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

## Table of Contents

- 1. H-Infinity synthesis of complementary filters
  - 1.1. Synthesis Architecture
  - 1.2. Design of Weighting Function
  - 1.3. Example
  - 1.4. H-Infinity Synthesis
  - 1.5. Obtained Complementary Filters
- 2. Design of complementary filters used in the Active Vibration Isolation System at the LIGO
  - 2.1. Specifications
  - 2.2. FIR Filter
  - 2.3. Weights
  - 2.4. H-Infinity Synthesis
  - 2.5. Compare FIR and H-Infinity Filters
- 3. “Closed-Loop” complementary filters
  - 3.1. Using Feedback architecture
- 4. Synthesis of three complementary filters
  - 4.1. Theory
  - 4.2. Weights
  - 4.3. H-Infinity Synthesis
  - 4.4. Obtained Complementary Filters
- 5. Functions
  - 5.1. `generateWF`: Generate Weighting Functions
    - Function description
    - Optional Parameters
    - Generate the Weighting function
    - Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains
  - 5.2. `generateCF`: Generate Complementary Filters
    - Function description
    - Optional Parameters
    - H-Infinity Synthesis

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

This document is divided into several sections:

- in section 1, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters
- in section 4, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters
- in section 2, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO
- in section 3

## 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.

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.

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);

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)

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

The specifications are translated in upper bounds of the complementary filters are shown on figure 5.

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;

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.

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.

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.

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)

## 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.

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);

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;

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