Last update before submission

matlab/dehaeze21_desig_compl_filte_matlab.org

@@ -113,12 +113,6 @@ freqs = logspace(-1, 3, 1000);
 
 #+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
 
 ** Design of Weighting Function - Proposed formula
@@ -407,12 +401,6 @@ freqs = logspace(-3, 0, 1000);
 
 #+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
 
 ** Specifications
@@ -949,12 +937,6 @@ freqs = logspace(-1, 3, 1000);
 
 #+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
 
 ** Weighting Function design
@@ -1126,11 +1108,6 @@ freqs = logspace(-2, 3, 1000);
 
 #+begin_src matlab :tangle no
 addpath('./matlab');
-addpath('./matlab/src');
-#+end_src
-
-#+begin_src matlab :eval no
-addpath('./src');
 #+end_src
 
 ** Synthesis Architecture
@@ -1365,7 +1342,7 @@ This scripts corresponds to section 5.2 of cite:dehaeze21_new_method_desig_compl
 
 ** =generateWF=: Generate Weighting Functions
 :PROPERTIES:
-:header-args:matlab+: :tangle matlab/src/generateWF.m
+:header-args:matlab+: :tangle matlab/generateWF.m
 :header-args:matlab+: :comments none :mkdirp yes :eval no
 :CUSTOM_ID: sec:generateWF
 :END:
@@ -1420,7 +1397,7 @@ function mustBeBetween(a,b,c)
 
 ** =generateCF=: Generate Complementary Filters
 :PROPERTIES:
-:header-args:matlab+: :tangle matlab/src/generateCF.m
+:header-args:matlab+: :tangle matlab/generateCF.m
 :header-args:matlab+: :comments none :mkdirp yes :eval no
 :CUSTOM_ID: sec:generateCF
 :END:
@@ -1463,5 +1440,8 @@ P = [W1 -W1;
 %% H1 is defined as the complementary of H2
 H1 = 1 - H2;
 #+end_src
+
 * Bibliography                                                        :ignore:
+bibliography:ref.bib
+
 #+latex: \printbibliography

matlab/dehaeze21_desig_compl_filte_matlab.tex

@@ -1,4 +1,4 @@
-% Created 2021-09-01 mer. 14:59
+% Created 2021-09-01 mer. 16:57
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full]{scrreprt}
 \usepackage[utf8]{inputenc}
@@ -47,7 +47,7 @@ This document is divided into the following sections also corresponding to the p
 \chapter{H-Infinity synthesis of complementary filters}
 \label{sec:h_inf_synthesis_complementary_filters}
 \section{Synthesis Architecture}
-\label{sec:orgbe50759}
+\label{sec:org7b3f474}
 In order to generate two complementary filters with a wanted shape, the generalized plant of Figure \ref{fig:h_infinity_robust_fusion_plant} can be used.
 The included weights \(W_1(s)\) and \(W_2(s)\) are used to specify the upper bounds of the complementary filters being generated.
 
@@ -93,7 +93,7 @@ The presented synthesis method therefore allows to shape two filters \(H_1(s)\)
 The complete Matlab script for this part is given in Section \ref{sec:1_synthesis_complementary_filters}.
 
 \section{Design of Weighting Function - Proposed formula}
-\label{sec:orgde64d6c}
+\label{sec:org69d0d73}
 A formula is proposed to help the design of the weighting functions:
 \begin{equation}
   W(s) = \left( \frac{
@@ -120,7 +120,7 @@ The general shape of a weighting function generated using the formula is shown i
 \end{figure}
 
 \section{Weighting functions for the design of two complementary filters}
-\label{sec:orgc65c135}
+\label{sec:org244c0a4}
 \label{sec:weighting_functions_example}
 
 The weighting function formula \eqref{eq:weighting_function_formula} is used to generate the upper bounds of two complementary filters that we wish to design.
@@ -142,7 +142,7 @@ The inverse magnitude of these two weighting functions are shown in Figure \ref{
 \end{figure}
 
 \section{Synthesis of the complementary filters}
-\label{sec:orgc8c8834}
+\label{sec:org3378b49}
 The generalized plant of Figure \ref{fig:h_infinity_robust_fusion_plant} is defined as follows:
 \begin{minted}[]{matlab}
 %% Generalized Plant
@@ -198,7 +198,7 @@ This function is described in Section \ref{sec:generateCF}.
 \end{minted}
 
 \section{Obtained Complementary Filters}
-\label{sec:orgc2b3de0}
+\label{sec:orgb52f55e}
 The obtained complementary filters are shown below and are found to be of order 5.
 Their bode plots are shown in figure \ref{fig:hinf_filters_results} and compare with the defined upper bounds.
 
@@ -231,7 +231,7 @@ In this section, the proposed method for the design of complementary filters is
 The complete Matlab script for this part is given in Section \ref{sec:2_ligo_complementary_filters}.
 
 \section{Specifications}
-\label{sec:orgeaf463d}
+\label{sec:orgc6bd7b2}
 The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}):
 \begin{itemize}
 \item From 0 to 0.008 Hz, the magnitude \(|L_1(j\omega)|\) should be less or equal to \(8 \times 10^{-4}\)
@@ -249,7 +249,7 @@ The specifications are translated into upper bounds of the complementary filters
 \end{figure}
 
 \section{FIR Filter}
-\label{sec:orge501c41}
+\label{sec:orgef72f87}
 To replicated the complementary filters developed in \cite{hua04_low_ligo}, the CVX Matlab toolbox \cite{grant14_cvx} is used.
 
 The CVX toolbox is initialized and the \texttt{SeDuMi} solver \cite{sturm99_using_sedum} is used.
@@ -409,7 +409,7 @@ H = [exp(-j*kron(w'.*2*pi,[0:n-1]))]*h;
 \end{figure}
 
 \section{Weighting function design}
-\label{sec:orge7a3d56}
+\label{sec:orgcbee7b7}
 The weightings function that will be used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters are now designed.
 
 These weights will determine the order of the obtained filters.
@@ -466,7 +466,7 @@ The inverse magnitude of the weighting functions are shown in Figure \ref{fig:li
 \end{figure}
 
 \section{Synthesis of the complementary filters}
-\label{sec:orge44276c}
+\label{sec:org8f9014e}
 The generalized plant of figure \ref{fig:h_infinity_robust_fusion_plant} is defined.
 \begin{minted}[]{matlab}
 %% Generalized plant for the H-infinity Synthesis
@@ -533,7 +533,7 @@ The magnitude of the obtained filters as well as the requirements are shown in F
 \end{figure}
 
 \section{Comparison of the FIR filters and synthesized filters}
-\label{sec:orgc915ecf}
+\label{sec:org6367ef0}
 
 Let's now compare the FIR filters designed in \cite{hua04_low_ligo} with the with complementary filters obtained with the \(\mathcal{H}_\infty\) synthesis.
 
@@ -558,7 +558,7 @@ In this section, the classical feedback architecture shown in Figure \ref{fig:fe
 The complete Matlab script for this part is given in Section \ref{sec:3_closed_loop_complementary_filters}.
 
 \section{Weighting Function design}
-\label{sec:org62f8027}
+\label{sec:org856a399}
 Weighting functions using the \texttt{generateWF} Matlab function are designed to specify the upper bounds of the complementary filters to be designed.
 These weighting functions are the same as the ones used in Section \ref{sec:weighting_functions_example}.
 
@@ -569,7 +569,7 @@ W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
 \end{minted}
 
 \section{Generalized plant}
-\label{sec:orgb431bb1}
+\label{sec:orgbba676f}
 The generalized plant of Figure \ref{fig:feedback_synthesis_architecture_generalized_plant} is defined below:
 \begin{minted}[]{matlab}
 %% Generalized plant for "closed-loop" complementary filter synthesis
@@ -584,7 +584,7 @@ P = [ W1 0   1;
 \end{figure}
 
 \section{Synthesis of the closed-loop complementary filters}
-\label{sec:orga0ede8a}
+\label{sec:org44dc95f}
 And the standard \(\mathcal{H}_\infty\) synthesis is performed.
 \begin{minted}[]{matlab}
 %% Standard H-Infinity Synthesis
@@ -614,7 +614,7 @@ And the standard \(\mathcal{H}_\infty\) synthesis is performed.
 \end{verbatim}
 
 \section{Synthesized filters}
-\label{sec:org174b330}
+\label{sec:org9e70c62}
 The obtained filter \(L(s)\) can then be included in the feedback architecture shown in Figure \ref{fig:hinf_filters_results_mixed_sensitivity}.
 
 The closed-loop transfer functions from \(\hat{x}_1\) to \(\hat{x}\) and from \(\hat{x}_2\) to \(\hat{x}\) corresponding respectively to the sensitivity and complementary sensitivity transfer functions are defined below:
@@ -651,7 +651,7 @@ In this section, the proposed synthesis method of complementary filters is gener
 The complete Matlab script for this part is given in Section \ref{sec:4_three_complementary_filters}.
 
 \section{Synthesis Architecture}
-\label{sec:org8851ed2}
+\label{sec:orgc4d55c9}
 The synthesis objective is to shape three filters that are complementary.
 This corresponds to the conditions \eqref{eq:obj_three_cf} where \(W_1(s)\), \(W_2(s)\) and \(W_3(s)\) are weighting functions used to specify the maximum wanted magnitude of the three complementary filters.
 
@@ -680,7 +680,7 @@ The last filter \(H_1(s)\) is defined as the complementary of the two others as
 \end{equation}
 
 \section{Weights}
-\label{sec:orgca84b75}
+\label{sec:orgf866c06}
 The three weighting functions are defined as shown below.
 
 \begin{minted}[]{matlab}
@@ -699,7 +699,7 @@ Their inverse magnitudes are displayed in Figure \ref{fig:three_weighting_functi
 \end{figure}
 
 \section{H-Infinity Synthesis}
-\label{sec:org65e90d4}
+\label{sec:org0db8a8c}
 
 The generalized plant in Figure \ref{fig:comp_filter_three_hinf} containing the weighting functions is defined below.
 
@@ -780,9 +780,6 @@ s = zpk('s');
 %% Initialize Frequency Vector
 freqs = logspace(-1, 3, 1000);
 
-%% Add functions to path
-addpath('./src');
-
 %% Weighting Function Design
 % Parameters
 n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
@@ -818,8 +815,8 @@ ylim([5e-4, 20]);
 yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
 
 %% Design of the Weighting Functions
-W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'G1', 1/10, 'Gc', 0.45);
+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, 'G1', 1000, 'Gc', 0.45);
+W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
 
 %% Plot of the Weighting function magnitude
 figure;
@@ -910,9 +907,6 @@ s = zpk('s');
 %% Initialize Frequency Vector
 freqs = logspace(-3, 0, 1000);
 
-%% Add functions to path
-addpath('./src');
-
 %% Upper bounds for the complementary filters
 figure;
 hold on;
@@ -1175,12 +1169,9 @@ s = zpk('s');
 %% Initialize Frequency Vector
 freqs = logspace(-1, 3, 1000);
 
-%% Add functions to path
-addpath('./src');
-
 %% Design of the Weighting Functions
-W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'G1', 1/10, 'Gc', 0.45);
+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, 'G1', 1000, 'Gc', 0.45);
+W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
 
 %% Generalized plant for "closed-loop" complementary filter synthesis
 P = [ W1 0   1;
@@ -1252,15 +1243,10 @@ s = zpk('s');
 
 freqs = logspace(-2, 3, 1000);
 
-addpath('./src');
-
-% Weights
-% First we define the weights.
-
 %% Design of the Weighting Functions
-W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'G1', 1000, 'Gc', 0.5);
+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, 'G1', 1/10, 'Gc', 0.5);
+W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.5);
 
 %% Inverse magnitude of the weighting functions
 figure;
@@ -1278,28 +1264,20 @@ xlim([freqs(1), freqs(end)]); ylim([2e-4, 1.3e1])
 leg = legend('location', 'northeast', 'FontSize', 8);
 leg.ItemTokenSize(1) = 18;
 
-% 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');
 
-% Obtained Complementary Filters
+%% Synthesized H2 and H3 filters
-% The obtained filters are:
-
-%%
 H2 = tf(H(1));
 H3 = tf(H(2));
+
+%% H1 is defined as the complementary filter of H2 and H3
 H1 = 1 - H2 - H3;
 
 %% Bode plot of the obtained complementary filters
@@ -1438,5 +1416,6 @@ P = [W1 -W1;
 %% H1 is defined as the complementary of H2
 H1 = 1 - H2;
 \end{minted}
+
 \printbibliography
 \end{document}

matlab/ref.bib

@@ -12,15 +12,13 @@
   organization    = {International Society for Optics and Photonics},
 }
 
-@inproceedings{dehaeze21_new_method_desig_compl_filter,
+@article{dehaeze21_new_method_desig_compl_filter,
   author          = {Dehaeze, Thomas and Vermat, Mohit and Collette, Christophe},
   title           = {A New Method of Designing Complementary Filters for Sensor
-                  Fusion Using the $\mathcal{H}_\infty$ Synthesis},
+                  Fusion Using the H-Infinity Synthesis},
-  booktitle       = {Gravitational Wave and Particle Astrophysics Detectors},
+  journal         = {Mechanical Systems and Signal Processing},
-  year            = 2004,
+  year            = 2021,
-  volume          = 5500,
+  month           = {Nov},
-  pages           = {194--205},
-  organization    = {International Society for Optics and Photonics},
 }
 
 @book{matlab20,