From 5d6df8df402b7f9956fb68fbde63363c7d36fbbc Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 26 Oct 2020 21:35:47 +0100 Subject: [PATCH] Rework all the Matlab computation --- matlab/figs-tikz | 1 + matlab/figs/comp_fir_ligo_hinf.pdf | Bin 0 -> 109170 bytes matlab/figs/comp_fir_ligo_hinf.png | Bin 0 -> 133036 bytes matlab/figs/fir_filter_ligo.pdf | Bin 0 -> 64385 bytes matlab/figs/fir_filter_ligo.png | Bin 0 -> 81574 bytes matlab/figs/hinf_filters_results.pdf | Bin 0 -> 87810 bytes matlab/figs/hinf_filters_results.png | Bin 0 -> 137192 bytes matlab/figs/hinf_synthesis_ligo_results.png | Bin 0 -> 76775 bytes matlab/figs/ligo_specifications.pdf | Bin 0 -> 23807 bytes matlab/figs/ligo_specifications.png | Bin 0 -> 70536 bytes matlab/figs/ligo_weights.png | Bin 0 -> 70492 bytes .../three_complementary_filters_results.pdf | 2111 +++++++++++++++++ .../three_complementary_filters_results.png | Bin 0 -> 186935 bytes matlab/figs/three_weighting_functions.pdf | Bin 0 -> 60972 bytes matlab/figs/three_weighting_functions.png | Bin 0 -> 103739 bytes matlab/figs/weights_W1_W2.pdf | Bin 0 -> 48719 bytes matlab/figs/weights_W1_W2.png | Bin 0 -> 78149 bytes matlab/index.html | 802 +++++++ matlab/index.org | 931 ++++++++ matlab/mat/comp_ligo_fir.csv | 1074 +++++++++ matlab/mat/comp_ligo_hinf.csv | 1001 ++++++++ matlab/mat/hinf_filters_results.csv | 1001 ++++++++ matlab/mat/hinf_three_results.csv | 1001 ++++++++ matlab/mat/hinf_three_weights.csv | 1001 ++++++++ matlab/mat/hinf_weights.csv | 1001 ++++++++ matlab/mat/ligo_weights.csv | 1001 ++++++++ matlab/mat/weight_formula.csv | 501 ++++ matlab/matlab.zip | Bin 0 -> 7701 bytes matlab/matlab/comp_filters_ligo.m | 395 +++ .../h_inf_synthesis_complementary_filters.m | 144 ++ matlab/matlab/three_comp_filters.m | 95 + matlab/ref.bib | 171 ++ paper/paper.org | 16 +- paper/paper.pdf | Bin 684567 -> 795403 bytes tikz/figs/comp_fir_ligo_hinf.pdf | Bin 132665 -> 109170 bytes tikz/figs/comp_fir_ligo_hinf.png | Bin 48823 -> 0 bytes tikz/figs/comp_fir_ligo_hinf.svg | 487 ---- tikz/figs/comp_fir_ligo_hinf.tex | 66 - tikz/figs/fir_filter_ligo.pdf | Bin 0 -> 64385 bytes tikz/figs/hinf_filters_results.pdf | Bin 0 -> 86757 bytes tikz/figs/hinf_synthesis_ligo_results.pdf | Bin 0 -> 44514 bytes tikz/figs/hinf_synthesis_results.pdf | Bin 103869 -> 0 bytes tikz/figs/hinf_synthesis_results.png | Bin 36489 -> 0 bytes tikz/figs/hinf_synthesis_results.svg | 408 ---- tikz/figs/hinf_synthesis_results.tex | 67 - tikz/figs/hinf_three_synthesis_results.pdf | Bin 123478 -> 0 bytes tikz/figs/hinf_three_synthesis_results.png | Bin 53659 -> 0 bytes tikz/figs/hinf_three_synthesis_results.svg | 442 ---- tikz/figs/hinf_three_synthesis_results.tex | 80 - tikz/figs/ligo_weights.pdf | Bin 76877 -> 41484 bytes tikz/figs/ligo_weights.png | Bin 21278 -> 0 bytes tikz/figs/ligo_weights.svg | 373 --- tikz/figs/ligo_weights.tex | 66 - .../three_complementary_filters_results.pdf | 2111 +++++++++++++++++ 54 files changed, 14350 insertions(+), 1997 deletions(-) create mode 120000 matlab/figs-tikz create mode 100644 matlab/figs/comp_fir_ligo_hinf.pdf create mode 100644 matlab/figs/comp_fir_ligo_hinf.png create mode 100644 matlab/figs/fir_filter_ligo.pdf create mode 100644 matlab/figs/fir_filter_ligo.png create mode 100644 matlab/figs/hinf_filters_results.pdf create mode 100644 matlab/figs/hinf_filters_results.png create mode 100644 matlab/figs/hinf_synthesis_ligo_results.png create mode 100644 matlab/figs/ligo_specifications.pdf create mode 100644 matlab/figs/ligo_specifications.png create mode 100644 matlab/figs/ligo_weights.png create mode 100644 matlab/figs/three_complementary_filters_results.pdf create mode 100644 matlab/figs/three_complementary_filters_results.png create mode 100644 matlab/figs/three_weighting_functions.pdf create mode 100644 matlab/figs/three_weighting_functions.png create mode 100644 matlab/figs/weights_W1_W2.pdf create mode 100644 matlab/figs/weights_W1_W2.png create mode 100644 matlab/index.html create mode 100644 matlab/index.org create mode 100644 matlab/mat/comp_ligo_fir.csv create mode 100644 matlab/mat/comp_ligo_hinf.csv create mode 100644 matlab/mat/hinf_filters_results.csv create mode 100644 matlab/mat/hinf_three_results.csv create mode 100644 matlab/mat/hinf_three_weights.csv create mode 100644 matlab/mat/hinf_weights.csv create mode 100644 matlab/mat/ligo_weights.csv create mode 100644 matlab/mat/weight_formula.csv create mode 100644 matlab/matlab.zip create mode 100644 matlab/matlab/comp_filters_ligo.m create mode 100644 matlab/matlab/h_inf_synthesis_complementary_filters.m create mode 100644 matlab/matlab/three_comp_filters.m create mode 100644 matlab/ref.bib delete mode 100644 tikz/figs/comp_fir_ligo_hinf.png delete mode 100644 tikz/figs/comp_fir_ligo_hinf.svg delete mode 100644 tikz/figs/comp_fir_ligo_hinf.tex create mode 100644 tikz/figs/fir_filter_ligo.pdf create mode 100644 tikz/figs/hinf_filters_results.pdf create mode 100644 tikz/figs/hinf_synthesis_ligo_results.pdf delete mode 100644 tikz/figs/hinf_synthesis_results.pdf delete mode 100644 tikz/figs/hinf_synthesis_results.png delete mode 100644 tikz/figs/hinf_synthesis_results.svg delete mode 100644 tikz/figs/hinf_synthesis_results.tex delete mode 100644 tikz/figs/hinf_three_synthesis_results.pdf delete mode 100644 tikz/figs/hinf_three_synthesis_results.png delete mode 100644 tikz/figs/hinf_three_synthesis_results.svg delete mode 100644 tikz/figs/hinf_three_synthesis_results.tex delete mode 100644 tikz/figs/ligo_weights.png delete mode 100644 tikz/figs/ligo_weights.svg delete mode 100644 tikz/figs/ligo_weights.tex create mode 100644 tikz/figs/three_complementary_filters_results.pdf diff --git a/matlab/figs-tikz b/matlab/figs-tikz new file mode 120000 index 0000000..e25bd77 --- /dev/null +++ b/matlab/figs-tikz @@ -0,0 +1 @@ +../tikz/figs/ \ No newline at end of file diff --git a/matlab/figs/comp_fir_ligo_hinf.pdf b/matlab/figs/comp_fir_ligo_hinf.pdf new file mode 100644 index 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    +

    Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Matlab Computation

    + + +

    +In this document, the design of complementary filters is studied. +

    + +

    +One use of complementary filter is described below: +

    +
    +

    +The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor, and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system. +To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth. +

    +
    + +

    +This document is divided into several sections: +

    +
      +
    • in section 1, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters
      • +
    • in section 2, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters
      • +
    • in section 3, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO
      • +
    + +
    +

    +Add the Matlab code use to obtain the results presented in the paper are accessible here and presented below. +

    + +
    + +
    +

    1 H-Infinity synthesis of complementary filters

    +
    +

    + +

    +
    +

    +The Matlab file corresponding to this section is accessible here. +

    + +
    +
    + +
    +

    1.1 Synthesis Architecture

    +
    +

    +We here synthesize two complementary filters using the \(\mathcal{H}_\infty\) synthesis. +The goal is to specify upper bounds on the norms of the two complementary filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property (\(H_1(s) + H_2(s) = 1\)). +

    + +

    +In order to do so, we use the generalized plant shown on figure 1 where \(W_1(s)\) and \(W_2(s)\) are weighting transfer functions that will be used to shape \(H_1(s)\) and \(H_2(s)\) respectively. +

    + + +
    +

    h_infinity_robust_fusion.png +

    +

    Figure 1: \(\mathcal{H}_\infty\) synthesis of the complementary filters

    +
    + +

    +The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_2\) (figure 1) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2]\) is less than one: +\[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \] +

    + +

    +Thus, if the above condition is verified, we can define \(H_1(s) = 1 - H_2(s)\) and we have that: +\[ \left\| \begin{array}{c} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \] +Which is almost (with an maximum error of \(\sqrt{2}\)) equivalent to: +

    +\begin{align*} + |H_1(j\omega)| &< \frac{1}{|W_1(j\omega)|}, \quad \forall \omega \\ + |H_2(j\omega)| &< \frac{1}{|W_2(j\omega)|}, \quad \forall \omega +\end{align*} + +

    +We then see that \(W_1(s)\) and \(W_2(s)\) can be used to shape both \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property by the definition of \(H_1(s) = 1 - H_2(s)\). +

    +
    +
    + +
    +

    1.2 Design of Weighting Function

    +
    +

    +A formula is proposed to help the design of the weighting functions: +

    +\begin{equation} + W(s) = \left( \frac{ + \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}} + }{ + \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}} + }\right)^n +\end{equation} + +

    +The parameters permits to specify: +

    +
      +
    • the low frequency gain: \(G_0 = lim_{\omega \to 0} |W(j\omega)|\)
      • +
    • the high frequency gain: \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\)
      • +
    • the absolute gain at \(\omega_0\): \(G_c = |W(j\omega_0)|\)
      • +
    • the absolute slope between high and low frequency: \(n\)
      • +
    + +

    +The general shape of a weighting function generated using the formula is shown in figure 2. +

    + + +
    +

    weight_formula.png +

    +

    Figure 2: Amplitude of the proposed formula for the weighting functions

    +
    + +
    +
    n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; 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 = 3; w0 = 2*pi*10; 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;
+
    +
    + + +
    +

    weights_W1_W2.png +

    +

    Figure 3: Weights on the complementary filters \(W_1\) and \(W_2\) and the associated performance weights

    +
    +
    +
    + +
    +

    1.3 H-Infinity Synthesis

    +
    +

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

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

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

    +
    +
    [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
    +
    + +
    +[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+Resetting value of Gamma min based on D_11, D_12, D_21 terms
+
+Test bounds:      0.1000 <  gamma  <=   1050.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+03   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  525.050   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  262.575   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+  131.337   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+   65.719   2.8e+01   2.4e-07   4.1e+00   -9.5e-14    0.0000    p
+   32.909   2.8e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+   16.505   2.8e+01   2.4e-07   4.1e+00   -1.0e-13    0.0000    p
+    8.302   2.8e+01   2.4e-07   4.1e+00   -7.2e-14    0.0000    p
+    4.201   2.8e+01   2.4e-07   4.1e+00   -2.5e-25    0.0000    p
+    2.151   2.7e+01   2.4e-07   4.1e+00   -3.8e-14    0.0000    p
+    1.125   2.6e+01   2.4e-07   4.1e+00   -5.4e-24    0.0000    p
+    0.613   2.3e+01 -3.7e+01#  4.1e+00    0.0e+00    0.0000    f
+    0.869   2.6e+01 -3.7e+02#  4.1e+00    0.0e+00    0.0000    f
+    0.997   2.6e+01 -1.1e+04#  4.1e+00    0.0e+00    0.0000    f
+    1.061   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.029   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.013   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.005   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+    1.001   2.6e+01 -3.1e+04#  4.1e+00   -3.8e-14    0.0000    f
+    1.003   2.6e+01 -2.8e+05#  4.1e+00    0.0e+00    0.0000    f
+    1.004   2.6e+01   2.4e-07   4.1e+00   -5.8e-24    0.0000    p
+    1.004   2.6e+01   2.4e-07   4.1e+00    0.0e+00    0.0000    p
+
+ Gamma value achieved:     1.0036
+
    + +

    +We then define the high pass filter \(H_1 = 1 - H_2\). The bode plot of both \(H_1\) and \(H_2\) is shown on figure 4. +

    + +
    +
    H1 = 1 - H2;
+
    +
    +
    +
    + +
    +

    1.4 Obtained Complementary Filters

    +
    +

    +The obtained complementary filters are shown on figure 4. +

    + + +
    +

    hinf_filters_results.png +

    +

    Figure 4: Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis

    +
    +
    +
    +
    + +
    +

    2 Generating 3 complementary filters

    +
    +

    + +

    +
    +

    +The Matlab file corresponding to this section is accessible here. +

    + +
    +
    + +
    +

    2.1 Theory

    +
    +

    +We want: +

    +\begin{align*} + & |H_1(j\omega)| < 1/|W_1(j\omega)|, \quad \forall\omega\\ + & |H_2(j\omega)| < 1/|W_2(j\omega)|, \quad \forall\omega\\ + & |H_3(j\omega)| < 1/|W_3(j\omega)|, \quad \forall\omega\\ + & H_1(s) + H_2(s) + H_3(s) = 1 +\end{align*} + +

    +For that, we use the \(\mathcal{H}_\infty\) synthesis with the architecture shown on figure 5. +

    + + +
    +

    comp_filter_three_hinf.png +

    +

    Figure 5: Generalized architecture for generating 3 complementary filters

    +
    + +

    +The \(\mathcal{H}_\infty\) objective is: +

    +\begin{align*} + & |(1 - H_2(j\omega) - H_3(j\omega)) W_1(j\omega)| < 1, \quad \forall\omega\\ + & |H_2(j\omega) W_2(j\omega)| < 1, \quad \forall\omega\\ + & |H_3(j\omega) W_3(j\omega)| < 1, \quad \forall\omega\\ +\end{align*} + +

    +And thus if we choose \(H_1 = 1 - H_2 - H_3\) we have solved the problem. +

    +
    +
    + +
    +

    2.2 Weights

    +
    +

    +First we define the weights. +

    +
    +
    n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; 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;
+
+W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
+
+n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2;
+W3 = (((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;
+
    +
    + + +
    +

    three_weighting_functions.png +

    +

    Figure 6: Three weighting functions used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters

    +
    +
    +
    + +
    +

    2.3 H-Infinity Synthesis

    +
    +

    +Then we create the generalized plant P. +

    +
    +
    P = [W1 -W1 -W1;
+     0   W2  0 ;
+     0   0   W3;
+     1   0   0];
+
    +
    + +

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

    +
    +
    [H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
    +
    + +
    +[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'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  <=   1050.0000
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+1.050e+03   3.2e+00   4.5e-13   6.3e-02   -1.2e-11    0.0000    p
+  525.050   3.2e+00   1.3e-13   6.3e-02    0.0e+00    0.0000    p
+  262.575   3.2e+00   2.1e-12   6.3e-02   -1.5e-13    0.0000    p
+  131.337   3.2e+00   1.1e-12   6.3e-02   -7.2e-29    0.0000    p
+   65.719   3.2e+00   2.0e-12   6.3e-02    0.0e+00    0.0000    p
+   32.909   3.2e+00   7.4e-13   6.3e-02   -5.9e-13    0.0000    p
+   16.505   3.2e+00   1.4e-12   6.3e-02    0.0e+00    0.0000    p
+    8.302   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    4.201   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    2.151   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    1.125   3.2e+00   2.8e-12   6.3e-02    0.0e+00    0.0000    p
+    0.613   3.0e+00 -2.5e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.869   3.1e+00 -2.9e+01#  6.3e-02    0.0e+00    0.0000    f
+    0.997   3.2e+00   1.9e-12   6.3e-02    0.0e+00    0.0000    p
+    0.933   3.1e+00 -6.9e+02#  6.3e-02    0.0e+00    0.0000    f
+    0.965   3.1e+00 -3.0e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.981   3.1e+00 -8.6e+03#  6.3e-02    0.0e+00    0.0000    f
+    0.989   3.2e+00 -2.7e+04#  6.3e-02    0.0e+00    0.0000    f
+    0.993   3.2e+00 -5.7e+05#  6.3e-02    0.0e+00    0.0000    f
+    0.995   3.2e+00   2.2e-12   6.3e-02    0.0e+00    0.0000    p
+    0.994   3.2e+00   1.6e-12   6.3e-02    0.0e+00    0.0000    p
+    0.994   3.2e+00   1.0e-12   6.3e-02    0.0e+00    0.0000    p
+
+ Gamma value achieved:     0.9936
+
    +
    +
    + +
    +

    2.4 Obtained Complementary Filters

    +
    +

    +The obtained filters are: +

    +
    +
    H2 = tf(H(1));
+H3 = tf(H(2));
+H1 = 1 - H2 - H3;
+
    +
    + + +
    +

    three_complementary_filters_results.png +

    +

    Figure 7: The three complementary filters obtained after \(\mathcal{H}_\infty\) synthesis

    +
    +
    +
    +
    + +
    +

    3 Implement complementary filters for LIGO

    +
    +

    + +

    +
    +

    +The Matlab file corresponding to this section is accessible here. +

    + +
    + +

    +Let’s try to design complementary filters that are corresponding to the complementary filters design for the LIGO and described in (Hua 2005). +

    + +

    +The FIR complementary filters designed in (Hua 2005) are of order 512. +

    +
    + +
    +

    3.1 Specifications

    +
    +

    +The specifications for the filters are: +

    +
      +
    1. From \(0\) to \(0.008\text{ Hz}\),the magnitude of the filter’s transfer function should be less than or equal to \(8 \times 10^{-3}\)
      2. +
    2. From \(0.008\text{ Hz}\) to \(0.04\text{ Hz}\), it attenuates the input signal proportional to frequency cubed
      3. +
    3. Between \(0.04\text{ Hz}\) and \(0.1\text{ Hz}\), the magnitude of the transfer function should be less than 3
      4. +
    4. Above \(0.1\text{ Hz}\), the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than \(0.1\). As the filters obtained in (Hua 2005) have a magnitude of \(0.045\), we will set that as our requirement
      5. +
    + +

    +The specifications are translated in upper bounds of the complementary filters are shown on figure 8. +

    + + +
    +

    ligo_specifications.png +

    +

    Figure 8: Specification for the LIGO complementary filters

    +
    +
    +
    + +
    +

    3.2 FIR Filter

    +
    +

    +We here try to implement the FIR complementary filter synthesis as explained in (Hua 2005). +For that, we use the CVX matlab Toolbox. +

    + +

    +We setup the CVX toolbox and use the SeDuMi solver. +

    +
    +
    cvx_startup;
+cvx_solver sedumi;
+
    +
    + +

    +We define the frequency vectors on which we will constrain the norm of the FIR filter. +

    +
    +
    w1 = 0:4.06e-4:0.008;
+w2 = 0.008:4.06e-4:0.04;
+w3 = 0.04:8.12e-4:0.1;
+w4 = 0.1:8.12e-4:0.83;
+
    +
    + +

    +We then define the order of the FIR filter. +

    +
    +
    n = 512;
+
    +
    + +
    +
    A1 = [ones(length(w1),1),  cos(kron(w1'.*(2*pi),[1:n-1]))];
+A2 = [ones(length(w2),1),  cos(kron(w2'.*(2*pi),[1:n-1]))];
+A3 = [ones(length(w3),1),  cos(kron(w3'.*(2*pi),[1:n-1]))];
+A4 = [ones(length(w4),1),  cos(kron(w4'.*(2*pi),[1:n-1]))];
+
+B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))];
+B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))];
+B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))];
+B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))];
+
    +
    + +

    +We run the convex optimization. +

    +
    +
    cvx_begin
+
+variable y(n+1,1)
+
+% t
+maximize(-y(1))
+
+for i = 1:length(w1)
+    norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
+end
+
+for  i = 1:length(w2)
+    norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+end
+
+for i = 1:length(w3)
+    norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
+end
+
+for i = 1:length(w4)
+    norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
+end
+
+cvx_end
+
+h = y(2:end);
+
    +
    + +
    +cvx_begin
+variable y(n+1,1)
+% t
+maximize(-y(1))
+for i = 1:length(w1)
+    norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3;
+end
+for  i = 1:length(w2)
+    norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3;
+end
+for i = 1:length(w3)
+    norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3;
+end
+for i = 1:length(w4)
+    norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1);
+end
+cvx_end
+
+Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints
+   For improved efficiency, SeDuMi is solving the dual problem.
+------------------------------------------------------------
+SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
+Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
+eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073
+nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190
+ it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
+  0 :            4.11E+02 0.000
+  1 :  -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000   4.87  1  1  3.0E+02
+  2 :  -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000   1.83  1  1  6.6E+01
+  3 :  -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000   1.76  1  1  1.5E+01
+  4 :  -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000   2.48  1  1  4.8E+00
+  5 :  -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000   3.12  1  1  2.5E+00
+  6 :  -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000   2.62  1  1  1.4E+00
+  7 :  -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000   2.14  1  1  8.5E-01
+  8 :  -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000   1.78  1  1  5.4E-01
+  9 :  -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000   1.51  1  1  3.3E-01
+ 10 :  -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000   1.31  1  1  2.0E-01
+ 11 :  -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000   1.17  1  1  1.0E-01
+ 12 :  -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000   1.08  1  1  5.9E-02
+ 13 :  -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000   1.05  1  1  3.1E-02
+ 14 :  -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000   1.02  1  1  1.4E-02
+ 15 :  -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000   1.01  1  1  6.0E-03
+ 16 :  -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000   1.00  1  1  2.3E-03
+ 17 :  -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000   1.00  1  1  9.5E-04
+ 18 :  -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000   1.00  1  1  4.7E-04
+ 19 :  -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000   1.00  1  1  2.8E-04
+ 20 :  -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000   1.00  1  1  1.6E-04
+ 21 :  -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000   1.00  1  1  9.9E-05
+ 22 :  -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000   1.00  1  1  6.6E-05
+ 23 :  -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000   1.00  1  1  3.9E-05
+ 24 :  -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000   1.00  1  1  2.6E-05
+ 25 :  -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000   1.00  1  1  1.6E-05
+ 26 :  -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000   1.00  1  1  8.6E-06
+ 27 :  -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000   1.00  1  1  4.5E-06
+ 28 :  -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000   1.00  1  1  1.5E-06
+ 29 :  -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000   1.00  2  2  5.3E-07
+ 30 :  -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000   1.00  2  2  1.6E-07
+ 31 :  -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000   1.00  2  2  4.7E-08
+ 32 :  -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000   1.00  2  2  8.8E-09
+
+iter seconds digits       c*x               b*y
+ 32     49.4   6.8 -4.3334083581e-02 -4.3334090214e-02
+|Ax-b| =   3.7e-09, [Ay-c]_+ =   1.1E-10, |x|=  1.0e+00, |y|=  2.6e+00
+
+Detailed timing (sec)
+   Pre          IPM          Post
+3.902E+00    4.576E+01    1.035E-02
+Max-norms: ||b||=1, ||c|| = 3,
+Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267.
+------------------------------------------------------------
+Status: Solved
+Optimal value (cvx_optval): -0.0433341
+h = y(2:end);
+
    + +

    +Finally, we compute the filter response over the frequency vector defined and the result is shown on figure 9 which is very close to the filters obtain in (Hua 2005). +

    + +
    +
    w = [w1 w2 w3 w4];
+H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
+
    +
    + + +
    +

    fir_filter_ligo.png +

    +

    Figure 9: FIR Complementary filters obtain after convex optimization (png, pdf)

    +
    +
    +
    + +
    +

    3.3 Weights

    +
    +

    +We design weights that will be used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters. +These weights will determine the order of the obtained filters. +Here are the requirements on the filters: +

    +
      +
    • reasonable order
      • +
    • to be as close as possible to the specified upper bounds
      • +
    • stable minimum phase
      • +
    + +

    +The bode plot of the weights is shown on figure 10. +

    + + +
    +

    ligo_weights.png +

    +

    Figure 10: Weights for the \(\mathcal{H}_\infty\) synthesis

    +
    +
    +
    + +
    +

    3.4 H-Infinity Synthesis

    +
    +

    +We define the generalized plant as shown on figure 1. +

    +
    +
    P = [0   wL;
+     wH -wH;
+     1   0];
+
    +
    + +

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

    +
    +
    [Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
    +
    + +
    +[Hl, ~, 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:      0.3276 <  gamma  <=      1.8063
+
+  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
+    1.806   1.4e-02 -1.7e-16   3.6e-03   -4.8e-12    0.0000    p
+    1.067   1.3e-02 -4.2e-14   3.6e-03   -1.9e-12    0.0000    p
+    0.697   1.3e-02 -3.0e-01#  3.6e-03   -3.5e-11    0.0000    f
+    0.882   1.3e-02 -9.5e-01#  3.6e-03   -1.2e-34    0.0000    f
+    0.975   1.3e-02 -2.7e+00#  3.6e-03   -1.6e-12    0.0000    f
+    1.021   1.3e-02 -8.7e+00#  3.6e-03   -4.5e-16    0.0000    f
+    1.044   1.3e-02 -6.5e-14   3.6e-03   -3.0e-15    0.0000    p
+    1.032   1.3e-02 -1.8e+01#  3.6e-03    0.0e+00    0.0000    f
+    1.038   1.3e-02 -3.8e+01#  3.6e-03    0.0e+00    0.0000    f
+    1.041   1.3e-02 -8.3e+01#  3.6e-03   -2.9e-33    0.0000    f
+    1.042   1.3e-02 -1.9e+02#  3.6e-03   -3.4e-11    0.0000    f
+    1.043   1.3e-02 -5.3e+02#  3.6e-03   -7.5e-13    0.0000    f
+
+ Gamma value achieved:     1.0439
+
    + +

    +The high pass filter is defined as \(H_H = 1 - H_L\). +

    +
    +
    Hh = 1 - Hl;
+
    +
    + +

    +The size of the filters is shown below. +

    + +
    +size(Hh), size(Hl)
+State-space model with 1 outputs, 1 inputs, and 27 states.
+State-space model with 1 outputs, 1 inputs, and 27 states.
+
    + +

    +The bode plot of the obtained filters as shown on figure 11. +

    + + +
    +

    hinf_synthesis_ligo_results.png +

    +

    Figure 11: Obtained complementary filters using the \(\mathcal{H}_\infty\) synthesis

    +
    +
    +
    + +
    +

    3.5 Compare FIR and H-Infinity Filters

    +
    +

    +Let’s now compare the FIR filters designed in (Hua 2005) and the one obtained with the \(\mathcal{H}_\infty\) synthesis on figure 12. +

    + + +
    +

    comp_fir_ligo_hinf.png +

    +

    Figure 12: Comparison between the FIR filters developped for LIGO and the \(\mathcal{H}_\infty\) complementary filters

    +
    +
    +
    +
    + +

    + +

    + +

    Bibliography

    +
    +
    Hua, Wensheng. 2005. “Low Frequency Vibration Isolation and Alignment System for Advanced LIGO.” stanford university.
    +
    +
    +
    +

    Author: Thomas Dehaeze

    +

    Created: 2020-10-26 lun. 18:59

    +
    + + diff --git a/matlab/index.org b/matlab/index.org new file mode 100644 index 0000000..a4a5f67 --- /dev/null +++ b/matlab/index.org @@ -0,0 +1,931 @@ +#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Matlab Computation +:DRAWER: +#+HTML_LINK_HOME: ../index.html +#+HTML_LINK_UP: ../index.html + +#+LATEX_CLASS: cleanreport +#+LATEX_CLASS_OPTIONS: [tocnp, secbreak, minted] + +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :tangle matlab/comp_filters_design.m +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :exports both +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :noweb yes +#+PROPERTY: header-args:matlab+ :mkdirp yes +#+PROPERTY: header-args:matlab+ :output-dir figs +:END: + +* Introduction :ignore: +In this document, the design of complementary filters is studied. + +One use of complementary filter is described below: +#+begin_quote + The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor, and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system. + To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth. +#+end_quote + +This document is divided into several sections: +- in section [[sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters +- in section [[sec:three_comp_filters]], a method using the $\mathcal{H}_\infty$ synthesis is proposed to shape three of more complementary filters +- in section [[sec:comp_filters_ligo]], the $\mathcal{H}_\infty$ synthesis is used and compared with FIR complementary filters used for LIGO + +#+begin_note + Add the Matlab code use to obtain the results presented in the paper are accessible [[file:matlab.zip][here]] and presented below. +#+end_note + +* H-Infinity synthesis of complementary filters +:PROPERTIES: +:header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** Introduction :ignore: +#+begin_note + The Matlab file corresponding to this section is accessible [[file:matlab/h_inf_synthesis_complementary_filters.m][here]]. +#+end_note + +** 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 + +** Synthesis Architecture +We here synthesize two complementary filters using the $\mathcal{H}_\infty$ synthesis. +The goal is to specify upper bounds on the norms of the two complementary filters $H_1(s)$ and $H_2(s)$ while ensuring their complementary property ($H_1(s) + H_2(s) = 1$). + +In order to do so, we use the generalized plant shown on figure [[fig:h_infinity_robst_fusion]] where $W_1(s)$ and $W_2(s)$ are weighting transfer functions that will be used to shape $H_1(s)$ and $H_2(s)$ respectively. + +#+name: fig:h_infinity_robst_fusion +#+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters +[[file:figs-tikz/h_infinity_robust_fusion.png]] + +The $\mathcal{H}_\infty$ synthesis applied on this generalized plant will give a transfer function $H_2$ (figure [[fig:h_infinity_robst_fusion]]) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2]$ is less than one: +\[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \] + +Thus, if the above condition is verified, we can define $H_1(s) = 1 - H_2(s)$ and we have that: +\[ \left\| \begin{array}{c} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \] +Which is almost (with an maximum error of $\sqrt{2}$) equivalent to: +\begin{align*} + |H_1(j\omega)| &< \frac{1}{|W_1(j\omega)|}, \quad \forall \omega \\ + |H_2(j\omega)| &< \frac{1}{|W_2(j\omega)|}, \quad \forall \omega +\end{align*} + +We then see that $W_1(s)$ and $W_2(s)$ can be used to shape both $H_1(s)$ and $H_2(s)$ while ensuring their complementary property by the definition of $H_1(s) = 1 - H_2(s)$. + +** Design of Weighting Function +A formula is proposed to help the design of the weighting functions: +\begin{equation} + W(s) = \left( \frac{ + \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}} + }{ + \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}} + }\right)^n +\end{equation} + +The parameters permits to specify: +- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$ +- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$ +- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$ +- the absolute slope between high and low frequency: $n$ + +The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]]. + +#+name: fig:weight_formula +#+caption: Amplitude of the proposed formula for the weighting functions +[[file:figs-tikz/weight_formula.png]] + +#+begin_src matlab + n = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; 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 = 3; w0 = 2*pi*10; 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; +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([5e-4, 20]); + xticks([0.1, 1, 10, 100, 1000]); + legend('location', 'northeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/weights_W1_W2.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:weights_W1_W2 +#+caption: Weights on the complementary filters $W_1$ and $W_2$ and the associated performance weights +#+RESULTS: +[[file:figs/weights_W1_W2.png]] + +** H-Infinity Synthesis +We define the generalized plant $P$ on matlab. +#+begin_src matlab + P = [W1 -W1; + 0 W2; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both + [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+RESULTS: +#+begin_example +[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +Resetting value of Gamma min based on D_11, D_12, D_21 terms + +Test bounds: 0.1000 < gamma <= 1050.0000 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f +1.050e+03 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 525.050 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 262.575 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 131.337 2.8e+01 2.4e-07 4.1e+00 -1.0e-13 0.0000 p + 65.719 2.8e+01 2.4e-07 4.1e+00 -9.5e-14 0.0000 p + 32.909 2.8e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 16.505 2.8e+01 2.4e-07 4.1e+00 -1.0e-13 0.0000 p + 8.302 2.8e+01 2.4e-07 4.1e+00 -7.2e-14 0.0000 p + 4.201 2.8e+01 2.4e-07 4.1e+00 -2.5e-25 0.0000 p + 2.151 2.7e+01 2.4e-07 4.1e+00 -3.8e-14 0.0000 p + 1.125 2.6e+01 2.4e-07 4.1e+00 -5.4e-24 0.0000 p + 0.613 2.3e+01 -3.7e+01# 4.1e+00 0.0e+00 0.0000 f + 0.869 2.6e+01 -3.7e+02# 4.1e+00 0.0e+00 0.0000 f + 0.997 2.6e+01 -1.1e+04# 4.1e+00 0.0e+00 0.0000 f + 1.061 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 1.029 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 1.013 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 1.005 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + 1.001 2.6e+01 -3.1e+04# 4.1e+00 -3.8e-14 0.0000 f + 1.003 2.6e+01 -2.8e+05# 4.1e+00 0.0e+00 0.0000 f + 1.004 2.6e+01 2.4e-07 4.1e+00 -5.8e-24 0.0000 p + 1.004 2.6e+01 2.4e-07 4.1e+00 0.0e+00 0.0000 p + + Gamma value achieved: 1.0036 +#+end_example + +We then define the high pass filter $H_1 = 1 - H_2$. The bode plot of both $H_1$ and $H_2$ is shown on figure [[fig:hinf_filters_results]]. + +#+begin_src matlab + H1 = 1 - H2; +#+end_src + +** Obtained Complementary Filters +The obtained complementary filters are shown on figure [[fig:hinf_filters_results]]. + +#+begin_src matlab :exports none + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile([2, 1]); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$'); + + set(gca,'ColorOrderIndex',1) + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([1e-4, 20]); + yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]); + legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + + % Phase + ax2 = nexttile; + 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([-180:90:180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:hinf_filters_results +#+caption: Obtained complementary filters using $\mathcal{H}_\infty$ synthesis +#+RESULTS: +[[file:figs/hinf_filters_results.png]] + +* Generating 3 complementary filters +:PROPERTIES: +:header-args:matlab+: :tangle matlab/three_comp_filters.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** Introduction :ignore: +#+begin_note + The Matlab file corresponding to this section is accessible [[file:matlab/three_comp_filters.m][here]]. +#+end_note + +** 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(-2, 4, 1000); +#+end_src + +** Theory +We want: +\begin{align*} + & |H_1(j\omega)| < 1/|W_1(j\omega)|, \quad \forall\omega\\ + & |H_2(j\omega)| < 1/|W_2(j\omega)|, \quad \forall\omega\\ + & |H_3(j\omega)| < 1/|W_3(j\omega)|, \quad \forall\omega\\ + & H_1(s) + H_2(s) + H_3(s) = 1 +\end{align*} + +For that, we use the $\mathcal{H}_\infty$ synthesis with the architecture shown on figure [[fig:comp_filter_three_hinf]]. + +#+name: fig:comp_filter_three_hinf +#+caption: Generalized architecture for generating 3 complementary filters +[[file:figs-tikz/comp_filter_three_hinf.png]] + +The $\mathcal{H}_\infty$ objective is: +\begin{align*} + & |(1 - H_2(j\omega) - H_3(j\omega)) W_1(j\omega)| < 1, \quad \forall\omega\\ + & |H_2(j\omega) W_2(j\omega)| < 1, \quad \forall\omega\\ + & |H_3(j\omega) W_3(j\omega)| < 1, \quad \forall\omega\\ +\end{align*} + +And thus if we choose $H_1 = 1 - H_2 - H_3$ we have solved the problem. + +** Weights +First we define the weights. +#+begin_src matlab + n = 2; w0 = 2*pi*1; G0 = 1/10; G1 = 1000; 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; + + W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2; + + n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 1/2; + W3 = (((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 :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$'); + set(gca,'ColorOrderIndex',3) + plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|W_3|^{-1}$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + legend('location', 'northeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/three_weighting_functions.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:three_weighting_functions +#+caption: Three weighting functions used for the $\mathcal{H}_\infty$ synthesis of the complementary filters +#+RESULTS: +[[file:figs/three_weighting_functions.png]] + +** H-Infinity Synthesis +Then we create the generalized plant =P=. +#+begin_src matlab + P = [W1 -W1 -W1; + 0 W2 0 ; + 0 0 W3; + 1 0 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis. +#+begin_src matlab :results output replace :exports both + [H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+RESULTS: +#+begin_example +[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'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 <= 1050.0000 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f +1.050e+03 3.2e+00 4.5e-13 6.3e-02 -1.2e-11 0.0000 p + 525.050 3.2e+00 1.3e-13 6.3e-02 0.0e+00 0.0000 p + 262.575 3.2e+00 2.1e-12 6.3e-02 -1.5e-13 0.0000 p + 131.337 3.2e+00 1.1e-12 6.3e-02 -7.2e-29 0.0000 p + 65.719 3.2e+00 2.0e-12 6.3e-02 0.0e+00 0.0000 p + 32.909 3.2e+00 7.4e-13 6.3e-02 -5.9e-13 0.0000 p + 16.505 3.2e+00 1.4e-12 6.3e-02 0.0e+00 0.0000 p + 8.302 3.2e+00 1.6e-12 6.3e-02 0.0e+00 0.0000 p + 4.201 3.2e+00 1.6e-12 6.3e-02 0.0e+00 0.0000 p + 2.151 3.2e+00 1.6e-12 6.3e-02 0.0e+00 0.0000 p + 1.125 3.2e+00 2.8e-12 6.3e-02 0.0e+00 0.0000 p + 0.613 3.0e+00 -2.5e+03# 6.3e-02 0.0e+00 0.0000 f + 0.869 3.1e+00 -2.9e+01# 6.3e-02 0.0e+00 0.0000 f + 0.997 3.2e+00 1.9e-12 6.3e-02 0.0e+00 0.0000 p + 0.933 3.1e+00 -6.9e+02# 6.3e-02 0.0e+00 0.0000 f + 0.965 3.1e+00 -3.0e+03# 6.3e-02 0.0e+00 0.0000 f + 0.981 3.1e+00 -8.6e+03# 6.3e-02 0.0e+00 0.0000 f + 0.989 3.2e+00 -2.7e+04# 6.3e-02 0.0e+00 0.0000 f + 0.993 3.2e+00 -5.7e+05# 6.3e-02 0.0e+00 0.0000 f + 0.995 3.2e+00 2.2e-12 6.3e-02 0.0e+00 0.0000 p + 0.994 3.2e+00 1.6e-12 6.3e-02 0.0e+00 0.0000 p + 0.994 3.2e+00 1.0e-12 6.3e-02 0.0e+00 0.0000 p + + Gamma value achieved: 0.9936 +#+end_example + +** Obtained Complementary Filters +The obtained filters are: +#+begin_src matlab + H2 = tf(H(1)); + H3 = tf(H(2)); + H1 = 1 - H2 - H3; +#+end_src + +#+begin_src matlab :exports none + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile([2, 1]); + hold on; + set(gca,'ColorOrderIndex',1) + plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$'); + set(gca,'ColorOrderIndex',2) + plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$'); + set(gca,'ColorOrderIndex',3) + plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'DisplayName', '$|W_3|^{-1}$'); + 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', '$H_2$'); + set(gca,'ColorOrderIndex',3) + plot(freqs, abs(squeeze(freqresp(H3, freqs, 'Hz'))), '-', 'DisplayName', '$H_3$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([1e-4, 20]); + legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + + % Phase + ax2 = nexttile; + 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')))); + set(gca,'ColorOrderIndex',3) + plot(freqs, 180/pi*phase(squeeze(freqresp(H3, freqs, 'Hz')))); + hold off; + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + set(gca, 'XScale', 'log'); + yticks([-360:90:360]); ylim([-270, 270]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/three_complementary_filters_results.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:three_complementary_filters_results +#+caption: The three complementary filters obtained after $\mathcal{H}_\infty$ synthesis +#+RESULTS: +[[file:figs/three_complementary_filters_results.png]] + +* Implement complementary filters for LIGO +:PROPERTIES: +:header-args:matlab+: :tangle matlab/comp_filters_ligo.m +:header-args:matlab+: :comments org :mkdirp yes +:END: +<> + +** Introduction :ignore: +#+begin_note + The Matlab file corresponding to this section is accessible [[file:matlab/comp_filters_ligo.m][here]]. +#+end_note + +Let's try to design complementary filters that are corresponding to the complementary filters design for the LIGO and described in cite:hua05_low_ligo. + +The FIR complementary filters designed in cite:hua05_low_ligo are of order 512. + +** 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(-3, 0, 1000); +#+end_src + +** Specifications +The specifications for the filters are: +1. From $0$ to $0.008\text{ Hz}$,the magnitude of the filter’s transfer function should be less than or equal to $8 \times 10^{-3}$ +2. From $0.008\text{ Hz}$ to $0.04\text{ Hz}$, it attenuates the input signal proportional to frequency cubed +3. Between $0.04\text{ Hz}$ and $0.1\text{ Hz}$, the magnitude of the transfer function should be less than 3 +4. Above $0.1\text{ Hz}$, the maximum of the magnitude of the complement filter should be as close to zero as possible. In our system, we would like to have the magnitude of the complementary filter to be less than $0.1$. As the filters obtained in cite:hua05_low_ligo have a magnitude of $0.045$, we will set that as our requirement + +The specifications are translated in upper bounds of the complementary filters are shown on figure [[fig:ligo_specifications]]. + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1) + plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$'); + set(gca,'ColorOrderIndex',1) + plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',1) + plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',2) + plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$'); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([1e-4, 10]); + legend('location', 'southeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/ligo_specifications.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:ligo_specifications +#+caption: Specification for the LIGO complementary filters +#+RESULTS: +[[file:figs/ligo_specifications.png]] + +** TODO FIR Filter +We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo. +For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]]. + +We setup the CVX toolbox and use the =SeDuMi= solver. +#+begin_src matlab + cvx_startup; + cvx_solver sedumi; +#+end_src + +We define the frequency vectors on which we will constrain the norm of the FIR filter. +#+begin_src matlab + w1 = 0:4.06e-4:0.008; + w2 = 0.008:4.06e-4:0.04; + w3 = 0.04:8.12e-4:0.1; + w4 = 0.1:8.12e-4:0.83; +#+end_src + +We then define the order of the FIR filter. +#+begin_src matlab + n = 512; +#+end_src + +#+begin_src matlab + A1 = [ones(length(w1),1), cos(kron(w1'.*(2*pi),[1:n-1]))]; + A2 = [ones(length(w2),1), cos(kron(w2'.*(2*pi),[1:n-1]))]; + A3 = [ones(length(w3),1), cos(kron(w3'.*(2*pi),[1:n-1]))]; + A4 = [ones(length(w4),1), cos(kron(w4'.*(2*pi),[1:n-1]))]; + + B1 = [zeros(length(w1),1), sin(kron(w1'.*(2*pi),[1:n-1]))]; + B2 = [zeros(length(w2),1), sin(kron(w2'.*(2*pi),[1:n-1]))]; + B3 = [zeros(length(w3),1), sin(kron(w3'.*(2*pi),[1:n-1]))]; + B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))]; +#+end_src + +We run the convex optimization. +#+begin_src matlab :results output replace :wrap example + cvx_begin + + variable y(n+1,1) + + % t + maximize(-y(1)) + + for i = 1:length(w1) + norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3; + end + + for i = 1:length(w2) + norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3; + end + + for i = 1:length(w3) + norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3; + end + + for i = 1:length(w4) + norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1); + end + + cvx_end + + h = y(2:end); +#+end_src + +#+RESULTS: +#+begin_example +cvx_begin +variable y(n+1,1) +% t +maximize(-y(1)) +for i = 1:length(w1) + norm([0 A1(i,:); 0 B1(i,:)]*y) <= 8e-3; +end +for i = 1:length(w2) + norm([0 A2(i,:); 0 B2(i,:)]*y) <= 8e-3*(2*pi*w2(i)/(0.008*2*pi))^3; +end +for i = 1:length(w3) + norm([0 A3(i,:); 0 B3(i,:)]*y) <= 3; +end +for i = 1:length(w4) + norm([[1 0]'- [0 A4(i,:); 0 B4(i,:)]*y]) <= y(1); +end +cvx_end + +Calling SeDuMi 1.34: 4291 variables, 1586 equality constraints + For improved efficiency, SeDuMi is solving the dual problem. +------------------------------------------------------------ +SeDuMi 1.34 (beta) by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003. +Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500 +eqs m = 1586, order n = 3220, dim = 4292, blocks = 1073 +nnz(A) = 1100727 + 0, nnz(ADA) = 1364794, nnz(L) = 683190 + it : b*y gap delta rate t/tP* t/tD* feas cg cg prec + 0 : 4.11E+02 0.000 + 1 : -2.58E+00 1.25E+02 0.000 0.3049 0.9000 0.9000 4.87 1 1 3.0E+02 + 2 : -2.36E+00 3.90E+01 0.000 0.3118 0.9000 0.9000 1.83 1 1 6.6E+01 + 3 : -1.69E+00 1.31E+01 0.000 0.3354 0.9000 0.9000 1.76 1 1 1.5E+01 + 4 : -8.60E-01 7.10E+00 0.000 0.5424 0.9000 0.9000 2.48 1 1 4.8E+00 + 5 : -4.91E-01 5.44E+00 0.000 0.7661 0.9000 0.9000 3.12 1 1 2.5E+00 + 6 : -2.96E-01 3.88E+00 0.000 0.7140 0.9000 0.9000 2.62 1 1 1.4E+00 + 7 : -1.98E-01 2.82E+00 0.000 0.7271 0.9000 0.9000 2.14 1 1 8.5E-01 + 8 : -1.39E-01 2.00E+00 0.000 0.7092 0.9000 0.9000 1.78 1 1 5.4E-01 + 9 : -9.99E-02 1.30E+00 0.000 0.6494 0.9000 0.9000 1.51 1 1 3.3E-01 + 10 : -7.57E-02 8.03E-01 0.000 0.6175 0.9000 0.9000 1.31 1 1 2.0E-01 + 11 : -5.99E-02 4.22E-01 0.000 0.5257 0.9000 0.9000 1.17 1 1 1.0E-01 + 12 : -5.28E-02 2.45E-01 0.000 0.5808 0.9000 0.9000 1.08 1 1 5.9E-02 + 13 : -4.82E-02 1.28E-01 0.000 0.5218 0.9000 0.9000 1.05 1 1 3.1E-02 + 14 : -4.56E-02 5.65E-02 0.000 0.4417 0.9045 0.9000 1.02 1 1 1.4E-02 + 15 : -4.43E-02 2.41E-02 0.000 0.4265 0.9004 0.9000 1.01 1 1 6.0E-03 + 16 : -4.37E-02 8.90E-03 0.000 0.3690 0.9070 0.9000 1.00 1 1 2.3E-03 + 17 : -4.35E-02 3.24E-03 0.000 0.3641 0.9164 0.9000 1.00 1 1 9.5E-04 + 18 : -4.34E-02 1.55E-03 0.000 0.4788 0.9086 0.9000 1.00 1 1 4.7E-04 + 19 : -4.34E-02 8.77E-04 0.000 0.5653 0.9169 0.9000 1.00 1 1 2.8E-04 + 20 : -4.34E-02 5.05E-04 0.000 0.5754 0.9034 0.9000 1.00 1 1 1.6E-04 + 21 : -4.34E-02 2.94E-04 0.000 0.5829 0.9136 0.9000 1.00 1 1 9.9E-05 + 22 : -4.34E-02 1.63E-04 0.015 0.5548 0.9000 0.0000 1.00 1 1 6.6E-05 + 23 : -4.33E-02 9.42E-05 0.000 0.5774 0.9053 0.9000 1.00 1 1 3.9E-05 + 24 : -4.33E-02 6.27E-05 0.000 0.6658 0.9148 0.9000 1.00 1 1 2.6E-05 + 25 : -4.33E-02 3.75E-05 0.000 0.5972 0.9187 0.9000 1.00 1 1 1.6E-05 + 26 : -4.33E-02 1.89E-05 0.000 0.5041 0.9117 0.9000 1.00 1 1 8.6E-06 + 27 : -4.33E-02 9.72E-06 0.000 0.5149 0.9050 0.9000 1.00 1 1 4.5E-06 + 28 : -4.33E-02 2.94E-06 0.000 0.3021 0.9194 0.9000 1.00 1 1 1.5E-06 + 29 : -4.33E-02 9.73E-07 0.000 0.3312 0.9189 0.9000 1.00 2 2 5.3E-07 + 30 : -4.33E-02 2.82E-07 0.000 0.2895 0.9063 0.9000 1.00 2 2 1.6E-07 + 31 : -4.33E-02 8.05E-08 0.000 0.2859 0.9049 0.9000 1.00 2 2 4.7E-08 + 32 : -4.33E-02 1.43E-08 0.000 0.1772 0.9059 0.9000 1.00 2 2 8.8E-09 + +iter seconds digits c*x b*y + 32 49.4 6.8 -4.3334083581e-02 -4.3334090214e-02 +|Ax-b| = 3.7e-09, [Ay-c]_+ = 1.1E-10, |x|= 1.0e+00, |y|= 2.6e+00 + +Detailed timing (sec) + Pre IPM Post +3.902E+00 4.576E+01 1.035E-02 +Max-norms: ||b||=1, ||c|| = 3, +Cholesky |add|=0, |skip| = 0, ||L.L|| = 4.26267. +------------------------------------------------------------ +Status: Solved +Optimal value (cvx_optval): -0.0433341 +h = y(2:end); +#+end_example + +Finally, we compute the filter response over the frequency vector defined and the result is shown on figure [[fig:fir_filter_ligo]] which is very close to the filters obtain in cite:hua05_low_ligo. + +#+begin_src matlab + w = [w1 w2 w3 w4]; + H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h; +#+end_src + +#+begin_src matlab :exports none + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile([2, 1]); + hold on; + plot(w, abs(H), 'k-'); + plot(w, abs(1-H), 'k--'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([5e-3, 5]); + + % Phase + ax2 = nexttile; + hold on; + plot(w, 180/pi*angle(H), 'k-'); + plot(w, 180/pi*angle(1-H), 'k--'); + hold off; + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + set(gca, 'XScale', 'log'); + yticks([-180:90:180]); ylim([-180, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([1e-3, 1]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/fir_filter_ligo.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:fir_filter_ligo +#+caption: FIR Complementary filters obtain after convex optimization +#+RESULTS: +[[file:figs/fir_filter_ligo.png]] + +** Weights +We design weights that will be used for the $\mathcal{H}_\infty$ synthesis of the complementary filters. +These weights will determine the order of the obtained filters. +Here are the requirements on the filters: +- reasonable order +- to be as close as possible to the specified upper bounds +- stable minimum phase + +The bode plot of the weights is shown on figure [[fig:ligo_weights]]. + +#+begin_src matlab :exports none + w1 = 2*pi*0.008; x1 = 0.35; + w2 = 2*pi*0.04; x2 = 0.5; + w3 = 2*pi*0.05; x3 = 0.5; + + % Slope of +3 from w1 + wH = 0.008*(s^2/w1^2 + 2*x1/w1*s + 1)*(s/w1 + 1); + % Little bump from w2 to w3 + wH = wH*(s^2/w2^2 + 2*x2/w2*s + 1)/(s^2/w3^2 + 2*x3/w3*s + 1); + % No Slope at high frequencies + wH = wH/(s^2/w3^2 + 2*x3/w3*s + 1)/(s/w3 + 1); + % Little bump between w2 and w3 + w0 = 2*pi*0.045; xi = 0.1; A = 2; n = 1; + wH = wH*((s^2 + 2*w0*xi*A^(1/n)*s + w0^2)/(s^2 + 2*w0*xi*s + w0^2))^n; + + wH = 1/wH; + wH = minreal(ss(wH)); +#+end_src + +#+begin_src matlab :exports none + n = 20; Rp = 1; Wp = 2*pi*0.102; + [b,a] = cheby1(n, Rp, Wp, 'high', 's'); + wL = 0.04*tf(a, b); + + wL = 1/wL; + wL = minreal(ss(wL)); +#+end_src + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(inv(wH), freqs, 'Hz'))), '-', 'DisplayName', '$|w_H|^{-1}$'); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(inv(wL), freqs, 'Hz'))), '-', 'DisplayName', '$|w_L|^{-1}$'); + + plot([0.0001, 0.008], [8e-3, 8e-3], 'k--', 'DisplayName', 'Spec.'); + plot([0.008 0.04], [8e-3, 1], 'k--', 'HandleVisibility', 'off'); + plot([0.04 0.1], [3, 3], 'k--', 'HandleVisibility', 'off'); + plot([0.1, 10], [0.045, 0.045], 'k--', 'HandleVisibility', 'off'); + + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([1e-3, 10]); + legend('location', 'southeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/ligo_weights.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:ligo_weights +#+caption: Weights for the $\mathcal{H}_\infty$ synthesis +#+RESULTS: +[[file:figs/ligo_weights.png]] + +** H-Infinity Synthesis +We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]]. +#+begin_src matlab + P = [0 wL; + wH -wH; + 1 0]; +#+end_src + +And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. +#+begin_src matlab :results output replace :exports both :wrap example + [Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); +#+end_src + +#+RESULTS: +#+begin_example +[Hl, ~, 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: 0.3276 < gamma <= 1.8063 + + gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f + 1.806 1.4e-02 -1.7e-16 3.6e-03 -4.8e-12 0.0000 p + 1.067 1.3e-02 -4.2e-14 3.6e-03 -1.9e-12 0.0000 p + 0.697 1.3e-02 -3.0e-01# 3.6e-03 -3.5e-11 0.0000 f + 0.882 1.3e-02 -9.5e-01# 3.6e-03 -1.2e-34 0.0000 f + 0.975 1.3e-02 -2.7e+00# 3.6e-03 -1.6e-12 0.0000 f + 1.021 1.3e-02 -8.7e+00# 3.6e-03 -4.5e-16 0.0000 f + 1.044 1.3e-02 -6.5e-14 3.6e-03 -3.0e-15 0.0000 p + 1.032 1.3e-02 -1.8e+01# 3.6e-03 0.0e+00 0.0000 f + 1.038 1.3e-02 -3.8e+01# 3.6e-03 0.0e+00 0.0000 f + 1.041 1.3e-02 -8.3e+01# 3.6e-03 -2.9e-33 0.0000 f + 1.042 1.3e-02 -1.9e+02# 3.6e-03 -3.4e-11 0.0000 f + 1.043 1.3e-02 -5.3e+02# 3.6e-03 -7.5e-13 0.0000 f + + Gamma value achieved: 1.0439 +#+end_example + +The high pass filter is defined as $H_H = 1 - H_L$. +#+begin_src matlab + Hh = 1 - Hl; +#+end_src + +#+begin_src matlab :exports none + Hh = minreal(Hh); + Hl = minreal(Hl); +#+end_src + +The size of the filters is shown below. + +#+begin_src matlab :exports results :results output replace :wrap example + size(Hh), size(Hl) +#+end_src + +#+RESULTS: +#+begin_example +size(Hh), size(Hl) +State-space model with 1 outputs, 1 inputs, and 27 states. +State-space model with 1 outputs, 1 inputs, and 27 states. +#+end_example + +The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_ligo_results]]. + +#+begin_src matlab :exports none + figure; + hold on; + set(gca,'ColorOrderIndex',1); + plot([0.0001, 0.008], [8e-3, 8e-3], ':', 'DisplayName', 'Spec. on $H_H$'); + set(gca,'ColorOrderIndex',1); + plot([0.008 0.04], [8e-3, 1], ':', 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',1); + plot([0.04 0.1], [3, 3], ':', 'HandleVisibility', 'off'); + + set(gca,'ColorOrderIndex',2); + plot([0.1, 10], [0.045, 0.045], ':', 'DisplayName', 'Spec. on $H_L$'); + + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'DisplayName', '$H_H$'); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'DisplayName', '$H_L$'); + + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Magnitude'); + hold off; + xlim([freqs(1), freqs(end)]); + ylim([1e-3, 10]); + legend('location', 'southeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/hinf_synthesis_ligo_results.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:hinf_synthesis_ligo_results +#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ synthesis +#+RESULTS: +[[file:figs/hinf_synthesis_ligo_results.png]] + +** TODO Compare FIR and H-Infinity Filters +Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]]. + +#+begin_src matlab :exports none + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + % Magnitude + ax1 = nexttile([2, 1]); + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', ... + 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$'); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', ... + 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$'); + + set(gca,'ColorOrderIndex',1); + plot(w, abs(H), '--', ... + 'DisplayName', '$H_H(s)$ - FIR'); + set(gca,'ColorOrderIndex',2); + plot(w, abs(1-H), '--', ... + 'DisplayName', '$H_L(s)$ - FIR'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); + set(gca, 'XTickLabel',[]); + ylim([5e-3, 10]); + legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + + + % Phase + ax2 = nexttile; + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Hh, freqs, 'Hz'))), '-'); + set(gca,'ColorOrderIndex',2); + plot(freqs, 180/pi*angle(squeeze(freqresp(Hl, freqs, 'Hz'))), '-'); + + set(gca,'ColorOrderIndex',1); + plot(w, 180/pi*angle(H), '--'); + set(gca,'ColorOrderIndex',2); + plot(w, 180/pi*angle(1-H), '--'); + set(gca, 'XScale', 'log'); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + yticks([-180:90:180]); ylim([-180, 180]); + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_fir_ligo_hinf +#+caption: Comparison between the FIR filters developped for LIGO and the $\mathcal{H}_\infty$ complementary filters +#+RESULTS: +[[file:figs/comp_fir_ligo_hinf.png]] + +* Bibliography :ignore: +bibliographystyle:unsrt +bibliography:ref.bib diff --git a/matlab/mat/comp_ligo_fir.csv 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H_3(s)$, the proposed $\mathcal{H}_\ *** Example of generated complementary filters :ignore: An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively). -Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:hinf_three_synthesis_results. -The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_three_synthesis_results. +Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:three_complementary_filters_results. +The bode plots of the obtained complementary filters are shown in Fig. ref:fig:three_complementary_filters_results. -#+name: fig:hinf_three_synthesis_results +#+name: fig:three_complementary_filters_results #+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis #+attr_latex: :scale 1 -[[file:figs/hinf_three_synthesis_results.pdf]] +[[file:figs/three_complementary_filters_results.pdf]] * Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO <> diff --git a/paper/paper.pdf b/paper/paper.pdf index 0183232107be1383506a42670d4cfde2491ef3ce..db9d4583ca035fd242ceaae5063db854322178e2 100644 GIT binary patch delta 369279 zcmZUbby!r}`}PeKX_0Of0g)V-q2nkZ9fEXs4&Aj?N~EM4M7q1XrKMwNq`PZ?cjGzw z{ax3~KkR$2^*rlyuf6AgfuuaxwyHs_&Q#e1J)%~a$+d8aT>J@)tHaX8|fkck0`!|{?rD$PSF>cKnZ9yCi z@eiLWPKF8$m@`lW`0KE&d~6)!7q(V0>Q>K(m)n!)9>vkWe<@Yc6Cd~_s7O8*ALU(2 zX^-K_v&Mw==PZTeqZa+%Ke}*Axs{;+r#=@80dONeiyx1fe79>* zp8KLdcfIq^kQJeX$n!S_I0|Du`1((5#0QoF1UR*AyT`wsH)?Ak9beB=_dj}B|4+%+ z*k9=)f7ISVG$yrT)>im~83SS5G@s_${#k8gkLqIr)sN`nOz{!uF9-%+ zgnz#K!12b3JqK06rtQfSWY*)){{DMSk3N!QY`eAj<5Hj|35EZ>i;a!`vgD3Yghm5h6#mC&sY!^5 z{Jp&v0if3%n}T}rmDX&J#;Wl}5wV)QPiWQkFU^eqCd(_dZIo}v(aeh#uktu 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