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Rework all the Matlab computation
2020-10-26 21:35:47 +01:00
#+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: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <script src="../js/jquery.min.js"></script>
#+HTML_HEAD: <script src="../js/bootstrap.min.js"></script>
#+HTML_HEAD: <script src="../js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script src="../js/readtheorg.js"></script>
#+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:
<<sec:h_inf_synthesis_complementary_filters>>
** 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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+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:
<<sec:three_comp_filters>>
** 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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+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:
<<sec:comp_filters_ligo>>
** 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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+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
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