - -

            -Similarly, we have that the maximum magnitude of \(|H_1|\) at frequencies where \(|N_1| > |N_2|\) is: +Similarly, we design \(H_1(s)\) such that at frequencies where \(|N_1| > |N_2|\): \[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \]

            -We could take \(\epsilon = 1\), then we increase the noise of the super sensor just by a factor \(\sqrt{2}\) over the all bandwidth from the idea case. +For instance, is we take \(\epsilon = 1\), then the PSD of \(\hat{x}\) is increased by just by a factor \(\sqrt{2}\) over the all frequencies from the idea case.

            -This could be used as weighting functions for the \(\mathcal{H}_\infty\) synthesis of the complementary filters. +We use this as the weighting functions for the \(\mathcal{H}_\infty\) synthesis of the complementary filters. +

            + +

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

            -
            omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
            -N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
            +
            epsilon = 2;
             
            -omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
            -N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
            +W1b = 1/epsilon*N1/N2;
            +W2b = 1/epsilon*N2/N1;
            +
            +W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper
             
            -
            W1 = N1/N2;
            -W2 = N2/N1;
            -
            -W1 = W1/(1 + s/2/pi/1000); % this is added so that it is proper
            -
            -
            - -
            -
            P = [W1 -W1;
            -     0   W2;
            -     1   0];
            +
            P = [W1b -W1b;
            +     0    W2b;
            +     1    0];
             
            @@ -838,49 +816,77 @@ W1 = W1/ -
            [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
            +
            [H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
             
            -[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
            -Test bounds:      0.0000 <  gamma  <=   2625.0000
            +[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
            +Test bounds:      0.0000 <  gamma  <=     32.8125
             
               gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
            -2.625e+03   1.8e+01 -2.3e-12   6.3e+00   -1.7e-14    0.0000    p
            -1.313e+03   1.8e+01 -3.3e-12   6.3e+00   -5.9e-13    0.0000    p
            -  656.250   1.8e+01 -1.8e-12   6.3e+00   -1.2e-13    0.0000    p
            -  328.125   1.8e+01 -8.3e-13   6.3e+00   -7.5e-14    0.0000    p
            -  164.063   1.8e+01 -3.8e-13   6.3e+00   -6.1e-14    0.0000    p
            -   82.031   1.8e+01 -3.3e-22   6.3e+00   -5.4e-14    0.0000    p
            -   41.016   1.8e+01 -3.4e-12   6.3e+00   -1.4e-16    0.0000    p
            -   20.508   1.8e+01 -6.4e-14   6.3e+00   -4.4e-13    0.0000    p
            -   10.254   1.8e+01 -1.4e-13   6.3e+00   -4.9e-14    0.0000    p
            -    5.127   1.7e+01 -1.1e-12   6.3e+00   -1.6e-13    0.0000    p
            -    2.563   1.7e+01 -6.5e-13   6.3e+00   -7.3e-14    0.0000    p
            -    1.282   1.4e+01 -8.9e+04#  6.3e+00   -1.6e-14    0.0000    f
            -    1.923   1.6e+01 -4.6e+05#  6.3e+00   -2.1e-13    0.0000    f
            -    2.243   1.7e+01 -1.9e+06#  6.3e+00   -9.9e-14    0.0000    f
            -    2.403   1.7e+01 -3.7e-11   6.3e+00   -2.1e-12    0.0000    p
            -    2.323   1.7e+01 -4.6e+06#  6.3e+00   -5.8e-14    0.0000    f
            -    2.363   1.7e+01 -1.2e+07#  6.3e+00   -3.3e-14    0.0000    f
            -    2.383   1.7e+01 -6.8e+07#  6.3e+00   -1.2e-13    0.0000    f
            -    2.393   1.7e+01 -5.5e-11   6.3e+00   -1.8e-12    0.0000    p
            -    2.388   1.7e+01   1.3e-22   6.3e+00   -2.5e-15    0.0000    p
            -    2.386   1.7e+01 -1.5e+08#  6.3e+00   -1.4e-13    0.0000    f
            -    2.387   1.7e+01 -4.0e+08#  6.3e+00   -1.0e-13    0.0000    f
            -    2.388   1.7e+01 -2.1e+09#  6.3e+00   -4.2e-13    0.0000    f
            +   32.812   1.8e+01   3.4e-10   6.3e+00   -2.9e-13    0.0000    p
            +   16.406   1.8e+01   3.4e-10   6.3e+00   -1.2e-15    0.0000    p
            +    8.203   1.8e+01   3.3e-10   6.3e+00   -2.6e-13    0.0000    p
            +    4.102   1.8e+01   3.3e-10   6.3e+00   -2.1e-13    0.0000    p
            +    2.051   1.7e+01   3.4e-10   6.3e+00   -7.2e-16    0.0000    p
            +    1.025   1.6e+01 -1.3e+06#  6.3e+00   -8.3e-14    0.0000    f
            +    1.538   1.7e+01   3.4e-10   6.3e+00   -2.0e-13    0.0000    p
            +    1.282   1.7e+01   3.4e-10   6.3e+00   -7.9e-17    0.0000    p
            +    1.154   1.7e+01   3.6e-10   6.3e+00   -1.8e-13    0.0000    p
            +    1.089   1.7e+01 -3.4e+06#  6.3e+00   -1.7e-13    0.0000    f
            +    1.122   1.7e+01 -1.0e+07#  6.3e+00   -3.2e-13    0.0000    f
            +    1.138   1.7e+01 -1.3e+08#  6.3e+00   -1.8e-13    0.0000    f
            +    1.146   1.7e+01   3.2e-10   6.3e+00   -3.0e-13    0.0000    p
            +    1.142   1.7e+01   5.5e-10   6.3e+00   -2.8e-13    0.0000    p
            +    1.140   1.7e+01 -1.5e-10   6.3e+00   -2.3e-13    0.0000    p
            +    1.139   1.7e+01 -4.8e+08#  6.3e+00   -6.2e-14    0.0000    f
            +    1.139   1.7e+01   1.3e-09   6.3e+00   -8.9e-17    0.0000    p
             
            - Gamma value achieved:     2.3882
            + Gamma value achieved:     1.1390
             
            -
            H1 = 1 - H2;
            +
            H1b = 1 - H2b;
             
            - +
            +

            weights_comp_filters_Hinfb.png +

            +

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

            +
            + +
            +
            PSD_Hb = abs(squeeze(freqresp(N1*H1b, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2b, freqs, 'Hz'))).^2;
            +CPS_Hb = 1/pi*cumtrapz(2*pi*freqs, PSD_Hb);
            +
            +
            + + + +
            +

            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 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. +

            + +
            +@@ -896,7 +902,7 @@ Test bounds: 0.0000 < gamma <= 2625.0000 - + @@ -905,40 +911,64 @@ Test bounds: 0.0000 < gamma <= 2625.0000 - - + + + + + + + + + + + +
            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
            Sensor 11.1e-021.3e-03
            Optimal Sensor Fusion1.9e-04H2 Fusion1.2e-04
            H-Infinity a2.4e-04
            H-Infinity b1.4e-04
            + + + +
            +

            comparison_psd_noise.png +

            +

            Figure 10: Comparison of the obtained Power Spectral Density using the three methods (png, pdf)

            +
            + + +
            +

            comparison_cps_noise.png +

            +

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

            +
            -
            -

            1.5.3 Conclusion

            -
            +
            +

            1.7 Conclusion

            +

            -From the above two experiments with \(\mathcal{H}_\infty\) it still seems that the \(\mathcal{H}_2\) synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the \(\mathcal{H}_2\) norm) +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).

            -
            -
            -

            2 Robustness to sensor dynamics uncertainty

            +
            +

            2 Robustness to sensor dynamics uncertainty

            - +

            -Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure 7). +Let's first consider ideal sensors where \(G_1 = 1\) and \(G_2 = 1\) (figure 12).

            -
            +

            fusion_two_noisy_sensors_with_dyn_bis.png

            -

            Figure 7: Fusion of two sensors

            +

            Figure 12: Fusion of two sensors

            @@ -965,19 +995,19 @@ 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 8. +Let's represent that with multiplicative input uncertainty as shown on figure 13.

            -
            +

            fusion_gain_mismatch.png

            -

            Figure 8: Fusion of two sensors with input multiplicative uncertainty

            +

            Figure 13: Fusion of two sensors with input multiplicative uncertainty

            @@ -1019,7 +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 9). +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).

            @@ -1028,10 +1058,10 @@ We then have that the angle introduced by the super sensor is bounded by \(\arcs

            -
            +

            uncertainty_gain_phase_variation.png

            -

            Figure 9: Maximum phase variation

            +

            Figure 14: Maximum phase variation

            @@ -1040,8 +1070,8 @@ 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 \(W_2\). @@ -1074,8 +1104,8 @@ 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 \(0.6\). @@ -1087,19 +1117,19 @@ G2 = 0.10. +Let's design two complementary filters as 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 10: The two complementary filters designed for the robustness test (png, pdf)

            +

            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 11). +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).

            @@ -1107,40 +1137,40 @@ We see that the blue complementary filters with a lower maximum norm permits to

            -
            +

            tf_super_sensor_comp.png

            -

            Figure 11: Comparison of the obtained super sensor transfer functions (png, pdf)

            +

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

            -
            -

            2.4 TODO More Complete example with model uncertainty

            +
            +

            2.4 TODO More Complete example with model uncertainty

            -
            -

            3 Equivalent Super Sensor

            +
            +

            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 12 where two sensors are merged. +Let consider figure 17 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 12: Sensor fusion architecture (png, pdf).

            +

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

            @@ -1167,28 +1197,28 @@ 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 13 is equivalent to the super sensor of figure 12. +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.

            -
            +

            sensor_fusion_equivalent.png

            -

            Figure 13: Equivalent Super Sensor (png, pdf).

            +

            Figure 18: 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 12 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 13 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 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.

            @@ -1203,11 +1233,11 @@ 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 12 is: +The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 17 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} \\ @@ -1215,7 +1245,7 @@ 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 13 is: +The PSD of the estimation \(\hat{x}\) when \(x = 0\) of the configuration shown on figure 18 is:

            \begin{align*} \Phi_{\hat{x}}(\omega) &= | N_{ss}(j\omega) |^2 \Phi_{\tilde{n}} \\ @@ -1234,8 +1264,8 @@ Ideally, we want to find a weight \(N_{ss}(s)\) such that:
            -
            -

            3.5 First guess

            +
            +

            3.5 First guess

            We could choose @@ -1263,8 +1293,8 @@ 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: @@ -1277,26 +1307,26 @@ 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 14. +Let's consider two sensors (sensor 1 and sensor 2) that are measuring the same quantity \(x\) as shown in figure 19.

    -
    +

    huddle_test.png

    -

    Figure 14: Huddle test block diagram

    +

    Figure 19: Huddle test block diagram

    @@ -1312,7 +1342,7 @@ We also assume that their dynamics is ideal: \(G_1(s) = G_2(s) = 1\). We then have:

    \begin{equation} -\label{orgdaace50} +\label{org456a8db} \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} @@ -1320,16 +1350,16 @@ We then have: Since the input signal \(x\) and the instrumental noise \(n\) are incoherent:

    \begin{equation} -\label{org4188c50} +\label{org0b69a39} |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)| \end{equation}

    -From equations \eqref{orgdaace50} and \eqref{org4188c50}, we finally obtain +From equations \eqref{org456a8db} and \eqref{org0b69a39}, we finally obtain

    \begin{equation} -\label{orgf2631e0} +\label{org47f030a} |\Phi_n(\omega)| = |\Phi_{\hat{x}}(\omega)| \left( 1 - \sqrt{\gamma_{\hat{x}_1\hat{x}_2}^2(\omega)} \right) \end{equation} @@ -1337,8 +1367,8 @@ From equations \eqref{orgdaace50} and \eqref{org4188c50}, 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. @@ -1357,20 +1387,20 @@ 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 15. +The sensor with a normalized noise input is shown in figure 20.

    -
    +

    one_sensor_normalized_noise.png

    -

    Figure 15: One sensor with normalized noise

    +

    Figure 20: 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. @@ -1386,8 +1416,8 @@ 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 @@ -1415,11 +1445,11 @@ 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 16. +Let's consider one sensor represented on figure 21.

    @@ -1431,16 +1461,16 @@ The goal is to accurately determine \(w(s)\) for the sensors that have to be mer

    -
    +

    one_sensor_dyn_uncertainty.png

    -

    Figure 16: Sensor with dynamic uncertainty

    +

    Figure 21: 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: @@ -1451,7 +1481,7 @@ 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 12 such that: +The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in figure 17 such that:

    • the uncertainty on the super sensor dynamics is minimized
    • @@ -1459,24 +1489,24 @@ The goal is to design complementary filters \(H_1(s)\) and \(H_2(s)\) shown in f
    -
    +

    sensor_fusion_full.png

    -

    Figure 17: Sensor fusion architecture with sensor dynamics uncertainty

    +

    Figure 22: 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

    - +

    @@ -1486,8 +1516,8 @@ 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: @@ -1498,7 +1528,7 @@ First order complementary filters are defined with following equations: \end{align}

    -Their bode plot is shown figure 18. +Their bode plot is shown figure 23.

    @@ -1510,16 +1540,16 @@ Hl1 = 1
    -
    +

    comp_filter_1st_order.png

    -

    Figure 18: Bode plot of first order complementary filter (png, pdf)

    +

    Figure 23: 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 \(H_H\). @@ -1545,15 +1575,15 @@ where:

    -This is illustrated on figure 19. +This is illustrated on figure 24. 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 19: Effect of the parameter \(\alpha\) on the shape of the generated second order complementary filters (png, pdf)

    +

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

    @@ -1568,16 +1598,16 @@ xlabel('$ +

    param_alpha_hinf_norm.png

    -

    Figure 20: Evolution of the H-Infinity norm of the complementary filters with the parameter \(\alpha\) (png, pdf)

    +

    Figure 25: 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 \(3\): @@ -1596,7 +1626,7 @@ The parameters are:

    -The filters are defined below and the result is shown on figure 21. +The filters are defined below and the result is shown on figure 26.

    @@ -1610,20 +1640,20 @@ Hl3_ana = ( +

    complementary_filters_third_order.png

    -

    Figure 21: Third order complementary filters using the analytical formula (png, pdf)

    +

    Figure 26: 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

    - +

    @@ -1633,8 +1663,8 @@ 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. @@ -1642,18 +1672,18 @@ 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 22 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 27 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 22: Generalized plant used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters

    +

    Figure 27: 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 23) 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 28) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_H,\ z_L]\) is less than one: \[ \left\| \begin{array}{c} H_L w_L \\ (1 - H_L) w_H \end{array} \right\|_\infty < 1 \]

    @@ -1672,16 +1702,16 @@ 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 23: \(\mathcal{H}_\infty\) synthesis of the complementary filters

    +

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

    -
    -

    5.2.2 Weights

    +
    +

    5.2.2 Weights

    omegab = 2*pi*9;
    @@ -1692,16 +1722,16 @@ wL = (s 
    +

    weights_wl_wh.png

    -

    Figure 24: Weights on the complementary filters \(w_L\) and \(w_H\) and the associated performance weights (png, pdf)

    +

    Figure 29: 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. @@ -1743,7 +1773,7 @@ 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 25. +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.

    Hh_hinf = 1 - Hl_hinf;
    @@ -1752,28 +1782,28 @@ 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 25. +The obtained complementary filters are shown on figure 30.

    -
    +

    hinf_filters_results.png

    -

    Figure 25: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis (png, pdf)

    +

    Figure 30: 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. @@ -1790,14 +1820,14 @@ All the files (data and Matlab scripts) are accessible -

    5.3.1 Architecture

    +
    +

    5.3.1 Architecture

    -
    +

    complementary_filters_feedback_architecture.png

    -

    Figure 26: Architecture used to generate the complementary filters

    +

    Figure 31: Architecture used to generate the complementary filters

    @@ -1821,8 +1851,8 @@ 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. @@ -1857,25 +1887,25 @@ Hh = 1/

    -
    +

    low_pass_high_pass_filters.png

    -

    Figure 28: Low pass and High pass filters \(H_L\) and \(H_H\) for different values of \(\alpha\) (png, pdf)

    +

    Figure 33: 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 @@ -1924,8 +1954,8 @@ Hh = 1/

    -
    -

    5.4.2 Matlab

    +
    +

    5.4.2 Matlab

    omega0 = 1*2*pi; % [rad/s]
    @@ -1943,16 +1973,16 @@ HL3 = (
    +

    comp_filters_literature.png

    -

    Figure 29: Comparison of some complementary filters found in the literature (png, pdf)

    +

    Figure 34: 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. @@ -1961,11 +1991,11 @@ 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:

      @@ -1991,7 +2021,7 @@ The generated complementary filters using \(\mathcal{H}_\infty\) and the analyti

    Author: Thomas Dehaeze

    -

    Created: 2019-08-29 jeu. 11:26

    +

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

    Validate

    diff --git a/matlab/index.org b/matlab/index.org index 130f37f..3ec04ab 100644 --- a/matlab/index.org +++ b/matlab/index.org @@ -86,61 +86,65 @@ The complementary filters have to be designed in order to minimize the effect no #+end_src ** Architecture -Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_with_dyn]] where two sensors 1 and 2 are measuring the same quantity $x$ with different noise characteristics determined by $W_1$ and $W_2$. +Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_weights]] where two sensors (sensor 1 and sensor 2) are measuring the same quantity $x$ with different noise characteristics determined by $N_1(s)$ and $N_2(s)$. -$n_1$ and $n_2$ are white noise (constant power spectral density over all frequencies). +$\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: +#+name: eq:normalized_noise +\begin{equation} + \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1 +\end{equation} -#+name: fig:fusion_two_noisy_sensors_with_dyn +#+name: fig:fusion_two_noisy_sensors_weights #+caption: Fusion of two sensors -[[file:figs-tikz/fusion_two_noisy_sensors_with_dyn.png]] +[[file:figs-tikz/fusion_two_noisy_sensors_weights.png]] -We consider that the two sensor dynamics $G_1$ and $G_2$ are ideal ($G_1 = G_2 = 1$). We obtain the architecture of figure [[fig:fusion_two_noisy_sensors]]. +We consider that the two sensor dynamics $G_1(s)$ and $G_2(s)$ are ideal: +#+name: eq:idea_dynamics +\begin{equation} + G_1(s) = G_2(s) = 1 +\end{equation} -#+name: fig:fusion_two_noisy_sensors +We obtain the architecture of figure [[fig:sensor_fusion_noisy_perfect_dyn]]. + +#+name: fig:sensor_fusion_noisy_perfect_dyn #+caption: Fusion of two sensors with ideal dynamics -[[file:figs-tikz/fusion_two_noisy_sensors.png]] +[[file:figs-tikz/sensor_fusion_noisy_perfect_dyn.png]] -$H_1$ and $H_2$ are complementary filters ($H_1 + H_2 = 1$). The goal is to design $H_1$ and $H_2$ such that the effect of the noise sources $n_1$ and $n_2$ has the smallest possible effect on the estimation $\hat{x}$. +$H_1(s)$ and $H_2(s)$ are complementary filters: +#+name: eq:comp_filters_property +\begin{equation} + H_1(s) + H_2(s) = 1 +\end{equation} + +The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise sources $\tilde{n}_1$ and $\tilde{n}_2$ has the smallest possible effect on the estimation $\hat{x}$. We have that the Power Spectral Density (PSD) of $\hat{x}$ is: -\[ \Gamma_{\hat{x}} = |H_1 W_1|^2 \Gamma_{n_1} + |H_2 W_2|^2 \Gamma_{n_2} \] +\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \] And the goal is the minimize the Root Mean Square (RMS) value of $\hat{x}$: -\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Gamma_{\hat{x}}(\omega) d\omega} \] - -As $n_1$ and $n_2$ are white noise: $\Gamma_{n_1} = \Gamma_{n_2} = 1$ and we have: -\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 W_1|^2(\omega) + |H_2 W_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2 \] - -Thus, the goal is to design $H_1$ and $H_2$ such that $H_1 + H_2 = 1$ and such that $\left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2$ is minimized. - -For that, we will use the $\mathcal{H}_2$ Synthesis. +#+name: eq:rms_value_estimation +\begin{equation} + \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega} +\end{equation} ** Noise of the sensors -Let's define the noise characteristics of the two sensors by choosing $W_1$ and $W_2$: -- Sensor 1 characterized by $W_1$ has low noise at low frequency (for instance a geophone) -- Sensor 2 characterized by $W_2$ has low noise at high frequency (for instance an accelerometer) - -#+begin_src matlab :exports none - omegac = 2*pi; G0 = 1e-2; Ginf = 1e-6; - W1 = ((sqrt(G0))/(s/omegac + 1))^2; - - omegac = 100*2*pi; G0 = 1e-6; Ginf = 1e-2; - W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; -#+end_src +Let's define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$: +- Sensor 1 characterized by $N_1(s)$ has low noise at low frequency (for instance a geophone) +- Sensor 2 characterized by $N_2(s)$ has low noise at high frequency (for instance an accelerometer) #+begin_src matlab omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; - W1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000); + N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100); omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; - W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; + N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; #+end_src #+begin_src matlab :exports none figure; hold on; - plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '-', 'DisplayName', '$W_1$'); - plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '-', 'DisplayName', '$W_2$'); + plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$N_1$'); + plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$N_2$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); hold off; @@ -149,15 +153,21 @@ Let's define the noise characteristics of the two sensors by choosing $W_1$ and #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/nosie_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/noise_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src -#+NAME: fig:nosie_characteristics_sensors -#+CAPTION: Noise Characteristics of the two sensors ([[./figs/nosie_characteristics_sensors.png][png]], [[./figs/nosie_characteristics_sensors.pdf][pdf]]) -[[file:figs/nosie_characteristics_sensors.png]] +#+NAME: fig:noise_characteristics_sensors +#+CAPTION: Noise Characteristics of the two sensors ([[./figs/noise_characteristics_sensors.png][png]], [[./figs/noise_characteristics_sensors.pdf][pdf]]) +[[file:figs/noise_characteristics_sensors.png]] ** H-Two Synthesis +As $\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: $\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1$ and we have: +\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \] +Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized. + +For that, we use the $\mathcal{H}_2$ Synthesis. + We use the generalized plant architecture shown on figure [[fig:h_infinity_optimal_comp_filters]]. #+name: fig:h_infinity_optimal_comp_filters @@ -165,16 +175,16 @@ We use the generalized plant architecture shown on figure [[fig:h_infinity_optim [[file:figs-tikz/h_infinity_optimal_comp_filters.png]] The transfer function from $[n_1, n_2]$ to $\hat{x}$ is: -\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \] +\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \] If we define $H_2 = 1 - H_1$, we obtain: -\[ \begin{bmatrix} W_1 H_1 \\ W_2 H_2 \end{bmatrix} \] +\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \] Thus, if we minimize the $\mathcal{H}_2$ norm of this transfer function, we minimize the RMS value of $\hat{x}$. We define the generalized plant $P$ on matlab as shown on figure [[fig:h_infinity_optimal_comp_filters]]. #+begin_src matlab - P = [0 W2 1; - W1 -W2 0]; + P = [0 N2 1; + N1 -N2 0]; #+end_src And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command. @@ -182,17 +192,18 @@ And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command. [H1, ~, gamma] = h2syn(P, 1, 1); #+end_src -What is minimized is =norm([W1*H1,W2*H2], 2)=. - -Finally, we define $H_2 = 1 - H_1$. +Finally, we define $H_2(s) = 1 - H_1(s)$. #+begin_src matlab H2 = 1 - H1; #+end_src -** Analysis -The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]. The PSD of the [[fig:psd_sensors_htwo_synthesis]]. -Finally, the RMS value of $\hat{x}$ is shown on table [[tab:rms_results]]. -The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor. +The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]. + +The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. [[fig:psd_sensors_htwo_synthesis]]. + +The Cumulative Power Spectrum (CPS) is shown on Fig. [[fig:cps_h2_synthesis]]. + +The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors. #+begin_src matlab :exports none figure; @@ -215,14 +226,20 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati #+CAPTION: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis ([[./figs/htwo_comp_filters.png][png]], [[./figs/htwo_comp_filters.pdf][pdf]]) [[file:figs/htwo_comp_filters.png]] +#+begin_src matlab + PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2; + PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2; + PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; +#+end_src + #+begin_src matlab :exports none figure; hold on; - plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|W_1|^2$'); - plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|W_2|^2$'); - plot(freqs, abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|W_1H_1|^2+|W_2H_2|^2$'); + plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$'); + plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$'); + plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Magnitude'); + xlabel('Frequency [Hz]'); ylabel('Power Spectral Density'); hold off; xlim([freqs(1), freqs(end)]); legend('location', 'northeast'); @@ -237,77 +254,76 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati #+CAPTION: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal ([[./figs/psd_sensors_htwo_synthesis.png][png]], [[./figs/psd_sensors_htwo_synthesis.pdf][pdf]]) [[file:figs/psd_sensors_htwo_synthesis.png]] -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([norm([W1], 2);norm([W2], 2);norm([W1*H1 + W2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e'); -#+end_src - -#+name: tab:rms_results -#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters -#+RESULTS: -| | rms value | -|-----------------------+-----------| -| Sensor 1 | 1.1e-02 | -| Sensor 2 | 1.3e-03 | -| Optimal Sensor Fusion | 1.5e-04 | - -** H-Infinity Synthesis -*** First idea -Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$: -\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \] - -In order to obtain that ideal case, we need that the complementary filters be designed such that: -\begin{align*} - & H_1(j\omega) = 1 \text{ and } H_2(j\omega) = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ - & H_1(j\omega) = 0 \text{ and } H_2(j\omega) = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) -\end{align*} - -Which is indeed *impossible in practice*. - -We could try to use filters with the highest possible order to approximate that. -However, we may have robustness and implementation problems. - #+begin_src matlab - omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; - N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000); - - omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; - N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; -#+end_src - -#+begin_src matlab - n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2; - W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; - - n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2; - W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; + CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1); + CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2); + CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2); #+end_src #+begin_src matlab :exports none figure; hold on; - plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$'); - plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Magnitude'); + plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end)))); + plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end)))); + plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end)))); + set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum'); hold off; - xlim([freqs(1), freqs(end)]); - legend('location', 'northeast'); + xlim([2e-1, freqs(end)]); + ylim([1e-10 1e-5]); + legend('location', 'southeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/cps_h2_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:cps_h2_synthesis +#+CAPTION: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis ([[./figs/cps_h2_synthesis.png][png]], [[./figs/cps_h2_synthesis.pdf][pdf]]) +[[file:figs/cps_h2_synthesis.png]] + +** H-Infinity Synthesis - method A +Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$: +\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \] + +In order to obtain that ideal case, we need that the complementary filters be designed such that: +\begin{align*} + & |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\ + & |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega) +\end{align*} + +Which is indeed impossible in practice. + +We could try to approach that with the $\mathcal{H}_\infty$ synthesis by using high order filters. + +As shown on Fig. [[fig:noise_characteristics_sensors]], the frequency where the two sensors have the same noise level is around 9Hz. +We will thus choose weighting functions such that the merging frequency is around 9Hz. + +The weighting functions used as well as the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfa]]. + +#+begin_src matlab + n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2; + W1a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; + + n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2; + W2a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n; #+end_src #+begin_src matlab - P = [W1 -W1; - 0 W2; - 1 0]; + P = [W1a -W1a; + 0 W2a; + 1 0]; #+end_src And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. #+begin_src matlab :results output replace :exports both - [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); + [H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); #+end_src #+RESULTS: #+begin_example -[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); Resetting value of Gamma min based on D_11, D_12, D_21 terms Test bounds: 0.1000 < gamma <= 10500.0000 @@ -343,7 +359,7 @@ Test bounds: 0.1000 < gamma <= 10500.0000 #+end_example #+begin_src matlab - H1 = 1 - H2; + H1a = 1 - H2a; #+end_src #+begin_src matlab :exports none @@ -352,14 +368,14 @@ Test bounds: 0.1000 < gamma <= 10500.0000 ax1 = subplot(2,1,1); hold on; set(gca,'ColorOrderIndex',1) - plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); + plot(freqs, 1./abs(squeeze(freqresp(W1a, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); set(gca,'ColorOrderIndex',2) - plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); + plot(freqs, 1./abs(squeeze(freqresp(W2a, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); set(gca,'ColorOrderIndex',1) - plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + plot(freqs, abs(squeeze(freqresp(H1a, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); set(gca,'ColorOrderIndex',2) - plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); + plot(freqs, abs(squeeze(freqresp(H2a, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); @@ -371,9 +387,9 @@ Test bounds: 0.1000 < gamma <= 10500.0000 ax2 = subplot(2,1,2); hold on; set(gca,'ColorOrderIndex',1) - plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-'); + plot(freqs, 180/pi*phase(squeeze(freqresp(H1a, freqs, 'Hz'))), '-'); set(gca,'ColorOrderIndex',2) - plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); + plot(freqs, 180/pi*phase(squeeze(freqresp(H2a, freqs, 'Hz'))), '-'); hold off; xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'XScale', 'log'); @@ -384,135 +400,100 @@ Test bounds: 0.1000 < gamma <= 10500.0000 xticks([0.1, 1, 10, 100, 1000]); #+end_src -#+begin_src matlab :exports none - figure; - hold on; - plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_1|^2$'); - plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_2|^2$'); - plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Magnitude'); - hold off; - xlim([freqs(1), freqs(end)]); - legend('location', 'northeast'); +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfa.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([norm([N1], 2);norm([N2], 2);norm([N1*H1 + N2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e'); +#+NAME: fig:weights_comp_filters_Hinfa +#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfa.png][png]], [[./figs/weights_comp_filters_Hinfa.pdf][pdf]]) +[[file:figs/weights_comp_filters_Hinfa.png]] + +We then compute the Power Spectral Density as well as the Cumulative Power Spectrum. + +#+begin_src matlab + PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2; + CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha); #+end_src -#+RESULTS: -| | rms value | -|-----------------------+-----------| -| Sensor 1 | 1.1e-02 | -| Sensor 2 | 1.3e-03 | -| Optimal Sensor Fusion | 3.2e-04 | - -*** Second idea +** H-Infinity Synthesis - method B We have that: -\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \] +\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \] -We model the Power Spectral Densities of the two sensors by weighting functions $N_1(s)$ and $N_2(s)$. -\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \] +Then, at frequencies where $|H_1(j\omega)| < |H_2(j\omega)|$ we would like that $|N_1(j\omega)| = 1$ and $|N_2(j\omega)| = 0$ as we discussed before. +Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2$. -Then, at frequencies where $|H_1| < |H_2|$ we would like that $|N_1| = 1$ and $|N_2| = 0$ as we discussed before. -Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2$. - -However, if we design $H_2$ such that when $|N_1| < |N_2|$, we have that: +We know that this is impossible in practice. A more realistic choice is to design $H_2(s)$ such that when $|N_2(j\omega)| > |N_1(j\omega)|$, we have that: \[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \] + +Which is equivalent to have (by supposing $|H_1| \approx 1$): +\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] + And we have: \begin{align*} - \Phi_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ - &= (1 + \epsilon)\left|H_1 N_1\right|^2 \\ - &\approx \left|H_1 N_1\right|^2 + \Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\ + &= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\ + &\approx \left|N_1\right|^2 \end{align*} -We have then that: -\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \] -We suppose that $|H_1| = 1$: -\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \] -We can use that for the maximum magnitude of $|H_2|$ at frequencies where $|N_1| < |N_2|$ - -Similarly, we have that the maximum magnitude of $|H_1|$ at frequencies where $|N_1| > |N_2|$ is: +Similarly, we design $H_1(s)$ such that at frequencies where $|N_1| > |N_2|$: \[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \] -We could take $\epsilon = 1$, then we increase the noise of the super sensor just by a factor $\sqrt{2}$ over the all bandwidth from the idea case. +For instance, is we take $\epsilon = 1$, then the PSD of $\hat{x}$ is increased by just by a factor $\sqrt{2}$ over the all frequencies from the idea case. -This could be used as weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters. +We use this as the weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters. + +The weighting function and the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfb]]. #+begin_src matlab - omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4; - N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000); + epsilon = 2; - omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8; - N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2; + W1b = 1/epsilon*N1/N2; + W2b = 1/epsilon*N2/N1; + + W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper #+end_src #+begin_src matlab - W1 = N1/N2; - W2 = N2/N1; - - W1 = W1/(1 + s/2/pi/1000); % this is added so that it is proper -#+end_src - -#+begin_src matlab :exports none - figure; - hold on; - plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$'); - plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Magnitude'); - hold off; - xlim([freqs(1), freqs(end)]); - legend('location', 'northeast'); -#+end_src - -#+begin_src matlab - P = [W1 -W1; - 0 W2; - 1 0]; + P = [W1b -W1b; + 0 W2b; + 1 0]; #+end_src And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. #+begin_src matlab :results output replace :exports both - [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); + [H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); #+end_src #+RESULTS: #+begin_example -[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -Test bounds: 0.0000 < gamma <= 2625.0000 +[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +Test bounds: 0.0000 < gamma <= 32.8125 gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f -2.625e+03 1.8e+01 -2.3e-12 6.3e+00 -1.7e-14 0.0000 p -1.313e+03 1.8e+01 -3.3e-12 6.3e+00 -5.9e-13 0.0000 p - 656.250 1.8e+01 -1.8e-12 6.3e+00 -1.2e-13 0.0000 p - 328.125 1.8e+01 -8.3e-13 6.3e+00 -7.5e-14 0.0000 p - 164.063 1.8e+01 -3.8e-13 6.3e+00 -6.1e-14 0.0000 p - 82.031 1.8e+01 -3.3e-22 6.3e+00 -5.4e-14 0.0000 p - 41.016 1.8e+01 -3.4e-12 6.3e+00 -1.4e-16 0.0000 p - 20.508 1.8e+01 -6.4e-14 6.3e+00 -4.4e-13 0.0000 p - 10.254 1.8e+01 -1.4e-13 6.3e+00 -4.9e-14 0.0000 p - 5.127 1.7e+01 -1.1e-12 6.3e+00 -1.6e-13 0.0000 p - 2.563 1.7e+01 -6.5e-13 6.3e+00 -7.3e-14 0.0000 p - 1.282 1.4e+01 -8.9e+04# 6.3e+00 -1.6e-14 0.0000 f - 1.923 1.6e+01 -4.6e+05# 6.3e+00 -2.1e-13 0.0000 f - 2.243 1.7e+01 -1.9e+06# 6.3e+00 -9.9e-14 0.0000 f - 2.403 1.7e+01 -3.7e-11 6.3e+00 -2.1e-12 0.0000 p - 2.323 1.7e+01 -4.6e+06# 6.3e+00 -5.8e-14 0.0000 f - 2.363 1.7e+01 -1.2e+07# 6.3e+00 -3.3e-14 0.0000 f - 2.383 1.7e+01 -6.8e+07# 6.3e+00 -1.2e-13 0.0000 f - 2.393 1.7e+01 -5.5e-11 6.3e+00 -1.8e-12 0.0000 p - 2.388 1.7e+01 1.3e-22 6.3e+00 -2.5e-15 0.0000 p - 2.386 1.7e+01 -1.5e+08# 6.3e+00 -1.4e-13 0.0000 f - 2.387 1.7e+01 -4.0e+08# 6.3e+00 -1.0e-13 0.0000 f - 2.388 1.7e+01 -2.1e+09# 6.3e+00 -4.2e-13 0.0000 f + 32.812 1.8e+01 3.4e-10 6.3e+00 -2.9e-13 0.0000 p + 16.406 1.8e+01 3.4e-10 6.3e+00 -1.2e-15 0.0000 p + 8.203 1.8e+01 3.3e-10 6.3e+00 -2.6e-13 0.0000 p + 4.102 1.8e+01 3.3e-10 6.3e+00 -2.1e-13 0.0000 p + 2.051 1.7e+01 3.4e-10 6.3e+00 -7.2e-16 0.0000 p + 1.025 1.6e+01 -1.3e+06# 6.3e+00 -8.3e-14 0.0000 f + 1.538 1.7e+01 3.4e-10 6.3e+00 -2.0e-13 0.0000 p + 1.282 1.7e+01 3.4e-10 6.3e+00 -7.9e-17 0.0000 p + 1.154 1.7e+01 3.6e-10 6.3e+00 -1.8e-13 0.0000 p + 1.089 1.7e+01 -3.4e+06# 6.3e+00 -1.7e-13 0.0000 f + 1.122 1.7e+01 -1.0e+07# 6.3e+00 -3.2e-13 0.0000 f + 1.138 1.7e+01 -1.3e+08# 6.3e+00 -1.8e-13 0.0000 f + 1.146 1.7e+01 3.2e-10 6.3e+00 -3.0e-13 0.0000 p + 1.142 1.7e+01 5.5e-10 6.3e+00 -2.8e-13 0.0000 p + 1.140 1.7e+01 -1.5e-10 6.3e+00 -2.3e-13 0.0000 p + 1.139 1.7e+01 -4.8e+08# 6.3e+00 -6.2e-14 0.0000 f + 1.139 1.7e+01 1.3e-09 6.3e+00 -8.9e-17 0.0000 p - Gamma value achieved: 2.3882 + Gamma value achieved: 1.1390 #+end_example #+begin_src matlab - H1 = 1 - H2; + H1b = 1 - H2b; #+end_src #+begin_src matlab :exports none @@ -521,14 +502,14 @@ Test bounds: 0.0000 < gamma <= 2625.0000 ax1 = subplot(2,1,1); hold on; set(gca,'ColorOrderIndex',1) - plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); + plot(freqs, 1./abs(squeeze(freqresp(W1b, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); set(gca,'ColorOrderIndex',2) - plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); + plot(freqs, 1./abs(squeeze(freqresp(W2b, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); set(gca,'ColorOrderIndex',1) - plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + plot(freqs, abs(squeeze(freqresp(H1b, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); set(gca,'ColorOrderIndex',2) - plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); + plot(freqs, abs(squeeze(freqresp(H2b, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); @@ -540,9 +521,9 @@ Test bounds: 0.0000 < gamma <= 2625.0000 ax2 = subplot(2,1,2); hold on; set(gca,'ColorOrderIndex',1) - plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-'); + plot(freqs, 180/pi*phase(squeeze(freqresp(H1b, freqs, 'Hz'))), '-'); set(gca,'ColorOrderIndex',2) - plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); + plot(freqs, 180/pi*phase(squeeze(freqresp(H2b, freqs, 'Hz'))), '-'); hold off; xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'XScale', 'log'); @@ -553,32 +534,96 @@ Test bounds: 0.0000 < gamma <= 2625.0000 xticks([0.1, 1, 10, 100, 1000]); #+end_src +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfb.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:weights_comp_filters_Hinfb +#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfb.png][png]], [[./figs/weights_comp_filters_Hinfb.pdf][pdf]]) +[[file:figs/weights_comp_filters_Hinfb.png]] + +#+begin_src matlab + PSD_Hb = abs(squeeze(freqresp(N1*H1b, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2b, freqs, 'Hz'))).^2; + CPS_Hb = 1/pi*cumtrapz(2*pi*freqs, PSD_Hb); +#+end_src + +** Comparison of the methods +The three methods are now compared. + +The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. [[fig:comparison_psd_noise]]. + +The Cumulative Power Spectrum for the same sensors are shown on Fig. [[fig:comparison_cps_noise]]. + +The RMS value of the obtained super sensors are shown on table [[tab:rms_results]]. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([norm([N1], 2) ; norm([N2], 2) ; norm([N1*H1, N2*H2], 2) ; norm([N1*H1a, N2*H2a], 2) ; norm([N1*H1b, N2*H2b], 2)], {'Sensor 1', 'Sensor 2', 'H2 Fusion', 'H-Infinity a', 'H-Infinity b'}, {'rms value'}, ' %.1e'); +#+end_src + +#+name: tab:rms_results +#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters +#+RESULTS: +| | rms value | +|--------------+-----------| +| Sensor 1 | 1.3e-03 | +| Sensor 2 | 1.3e-03 | +| H2 Fusion | 1.2e-04 | +| H-Infinity a | 2.4e-04 | +| H-Infinity b | 1.4e-04 | + + #+begin_src matlab :exports none figure; hold on; - plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_1|^2$'); - plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_2|^2$'); - plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$'); + plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$'); + plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$'); + plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$'); + plot(freqs, PSD_Ha, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},a}$'); + plot(freqs, PSD_Hb, 'k-.', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},b}$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Magnitude'); + xlabel('Frequency [Hz]'); ylabel('Power Spectral Density'); hold off; xlim([freqs(1), freqs(end)]); legend('location', 'northeast'); #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([norm([N1], 2);norm([N2], 2);norm([N1*H1 + N2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e'); +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comparison_psd_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -#+RESULTS: -| | rms value | -|-----------------------+-----------| -| Sensor 1 | 1.1e-02 | -| Sensor 2 | 1.3e-03 | -| Optimal Sensor Fusion | 1.9e-04 | +#+NAME: fig:comparison_psd_noise +#+CAPTION: Comparison of the obtained Power Spectral Density using the three methods ([[./figs/comparison_psd_noise.png][png]], [[./figs/comparison_psd_noise.pdf][pdf]]) +[[file:figs/comparison_psd_noise.png]] -*** Conclusion -From the above two experiments with $\mathcal{H}_\infty$ it still seems that the $\mathcal{H}_2$ synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the $\mathcal{H}_2$ norm) +#+begin_src matlab :exports none + figure; + hold on; + plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end)))); + plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end)))); + plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end)))); + plot(freqs, CPS_Ha, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, a}} = %.1e$', sqrt(CPS_Ha(end)))); + plot(freqs, CPS_Hb, 'k-.', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, b}} = %.1e$', sqrt(CPS_Hb(end)))); + set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum'); + hold off; + xlim([2e-1, freqs(end)]); + ylim([1e-10 1e-5]); + legend('location', 'southeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/comparison_cps_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:comparison_cps_noise +#+CAPTION: Comparison of the obtained Cumulative Power Spectrum using the three methods ([[./figs/comparison_cps_noise.png][png]], [[./figs/comparison_cps_noise.pdf][pdf]]) +[[file:figs/comparison_cps_noise.png]] + +** Conclusion +From the above complementary filter design with the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ synthesis, it still seems that the $\mathcal{H}_2$ synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the $\mathcal{H}_2$ norm). * Robustness to sensor dynamics uncertainty :PROPERTIES: diff --git 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