From c520f7903e1b42a5d98ad4954ba6de7e6093adda Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 1 Sep 2021 11:35:04 +0200 Subject: [PATCH] Update all matlab figures --- .../dehaeze21_desig_compl_filte_matlab.html | 1109 ++++++++++ matlab/dehaeze21_desig_compl_filte_matlab.org | 1954 +++-------------- matlab/{figs => figs-journal} | 0 matlab/figs/comp_fir_ligo_hinf.pdf | Bin 0 -> 108099 bytes matlab/figs/comp_fir_ligo_hinf.png | Bin 0 -> 122021 bytes matlab/figs/fir_filter_ligo.pdf | Bin 0 -> 68057 bytes matlab/figs/fir_filter_ligo.png | Bin 0 -> 114471 bytes matlab/figs/hinf_filters_results.pdf | Bin 0 -> 89466 bytes matlab/figs/hinf_filters_results.png | Bin 0 -> 129086 bytes ...hinf_filters_results_mixed_sensitivity.pdf | 1815 +++++++++++++++ ...hinf_filters_results_mixed_sensitivity.png | Bin 0 -> 142079 bytes matlab/figs/hinf_synthesis_ligo_results.pdf | Bin 0 -> 46817 bytes matlab/figs/hinf_synthesis_ligo_results.png | Bin 0 -> 71781 bytes matlab/figs/ligo_specifications.pdf | Bin 0 -> 23931 bytes matlab/figs/ligo_specifications.png | Bin 0 -> 71449 bytes matlab/figs/ligo_weights.pdf | Bin 0 -> 41346 bytes matlab/figs/ligo_weights.png | Bin 0 -> 69496 bytes .../three_complementary_filters_results.pdf | Bin 0 -> 118775 bytes .../three_complementary_filters_results.png | Bin 0 -> 175321 bytes matlab/figs/three_weighting_functions.pdf | 1288 +++++++++++ matlab/figs/three_weighting_functions.png | Bin 0 -> 87290 bytes matlab/figs/weight_formula.pdf | Bin 0 -> 41034 bytes matlab/figs/weight_formula.png | Bin 0 -> 78360 bytes matlab/figs/weights_W1_W2.pdf | Bin 0 -> 48737 bytes matlab/figs/weights_W1_W2.png | Bin 0 -> 83624 bytes matlab/index.html | 1603 -------------- 26 files changed, 4477 insertions(+), 3292 deletions(-) create mode 100644 matlab/dehaeze21_desig_compl_filte_matlab.html rename matlab/{figs => figs-journal} (100%) 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_filters_results_mixed_sensitivity.pdf create mode 100644 matlab/figs/hinf_filters_results_mixed_sensitivity.png create mode 100644 matlab/figs/hinf_synthesis_ligo_results.pdf 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.pdf 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/weight_formula.pdf create mode 100644 matlab/figs/weight_formula.png create mode 100644 matlab/figs/weights_W1_W2.pdf create mode 100644 matlab/figs/weights_W1_W2.png delete mode 100644 matlab/index.html diff --git a/matlab/dehaeze21_desig_compl_filte_matlab.html b/matlab/dehaeze21_desig_compl_filte_matlab.html new file mode 100644 index 0000000..5916be9 --- /dev/null +++ b/matlab/dehaeze21_desig_compl_filte_matlab.html @@ -0,0 +1,1109 @@ + + + + + + +A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis - Matlab Computation + + + + + + + + +
+ UP + | + HOME +
+

A new method of designing complementary filters for sensor fusion using the \(\mathcal{H}_\infty\) synthesis - Matlab Computation

+ + +

+This file is the Matlab file for the paper (Dehaeze, Vermat, and Collette 2021). +

+ +

+This document is divided into several sections: +

+ + +
+

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_plant.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: Gain of the Weighting Function formula

+
+
+
+ +
+

1.3. Example

+
+
+
%% Design of the Weighting Functions
+W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
+W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
+
+
+ + +
+

weights_W1_W2.png +

+

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

+
+
+
+ +
+

1.4. H-Infinity Synthesis

+
+

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

+
+
%% Generalized Plant
+P = [W1 -W1;
+     0   W2;
+     1   0];
+
+
+ +

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

+
+
%% H-Infinity Synthesis
+[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');
+
+  Test bounds:  0.3223 <=  gamma  <=  1000
+
+    gamma        X>=0        Y>=0       rho(XY)<1    p/f
+  1.795e+01     1.4e-07     0.0e+00     1.481e-16     p
+  2.406e+00     1.4e-07     0.0e+00     3.604e-15     p
+  8.806e-01    -3.1e+02 #  -1.4e-16     7.370e-19     f
+  1.456e+00     1.4e-07     0.0e+00     1.499e-18     p
+  1.132e+00     1.4e-07     0.0e+00     8.587e-15     p
+  9.985e-01     1.4e-07     0.0e+00     2.331e-13     p
+  9.377e-01    -7.7e+02 #  -6.6e-17     3.744e-14     f
+  9.676e-01    -2.0e+03 #  -5.7e-17     1.046e-13     f
+  9.829e-01    -6.6e+03 #  -1.1e-16     2.949e-13     f
+  9.907e-01     1.4e-07     0.0e+00     2.374e-19     p
+  9.868e-01    -1.6e+04 #  -6.4e-17     5.331e-14     f
+  9.887e-01    -5.1e+04 #  -1.5e-17     2.703e-19     f
+  9.897e-01     1.4e-07     0.0e+00     1.583e-11     p
+  Limiting gains...
+  9.897e-01     1.5e-07     0.0e+00     1.183e-12     p
+  9.897e-01     6.9e-07     0.0e+00     1.365e-12     p
+
+  Best performance (actual): 0.9897
+
+ +

+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. +

+ +
+
%% Define H1 to be the complementary of H2
+H1 = 1 - H2;
+
+
+ +

+Or one can just used to generateCF Matlab function: +

+
+
[H1, H2] = generateCF(W1, W2);
+
+
+
+
+ +
+

1.5. Obtained Complementary Filters

+
+

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

+ +
+zpk(H1)
+ans =
+
+            (s+1.289e05) (s+153.6) (s+3.842)^3
+  -------------------------------------------------------
+  (s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
+
+zpk(H2)
+ans =
+
+         125.61 (s+3358)^2 (s^2 + 46.61s + 813.8)
+  -------------------------------------------------------
+  (s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
+
+ + +
+

hinf_filters_results.png +

+

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

+
+
+
+
+ +
+

2. Design of complementary filters used in the Active Vibration Isolation System at the 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. +

+
+ +
+

2.1. 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 5. +

+ + +
+

ligo_specifications.png +

+

Figure 5: Specification for the LIGO complementary filters

+
+
+
+ +
+

2.2. 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. +

+
+
%% Initialized CVX
+cvx_startup;
+cvx_solver sedumi;
+
+
+ +

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

+
+
%% Frequency vectors
+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. +

+
+
%% Filter order
+n = 512;
+
+
+ +
+
%% Initialization of filter responses
+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. +

+
+
%% 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 6 which is very close to the filters obtain in (Hua 2005). +

+ +
+
%% Combine the frequency vectors to form the obtained filter
+w = [w1 w2 w3 w4];
+H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
+
+
+ + +
+

fir_filter_ligo.png +

+

Figure 6: FIR Complementary filters obtain after convex optimization

+
+
+
+ +
+

2.3. 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 7. +

+ + +
+

ligo_weights.png +

+

Figure 7: Weights for the \(\mathcal{H}_\infty\) synthesis

+
+
+
+ +
+

2.4. H-Infinity Synthesis

+
+

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

+
+
%% Generalized plant for the H-infinity Synthesis
+P = [0   wL;
+     wH -wH;
+     1   0];
+
+
+ +

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

+
+
%% Standard H-Infinity synthesis
+[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 \(H_H = 1 - H_L\). +

+
+
%% High pass filter as the complementary of the low pass filter
+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 8. +

+ + +
+

hinf_synthesis_ligo_results.png +

+

Figure 8: Obtained complementary filters using the \(\mathcal{H}_\infty\) synthesis

+
+
+
+ +
+

2.5. 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 9. +

+ + +
+

comp_fir_ligo_hinf.png +

+

Figure 9: Comparison between the FIR filters developped for LIGO and the \(\mathcal{H}_\infty\) complementary filters

+
+
+
+
+ +
+

3. “Closed-Loop” complementary filters

+
+
+

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

+ +
+
+ +
+

3.1. Using Feedback architecture

+
+
+
%% Design of the Weighting Functions
+W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
+W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
+
+
+ +

+Let’s first synthesize \(H_1(s)\): +

+
+
%% Generalized plant for "closed-loop" complementary filter synthesis
+P = [ W1 0   1;
+     -W1 W2 -1];
+
+
+ +
+
%% Standard H-Infinity Synthesis
+[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
+
+
+ + +
+
%% Complementary filters
+H1 = inv(1 + L);
+H2 = 1 - H1;
+
+
+ +
+zpk(H1) =
+            (s+3.842)^3 (s+153.6) (s+1.289e05)
+  -------------------------------------------------------
+  (s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
+
+zpk(H2) =
+         125.61 (s+3358)^2 (s^2 + 46.61s + 813.8)
+  -------------------------------------------------------
+  (s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272)
+
+ + +
+

hinf_filters_results_mixed_sensitivity.png +

+
+
+
+
+ +
+

4. Synthesis of three complementary filters

+
+
+

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

+ +
+
+ +
+

4.1. 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 11. +

+ + +
+

comp_filter_three_hinf_fb.png +

+

Figure 11: 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. +

+
+
+ +
+

4.2. Weights

+
+

+First we define the weights. +

+
+
%% Design of the Weighting Functions
+W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.5);
+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;
+W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.5);
+
+
+ + +
+

three_weighting_functions.png +

+

Figure 12: Three weighting functions used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters

+
+
+
+ +
+

4.3. H-Infinity Synthesis

+
+

+Then we create the generalized plant P. +

+
+
%% Generalized plant for the synthesis of 3 complementary filters
+P = [W1 -W1 -W1;
+     0   W2  0 ;
+     0   0   W3;
+     1   0   0];
+
+
+ +

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

+
+
%% Standard H-Infinity 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
+
+
+
+ +
+

4.4. 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 13: The three complementary filters obtained after \(\mathcal{H}_\infty\) synthesis

+
+
+
+
+ +

+ +

+ +

Bibliography

+
+
Dehaeze, Thomas, Mohit Vermat, and Christophe Collette. 2021. “A New Method of Designing Complementary Filters for Sensor Fusion Using the $H_\Infty$ Synthesis.” Mechanical Systems and Signal Processing, November.
+
Hua, Wensheng. 2005. “Low Frequency Vibration Isolation and Alignment System for Advanced LIGO.” stanford university.
+
+ + +
+

5. Functions

+
+
+ +
+

5.1. generateWF: Generate Weighting Functions

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [W] = generateWF(args)
+% createWeight -
+%
+% Syntax: [W] = generateWeight(args)
+%
+% Inputs:
+%    - n  - Weight Order (integer)
+%    - G0 - Low frequency Gain
+%    - G1 - High frequency Gain
+%    - Gc - Gain of the weight at frequency w0
+%    - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
+%
+% Outputs:
+%    - W - Generated Weighting Function
+
+
+
+
+ +
+

Optional Parameters

+
+
+
%% Argument validation
+arguments
+    args.n    (1,1) double {mustBeInteger, mustBePositive} = 1
+    args.G0   (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+    args.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
+    args.Gc   (1,1) double {mustBeNumeric, mustBePositive} = 1
+    args.w0   (1,1) double {mustBeNumeric, mustBePositive} = 1
+end
+
+
+ +

+Verification that the parameters \(G_0\), \(G_c\) and \(G_\infty\) are satisfy condition \eqref{eq:cond_formula_1} or \eqref{eq:cond_formula_2}. +

+\begin{equation} + G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1} +\end{equation} +\begin{equation} + G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2} +\end{equation} + +
+
% Verification of correct relation between G0, Gc and Ginf
+mustBeBetween(args.G0, args.Gc, args.Ginf);
+
+
+
+
+ +
+

Generate the Weighting function

+
+
+
%% Initialize the Laplace variable
+s = zpk('s');
+
+
+ +

+The weighting function formula use is: +

+\begin{equation} +\label{orge02c446} + W(s) = \left( \frac{ + \frac{1}{\omega_c} \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_c} \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} + +
+
%% Create the weighting function according to formula
+W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
+      (args.G0/args.Gc)^(1/args.n))/...
+     ((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
+      (1/args.Gc)^(1/args.n)))^args.n;
+
+
+
+
+ + +
+

Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains

+
+
+
%% Custom validation function
+function mustBeBetween(a,b,c)
+    if ~((a > b && b > c) || (c > b && b > a))
+        eid = 'createWeight:inputError';
+        msg = 'Gc should be between G0 and Ginf.';
+        throwAsCaller(MException(eid,msg))
+    end
+
+
+
+
+
+ +
+

5.2. generateCF: Generate Complementary Filters

+
+

+ +

+ +

+This Matlab function is accessible here. +

+
+ +
+

Function description

+
+
+
function [H1, H2] = generateCF(W1, W2, args)
+% createWeight -
+%
+% Syntax: [H1, H2] = generateCF(W1, W2, args)
+%
+% Inputs:
+%    - W1 - Weighting Function for H1
+%    - W2 - Weighting Function for H2
+%    - args:
+%      - method  - H-Infinity solver ('lmi' or 'ric')
+%      - display - Display synthesis results ('on' or 'off')
+%
+% Outputs:
+%    - H1 - Generated H1 Filter
+%    - H2 - Generated H2 Filter
+
+
+
+
+ +
+

Optional Parameters

+
+
+
%% Argument validation
+arguments
+    W1
+    W2
+    args.method  char {mustBeMember(args.method,{'lmi', 'ric'})} = 'ric'
+    args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
+end
+
+
+
+
+ +
+

H-Infinity Synthesis

+
+
+
%% The generalized plant is defined
+P = [W1 -W1;
+     0   W2;
+     1   0];
+
+
+ +
+
%% The standard H-infinity synthesis is performed
+[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
+
+
+ +
+
%% H1 is defined as the complementary of H2
+H1 = 1 - H2;
+
+
+
+
+
+
+
+
+

Author: Thomas Dehaeze

+

Created: 2021-09-01 mer. 11:26

+
+ + diff --git a/matlab/dehaeze21_desig_compl_filte_matlab.org b/matlab/dehaeze21_desig_compl_filte_matlab.org index a5e5bde..6316ea0 100644 --- a/matlab/dehaeze21_desig_compl_filte_matlab.org +++ b/matlab/dehaeze21_desig_compl_filte_matlab.org @@ -10,46 +10,34 @@ #+LaTeX_HEADER_EXTRA: \input{/home/thomas/Cloud/org-theme/preamble.tex} #+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+ :comments none #+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 - -#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") -#+PROPERTY: header-args:latex+ :imagemagick t :fit yes -#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 -#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 -#+PROPERTY: header-args:latex+ :results file raw replace -#+PROPERTY: header-args:latex+ :buffer no -#+PROPERTY: header-args:latex+ :tangle no -#+PROPERTY: header-args:latex+ :eval no-export -#+PROPERTY: header-args:latex+ :exports results -#+PROPERTY: header-args:latex+ :mkdirp yes -#+PROPERTY: header-args:latex+ :output-dir figs -#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") :END: * Introduction :ignore: +This file is the Matlab file for the paper cite:dehaeze21_new_method_desig_compl_filter. + 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 +- 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 +- in section [[#sec:closed_loop_complementary_filters]] * 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 +:header-args:matlab+: :tangle matlab/1_synthesis_complementary_filters.m :CUSTOM_ID: sec:h_inf_synthesis_complementary_filters :END: ** Introduction :ignore: #+begin_note - The Matlab file corresponding to this section is accessible [[file:matlab/h_inf_synthesis_complementary_filters.m][here]]. + The Matlab file corresponding to this section is accessible [[file:matlab/1_synthesis_complementary_filters.m][here]]. #+end_note ** Matlab Init :noexport:ignore: @@ -62,6 +50,7 @@ This document is divided into several sections: #+end_src #+begin_src matlab +%% Initialize Frequency Vector freqs = logspace(-1, 3, 1000); #+end_src @@ -70,7 +59,8 @@ addpath('./matlab'); addpath('./matlab/src'); #+end_src -#+begin_src matlab :exec no +#+begin_src matlab :eval no +%% Add functions to path addpath('./src'); #+end_src @@ -82,7 +72,7 @@ In order to do so, we use the generalized plant shown on figure [[fig:h_infinity #+name: fig:h_infinity_robst_fusion #+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters -[[file:figs/h_infinity_robust_fusion.png]] +[[file:figs-journal/h_infinity_robust_fusion_plant.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: \[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \] @@ -116,10 +106,16 @@ The parameters permits to specify: The general shape of a weighting function generated using the formula is shown in figure [[fig:weight_formula]]. #+begin_src matlab :exports none +%% Weighting Function Design +% Parameters n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2; +% Formulas W = (((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 +%% Magnitude of the weighting function with parameters figure; hold on; plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-'); @@ -147,6 +143,10 @@ ylim([5e-4, 20]); yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]); #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/weight_formula.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/weight_formula.pdf', 'width', 'wide', 'height', 'normal'); #+end_src @@ -156,15 +156,16 @@ exportFig('figs/weight_formula.pdf', 'width', 'wide', 'height', 'normal'); #+RESULTS: [[file:figs/weight_formula.png]] -#+begin_src matlab -n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 0.45; -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; +** Example -n = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.45; -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; +#+begin_src matlab +%% Design of the Weighting Functions +W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45); +W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45); #+end_src #+begin_src matlab :exports none +%% Plot of the Weighting function magnitude figure; tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile(); @@ -186,8 +187,12 @@ leg = legend('location', 'south', 'FontSize', 8); leg.ItemTokenSize(1) = 18; #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/weights_W1_W2.pdf', 'width', 'half', 'height', 350); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/weights_W1_W2.pdf', 'width', 'half', 'height', 350); +exportFig('figs/weights_W1_W2.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:weights_W1_W2 @@ -198,6 +203,7 @@ exportFig('figs/weights_W1_W2.pdf', 'width', 'half', 'height', 350); ** H-Infinity Synthesis We define the generalized plant $P$ on matlab. #+begin_src matlab +%% Generalized Plant P = [W1 -W1; 0 W2; 1 0]; @@ -205,6 +211,7 @@ P = [W1 -W1; And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. #+begin_src matlab :results output replace :exports both +%% H-Infinity Synthesis [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); #+end_src @@ -238,9 +245,15 @@ And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. 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 +%% Define H1 to be the complementary of H2 H1 = 1 - H2; #+end_src +Or one can just used to =generateCF= Matlab function: +#+begin_src matlab :tangle no :eval no +[H1, H2] = generateCF(W1, W2); +#+end_src + ** Obtained Complementary Filters The obtained complementary filters are shown on figure [[fig:hinf_filters_results]]. @@ -267,6 +280,7 @@ ans = #+end_example #+begin_src matlab :exports none +%% Bode plot of the complementary filters figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -309,8 +323,12 @@ linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/hinf_filters_results.pdf', 'width', 700, 'height', 450); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/hinf_filters_results.pdf', 'width', 700, 'height', 450); +exportFig('figs/hinf_filters_results.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:hinf_filters_results @@ -318,223 +336,15 @@ exportFig('figs/hinf_filters_results.pdf', 'width', 700, 'height', 450); #+RESULTS: [[file:figs/hinf_filters_results.png]] -* Generating 3 complementary filters +* Design of complementary filters used in the Active Vibration Isolation System at the LIGO :PROPERTIES: -:header-args:matlab+: :tangle matlab/three_comp_filters.m -:header-args:matlab+: :comments org :mkdirp yes +:header-args:matlab+: :tangle matlab/2_ligo_complementary_filters.m +:CUSTOM_ID: sec:comp_filters_ligo :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, 3, 1000); -#+end_src - -#+begin_src matlab :tangle no -addpath('./matlab'); -addpath('./matlab/src'); -#+end_src - -#+begin_src matlab :exec no -addpath('./src'); -#+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/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)]); ylim([2e-4, 1.3e1]) -leg = legend('location', 'northeast', 'FontSize', 8); -leg.ItemTokenSize(1) = 18; -#+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]); -leg = legend('location', 'northeast', 'FontSize', 8); -leg.ItemTokenSize(1) = 18; - -% 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([-180:90:180]); ylim([-220, 220]); - -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', 600); -#+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]]. + The Matlab file corresponding to this section is accessible [[file:matlab/2_ligo_complementary_filters.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. @@ -551,6 +361,7 @@ The FIR complementary filters designed in cite:hua05_low_ligo are of order 512. #+end_src #+begin_src matlab +%% Initialize Frequency Vector freqs = logspace(-3, 0, 1000); #+end_src @@ -559,7 +370,8 @@ addpath('./matlab'); addpath('./matlab/src'); #+end_src -#+begin_src matlab :exec no +#+begin_src matlab :eval no +%% Add functions to path addpath('./src'); #+end_src @@ -573,6 +385,7 @@ The specifications for the filters are: The specifications are translated in upper bounds of the complementary filters are shown on figure [[fig:ligo_specifications]]. #+begin_src matlab :exports none +%% Upper bounds for the complementary filters figure; hold on; set(gca,'ColorOrderIndex',1) @@ -607,12 +420,14 @@ 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 +%% Initialized CVX 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 +%% Frequency vectors w1 = 0:4.06e-4:0.008; w2 = 0.008:4.06e-4:0.04; w3 = 0.04:8.12e-4:0.1; @@ -621,10 +436,12 @@ w4 = 0.1:8.12e-4:0.83; We then define the order of the FIR filter. #+begin_src matlab +%% Filter order n = 512; #+end_src #+begin_src matlab +%% Initialization of filter responses 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]))]; @@ -638,6 +455,7 @@ B4 = [zeros(length(w4),1), sin(kron(w4'.*(2*pi),[1:n-1]))]; We run the convex optimization. #+begin_src matlab :results output replace :wrap example +%% Convex optimization cvx_begin variable y(n+1,1) @@ -746,11 +564,13 @@ h = y(2:end); 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 +%% Combine the frequency vectors to form the obtained filter w = [w1 w2 w3 w4]; H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h; #+end_src #+begin_src matlab :exports none +%% Bode plot of the obtained complementary filters figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -788,8 +608,12 @@ linkaxes([ax1,ax2],'x'); xlim([1e-3, 1]); #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/fir_filter_ligo.pdf', 'width', 'wide', 'height', 600); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/fir_filter_ligo.pdf', 'width', 'wide', 'height', 600); +exportFig('figs/fir_filter_ligo.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:fir_filter_ligo @@ -808,6 +632,7 @@ Here are the requirements on the filters: The bode plot of the weights is shown on figure [[fig:ligo_weights]]. #+begin_src matlab :exports none +%% Design of the weight for the high pass filter w1 = 2*pi*0.008; x1 = 0.35; w2 = 2*pi*0.04; x2 = 0.5; w3 = 2*pi*0.05; x3 = 0.5; @@ -827,7 +652,12 @@ wH = minreal(ss(wH)); #+end_src #+begin_src matlab :exports none -n = 20; Rp = 1; Wp = 2*pi*0.102; +%% Design of the weight for the low pass filter +n = 20; % Filter order +Rp = 1; % Peak to peak passband ripple +Wp = 2*pi*0.102; % Edge frequency + +% Chebyshev Type I filter design [b,a] = cheby1(n, Rp, Wp, 'high', 's'); wL = 0.04*tf(a, b); @@ -836,6 +666,7 @@ wL = minreal(ss(wL)); #+end_src #+begin_src matlab :exports none +%% Magnitude of the designed Weights and initial specifications figure; hold on; set(gca,'ColorOrderIndex',1); @@ -857,6 +688,10 @@ leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); leg.ItemTokenSize(1) = 16; #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/ligo_weights.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/ligo_weights.pdf', 'width', 'wide', 'height', 'normal'); #+end_src @@ -869,6 +704,7 @@ exportFig('figs/ligo_weights.pdf', 'width', 'wide', 'height', 'normal'); ** H-Infinity Synthesis We define the generalized plant as shown on figure [[fig:h_infinity_robst_fusion]]. #+begin_src matlab +%% Generalized plant for the H-infinity Synthesis P = [0 wL; wH -wH; 1 0]; @@ -876,6 +712,7 @@ P = [0 wL; And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. #+begin_src matlab :results output replace :exports both :wrap example +%% Standard H-Infinity synthesis [Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); #+end_src @@ -905,17 +742,19 @@ Test bounds: 0.3276 < gamma <= 1.8063 The high pass filter is defined as $H_H = 1 - H_L$. #+begin_src matlab +%% High pass filter as the complementary of the low pass filter Hh = 1 - Hl; #+end_src #+begin_src matlab :exports none +%% Minimum realization of the filters 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 +#+begin_src matlab :exports results :results output replace :wrap example :tangle no size(Hh), size(Hl) #+end_src @@ -929,6 +768,7 @@ State-space model with 1 outputs, 1 inputs, and 27 states. The bode plot of the obtained filters as shown on figure [[fig:hinf_synthesis_ligo_results]]. #+begin_src matlab :exports none +%% Bode plot of the obtained filters and comparison with the upper bounds figure; hold on; set(gca,'ColorOrderIndex',1); @@ -963,6 +803,7 @@ exportFig('figs/hinf_synthesis_ligo_results.pdf', 'width', 'wide', 'height', 'no 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 +%% Comparison of the complementary filters obtained with H-infinity and with CVX figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -1011,8 +852,12 @@ linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 600); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 600); +exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:comp_fir_ligo_hinf @@ -1020,8 +865,17 @@ exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 600); #+RESULTS: [[file:figs/comp_fir_ligo_hinf.png]] -* Alternative Synthesis +* "Closed-Loop" complementary filters +:PROPERTIES: +:header-args:matlab+: :tangle matlab/3_closed_loop_complementary_filters.m +:CUSTOM_ID: sec:closed_loop_complementary_filters +:END: + ** Introduction :ignore: +#+begin_note + The Matlab file corresponding to this section is accessible [[file:matlab/3_closed_loop_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) <> @@ -1031,327 +885,43 @@ exportFig('figs/comp_fir_ligo_hinf.pdf', 'width', 'wide', 'height', 600); <> #+end_src +#+begin_src matlab +%% Initialize Frequency Vector +freqs = logspace(-1, 3, 1000); +#+end_src + #+begin_src matlab :tangle no addpath('./matlab'); addpath('./matlab/src'); #+end_src #+begin_src matlab :eval no +%% Add functions to path addpath('./src'); #+end_src -** Two generalized plants -In order to synthesize the complementary filter using the proposed method, we can use two alternative generalized plant as shown in Figures [[fig:h_infinity_arch_1]] and [[fig:h_infinity_arch_2]]. - -\begin{equation} - P_1 = \begin{bmatrix} W_1 & -W_1 \\ 0 & W_2 \\ 1 & 0 \end{bmatrix} -\end{equation} - - -#+begin_src latex :file h_infinity_arch_1.pdf - \begin{tikzpicture} - \node[block={4.5cm}{3.0cm}, fill=black!20!white, dashed] (P) {}; - \node[above] at (P.north) {$P_1(s)$}; - - \coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$); - \coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$); - - \coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$); - \coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$); - \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$); - - \node[block, left=1.4 of output1] (W1){$W_1(s)$}; - \node[block, left=1.4 of output2] (W2){$W_2(s)$}; - \node[addb={+}{}{}{}{-}, left=of W1] (sub) {}; - - \node[block, below=0.3 of P] (H2) {$H_2(s)$}; - - \draw[->] (inputw) node[above right]{$w$} -- (sub.west); - \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west); - \draw[->] (inputu-|sub) node[branch]{} -- (sub.south); - \draw[->] (sub.east) -- (W1.west); - \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east); - \draw[->] (W1.east) -- (output1)node[above left]{$z_1$}; - \draw[->] (W2.east) -- (output2)node[above left]{$z_2$}; - \end{tikzpicture} -#+end_src - -#+name: fig:h_infinity_arch_1 -#+caption: Complementary Filter Synthesis - Conf 1 -#+RESULTS: -[[file:figs/h_infinity_arch_1.png]] - -\begin{equation} - P_2 = \begin{bmatrix} 0 & W_1 & 1 \\ W_2 & -W_1 & 0 \end{bmatrix} -\end{equation} - -#+begin_src latex :file h_infinity_arch_2.pdf -\begin{tikzpicture} - \node[block={4.5cm}{4.5cm}, fill=black!20!white, dashed] (P) {}; - \node[above] at (P.north) {$P_2(s)$}; - - \coordinate[] (input2) at ($(P.south west)!0.85!(P.north west) + (-0.7, 0)$); - \coordinate[] (input1) at ($(P.south west)!0.55!(P.north west) + (-0.7, 0)$); - \coordinate[] (inputu) at ($(P.south west)!0.3!( P.north west) + (-0.7, 0)$); - - \coordinate[] (outputz) at ($(P.south east)!0.3!(P.north east) + (0.7, 0)$); - \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + (0.7, 0)$); - - \node[block, right=1.4 of input2] (W2){$W_2(s)$}; - \node[block, right=1.4 of input1] (W1){$W_1(s)$}; - \node[addb={+}{-}{}{}{}, right=of W1] (sub) {}; - \node[addb, left=2.5 of outputz] (add) {}; - - \node[block, below=0.3 of P] (H2) {$H_2(s)$}; - - \draw[->] (input2) node[above right]{$w_2$} -- (W2.west); - \draw[->] (input1) node[above right]{$w_1$} -- (W1.west); - \draw[->] (W2.east) -| (sub.north); - \draw[->] (W1.east) -- (sub.west); - \draw[->] (W1-|add)node[branch]{} -- (add.north); - \draw[->] (sub.south) |- (outputv) node[above left]{$v$} |- (H2.east); - \draw[->] (H2.west) -| (inputu) node[above right]{$u$} -- (add.west); - \draw[->] (add.east) -- (outputz) node[above left]{$z$}; -\end{tikzpicture} -#+end_src - -#+name: fig:h_infinity_arch_2 -#+caption: Complementary Filter Synthesis - Conf 2 -#+RESULTS: -[[file:figs/h_infinity_arch_2.png]] - -Let's run the $\mathcal{H}_\infty$ synthesis for both generalized plant using the same weights and see if the obtained filters are the same: -#+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 -P1 = [W1 -W1; - 0 W2; - 1 0]; -#+end_src - -#+begin_src matlab :results output replace :exports both -[H2, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -#+end_src - -#+RESULTS: -#+begin_example -[H2, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); - - Test bounds: 0.3263 <= gamma <= 1000 - - gamma X>=0 Y>=0 rho(XY)<1 p/f - 1.807e+01 1.4e-07 0.0e+00 1.185e-18 p - 2.428e+00 1.5e-07 0.0e+00 1.285e-18 p - 8.902e-01 -2.9e+02 # -7.1e-17 5.168e-19 f - 1.470e+00 1.5e-07 0.0e+00 1.462e-14 p - 1.144e+00 1.5e-07 0.0e+00 1.260e-14 p - 1.009e+00 1.5e-07 0.0e+00 4.120e-13 p - 9.478e-01 -6.8e+02 # -2.4e-17 1.449e-14 f - 9.780e-01 -1.6e+03 # -7.3e-17 6.791e-14 f - 9.934e-01 -4.2e+03 # -1.2e-16 3.524e-14 f - 1.001e+00 -2.0e+04 # -2.3e-17 5.717e-20 f - 1.005e+00 1.5e-07 0.0e+00 8.953e-18 p - 1.003e+00 -2.2e+05 # -1.8e-17 3.225e-12 f - 1.004e+00 1.5e-07 0.0e+00 2.445e-12 p - Limiting gains... - 1.004e+00 1.6e-07 0.0e+00 5.811e-18 p - - Best performance (actual): 1.004 -#+end_example - -#+begin_src matlab -P2 = [0 W1 1; - W2 -W1 0]; -#+end_src - -#+begin_src matlab :results output replace :exports both -[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -#+end_src - -#+RESULTS: -#+begin_example -[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); - - Test bounds: 0.3263 <= gamma <= 1000 - - gamma X>=0 Y>=0 rho(XY)<1 p/f - 1.807e+01 0.0e+00 1.4e-07 2.055e-16 p - 2.428e+00 0.0e+00 1.4e-07 1.894e-18 p - 8.902e-01 -2.1e-16 -2.7e+02 # 1.466e-16 f - 1.470e+00 0.0e+00 1.4e-07 4.118e-16 p - 1.144e+00 0.0e+00 1.5e-07 2.105e-18 p - 1.009e+00 0.0e+00 1.5e-07 2.590e-13 p - 9.478e-01 -9.5e-17 -6.3e+02 # 1.663e-19 f - 9.780e-01 -1.1e-16 -1.5e+03 # 1.546e-14 f - 9.934e-01 -2.8e-17 -4.0e+03 # 3.934e-14 f - 1.001e+00 -3.1e-17 -1.9e+04 # 1.191e-19 f - 1.005e+00 0.0e+00 1.5e-07 1.443e-12 p - 1.003e+00 -8.3e-17 -2.1e+05 # 8.807e-13 f - 1.004e+00 0.0e+00 1.5e-07 1.459e-15 p - Limiting gains... - 1.004e+00 0.0e+00 1.5e-07 9.086e-19 p - - Best performance (actual): 1.004 -#+end_example - -And indeed, we can see that the exact same filters are obtained (Figure [[fig:hinf_comp_P1_P2_syn]]). -#+begin_src matlab :exports none -freqs = logspace(-2, 4, 1000); - -figure; -hold on; -plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); -plot(freqs, abs(squeeze(freqresp(H2b, freqs, 'Hz'))), '--', 'DisplayName', '$H_{2b}$'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude'); -legend('location', 'southeast', 'FontSize', 8); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/hinf_comp_P1_P2_syn.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:hinf_comp_P1_P2_syn -#+caption: Comparison of $H_2(s)$ when using $P_1(s)$ or $P_2(s)$ -#+RESULTS: -[[file:figs/hinf_comp_P1_P2_syn.png]] - -** Shaping the Low pass filter or the high pass filter? -Let's see if there is a difference by explicitly shaping $H_1(s)$ or $H_2(s)$. - -#+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 - -Let's first synthesize $H_1(s)$: -#+begin_src matlab -P1 = [W2 -W2; - 0 W1; - 1 0]; -#+end_src - -#+begin_src matlab :results output replace :exports both -[H1, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -#+end_src - -#+RESULTS: -#+begin_example -[H1, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); - - Test bounds: 0.3263 <= gamma <= 1.712 - - gamma X>=0 Y>=0 rho(XY)<1 p/f - 7.476e-01 -2.5e+01 # -8.3e-18 4.938e-20 f - 1.131e+00 1.9e-07 0.0e+00 1.566e-16 p - 9.197e-01 -1.4e+02 # -7.9e-17 4.241e-17 f - 1.020e+00 1.9e-07 0.0e+00 2.095e-16 p - 9.686e-01 -3.8e+02 # -7.0e-17 1.463e-23 f - 9.940e-01 -1.5e+03 # -1.3e-17 3.168e-19 f - 1.007e+00 1.9e-07 0.0e+00 1.696e-15 p - 1.000e+00 -4.8e+03 # -7.1e-18 7.203e-20 f - 1.004e+00 1.9e-07 0.0e+00 1.491e-14 p - 1.002e+00 -1.1e+04 # -2.6e-16 2.579e-14 f - 1.003e+00 -2.8e+04 # -6.0e-18 8.558e-20 f - Limiting gains... - 1.004e+00 2.0e-07 0.0e+00 5.647e-18 p - 1.004e+00 1.0e-06 0.0e+00 5.648e-18 p - - Best performance (actual): 1.004 -#+end_example - -And now $H_2(s)$: -#+begin_src matlab -P2 = [W1 -W1; - 0 W2; - 1 0]; -#+end_src - -#+begin_src matlab :results output replace :exports both -[H2, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -#+end_src - -#+RESULTS: -#+begin_example -[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); - - Test bounds: 0.3263 <= gamma <= 1000 - - gamma X>=0 Y>=0 rho(XY)<1 p/f - 1.807e+01 1.4e-07 0.0e+00 1.185e-18 p - 2.428e+00 1.5e-07 0.0e+00 1.285e-18 p - 8.902e-01 -2.9e+02 # -7.1e-17 5.168e-19 f - 1.470e+00 1.5e-07 0.0e+00 1.462e-14 p - 1.144e+00 1.5e-07 0.0e+00 1.260e-14 p - 1.009e+00 1.5e-07 0.0e+00 4.120e-13 p - 9.478e-01 -6.8e+02 # -2.4e-17 1.449e-14 f - 9.780e-01 -1.6e+03 # -7.3e-17 6.791e-14 f - 9.934e-01 -4.2e+03 # -1.2e-16 3.524e-14 f - 1.001e+00 -2.0e+04 # -2.3e-17 5.717e-20 f - 1.005e+00 1.5e-07 0.0e+00 8.953e-18 p - 1.003e+00 -2.2e+05 # -1.8e-17 3.225e-12 f - 1.004e+00 1.5e-07 0.0e+00 2.445e-12 p - Limiting gains... - 1.004e+00 1.6e-07 0.0e+00 5.811e-18 p - - Best performance (actual): 1.004 -#+end_example - -And compare $H_1(s)$ with $1 - H_2(s)$ and $H_2(s)$ with $1 - H_1(s)$ in Figure [[fig:hinf_comp_H1_H2_syn]]. -#+begin_src matlab :exports none -freqs = logspace(-2, 4, 1000); - -figure; -hold on; -plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); -plot(freqs, abs(squeeze(freqresp(1-H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); -plot(freqs, abs(squeeze(freqresp(1-H2, freqs, 'Hz'))), '--', 'DisplayName', '$H_2$'); -plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '--', 'DisplayName', '$H_1$'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude'); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/hinf_comp_H1_H2_syn.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:hinf_comp_H1_H2_syn -#+caption: Comparison of $H_1(s)$ with $1-H_2(s)$, and $H_2(s)$ with $1-H_1(s)$ -#+RESULTS: -[[file:figs/hinf_comp_H1_H2_syn.png]] - ** Using Feedback architecture #+begin_src matlab -n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 0.45; -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 = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.45; -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; +%% Design of the Weighting Functions +W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45); +W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45); #+end_src Let's first synthesize $H_1(s)$: #+begin_src matlab +%% Generalized plant for "closed-loop" complementary filter synthesis P = [ W1 0 1; -W1 W2 -1]; #+end_src #+begin_src matlab :results output replace :exports both +%% Standard H-Infinity Synthesis [L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); #+end_src + #+begin_src matlab +%% Complementary filters H1 = inv(1 + L); H2 = 1 - H1; #+end_src @@ -1363,27 +933,19 @@ zpk(H2) #+RESULTS: #+begin_example -zpk(H1) - -ans = - +zpk(H1) = (s+3.842)^3 (s+153.6) (s+1.289e05) ------------------------------------------------------- (s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272) -Continuous-time zero/pole/gain model. -zpk(H2) - -ans = - +zpk(H2) = 125.61 (s+3358)^2 (s^2 + 46.61s + 813.8) ------------------------------------------------------- (s+1.29e05) (s^2 + 102.1s + 2733) (s^2 + 69.45s + 3272) - -Continuous-time zero/pole/gain model. #+end_example #+begin_src matlab :exports none +%% Bode plot of the obtained Complementary filters with upper-bounds freqs = logspace(-1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -1429,8 +991,12 @@ linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/hinf_filters_results_mixed_sensitivity.pdf', 'width', 700, 'height', 600); +#+end_src + #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/hinf_filters_results_mixed_sensitivity.pdf', 'width', 700 , 'height', 600); +exportFig('figs/hinf_filters_results_mixed_sensitivity.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:hinf_filters_results_mixed_sensitivity @@ -1438,846 +1004,17 @@ exportFig('figs/hinf_filters_results_mixed_sensitivity.pdf', 'width', 700 , 'h #+RESULTS: [[file:figs/hinf_filters_results_mixed_sensitivity.png]] -#+begin_src matlab :exports none -freqs = logspace(-2, 4, 1000); +* Synthesis of three complementary filters +:PROPERTIES: +:header-args:matlab+: :tangle matlab/4_three_complementary_filters.m +:header-args:matlab+: :comments org :mkdirp yes +:CUSTOM_ID: sec:three_comp_filters +:END: -figure; -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$'); -plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$'); -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$|H_1|$'); -plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$|H_2|$'); -plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), 'k--', 'DisplayName', '$|L|$'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude'); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); -#+end_src - -** Adding feature in the filters - -#+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 -Wf = (1 + s/2/pi/1)/s; -Wf = s/(1 + s/2/pi/1e2); - -% W2 = W2/Wf/(1 + s/2/pi/1e3); -#+end_src - -#+begin_src matlab -P = [W1 -Wf*W1; - 0 Wf*W2; - 1 -Wf]; -#+end_src - -#+begin_src matlab :results output replace :exports both -[Ka, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -#+end_src - -#+begin_src matlab -K = Ka*Wf; -#+end_src - -#+begin_src matlab -H1 = inv(1 + K); -H2 = 1 - H1; -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(-2, 4, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); -plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); -set(gca,'ColorOrderIndex',1); -plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$'); -plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude'); -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'))), '-'); -hold off; -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -set(gca, 'XScale', 'log'); -yticks([-180:90:180]); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - - -** Compare "open-loop" shaping with "close-loop" shaping -*** Simple weights -#+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 -Pol = [0 W1 1; - W2 -W1 0]; -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('The number of states of Pol is %i', length(Pol.StateName)) -#+end_src - -#+RESULTS: -: The number of states of Pol is 7 - -#+begin_src matlab :results output replace :exports both -tic; -[Hol, ~, gamma, ~] = hinfsyn(Pol, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hol_2 = Hol; -Hol_1 = 1 - Hol; -#+end_src - -#+begin_src matlab -Pcl = [0 W2 1 - W1 -W2 -1]; -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('The number of states of Pcl is %i', length(Pcl.StateName)) -#+end_src - -#+RESULTS: -: The number of states of Pcl is 8 - -#+begin_src matlab :results output replace :exports both -tic; -[Hcl, ~, gamma, ~] = hinfsyn(Pcl, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hcl_1 = 1 - 1/(1 + Hcl); -Hcl_2 = 1/(1 + Hcl); -#+end_src - -#+begin_src matlab :results output replace :exports results -size(Hol_1) -size(Hol_2) -size(Hcl_1) -size(Hcl_2) -#+end_src - -#+RESULTS: -#+begin_example -size(Hol_1) -State-space model with 1 outputs, 1 inputs, and 5 states. -size(Hol_2) -State-space model with 1 outputs, 1 inputs, and 5 states. -size(Hcl_1) -State-space model with 1 outputs, 1 inputs, and 5 states. -size(Hcl_2) -State-space model with 1 outputs, 1 inputs, and 5 states. -'org_babel_eoe' -ans = - 'org_babel_eoe' -#+end_example - -#+begin_src matlab :exports none -freqs = logspace(-1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hol_1, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ OL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hol_2, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ OL'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hcl_1, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ CL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hcl_2, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ CL'); - -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Magnitude'); -set(gca, 'XTickLabel',[]); -leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 16; - -% Phase -ax2 = nexttile; -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_1, freqs, 'Hz')))), '-'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_2, freqs, 'Hz')))), '-'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_1, freqs, 'Hz')))), '--'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_2, freqs, 'Hz')))), '--'); -set(gca, 'XScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks([-450:90:180]); ylim([-450, 200]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -*** Simple weights with LMI -#+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 -Pol = [0 W1 1; - W2 -W1 0]; -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('The number of states of Pol is %i', length(Pol.StateName)) -#+end_src - -#+RESULTS: -: The number of states of Pol is 7 - -#+begin_src matlab :results output replace :exports both -tic; -[Hol, ~, gamma, ~] = hinfsyn(Pol, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hol_2 = Hol; -Hol_1 = 1 - Hol; -#+end_src - -#+begin_src matlab -Pcl = [0 W2 1 - W1 -W2 -1]; -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -sprintf('The number of states of Pcl is %i', length(Pcl.StateName)) -#+end_src - -#+RESULTS: -: The number of states of Pcl is 8 - -#+begin_src matlab :results output replace :exports both -tic; -[Hcl, ~, gamma, ~] = hinfsyn(Pcl, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hcl_1 = 1 - 1/(1 + Hcl); -Hcl_2 = 1/(1 + Hcl); -#+end_src - -#+begin_src matlab :results output replace :exports results -size(Hol_1) -size(Hol_2) -size(Hcl_1) -size(Hcl_2) -#+end_src - -#+RESULTS: -#+begin_example -size(Hol_1) -State-space model with 1 outputs, 1 inputs, and 5 states. -size(Hol_2) -State-space model with 1 outputs, 1 inputs, and 5 states. -size(Hcl_1) -State-space model with 1 outputs, 1 inputs, and 5 states. -size(Hcl_2) -State-space model with 1 outputs, 1 inputs, and 5 states. -'org_babel_eoe' -ans = - 'org_babel_eoe' -#+end_example - -#+begin_src matlab :exports none -freqs = logspace(-1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hol_1, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ OL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hol_2, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ OL'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hcl_1, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ CL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hcl_2, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ CL'); - -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Magnitude'); -set(gca, 'XTickLabel',[]); -leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 16; - -% Phase -ax2 = nexttile; -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_1, freqs, 'Hz')))), '-'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_2, freqs, 'Hz')))), '-'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_1, freqs, 'Hz')))), '--'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_2, freqs, 'Hz')))), '--'); -set(gca, 'XScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks([-450:90:180]); ylim([-450, 200]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -*** Complex 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; -W1 = 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; -W2 = wL; -#+end_src - -#+begin_src matlab -Pol = ss([0 W1 1; - W2 -W1 0]); -#+end_src - -#+begin_src matlab -Pcl = ss([0 W2 1 - W1 -W2 -1]); -#+end_src - -#+begin_src matlab :results output replace :exports results -size(Pol) -size(Pcl) -#+end_src - -#+RESULTS: -: size(Pol) -: State-space model with 2 outputs, 3 inputs, and 27 states. -: size(Pcl) -: State-space model with 2 outputs, 3 inputs, and 27 states. - -#+begin_src matlab :results output replace :exports both -tic; -for iter = 1:10 - [Hol, ~, gamma, ~] = hinfsyn(Pol, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'off'); -end; -toc; -#+end_src - -#+begin_src matlab -Hol_1 = 1 - Hol; -Hol_2 = Hol; -#+end_src - -#+begin_src matlab :results output replace :exports both -tic; -for iter = 1:10 - [Hcl, ~, gamma, ~] = hinfsyn(Pcl, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'off'); -end -toc; -#+end_src - -#+begin_src matlab -Hcl_1 = 1 - 1/(1 + Hcl); -Hcl_2 = 1/(1 + Hcl); -#+end_src - -#+begin_src matlab :results output replace :exports results -size(Hol_1) -size(Hol_2) -size(Hcl_1) -size(Hcl_2) -#+end_src - -#+RESULTS: -#+begin_example -size(Hol_1) -State-space model with 1 outputs, 1 inputs, and 27 states. -size(Hol_2) -State-space model with 1 outputs, 1 inputs, and 27 states. -size(Hcl_1) -State-space model with 1 outputs, 1 inputs, and 27 states. -size(Hcl_2) -State-space model with 1 outputs, 1 inputs, and 27 states. -'org_babel_eoe' -ans = - 'org_babel_eoe' -#+end_example - -#+begin_src matlab :exports none -freqs = logspace(-3, 1, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hol_1, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ OL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hol_2, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ OL'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hcl_1, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ CL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hcl_2, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ CL'); - -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Magnitude'); -set(gca, 'XTickLabel',[]); -ylim([5e-3, 10]); -leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 16; - -% Phase -ax2 = nexttile; -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_1, freqs, 'Hz')))), '-'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_2, freqs, 'Hz')))), '-'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_1, freqs, 'Hz')))), '--'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_2, freqs, 'Hz')))), '--'); -set(gca, 'XScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks([-450:90:180]); ylim([-450, 200]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -*** Complex weights with alternative conf - -#+begin_src matlab -Pcl = ss([W1 -W1; - 0 W2; - 1 -1]); -#+end_src - -#+begin_src matlab :results output replace :exports both -tic; -[Hcl, ~, gamma, ~] = hinfsyn(Pcl, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hcl_1 = 1 - 1/(1 + Hcl); -Hcl_2 = 1/(1 + Hcl); -#+end_src - - - -*** Complex weights with LMI -#+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; -W1 = ss(minreal(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; -W2 = ss(minreal(wL)); -#+end_src - -#+begin_src matlab -Pol = [0 W1 1; - W2 -W1 0]; -#+end_src - -#+begin_src matlab -Pcl = [0 W2 1 - W1 -W2 -1]; -#+end_src - -#+begin_src matlab :results output replace :exports both -tic; -[Hol, ~, gamma, ~] = hinfsyn(Pol, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hol_1 = 1 - Hol; -Hol_2 = Hol; -#+end_src - -#+begin_src matlab :results output replace :exports both -tic; -[Hcl, ~, gamma, ~] = hinfsyn(Pcl, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -toc; -#+end_src - -#+begin_src matlab -Hcl_1 = 1 - 1/(1 + Hcl); -Hcl_2 = 1/(1 + Hcl); -#+end_src - -#+begin_src matlab :results output replace :exports results -size(Hol_1) -size(Hol_2) -size(Hcl_1) -size(Hcl_2) -#+end_src - -#+RESULTS: -#+begin_example -size(Hol_1) -State-space model with 1 outputs, 1 inputs, and 34 states. -size(Hol_2) -State-space model with 1 outputs, 1 inputs, and 34 states. -size(Hcl_1) -State-space model with 1 outputs, 1 inputs, and 47 states. -size(Hcl_2) -State-space model with 1 outputs, 1 inputs, and 47 states. -'org_babel_eoe' -ans = - 'org_babel_eoe' -#+end_example - -#+begin_src matlab :exports none -freqs = logspace(-3, 1, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hol_1, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ OL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hol_2, freqs, 'Hz'))), '-', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ OL'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, abs(squeeze(freqresp(Hcl_1, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_H(s)$ - $\mathcal{H}_\infty$ CL'); -set(gca,'ColorOrderIndex',2); -plot(freqs, abs(squeeze(freqresp(Hcl_2, freqs, 'Hz'))), '--', ... - 'DisplayName', '$H_L(s)$ - $\mathcal{H}_\infty$ CL'); - -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Magnitude'); -set(gca, 'XTickLabel',[]); -ylim([5e-3, 10]); -leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 16; - -% Phase -ax2 = nexttile; -hold on; -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_1, freqs, 'Hz')))), '-'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hol_2, freqs, 'Hz')))), '-'); - -set(gca,'ColorOrderIndex',1); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_1, freqs, 'Hz')))), '--'); -set(gca,'ColorOrderIndex',2); -plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hcl_2, freqs, 'Hz')))), '--'); -set(gca, 'XScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks([-450:90:180]); ylim([-450, 200]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -*** Conclusion -#+begin_important -There is no difference between " open-loop" shaping and "close-loop" shaping: -- same " solving" time -- same obtained filter orders -#+end_important - -** Integral Action -*** Test -#+begin_src matlab -freqs = logspace(-2, 3, 1000); -#+end_src - -\begin{equation} -W_1(s) = \frac{W_{10}(s)}{s} -\end{equation} -with $W_{10}(0) \neq 0$ - -#+begin_src matlab -W10 = 0.1*(s+125.7)^2/(s+1.257)^2*(s + 0.0001)/(1 + s/1000); -W1 = W10/s; -#+end_src - -#+begin_src matlab -n = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.5; -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([1e-4, 20]); -xticks([0.1, 1, 10, 100, 1000]); -leg = legend('location', 'southeast', 'FontSize', 8); -leg.ItemTokenSize(1) = 18; -#+end_src - -#+begin_src matlab -V = 1; -#+end_src - -#+begin_src matlab -P = [ V*W10 W10; - 0 W2; - -V -1]; -#+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', 'lmi', 'DISPLAY', 'on'); -#+end_src - -#+begin_src matlab -H1 = 1 - H2; -#+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', '$w_1$'); -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]); -leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 18; - -% 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 - -*** Example 6.7.3 -#+begin_src matlab -n = 2; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; Gc = 0.45; -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 = 2; w0 = 2*pi*10; G0 = 1/10; G1 = 1000; Gc = 0.45; -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 -V = 1/(1 + s/2/pi/100); -#+end_src - -#+begin_src matlab -V = W2; -W2 = tf(1); -#+end_src - -#+begin_src matlab -P = [W1 -W1; - 0 W2; - V 0]; -#+end_src - -#+begin_src matlab :results output replace :exports both -[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); -#+end_src - -#+begin_src matlab -H2 = H2*V; -#+end_src - -#+begin_src matlab -H1 = 1 - H2; -#+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', '$w_1$'); -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]); -leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 18; - -% 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 - -* Impose a positive slope at DC or a negative slope at infinite frequency ** Introduction :ignore: +#+begin_note + The Matlab file corresponding to this section is accessible [[file:matlab/4_three_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) @@ -2288,82 +1025,140 @@ xlim([freqs(1), freqs(end)]); <> #+end_src +#+begin_src matlab +freqs = logspace(-2, 3, 1000); +#+end_src + #+begin_src matlab :tangle no addpath('./matlab'); addpath('./matlab/src'); #+end_src -#+begin_src matlab :exec no +#+begin_src matlab :eval no addpath('./src'); #+end_src -** Manually shift zeros to the origin after synthesis -Suppose we want $H_2(s)$ to be an high pass filter with a slope of +2 at low frequency (from 0Hz). +** 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*} -We cannot impose that using the weight $W_2(s)$ as it would be improper. +For that, we use the $\mathcal{H}_\infty$ synthesis with the architecture shown on figure [[fig:comp_filter_three_hinf]]. -However, we may manually shift 2 of the low frequency zeros to the origin. +#+name: fig:comp_filter_three_hinf +#+caption: Generalized architecture for generating 3 complementary filters +[[file:figs-journal/comp_filter_three_hinf_fb.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*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 = 1e4; 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; +%% Design of the Weighting Functions +W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.5); +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; +W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.5); #+end_src -#+begin_src matlab -P = [W1 -W1; - 0 W2; - 1 0]; +#+begin_src matlab :exports none +%% Inverse magnitude of the weighting functions +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)]); ylim([2e-4, 1.3e1]) +leg = legend('location', 'northeast', 'FontSize', 8); +leg.ItemTokenSize(1) = 18; #+end_src -And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command: -#+begin_src matlab -[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); +#+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 -[z,p,k] = zpkdata(H2) +%% Generalized plant for the synthesis of 3 complementary filters +P = [W1 -W1 -W1; + 0 W2 0 ; + 0 0 W3; + 1 0 0]; #+end_src -Looking at the zeros, we see two low frequency complex conjugate zeros. -#+begin_src matlab :results output replace :exports results -z{1} +And we do the $\mathcal{H}_\infty$ synthesis. +#+begin_src matlab :results output replace :exports both +%% Standard H-Infinity Synthesis +[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on'); #+end_src #+RESULTS: #+begin_example -z{1} -ans = - -4690930.24283199 + 0i - -163.420524657426 + 0i - -0.853192261081498 + 0.713416012479897i - -0.853192261081498 - 0.713416012479897i - -3.15812268762265 + 0i +[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 -We manually put these zeros at the origin: +** Obtained Complementary Filters +The obtained filters are: #+begin_src matlab -z{1}([3,4]) = 0; +%% +H2 = tf(H(1)); +H3 = tf(H(2)); +H1 = 1 - H2 - H3; #+end_src -And we create a modified filter $H_{2z}(s)$: -#+begin_src matlab -H2z = zpk(z,p,k); -#+end_src - -And as usual, $H_{1z}(s)$ is defined as the complementary of $H_{2z}(s)$: -#+begin_src matlab -H1z = 1 - H2z; -#+end_src - -The bode plots of $H_1(s)$, $H_2(s)$, $H_{1z}(s)$ and $H_{2z}(s)$ are shown in Figure [[fig:comp_filters_shift_zero]]. -And we see that $H_{1z}(s)$ is slightly modified when setting the zeros at the origin for $H_{2z}(s)$. - #+begin_src matlab :exports none -freqs = logspace(-2, 4, 1000); - +%% Bode plot of the obtained complementary filters figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -2371,287 +1166,69 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2, 1]); hold on; set(gca,'ColorOrderIndex',1) -plot(freqs, abs(squeeze(freqresp(1-H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); -plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); - -set(gca,'ColorOrderIndex',1) -plot(freqs, abs(squeeze(freqresp(1-H2z, freqs, 'Hz'))), '--', 'DisplayName', '$H_{1z}$'); -plot(freqs, abs(squeeze(freqresp(H2z, freqs, 'Hz'))), '--', 'DisplayName', '$H_{2z}$'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Magnitude'); -set(gca, 'XTickLabel',[]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); - -% Phase -ax2 = nexttile; -hold on; -set(gca,'ColorOrderIndex',1) -plot(freqs, 180/pi*phase(squeeze(freqresp(1-H2, freqs, 'Hz'))), '-'); -plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); -set(gca,'ColorOrderIndex',1) -plot(freqs, 180/pi*phase(squeeze(freqresp(H1z, freqs, 'Hz'))), '--'); -plot(freqs, 360+180/pi*phase(squeeze(freqresp(H2z, 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/comp_filters_shift_zero.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:comp_filters_shift_zero -#+caption: Bode plots of $H_1(s)$, $H_2(s)$, $H_{1z}(s)$ and $H_{2z}(s)$ -#+RESULTS: -[[file:figs/comp_filters_shift_zero.png]] - -** Imposing a positive slope at DC during the synthesis phase - -Suppose we want to synthesize $H_2(s)$ such that it has a slope of +2 from DC. -We can include this "feature" in the generalized plant as shown in Figure [[fig:h_infinity_arch_H2_feature]]. - -#+begin_src latex :file h_infinity_arch_H2_feature.pdf - \begin{tikzpicture} - \node[block={4.5cm}{4.0cm}, fill=black!20!white, dashed] (P) {}; - \node[above] at (P.north) {$P(s)$}; - - \coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$); - \coordinate[] (inputu) at ($(P.south west)!0.5!(P.north west) + (-0.7, 0)$); - - \coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + ( 0.7, 0)$); - \coordinate[] (output2) at ($(P.south east)!0.5!(P.north east) + ( 0.7, 0)$); - \coordinate[] (outputv) at ($(P.south east)!0.2!(P.north east) + ( 0.7, 0)$); - - \node[block, left=1.4 of output1] (W1){$W_1(s)$}; - \node[block, left=1.4 of output2] (W2){$W_2(s)$}; - \node[block, left=1.4 of outputv] (Hw){$H_{2w}(s)$}; - \node[addb={+}{}{}{}{-}, left=of W1] (sub) {}; - - \node[block, below=0.3 of P] (H2) {$H_2^\prime(s)$}; - - \draw[->] (inputw) node[above right]{$w$} -- (sub.west); - \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west); - \draw[->] (inputu-|sub) node[branch]{} -- (sub.south); - \draw[->] (sub.east) -- (W1.west); - \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (Hw.west); - \draw[->] (Hw.east) -- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east); - \draw[->] (W1.east) -- (output1)node[above left]{$z_1$}; - \draw[->] (W2.east) -- (output2)node[above left]{$z_2$}; - \end{tikzpicture} -#+end_src - -#+name: fig:h_infinity_arch_H2_feature -#+caption: Generalized plant with included wanted feature represented by $H_{2w}(s)$ -#+RESULTS: -[[file:figs/h_infinity_arch_H2_feature.png]] - -After synthesis, the obtained filter will be: -\begin{equation} - H_2(s) = H_2^\prime(s) H_{2w}(s) -\end{equation} -and therefore the "feature" will be included in the filter. - -For $H_1(s)$ nothing is changed: $H_1(s) = 1 - H_2(s)$. - -The weighting functions are defined as usual: -#+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 = 1e4; 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 - -The wanted feature here is a +2 slope at low frequency. -For that, we use an high pass filter with a slope of +2 at low frequency. -#+begin_src matlab -w0 = 2*pi*50; -H2w = (s/w0/(s/w0+1))^2; -#+end_src - -We define the generalized plant as shown in Figure [[fig:h_infinity_arch_H2_feature]]. -#+begin_src matlab -P = [W1 -W1; - 0 W2; - H2w 0]; -#+end_src - -And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. -#+begin_src matlab :results output replace :exports both -[H2p, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -#+end_src - -Finally, we define $H_2(s)$ as the product of the synthesized filter and the wanted "feature": -#+begin_src matlab -H2 = H2p*H2w; -#+end_src - -And we define $H_1(s)$ to be the complementary of $H_2(s)$: -#+begin_src matlab -H1 = 1 - H2; -#+end_src - -The obtained complementary filters are shown in Figure [[fig:comp_filters_H2_feature]]. - -#+begin_src matlab :exports none -freqs = logspace(-3, 3, 1000); - -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', '$w_1$'); +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', '$w_2$'); - +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',[]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-4, 20]); +leg = legend('location', 'northeast', 'FontSize', 8); +leg.ItemTokenSize(1) = 18; % Phase ax2 = nexttile; hold on; set(gca,'ColorOrderIndex',1) -plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-'); +plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz')))); set(gca,'ColorOrderIndex',2) -plot(freqs, 360+180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); +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([-180:90:180]); +yticks([-180:90:180]); ylim([-220, 220]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_filters_H2_feature.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:comp_filters_H2_feature -#+caption: Obtained complementary fitlers -#+RESULTS: -[[file:figs/comp_filters_H2_feature.png]] - -** Imposing a negative slope at infinity frequency during the synthesis phase -Let's suppose we now want to shape a low pass filter that as a negative slope until infinite frequency. - -The used technique is the same as in the previous section, and the generalized plant is shown in Figure [[fig:h_infinity_arch_H2_feature]]. - -The weights are defined as usual. -#+begin_src matlab -n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; 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 = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; 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 - -This time, the feature is a low pass filter with a slope of -2 at high frequency. -#+begin_src matlab -H2w = 1/(s/(2*pi*10) + 1)^2; -#+end_src - -The generalized plant is defined: -#+begin_src matlab -P = [W1 -W1; - 0 W2; - H2w 0]; -#+end_src - -And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command. -#+begin_src matlab :results output replace :exports both -[H2p, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on'); -#+end_src - -The feature is added to the synthesized filter: -#+begin_src matlab -H2 = H2p*H2w; -#+end_src - -And $H_1(s)$ is defined as follows: -#+begin_src matlab -H1 = 1 - H2; -#+end_src - -The obtained complementary filters are shown in Figure [[fig:comp_filters_H2_feature_neg_slope]]. - -#+begin_src matlab :exports none -freqs = logspace(-1, 4, 1000); - -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',[]); -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)]); +#+begin_src matlab :tangle no :exports none +exportFig('figs-journal/three_complementary_filters_results.pdf', 'width', 'wide', 'height', 600); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_filters_H2_feature_neg_slope.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/three_complementary_filters_results.pdf', 'width', 'wide', 'height', 'tall'); #+end_src -#+name: fig:comp_filters_H2_feature_neg_slope -#+caption: Obtained complementary fitlers +#+name: fig:three_complementary_filters_results +#+caption: The three complementary filters obtained after $\mathcal{H}_\infty$ synthesis #+RESULTS: -[[file:figs/comp_filters_H2_feature_neg_slope.png]] - +[[file:figs/three_complementary_filters_results.png]] * Bibliography :ignore: +:PROPERTIES: +:CUSTOM_ID: sec:bibliography +:END: bibliographystyle:unsrt bibliography:ref.bib * Functions :PROPERTIES: :header-args:matlab+: :comments none :mkdirp yes :eval no +:CUSTOM_ID: sec:functions :END: -<> ** =generateWF=: Generate Weighting Functions :PROPERTIES: @@ -2699,13 +1276,12 @@ end #+end_src Verification that the parameters $G_0$, $G_c$ and $G_\infty$ are satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2. -#+name: eq:condition_params_formula -\begin{subequations} - \begin{align} - G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\ - G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2} - \end{align} -\end{subequations} +\begin{equation} + G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1} +\end{equation} +\begin{equation} + G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2} +\end{equation} #+begin_src matlab % Verification of correct relation between G0, Gc and Ginf @@ -2725,7 +1301,7 @@ The weighting function formula use is: #+name: eq:weight_formula \begin{equation} W(s) = \left( \frac{ - \hfill{} \frac{1}{\omega_c} \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}} + \frac{1}{\omega_c} \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_c} \sqrt{\frac{1 - 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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

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    -
    -

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

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    -
    - -
    -

    1.1 Synthesis Architecture

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    -

    -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 -

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    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: Gain of the Weighting Function formula

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    - -
    -
    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;
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    -
    - - -
    -

    weights_W1_W2.png -

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

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    1.3 H-Infinity Synthesis

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

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

    -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');
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    -
    - -
    -[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
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    - -

    -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. -

    - -
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    H1 = 1 - H2;
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    1.4 Obtained Complementary Filters

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

    - - -
    -

    hinf_filters_results.png -

    -

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

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    2 Generating 3 complementary filters

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    - -

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

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    2.1 Theory

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    -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

    -
    -
    -
    - -
    -

    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

    -
    -
    -
    -
    - -
    -

    4 Alternative Synthesis

    -
    -
    -
    -

    4.1 Two generalized plants

    -
    -

    -In order to synthesize the complementary filter using the proposed method, we can use two alternative generalized plant as shown in Figures 13 and 14. -

    - -\begin{equation} - P_1 = \begin{bmatrix} W_1 & -W_1 \\ 0 & W_2 \\ 1 & 0 \end{bmatrix} -\end{equation} - - - -
    -

    h_infinity_arch_1.png -

    -

    Figure 13: Complementary Filter Synthesis - Conf 1

    -
    - -\begin{equation} - P_2 = \begin{bmatrix} 0 & W_1 & 1 \\ W_2 & -W_1 & 0 \end{bmatrix} -\end{equation} - - -
    -

    h_infinity_arch_2.png -

    -

    Figure 14: Complementary Filter Synthesis - Conf 2

    -
    - -

    -Let’s run the \(\mathcal{H}_\infty\) synthesis for both generalized plant using the same weights and see if the obtained filters are the same: -

    -
    -
    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;
-
    -
    - -
    -
    P1 = [W1 -W1;
-      0   W2;
-      1   0];
-
    -
    - -
    -
    [H2, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
    -
    - -
    -[H2, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
-  Test bounds:  0.3263 <=  gamma  <=  1000
-
-    gamma        X>=0        Y>=0       rho(XY)<1    p/f
-  1.807e+01     1.4e-07     0.0e+00     1.185e-18     p
-  2.428e+00     1.5e-07     0.0e+00     1.285e-18     p
-  8.902e-01    -2.9e+02 #  -7.1e-17     5.168e-19     f
-  1.470e+00     1.5e-07     0.0e+00     1.462e-14     p
-  1.144e+00     1.5e-07     0.0e+00     1.260e-14     p
-  1.009e+00     1.5e-07     0.0e+00     4.120e-13     p
-  9.478e-01    -6.8e+02 #  -2.4e-17     1.449e-14     f
-  9.780e-01    -1.6e+03 #  -7.3e-17     6.791e-14     f
-  9.934e-01    -4.2e+03 #  -1.2e-16     3.524e-14     f
-  1.001e+00    -2.0e+04 #  -2.3e-17     5.717e-20     f
-  1.005e+00     1.5e-07     0.0e+00     8.953e-18     p
-  1.003e+00    -2.2e+05 #  -1.8e-17     3.225e-12     f
-  1.004e+00     1.5e-07     0.0e+00     2.445e-12     p
-  Limiting gains...
-  1.004e+00     1.6e-07     0.0e+00     5.811e-18     p
-
-  Best performance (actual): 1.004
-
    - -
    -
    P2 = [0   W1 1;
-      W2 -W1 0];
-
    -
    - -
    -
    [H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
    -
    - -
    -[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
-  Test bounds:  0.3263 <=  gamma  <=  1000
-
-    gamma        X>=0        Y>=0       rho(XY)<1    p/f
-  1.807e+01     0.0e+00     1.4e-07     2.055e-16     p
-  2.428e+00     0.0e+00     1.4e-07     1.894e-18     p
-  8.902e-01    -2.1e-16    -2.7e+02 #   1.466e-16     f
-  1.470e+00     0.0e+00     1.4e-07     4.118e-16     p
-  1.144e+00     0.0e+00     1.5e-07     2.105e-18     p
-  1.009e+00     0.0e+00     1.5e-07     2.590e-13     p
-  9.478e-01    -9.5e-17    -6.3e+02 #   1.663e-19     f
-  9.780e-01    -1.1e-16    -1.5e+03 #   1.546e-14     f
-  9.934e-01    -2.8e-17    -4.0e+03 #   3.934e-14     f
-  1.001e+00    -3.1e-17    -1.9e+04 #   1.191e-19     f
-  1.005e+00     0.0e+00     1.5e-07     1.443e-12     p
-  1.003e+00    -8.3e-17    -2.1e+05 #   8.807e-13     f
-  1.004e+00     0.0e+00     1.5e-07     1.459e-15     p
-  Limiting gains...
-  1.004e+00     0.0e+00     1.5e-07     9.086e-19     p
-
-  Best performance (actual): 1.004
-
    - -

    -And indeed, we can see that the exact same filters are obtained (Figure 15). -

    - -
    -

    hinf_comp_P1_P2_syn.png -

    -

    Figure 15: Comparison of \(H_2(s)\) when using \(P_1(s)\) or \(P_2(s)\)

    -
    -
    -
    - -
    -

    4.2 Shaping the Low pass filter or the high pass filter?

    -
    -

    -Let’s see if there is a difference by explicitly shaping \(H_1(s)\) or \(H_2(s)\). -

    - -
    -
    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;
-
    -
    - -

    -Let’s first synthesize \(H_1(s)\): -

    -
    -
    P1 = [W2 -W2;
-      0   W1;
-      1   0];
-
    -
    - -
    -
    [H1, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
    -
    - -
    -[H1, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
-  Test bounds:  0.3263 <=  gamma  <=  1.712
-
-    gamma        X>=0        Y>=0       rho(XY)<1    p/f
-  7.476e-01    -2.5e+01 #  -8.3e-18     4.938e-20     f
-  1.131e+00     1.9e-07     0.0e+00     1.566e-16     p
-  9.197e-01    -1.4e+02 #  -7.9e-17     4.241e-17     f
-  1.020e+00     1.9e-07     0.0e+00     2.095e-16     p
-  9.686e-01    -3.8e+02 #  -7.0e-17     1.463e-23     f
-  9.940e-01    -1.5e+03 #  -1.3e-17     3.168e-19     f
-  1.007e+00     1.9e-07     0.0e+00     1.696e-15     p
-  1.000e+00    -4.8e+03 #  -7.1e-18     7.203e-20     f
-  1.004e+00     1.9e-07     0.0e+00     1.491e-14     p
-  1.002e+00    -1.1e+04 #  -2.6e-16     2.579e-14     f
-  1.003e+00    -2.8e+04 #  -6.0e-18     8.558e-20     f
-  Limiting gains...
-  1.004e+00     2.0e-07     0.0e+00     5.647e-18     p
-  1.004e+00     1.0e-06     0.0e+00     5.648e-18     p
-
-  Best performance (actual): 1.004
-
    - -

    -And now \(H_2(s)\): -

    -
    -
    P2 = [W1 -W1;
-      0   W2;
-      1   0];
-
    -
    - -
    -
    [H2, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
    -
    - -
    -[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
-
-  Test bounds:  0.3263 <=  gamma  <=  1000
-
-    gamma        X>=0        Y>=0       rho(XY)<1    p/f
-  1.807e+01     1.4e-07     0.0e+00     1.185e-18     p
-  2.428e+00     1.5e-07     0.0e+00     1.285e-18     p
-  8.902e-01    -2.9e+02 #  -7.1e-17     5.168e-19     f
-  1.470e+00     1.5e-07     0.0e+00     1.462e-14     p
-  1.144e+00     1.5e-07     0.0e+00     1.260e-14     p
-  1.009e+00     1.5e-07     0.0e+00     4.120e-13     p
-  9.478e-01    -6.8e+02 #  -2.4e-17     1.449e-14     f
-  9.780e-01    -1.6e+03 #  -7.3e-17     6.791e-14     f
-  9.934e-01    -4.2e+03 #  -1.2e-16     3.524e-14     f
-  1.001e+00    -2.0e+04 #  -2.3e-17     5.717e-20     f
-  1.005e+00     1.5e-07     0.0e+00     8.953e-18     p
-  1.003e+00    -2.2e+05 #  -1.8e-17     3.225e-12     f
-  1.004e+00     1.5e-07     0.0e+00     2.445e-12     p
-  Limiting gains...
-  1.004e+00     1.6e-07     0.0e+00     5.811e-18     p
-
-  Best performance (actual): 1.004
-
    - -

    -And compare \(H_1(s)\) with \(1 - H_2(s)\) and \(H_2(s)\) with \(1 - H_1(s)\) in Figure 16. -

    - -
    -

    hinf_comp_H1_H2_syn.png -

    -

    Figure 16: Comparison of \(H_1(s)\) with \(1-H_2(s)\), and \(H_2(s)\) with \(1-H_1(s)\)

    -
    -
    -
    - -
    -

    4.3 Using Feedback architecture

    -
    -
    -
    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;
-
    -
    - -

    -Let’s first synthesize \(H_1(s)\): -

    -
    -
    P = [W1 -W1;
-     0   W2;
-     1  -1];
-
    -
    - -
    -
    [K, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on');
-
    -
    - -
    -
    H1 = inv(1 + K);
-H2 = 1 - H1;
-
    -
    -
    -
    - -
    -

    4.4 Adding feature in the filters

    -
    -
    -
    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;
-
    -
    - -
    -
    Wf = (1 + s/2/pi/1)/s;
-Wf = s/(1 + s/2/pi/1e2);
-
-% W2 = W2/Wf/(1 + s/2/pi/1e3);
-
    -
    - -
    -
    P = [W1 -Wf*W1;
-     0   Wf*W2;
-     1  -Wf];
-
    -
    - -
    -
    [Ka, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on');
-
    -
    - -
    -
    K = Ka*Wf;
-
    -
    - -
    -
    H1 = inv(1 + K);
-H2 = 1 - H1;
-
    -
    -
    -
    -
    - -
    -

    5 Impose a positive slope at DC or a negative slope at infinite frequency

    -
    -
    -
    -

    5.1 Manually shift zeros to the origin after synthesis

    -
    -

    -Suppose we want \(H_2(s)\) to be an high pass filter with a slope of +2 at low frequency (from 0Hz). -

    - -

    -We cannot impose that using the weight \(W_2(s)\) as it would be improper. -

    - -

    -However, we may manually shift 2 of the low frequency zeros to the origin. -

    - -
    -
    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 = 1e4; 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;
-
    -
    - -
    -
    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', 'lmi', 'DISPLAY', 'on');
-
    -
    - -
    -
    [z,p,k] = zpkdata(H2)
-
    -
    - -

    -Looking at the zeros, we see two low frequency complex conjugate zeros. -

    -
    -z{1}
-ans =
-          -4690930.24283199 +                     0i
-          -163.420524657426 +                     0i
-         -0.853192261081498 +     0.713416012479897i
-         -0.853192261081498 -     0.713416012479897i
-          -3.15812268762265 +                     0i
-
    - -

    -We manually put these zeros at the origin: -

    -
    -
    z{1}([3,4]) = 0;
-
    -
    - -

    -And we create a modified filter \(H_{2z}(s)\): -

    -
    -
    H2z = zpk(z,p,k);
-
    -
    - -

    -And as usual, \(H_{1z}(s)\) is defined as the complementary of \(H_{2z}(s)\): -

    -
    -
    H1z = 1 - H2z;
-
    -
    - -

    -The bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\) are shown in Figure 17. -And we see that \(H_{1z}(s)\) is slightly modified when setting the zeros at the origin for \(H_{2z}(s)\). -

    - - -
    -

    comp_filters_shift_zero.png -

    -

    Figure 17: Bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\)

    -
    -
    -
    - -
    -

    5.2 Imposing a positive slope at DC during the synthesis phase

    -
    -

    -Suppose we want to synthesize \(H_2(s)\) such that it has a slope of +2 from DC. -We can include this “feature” in the generalized plant as shown in Figure 18. -

    - - -
    -

    h_infinity_arch_H2_feature.png -

    -

    Figure 18: Generalized plant with included wanted feature represented by \(H_{2w}(s)\)

    -
    - -

    -After synthesis, the obtained filter will be: -

    -\begin{equation} - H_2(s) = H_2^\prime(s) H_{2w}(s) -\end{equation} -

    -and therefore the “feature” will be included in the filter. -

    - -

    -For \(H_1(s)\) nothing is changed: \(H_1(s) = 1 - H_2(s)\). -

    - -

    -The weighting functions are defined as usual: -

    -
    -
    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 = 1e4; 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;
-
    -
    - -

    -The wanted feature here is a +2 slope at low frequency. -For that, we use an high pass filter with a slope of +2 at low frequency. -

    -
    -
    w0 = 2*pi*50;
-H2w = (s/w0/(s/w0+1))^2;
-
    -
    - -

    -We define the generalized plant as shown in Figure 18. -

    -
    -
    P = [W1 -W1;
-     0   W2;
-     H2w 0];
-
    -
    - -

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

    -
    -
    [H2p, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on');
-
    -
    - -

    -Finally, we define \(H_2(s)\) as the product of the synthesized filter and the wanted “feature”: -

    -
    -
    H2 = H2p*H2w;
-
    -
    - -

    -And we define \(H_1(s)\) to be the complementary of \(H_2(s)\): -

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

    -The obtained complementary filters are shown in Figure 19. -

    - - -
    -

    comp_filters_H2_feature.png -

    -

    Figure 19: Obtained complementary fitlers

    -
    -
    -
    - -
    -

    5.3 Imposing a negative slope at infinity frequency during the synthesis phase

    -
    -

    -Let’s suppose we now want to shape a low pass filter that as a negative slope until infinite frequency. -

    - -

    -The used technique is the same as in the previous section, and the generalized plant is shown in Figure 18. -

    - -

    -The weights are defined as usual. -

    -
    -
    n = 3; w0 = 2*pi*10; G0 = 1000; G1 = 0.1; 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 = 2; w0 = 2*pi*11; G0 = 1/10; G1 = 1000; 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;
-
    -
    - -

    -This time, the feature is a low pass filter with a slope of -2 at high frequency. -

    -
    -
    H2w = 1/(s/(2*pi*10) + 1)^2;
-
    -
    - -

    -The generalized plant is defined: -

    -
    -
    P = [W1 -W1;
-     0   W2;
-     H2w 0];
-
    -
    - -

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

    -
    -
    [H2p, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'lmi', 'DISPLAY', 'on');
-
    -
    - -

    -The feature is added to the synthesized filter: -

    -
    -
    H2 = H2p*H2w;
-
    -
    - -

    -And \(H_1(s)\) is defined as follows: -

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

    -The obtained complementary filters are shown in Figure 20. -

    - - -
    -

    comp_filters_H2_feature_neg_slope.png -

    -

    Figure 20: Obtained complementary fitlers

    -
    -
    -
    -
    - - -

    - -

    - -

    Bibliography

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

    6 Functions

    -
    -
    -
    -

    6.1 generateWF: Generate Weighting Functions

    -
    -

    - -

    - -

    -This Matlab function is accessible here. -

    -
    - -
    -

    Function description

    -
    -
    -
    function [W] = generateWF(args)
-% createWeight -
-%
-% Syntax: [W] = generateWeight(args)
-%
-% Inputs:
-%    - n  - Weight Order (integer)
-%    - G0 - Low frequency Gain
-%    - G1 - High frequency Gain
-%    - Gc - Gain of the weight at frequency w0
-%    - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
-%
-% Outputs:
-%    - W - Generated Weight
-
    -
    -
    -
    - -
    -

    Optional Parameters

    -
    -
    -
    arguments
-    args.n    (1,1) double {mustBeInteger, mustBePositive} = 1
-    args.G0   (1,1) double {mustBeNumeric, mustBePositive} = 0.1
-    args.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
-    args.Gc   (1,1) double {mustBeNumeric, mustBePositive} = 1
-    args.w0   (1,1) double {mustBeNumeric, mustBePositive} = 1
-end
-
-mustBeBetween(args.G0, args.Gc, args.Ginf);
-
    -
    -
    -
    - -
    -

    Generate the Weighting function

    -
    -
    -
    s = zpk('s');
-
    -
    - -
    -
    W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
-      (args.G0/args.Gc)^(1/args.n))/...
-     ((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
-      (1/args.Gc)^(1/args.n)))^args.n;
-
    -
    -
    -
    - - -
    -

    Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains

    -
    -
    -
    % Custom validation function
-function mustBeBetween(a,b,c)
-    if ~((a > b && b > c) || (c > b && b > a))
-        eid = 'createWeight:inputError';
-        msg = 'Gc should be between G0 and Ginf.';
-        throwAsCaller(MException(eid,msg))
-    end
-
    -
    -
    -
    -
    - -
    -

    6.2 generateCF: Generate Complementary Filters

    -
    -

    - -

    - -

    -This Matlab function is accessible here. -

    -
    - -
    -

    Function description

    -
    -
    -
    function [H1, H2] = generateCF(W1, W2, args)
-% createWeight -
-%
-% Syntax: [W] = generateCF(args)
-%
-% Inputs:
-%    - W1 - Weighting Function for H1
-%    - W2 - Weighting Function for H2
-%    - args:
-%      - method  - H-Infinity solver ('lmi' or 'ric')
-%      - display - Display synthesis results ('on' or 'off')
-%
-% Outputs:
-%    - H1 - Generated H1 Filter
-%    - H2 - Generated H2 Filter
-
    -
    -
    -
    - -
    -

    Optional Parameters

    -
    -
    -
    arguments
-    W1
-    W2
-    args.method  char {mustBeMember(args.method,{'lmi', 'ric'})} = 'ric'
-    args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
-end
-
    -
    -
    -
    - -
    -

    H-Infinity Synthesis

    -
    -
    -
    P = [W1 -W1;
-     0   W2;
-     1   0];
-
    -
    - -
    -
    [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
-
    -
    - -
    -
    H1 = 1 - H2;
-
    -
    -
    -
    -
    -
    -
    -
    -

    Author: Thomas Dehaeze

    -

    Created: 2021-04-29 jeu. 17:25

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    - -

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