Rework figures, add colors

journal/journal.org

@@ -254,12 +254,27 @@ Finally, concluding remarks are presented in Section [[*Concluding remarks][5]].
 
 * Complementary Filters Requirements
 <<sec:requirements>>
+** Introduction                                                      :ignore:
+
+Complementary filters provides a framework for fusing signals from different sensors.
+As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
+These requirements are discussed in this section.
+
 ** Sensor Models
 <<sec:sensor_models>>
 
-- Noise + dynamical uncertainty
+- Noise + dynamics
+
+#+name: fig:sensor_model
+#+caption: Basic Sensor Model
+[[file:figs/sensor_model.pdf]]
+
 - Suppose we calibrate the sensors
 
+#+name: fig:sensor_model_calibrated
+#+caption: Calibrated Sensor
+[[file:figs/sensor_model_calibrated.pdf]]
+
 ** Sensor Fusion Architecture
 <<sec:sensor_fusion>>
 
@@ -322,6 +337,15 @@ In practical systems the sensor dynamics is not perfect and eqref:eq:perfect_dyn
 In such case, one can use an inversion filter $\hat{G}_i^{-1}(s)$ to normalize the sensor dynamics, where $\hat{G}_i(s)$ is an estimate of the sensor dynamics $G_i(s)$.
 However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and $\hat{G}_i^{-1}(s) G_i(s) \neq 1$.
 
+#+name: fig:sensor_model_uncertainty
+#+caption: Input Uncertainty
+[[file:figs/sensor_model_uncertainty.png]]
+
+#+name: fig:sensor_model_uncertainty_simplified
+#+caption: Input Uncertainty
+#+RESULTS:
+[[file:figs/sensor_model_uncertainty_simplified.png]]
+
 Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. ref:fig:sensor_fusion_dynamic_uncertainty where $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $|w_i(s)|$ is a weight representing the magnitude of the uncertainty.
 
 #+name: fig:sensor_fusion_dynamic_uncertainty
@@ -462,7 +486,7 @@ Let's validate the proposed design method of complementary filters with a simple
 - the gain of both filters is equal to $10^{-3}$ away from the merging frequency
 
 The weighting functions $W_1(s)$ and $W_2(s)$ are designed using eqref:eq:weight_formula.
-The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_synthesis_results.
+The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_filters_results.
 
 #+name: tab:weights_params
 #+caption: Parameters used for $W_1(s)$ and $W_2(s)$
@@ -476,16 +500,15 @@ The parameters used are summarized in table ref:tab:weights_params and the magni
 | $G_c$                  | $0.5$    | $0.5$    |
 | $n$                    | $2$      | $3$      |
 
-The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_results and their transfer functions in the Laplace domain are given below.
+The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_filters_results and their transfer functions in the Laplace domain are given below.
 \begin{align*}
   H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
   H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
 \end{align*}
 
-#+name: fig:hinf_synthesis_results
+#+name: fig:hinf_filters_results
 #+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
-#+attr_latex: :scale 1
+[[file:figs/hinf_filters_results.pdf]]
-[[file:figs/hinf_synthesis_results.pdf]]
 
 * Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
 <<sec:application_ligo>>

journal/journal.tex

@@ -1,28 +1,21 @@
-% Created 2021-04-28 mer. 15:56
+% Created 2021-04-30 ven. 11:16
 % Intended LaTeX compiler: pdflatex
 \documentclass[preprint, sort&compress]{elsarticle}
-
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{graphicx}
 \usepackage{grffile}
+\usepackage{longtable}
+\usepackage{wrapfig}
 \usepackage{rotating}
 \usepackage[normalem]{ulem}
+\usepackage{amsmath}
+\usepackage{textcomp}
+\usepackage{amssymb}
 \usepackage{capt-of}
 \usepackage{hyperref}
-\usepackage{bm}
+\usepackage{amsfonts}
-\usepackage{array}
+\usepackage{siunitx}
-\usepackage{amsmath,amssymb,amsfonts}
-\usepackage{algorithmic}
-\usepackage{textcomp}
-\usepackage{cases}
-\usepackage{tabularx,siunitx,booktabs}
-\usepackage{algorithmic}
-\usepackage{import}
-\usepackage{hyperref}
-\usepackage[hyperref]{xcolor}
-\hypersetup{colorlinks=true}
-\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
 \journal{Mechanical Systems and Signal Processing}
 \author[a1,a2]{Thomas Dehaeze\corref{cor1}}
 \author[a3,a4]{Mohit Verma}
@@ -32,6 +25,12 @@
 \address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
 \address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
 \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
+\usepackage{tabularx}
+\usepackage{booktabs}
+\usepackage{array}
+\usepackage[hyperref]{xcolor}
+\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
+\hypersetup{colorlinks=true}
 \date{}
 \title{A new method of designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis}
 \begin{document}
@@ -58,7 +57,7 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
 \end{frontmatter}
 
 \section{Introduction}
-\label{sec:orgf244196}
+\label{sec:org341e767}
 \label{sec:introduction}
 \begin{itemize}
 \item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
@@ -78,30 +77,64 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
 \item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering
 \end{itemize}
 \begin{itemize}
-\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
+\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
-\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
+
-\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory
-\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
-\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
 \item Analytical methods:
 \begin{itemize}
 \item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic}
 \item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas}
 \item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo}
 \end{itemize}
-\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
+\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
+\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
+\item Similar to feedback system:
+\begin{itemize}
+\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
+\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory
+\end{itemize}
+
+\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
 \end{itemize}
 \begin{itemize}
 \item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr} change of phase near the merging frequency
 \item Trial and error
 \item Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available.
 \end{itemize}
+Most of the requirements => shape of the complementary filters
+=> propose a way to shape complementary filters.
 
 \section{Complementary Filters Requirements}
-\label{sec:org2279a0f}
+\label{sec:org77471d1}
 \label{sec:requirements}
+Complementary filters provides a framework for fusing signals from different sensors.
+As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
+These requirements are discussed in this section.
+\subsection{Sensor Models}
+\label{sec:org363af04}
+\label{sec:sensor_models}
+
+\begin{itemize}
+\item Noise + dynamics
+\end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model.pdf}
+\caption{\label{fig:sensor_model}Basic Sensor Model}
+\end{figure}
+
+\begin{itemize}
+\item Suppose we calibrate the sensors
+\end{itemize}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
+\caption{\label{fig:sensor_model_calibrated}Calibrated Sensor}
+\end{figure}
+
 \subsection{Sensor Fusion Architecture}
-\label{sec:org60aa99d}
+\label{sec:org240da2b}
 \label{sec:sensor_fusion}
 
 Let's consider two sensors measuring the same physical quantity \(x\) with dynamics \(G_1(s)\) and \(G_2(s)\), and with uncorrelated noise characteristics \(n_1\) and \(n_2\).
@@ -125,7 +158,7 @@ The complementary property of \(H_1(s)\) and \(H_2(s)\) implies that their trans
 \end{equation}
 
 \subsection{Noise Sensor Filtering}
-\label{sec:orgf9a3723}
+\label{sec:orgc5064da}
 \label{sec:noise_filtering}
 
 Let's first consider sensors with perfect dynamics
@@ -159,13 +192,25 @@ Usually, the two sensors have high noise levels over distinct frequency regions.
 In order to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\).
 
 \subsection{Robustness of the Fusion}
-\label{sec:org227af67}
+\label{sec:orgfc9ea9e}
 \label{sec:fusion_robustness}
 
 In practical systems the sensor dynamics is not perfect and \eqref{eq:perfect_dynamics} is not verified.
 In such case, one can use an inversion filter \(\hat{G}_i^{-1}(s)\) to normalize the sensor dynamics, where \(\hat{G}_i(s)\) is an estimate of the sensor dynamics \(G_i(s)\).
 However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and \(\hat{G}_i^{-1}(s) G_i(s) \neq 1\).
 
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_uncertainty.png}
+\caption{\label{fig:sensor_model_uncertainty}Input Uncertainty}
+\end{figure}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_uncertainty_simplified.png}
+\caption{\label{fig:sensor_model_uncertainty_simplified}Input Uncertainty}
+\end{figure}
+
 Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. \ref{fig:sensor_fusion_dynamic_uncertainty} where \(\Delta_i\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(|w_i(s)|\) is a weight representing the magnitude of the uncertainty.
 
 \begin{figure}[htbp]
@@ -204,13 +249,13 @@ where \(\Delta \phi_\text{max}\) is the maximum allowed added phase.
 Thus the norm of the complementary filter \(|H_i|\) should be made small at frequencies where \(|w_i|\) is large.
 
 \section{Complementary Filters Shaping using \(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org10fbb17}
+\label{sec:org678f099}
 \label{sec:hinf_method}
 As shown in Sec. \ref{sec:requirements}, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms.
 Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary.
-\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
+\subsection{Synthesis Objective}
-\label{sec:org933b14f}
+\label{sec:orgf726b5b}
-\label{sec:hinf_synthesis}
+\label{sec:synthesis_objective}
 The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}.
 This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:comp_filter_problem_form} are satisfied.
 \begin{subequations}
@@ -223,6 +268,9 @@ This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(
 \end{subequations}
 where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to shape the norms of the corresponding filters.
 
+\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
+\label{sec:orga266a36}
+\label{sec:hinf_synthesis}
 In order to express this optimization problem as a standard \(\mathcal{H}_\infty\) problem, the architecture shown in Fig. \ref{fig:h_infinity_robust_fusion} is used where the generalized plant \(P\) is described by \eqref{eq:generalized_plant}.
 \begin{equation}
 \label{eq:generalized_plant}
@@ -257,7 +305,7 @@ The conditions \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} on the filter
 Therefore, all the conditions \eqref{eq:comp_filter_problem_form} are satisfied using this synthesis method based on \(\mathcal{H}_\infty\) synthesis, and thus it permits to shape complementary filters as desired.
 
 \subsection{Weighting Functions Design}
-\label{sec:org3bb6eca}
+\label{sec:org911c399}
 \label{sec:hinf_weighting_func}
 The proper design of the weighting functions is of primary importance for the success of the presented complementary filters \(\mathcal{H}_\infty\) synthesis.
 
@@ -302,7 +350,7 @@ The general shape of a weighting function generated using \eqref{eq:weight_formu
 \end{figure}
 
 \subsection{Validation of the proposed synthesis method}
-\label{sec:org5517901}
+\label{sec:org6867aff}
 \label{sec:hinf_example}
 Let's validate the proposed design method of complementary filters with a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
 \begin{itemize}
@@ -313,7 +361,7 @@ Let's validate the proposed design method of complementary filters with a simple
 \end{itemize}
 
 The weighting functions \(W_1(s)\) and \(W_2(s)\) are designed using \eqref{eq:weight_formula}.
-The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_synthesis_results}.
+The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_filters_results}.
 
 \begin{table}[htbp]
 \caption{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\)}
@@ -331,7 +379,7 @@ Parameter & \(W_1(s)\) & \(W_2(s)\)\\
 \end{tabularx}
 \end{table}
 
-The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_synthesis_results} and their transfer functions in the Laplace domain are given below.
+The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_filters_results} and their transfer functions in the Laplace domain are given below.
 \begin{align*}
   H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
   H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
@@ -339,50 +387,12 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,scale=1]{figs/hinf_synthesis_results.pdf}
+\includegraphics[scale=1]{figs/hinf_filters_results.pdf}
-\caption{\label{fig:hinf_synthesis_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
+\caption{\label{fig:hinf_filters_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
-\end{figure}
-
-\subsection{Synthesis of Three Complementary Filters}
-\label{sec:org1fd1484}
-\label{sec:hinf_three_comp_filters}
-Some applications may require to merge more than two sensors.
-In such a case, it is necessary to design as many complementary filters as the number of sensors used.
-The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s)\) such that \eqref{eq:hinf_problem_gen} is satisfied.
-\begin{subequations}
-\label{eq:hinf_problem_gen}
-  \begin{align}
-  & \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
-  & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
-  \end{align}
-\end{subequations}
-The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. \ref{fig:comp_filter_three_hinf}.
-
-The \(\mathcal{H}_\infty\) synthesis objective applied on \(P(s)\) is to design two stable filters \(H_2(s)\) and \(H_3(s)\) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one \eqref{eq:hinf_syn_obj_three}.
-\begin{equation}
-\label{eq:hinf_syn_obj_three}
-  \left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
-\end{equation}
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,scale=1]{figs/comp_filter_three_hinf.pdf}
-\caption{\label{fig:comp_filter_three_hinf}Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters}
-\end{figure}
-
-By choosing \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen}. \par
-An example is given to validate the method where three sensors are used in different frequency bands (up to \(\SI{1}{Hz}\), from \(1\) to \(\SI{10}{Hz}\) and above \(\SI{10}{Hz}\) respectively).
-Three weighting functions are designed using \eqref{eq:weight_formula} and shown by dashed curves in Fig. \ref{fig:hinf_three_synthesis_results}.
-The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_three_synthesis_results}.
-
-\begin{figure}[htbp]
-\centering
-\includegraphics[scale=1,scale=1]{figs/hinf_three_synthesis_results.pdf}
-\caption{\label{fig:hinf_three_synthesis_results}Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
 \end{figure}
 
 \section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
-\label{sec:org925754e}
+\label{sec:org377e66e}
 \label{sec:application_ligo}
 Several complementary filters are used in the active isolation system at the LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}.
 The requirements on those filters are very tight and thus their design is complex.
@@ -391,7 +401,7 @@ The obtained FIR filters are compliant with the requirements. However they are o
 
 The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in \cite{hua05_low_ligo}.
 \subsection{Complementary Filters Specifications}
-\label{sec:org4ade5f6}
+\label{sec:org75813ae}
 \label{sec:ligo_specifications}
 The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}) and shown in Fig. \ref{fig:ligo_weights}:
 \begin{itemize}
@@ -402,7 +412,7 @@ The specifications for one pair of complementary filters used at the LIGO are su
 \end{itemize}
 
 \subsection{Weighting Functions Design}
-\label{sec:org8c85120}
+\label{sec:org7015511}
 \label{sec:ligo_weights}
 The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
 However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
@@ -418,7 +428,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
 \end{figure}
 
 \subsection{\(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org89d27e1}
+\label{sec:orge5b6fe6}
 \label{sec:ligo_results}
 \(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}.
 The complementary filters obtained are of order \(27\).
@@ -432,7 +442,7 @@ They are found to be very close to each other and this shows the effectiveness o
 \end{figure}
 
 \section{Conclusion}
-\label{sec:org153df0f}
+\label{sec:org39b90d9}
 \label{sec:conclusion}
 This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
 Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters.
@@ -440,7 +450,7 @@ Therefore, a synthesis method that permits the shaping of the complementary filt
 Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.
 
 \section*{Acknowledgment}
-\label{sec:org6c8e8b0}
+\label{sec:org1ece332}
 This research benefited from a FRIA grant from the French Community of Belgium.
 
 \bibliographystyle{elsarticle-num}

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@@ -30,42 +39,63 @@
 <h2>Table of Contents</h2>
 <div id="text-table-of-contents">
 <ul>
-<li><a href="#orga86008b">1. H-Infinity synthesis of complementary filters</a>
+<li><a href="#sec:h_inf_synthesis_complementary_filters">1. H-Infinity synthesis of complementary filters</a>
 <ul>
-<li><a href="#orga8d0882">1.1. Synthesis Architecture</a></li>
+<li><a href="#org38a6275">1.1. Synthesis Architecture</a></li>
-<li><a href="#orgf309349">1.2. Design of Weighting Function</a></li>
+<li><a href="#org8d0a2ea">1.2. Design of Weighting Function</a></li>
-<li><a href="#org3d519e3">1.3. H-Infinity Synthesis</a></li>
+<li><a href="#orgc235246">1.3. H-Infinity Synthesis</a></li>
-<li><a href="#org42911cd">1.4. Obtained Complementary Filters</a></li>
+<li><a href="#org326a6a1">1.4. Obtained Complementary Filters</a></li>
 </ul>
 </li>
-<li><a href="#org1cd882b">2. Generating 3 complementary filters</a>
+<li><a href="#orgb616673">2. Generating 3 complementary filters</a>
 <ul>
-<li><a href="#org201b962">2.1. Theory</a></li>
+<li><a href="#orge13ec24">2.1. Theory</a></li>
-<li><a href="#orgbb81a3a">2.2. Weights</a></li>
+<li><a href="#org0043ce8">2.2. Weights</a></li>
-<li><a href="#orgc782a41">2.3. H-Infinity Synthesis</a></li>
+<li><a href="#orge8e2214">2.3. H-Infinity Synthesis</a></li>
-<li><a href="#orgbe6c26a">2.4. Obtained Complementary Filters</a></li>
+<li><a href="#org3e0db09">2.4. Obtained Complementary Filters</a></li>
 </ul>
 </li>
-<li><a href="#orgfb4a473">3. Implement complementary filters for LIGO</a>
+<li><a href="#orgb0c5eb8">3. Implement complementary filters for LIGO</a>
 <ul>
-<li><a href="#org0a64590">3.1. Specifications</a></li>
+<li><a href="#org68b2264">3.1. Specifications</a></li>
-<li><a href="#org5187f2d">3.2. FIR Filter</a></li>
+<li><a href="#org639758d">3.2. FIR Filter</a></li>
-<li><a href="#org6e83a71">3.3. Weights</a></li>
+<li><a href="#org3361251">3.3. Weights</a></li>
-<li><a href="#org56349cf">3.4. H-Infinity Synthesis</a></li>
+<li><a href="#org3ea9781">3.4. H-Infinity Synthesis</a></li>
-<li><a href="#org3ef818f">3.5. Compare FIR and H-Infinity Filters</a></li>
+<li><a href="#org2e38aee">3.5. Compare FIR and H-Infinity Filters</a></li>
 </ul>
 </li>
-<li><a href="#org6fa1123">4. Alternative Synthesis</a>
+<li><a href="#org278cb17">4. Alternative Synthesis</a>
 <ul>
-<li><a href="#org1bb8ee7">4.1. Two generalized plants</a></li>
+<li><a href="#orgc2f7629">4.1. Two generalized plants</a></li>
-<li><a href="#orga117463">4.2. Shaping the Low pass filter or the high pass filter?</a></li>
+<li><a href="#org14ddd98">4.2. Shaping the Low pass filter or the high pass filter?</a></li>
+<li><a href="#orgc9d9779">4.3. Using Feedback architecture</a></li>
+<li><a href="#org0aa4270">4.4. Adding feature in the filters</a></li>
 </ul>
 </li>
-<li><a href="#orgf082879">5. Impose a positive slope at DC or a negative slope at infinite frequency</a>
+<li><a href="#orgb026d30">5. Impose a positive slope at DC or a negative slope at infinite frequency</a>
 <ul>
-<li><a href="#org96df1d1">5.1. Manually shift zeros to the origin after synthesis</a></li>
+<li><a href="#org4d4e2ad">5.1. Manually shift zeros to the origin after synthesis</a></li>
-<li><a href="#org6b92ce0">5.2. Imposing a positive slope at DC during the synthesis phase</a></li>
+<li><a href="#orgf1693db">5.2. Imposing a positive slope at DC during the synthesis phase</a></li>
-<li><a href="#org71e3235">5.3. Imposing a negative slope at infinity frequency during the synthesis phase</a></li>
+<li><a href="#org52c02ef">5.3. Imposing a negative slope at infinity frequency during the synthesis phase</a></li>
+</ul>
+</li>
+<li><a href="#org29e79e7">6. Functions</a>
+<ul>
+<li><a href="#orgfbca8a7">6.1. <code>generateWF</code>: Generate Weighting Functions</a>
+<ul>
+<li><a href="#org95965a4">Function description</a></li>
+<li><a href="#orgaeb61de">Optional Parameters</a></li>
+<li><a href="#orgd91e493">Generate the Weighting function</a></li>
+<li><a href="#org84eb536">Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains</a></li>
+</ul>
+</li>
+<li><a href="#org2f9ac50">6.2. <code>generateCF</code>: Generate Complementary Filters</a>
+<ul>
+<li><a href="#org7c373f8">Function description</a></li>
+<li><a href="#org00c9d5d">Optional Parameters</a></li>
+<li><a href="#org92a7c04">H-Infinity Synthesis</a></li>
+</ul>
+</li>
 </ul>
 </li>
 </ul>
@@ -90,25 +120,22 @@ To achieve this, the sensors included in the filter should complement one anothe
 This document is divided into several sections:
 </p>
 <ul class="org-ul">
-<li>in section <a href="#orge265c61">1</a>, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters</li>
+<li>in section <a href="#sec:h_inf_synthesis_complementary_filters">1</a>, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters</li>
-<li>in section <a href="#org0f5d922">2</a>, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters</li>
+<li>in section <a href="#orgd4d516e">2</a>, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters</li>
-<li>in section <a href="#org2c84916">3</a>, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO</li>
+<li>in section <a href="#org9327342">3</a>, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO</li>
 </ul>
 
-<div class="note" id="org9abbfdc">
+<div class="note" id="orgad6d854">
 <p>
 Add the Matlab code use to obtain the results presented in the paper are accessible <a href="matlab.zip">here</a> and presented below.
 </p>
 
 </div>
 
-<div id="outline-container-orga86008b" class="outline-2">
+<div id="outline-container-sec:h_inf_synthesis_complementary_filters" class="outline-2">
-<h2 id="orga86008b"><span class="section-number-2">1</span> H-Infinity synthesis of complementary filters</h2>
+<h2 id="sec:h_inf_synthesis_complementary_filters"><span class="section-number-2">1</span> H-Infinity synthesis of complementary filters</h2>
-<div class="outline-text-2" id="text-1">
+<div class="outline-text-2" id="text-sec:h_inf_synthesis_complementary_filters">
-<p>
+<div class="note" id="org65d886e">
-<a id="orge265c61"></a>
-</p>
-<div class="note" id="org7b6c965">
 <p>
 The Matlab file corresponding to this section is accessible <a href="matlab/h_inf_synthesis_complementary_filters.m">here</a>.
 </p>
@@ -116,8 +143,8 @@ The Matlab file corresponding to this section is accessible <a href="matlab/h_in
 </div>
 </div>
 
-<div id="outline-container-orga8d0882" class="outline-3">
+<div id="outline-container-org38a6275" class="outline-3">
-<h3 id="orga8d0882"><span class="section-number-3">1.1</span> Synthesis Architecture</h3>
+<h3 id="org38a6275"><span class="section-number-3">1.1</span> Synthesis Architecture</h3>
 <div class="outline-text-3" id="text-1-1">
 <p>
 We here synthesize two complementary filters using the \(\mathcal{H}_\infty\) synthesis.
@@ -125,18 +152,18 @@ The goal is to specify upper bounds on the norms of the two complementary filter
 </p>
 
 <p>
-In order to do so, we use the generalized plant shown on figure <a href="#orge741156">1</a> 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.
+In order to do so, we use the generalized plant shown on figure <a href="#org49ae644">1</a> 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.
 </p>
 
 
-<div id="orge741156" class="figure">
+<div id="org49ae644" class="figure">
 <p><img src="figs-tikz/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
 </p>
 <p><span class="figure-number">Figure 1: </span>\(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
 </div>
 
 <p>
-The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_2\) (figure <a href="#orge741156">1</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2]\) is less than one:
+The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_2\) (figure <a href="#org49ae644">1</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2]\) is less than one:
 \[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
 </p>
 
@@ -156,8 +183,8 @@ We then see that \(W_1(s)\) and \(W_2(s)\) can be used to shape both \(H_1(s)\)
 </div>
 </div>
 
-<div id="outline-container-orgf309349" class="outline-3">
+<div id="outline-container-org8d0a2ea" class="outline-3">
-<h3 id="orgf309349"><span class="section-number-3">1.2</span> Design of Weighting Function</h3>
+<h3 id="org8d0a2ea"><span class="section-number-3">1.2</span> Design of Weighting Function</h3>
 <div class="outline-text-3" id="text-1-2">
 <p>
 A formula is proposed to help the design of the weighting functions:
@@ -181,11 +208,11 @@ The parameters permits to specify:
 </ul>
 
 <p>
-The general shape of a weighting function generated using the formula is shown in figure <a href="#org18f93a4">2</a>.
+The general shape of a weighting function generated using the formula is shown in figure <a href="#orgca3464c">2</a>.
 </p>
 
 
-<div id="org18f93a4" class="figure">
+<div id="orgca3464c" class="figure">
 <p><img src="figs/weight_formula.png" alt="weight_formula.png" />
 </p>
 <p><span class="figure-number">Figure 2: </span>Gain of the Weighting Function formula</p>
@@ -201,7 +228,7 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </div>
 
 
-<div id="org54a46b0" class="figure">
+<div id="org1ce2cf7" class="figure">
 <p><img src="figs/weights_W1_W2.png" alt="weights_W1_W2.png" />
 </p>
 <p><span class="figure-number">Figure 3: </span>Weights on the complementary filters \(W_1\) and \(W_2\) and the associated performance weights</p>
@@ -209,8 +236,8 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </div>
 </div>
 
-<div id="outline-container-org3d519e3" class="outline-3">
+<div id="outline-container-orgc235246" class="outline-3">
-<h3 id="org3d519e3"><span class="section-number-3">1.3</span> H-Infinity Synthesis</h3>
+<h3 id="orgc235246"><span class="section-number-3">1.3</span> H-Infinity Synthesis</h3>
 <div class="outline-text-3" id="text-1-3">
 <p>
 We define the generalized plant \(P\) on matlab.
@@ -230,7 +257,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> co
 </pre>
 </div>
 
-<pre class="example" id="orgb213b31">
+<pre class="example" id="org44ccac0">
 [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
 
@@ -264,7 +291,7 @@ Test bounds:      0.1000 &lt;  gamma  &lt;=   1050.0000
 </pre>
 
 <p>
-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 <a href="#orgc79ce80">4</a>.
+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 <a href="#orgdd6084f">4</a>.
 </p>
 
 <div class="org-src-container">
@@ -274,15 +301,15 @@ We then define the high pass filter \(H_1 = 1 - H_2\). The bode plot of both \(H
 </div>
 </div>
 
-<div id="outline-container-org42911cd" class="outline-3">
+<div id="outline-container-org326a6a1" class="outline-3">
-<h3 id="org42911cd"><span class="section-number-3">1.4</span> Obtained Complementary Filters</h3>
+<h3 id="org326a6a1"><span class="section-number-3">1.4</span> Obtained Complementary Filters</h3>
 <div class="outline-text-3" id="text-1-4">
 <p>
-The obtained complementary filters are shown on figure <a href="#orgc79ce80">4</a>.
+The obtained complementary filters are shown on figure <a href="#orgdd6084f">4</a>.
 </p>
 
 
-<div id="orgc79ce80" class="figure">
+<div id="orgdd6084f" class="figure">
 <p><img src="figs/hinf_filters_results.png" alt="hinf_filters_results.png" />
 </p>
 <p><span class="figure-number">Figure 4: </span>Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis</p>
@@ -291,13 +318,13 @@ The obtained complementary filters are shown on figure <a href="#orgc79ce80">4</
 </div>
 </div>
 
-<div id="outline-container-org1cd882b" class="outline-2">
+<div id="outline-container-orgb616673" class="outline-2">
-<h2 id="org1cd882b"><span class="section-number-2">2</span> Generating 3 complementary filters</h2>
+<h2 id="orgb616673"><span class="section-number-2">2</span> Generating 3 complementary filters</h2>
 <div class="outline-text-2" id="text-2">
 <p>
-<a id="org0f5d922"></a>
+<a id="orgd4d516e"></a>
 </p>
-<div class="note" id="org7c35287">
+<div class="note" id="org1a1e099">
 <p>
 The Matlab file corresponding to this section is accessible <a href="matlab/three_comp_filters.m">here</a>.
 </p>
@@ -305,8 +332,8 @@ The Matlab file corresponding to this section is accessible <a href="matlab/thre
 </div>
 </div>
 
-<div id="outline-container-org201b962" class="outline-3">
+<div id="outline-container-orge13ec24" class="outline-3">
-<h3 id="org201b962"><span class="section-number-3">2.1</span> Theory</h3>
+<h3 id="orge13ec24"><span class="section-number-3">2.1</span> Theory</h3>
 <div class="outline-text-3" id="text-2-1">
 <p>
 We want:
@@ -319,11 +346,11 @@ We want:
 \end{align*}
 
 <p>
-For that, we use the \(\mathcal{H}_\infty\) synthesis with the architecture shown on figure <a href="#org86ebebf">5</a>.
+For that, we use the \(\mathcal{H}_\infty\) synthesis with the architecture shown on figure <a href="#org38edab5">5</a>.
 </p>
 
 
-<div id="org86ebebf" class="figure">
+<div id="org38edab5" class="figure">
 <p><img src="figs-tikz/comp_filter_three_hinf.png" alt="comp_filter_three_hinf.png" />
 </p>
 <p><span class="figure-number">Figure 5: </span>Generalized architecture for generating 3 complementary filters</p>
@@ -344,8 +371,8 @@ And thus if we choose \(H_1 = 1 - H_2 - H_3\) we have solved the problem.
 </div>
 </div>
 
-<div id="outline-container-orgbb81a3a" class="outline-3">
+<div id="outline-container-org0043ce8" class="outline-3">
-<h3 id="orgbb81a3a"><span class="section-number-3">2.2</span> Weights</h3>
+<h3 id="org0043ce8"><span class="section-number-3">2.2</span> Weights</h3>
 <div class="outline-text-3" id="text-2-2">
 <p>
 First we define the weights.
@@ -362,7 +389,7 @@ W3 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </div>
 
 
-<div id="org6ef9224" class="figure">
+<div id="orgf1d851e" class="figure">
 <p><img src="figs/three_weighting_functions.png" alt="three_weighting_functions.png" />
 </p>
 <p><span class="figure-number">Figure 6: </span>Three weighting functions used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
@@ -370,8 +397,8 @@ W3 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </div>
 </div>
 
-<div id="outline-container-orgc782a41" class="outline-3">
+<div id="outline-container-orge8e2214" class="outline-3">
-<h3 id="orgc782a41"><span class="section-number-3">2.3</span> H-Infinity Synthesis</h3>
+<h3 id="orge8e2214"><span class="section-number-3">2.3</span> H-Infinity Synthesis</h3>
 <div class="outline-text-3" id="text-2-3">
 <p>
 Then we create the generalized plant <code>P</code>.
@@ -392,7 +419,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis.
 </pre>
 </div>
 
-<pre class="example" id="orga35c0a2">
+<pre class="example" id="org65963f1">
 [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
 
@@ -427,8 +454,8 @@ Test bounds:      0.1000 &lt;  gamma  &lt;=   1050.0000
 </div>
 </div>
 
-<div id="outline-container-orgbe6c26a" class="outline-3">
+<div id="outline-container-org3e0db09" class="outline-3">
-<h3 id="orgbe6c26a"><span class="section-number-3">2.4</span> Obtained Complementary Filters</h3>
+<h3 id="org3e0db09"><span class="section-number-3">2.4</span> Obtained Complementary Filters</h3>
 <div class="outline-text-3" id="text-2-4">
 <p>
 The obtained filters are:
@@ -441,7 +468,7 @@ H1 = 1 <span class="org-type">-</span> H2 <span class="org-type">-</span> H3;
 </div>
 
 
-<div id="orga85736d" class="figure">
+<div id="orgd968c86" class="figure">
 <p><img src="figs/three_complementary_filters_results.png" alt="three_complementary_filters_results.png" />
 </p>
 <p><span class="figure-number">Figure 7: </span>The three complementary filters obtained after \(\mathcal{H}_\infty\) synthesis</p>
@@ -450,13 +477,13 @@ H1 = 1 <span class="org-type">-</span> H2 <span class="org-type">-</span> H3;
 </div>
 </div>
 
-<div id="outline-container-orgfb4a473" class="outline-2">
+<div id="outline-container-orgb0c5eb8" class="outline-2">
-<h2 id="orgfb4a473"><span class="section-number-2">3</span> Implement complementary filters for LIGO</h2>
+<h2 id="orgb0c5eb8"><span class="section-number-2">3</span> Implement complementary filters for LIGO</h2>
 <div class="outline-text-2" id="text-3">
 <p>
-<a id="org2c84916"></a>
+<a id="org9327342"></a>
 </p>
-<div class="note" id="org4890f37">
+<div class="note" id="orgd570d71">
 <p>
 The Matlab file corresponding to this section is accessible <a href="matlab/comp_filters_ligo.m">here</a>.
 </p>
@@ -472,8 +499,8 @@ The FIR complementary filters designed in (<a href="#citeproc_bib_item_1">Hua 20
 </p>
 </div>
 
-<div id="outline-container-org0a64590" class="outline-3">
+<div id="outline-container-org68b2264" class="outline-3">
-<h3 id="org0a64590"><span class="section-number-3">3.1</span> Specifications</h3>
+<h3 id="org68b2264"><span class="section-number-3">3.1</span> Specifications</h3>
 <div class="outline-text-3" id="text-3-1">
 <p>
 The specifications for the filters are:
@@ -486,11 +513,11 @@ The specifications for the filters are:
 </ol>
 
 <p>
-The specifications are translated in upper bounds of the complementary filters are shown on figure <a href="#org40b6368">8</a>.
+The specifications are translated in upper bounds of the complementary filters are shown on figure <a href="#orge912401">8</a>.
 </p>
 
 
-<div id="org40b6368" class="figure">
+<div id="orge912401" class="figure">
 <p><img src="figs/ligo_specifications.png" alt="ligo_specifications.png" />
 </p>
 <p><span class="figure-number">Figure 8: </span>Specification for the LIGO complementary filters</p>
@@ -498,8 +525,8 @@ The specifications are translated in upper bounds of the complementary filters a
 </div>
 </div>
 
-<div id="outline-container-org5187f2d" class="outline-3">
+<div id="outline-container-org639758d" class="outline-3">
-<h3 id="org5187f2d"><span class="section-number-3">3.2</span> FIR Filter</h3>
+<h3 id="org639758d"><span class="section-number-3">3.2</span> FIR Filter</h3>
 <div class="outline-text-3" id="text-3-2">
 <p>
 We here try to implement the FIR complementary filter synthesis as explained in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
@@ -580,7 +607,7 @@ h = y(2<span class="org-type">:</span>end);
 </pre>
 </div>
 
-<pre class="example" id="org199d33a">
+<pre class="example" id="org1229f63">
 cvx_begin
 variable y(n+1,1)
 % t
@@ -657,7 +684,7 @@ h = y(2:end);
 </pre>
 
 <p>
-Finally, we compute the filter response over the frequency vector defined and the result is shown on figure <a href="#org1807c4b">9</a> which is very close to the filters obtain in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
+Finally, we compute the filter response over the frequency vector defined and the result is shown on figure <a href="#org13d9ffd">9</a> which is very close to the filters obtain in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
 </p>
 
 <div class="org-src-container">
@@ -667,7 +694,7 @@ H = [exp(<span class="org-type">-</span><span class="org-constant">j</span><span
 </div>
 
 
-<div id="org1807c4b" class="figure">
+<div id="org13d9ffd" class="figure">
 <p><img src="figs/fir_filter_ligo.png" alt="fir_filter_ligo.png" />
 </p>
 <p><span class="figure-number">Figure 9: </span>FIR Complementary filters obtain after convex optimization</p>
@@ -675,8 +702,8 @@ H = [exp(<span class="org-type">-</span><span class="org-constant">j</span><span
 </div>
 </div>
 
-<div id="outline-container-org6e83a71" class="outline-3">
+<div id="outline-container-org3361251" class="outline-3">
-<h3 id="org6e83a71"><span class="section-number-3">3.3</span> Weights</h3>
+<h3 id="org3361251"><span class="section-number-3">3.3</span> Weights</h3>
 <div class="outline-text-3" id="text-3-3">
 <p>
 We design weights that will be used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters.
@@ -690,11 +717,11 @@ Here are the requirements on the filters:
 </ul>
 
 <p>
-The bode plot of the weights is shown on figure <a href="#org8999a4f">10</a>.
+The bode plot of the weights is shown on figure <a href="#org17273a4">10</a>.
 </p>
 
 
-<div id="org8999a4f" class="figure">
+<div id="org17273a4" class="figure">
 <p><img src="figs/ligo_weights.png" alt="ligo_weights.png" />
 </p>
 <p><span class="figure-number">Figure 10: </span>Weights for the \(\mathcal{H}_\infty\) synthesis</p>
@@ -702,11 +729,11 @@ The bode plot of the weights is shown on figure <a href="#org8999a4f">10</a>.
 </div>
 </div>
 
-<div id="outline-container-org56349cf" class="outline-3">
+<div id="outline-container-org3ea9781" class="outline-3">
-<h3 id="org56349cf"><span class="section-number-3">3.4</span> H-Infinity Synthesis</h3>
+<h3 id="org3ea9781"><span class="section-number-3">3.4</span> H-Infinity Synthesis</h3>
 <div class="outline-text-3" id="text-3-4">
 <p>
-We define the generalized plant as shown on figure <a href="#orge741156">1</a>.
+We define the generalized plant as shown on figure <a href="#org49ae644">1</a>.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">P = [0   wL;
@@ -723,7 +750,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> co
 </pre>
 </div>
 
-<pre class="example" id="orgd8777e8">
+<pre class="example" id="org82970b4">
 [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
 
@@ -758,18 +785,18 @@ The high pass filter is defined as \(H_H = 1 - H_L\).
 The size of the filters is shown below.
 </p>
 
-<pre class="example" id="orgc30176a">
+<pre class="example" id="org5988dfe">
 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.
 </pre>
 
 <p>
-The bode plot of the obtained filters as shown on figure <a href="#orgf3626bd">11</a>.
+The bode plot of the obtained filters as shown on figure <a href="#org1a89dc1">11</a>.
 </p>
 
 
-<div id="orgf3626bd" class="figure">
+<div id="org1a89dc1" class="figure">
 <p><img src="figs/hinf_synthesis_ligo_results.png" alt="hinf_synthesis_ligo_results.png" />
 </p>
 <p><span class="figure-number">Figure 11: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) synthesis</p>
@@ -777,15 +804,15 @@ The bode plot of the obtained filters as shown on figure <a href="#orgf3626bd">1
 </div>
 </div>
 
-<div id="outline-container-org3ef818f" class="outline-3">
+<div id="outline-container-org2e38aee" class="outline-3">
-<h3 id="org3ef818f"><span class="section-number-3">3.5</span> Compare FIR and H-Infinity Filters</h3>
+<h3 id="org2e38aee"><span class="section-number-3">3.5</span> Compare FIR and H-Infinity Filters</h3>
 <div class="outline-text-3" id="text-3-5">
 <p>
-Let&rsquo;s now compare the FIR filters designed in (<a href="#citeproc_bib_item_1">Hua 2005</a>) and the one obtained with the \(\mathcal{H}_\infty\) synthesis on figure <a href="#orga60b7f8">12</a>.
+Let&rsquo;s now compare the FIR filters designed in (<a href="#citeproc_bib_item_1">Hua 2005</a>) and the one obtained with the \(\mathcal{H}_\infty\) synthesis on figure <a href="#orgf51384d">12</a>.
 </p>
 
 
-<div id="orga60b7f8" class="figure">
+<div id="orgf51384d" class="figure">
 <p><img src="figs/comp_fir_ligo_hinf.png" alt="comp_fir_ligo_hinf.png" />
 </p>
 <p><span class="figure-number">Figure 12: </span>Comparison between the FIR filters developped for LIGO and the \(\mathcal{H}_\infty\) complementary filters</p>
@@ -794,15 +821,15 @@ Let&rsquo;s now compare the FIR filters designed in (<a href="#citeproc_bib_item
 </div>
 </div>
 
-<div id="outline-container-org6fa1123" class="outline-2">
+<div id="outline-container-org278cb17" class="outline-2">
-<h2 id="org6fa1123"><span class="section-number-2">4</span> Alternative Synthesis</h2>
+<h2 id="org278cb17"><span class="section-number-2">4</span> Alternative Synthesis</h2>
 <div class="outline-text-2" id="text-4">
 </div>
-<div id="outline-container-org1bb8ee7" class="outline-3">
+<div id="outline-container-orgc2f7629" class="outline-3">
-<h3 id="org1bb8ee7"><span class="section-number-3">4.1</span> Two generalized plants</h3>
+<h3 id="orgc2f7629"><span class="section-number-3">4.1</span> Two generalized plants</h3>
 <div class="outline-text-3" id="text-4-1">
 <p>
-In order to synthesize the complementary filter using the proposed method, we can use two alternative generalized plant as shown in Figures <a href="#orgdfb88a5">13</a> and <a href="#orgb41d84e">14</a>.
+In order to synthesize the complementary filter using the proposed method, we can use two alternative generalized plant as shown in Figures <a href="#orgdc5c349">13</a> and <a href="#orgf13eaf1">14</a>.
 </p>
 
 \begin{equation}
@@ -811,7 +838,7 @@ In order to synthesize the complementary filter using the proposed method, we ca
 
 
 
-<div id="orgdfb88a5" class="figure">
+<div id="orgdc5c349" class="figure">
 <p><img src="figs/h_infinity_arch_1.png" alt="h_infinity_arch_1.png" />
 </p>
 <p><span class="figure-number">Figure 13: </span>Complementary Filter Synthesis - Conf 1</p>
@@ -822,7 +849,7 @@ In order to synthesize the complementary filter using the proposed method, we ca
 \end{equation}
 
 
-<div id="orgb41d84e" class="figure">
+<div id="orgf13eaf1" class="figure">
 <p><img src="figs/h_infinity_arch_2.png" alt="h_infinity_arch_2.png" />
 </p>
 <p><span class="figure-number">Figure 14: </span>Complementary Filter Synthesis - Conf 2</p>
@@ -852,7 +879,7 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </pre>
 </div>
 
-<pre class="example" id="org80f6d17">
+<pre class="example" id="org14b6e96">
 [H2, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 
   Test bounds:  0.3263 &lt;=  gamma  &lt;=  1000
@@ -888,7 +915,7 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </pre>
 </div>
 
-<pre class="example" id="org7155dd4">
+<pre class="example" id="org594f397">
 [H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 
   Test bounds:  0.3263 &lt;=  gamma  &lt;=  1000
@@ -914,10 +941,10 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
 </pre>
 
 <p>
-And indeed, we can see that the exact same filters are obtained (Figure <a href="#org4ff7339">15</a>).
+And indeed, we can see that the exact same filters are obtained (Figure <a href="#org806650d">15</a>).
 </p>
 
-<div id="org4ff7339" class="figure">
+<div id="org806650d" class="figure">
 <p><img src="figs/hinf_comp_P1_P2_syn.png" alt="hinf_comp_P1_P2_syn.png" />
 </p>
 <p><span class="figure-number">Figure 15: </span>Comparison of \(H_2(s)\) when using \(P_1(s)\) or \(P_2(s)\)</p>
@@ -925,8 +952,8 @@ And indeed, we can see that the exact same filters are obtained (Figure <a href=
 </div>
 </div>
 
-<div id="outline-container-orga117463" class="outline-3">
+<div id="outline-container-org14ddd98" class="outline-3">
-<h3 id="orga117463"><span class="section-number-3">4.2</span> Shaping the Low pass filter or the high pass filter?</h3>
+<h3 id="org14ddd98"><span class="section-number-3">4.2</span> Shaping the Low pass filter or the high pass filter?</h3>
 <div class="outline-text-3" id="text-4-2">
 <p>
 Let&rsquo;s see if there is a difference by explicitly shaping \(H_1(s)\) or \(H_2(s)\).
@@ -956,7 +983,7 @@ Let&rsquo;s first synthesize \(H_1(s)\):
 </pre>
 </div>
 
-<pre class="example" id="org45692a7">
+<pre class="example" id="org661bf48">
 [H1, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 
   Test bounds:  0.3263 &lt;=  gamma  &lt;=  1.712
@@ -995,7 +1022,7 @@ And now \(H_2(s)\):
 </pre>
 </div>
 
-<pre class="example" id="org6543cab">
+<pre class="example" id="org5a20bfb">
 [H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
 
   Test bounds:  0.3263 &lt;=  gamma  &lt;=  1000
@@ -1021,24 +1048,104 @@ And now \(H_2(s)\):
 </pre>
 
 <p>
-And compare \(H_1(s)\) with \(1 - H_2(s)\) and \(H_2(s)\) with \(1 - H_1(s)\) in Figure <a href="#org4a9724c">16</a>.
+And compare \(H_1(s)\) with \(1 - H_2(s)\) and \(H_2(s)\) with \(1 - H_1(s)\) in Figure <a href="#orgb9c9d7c">16</a>.
 </p>
 
-<div id="org4a9724c" class="figure">
+<div id="orgb9c9d7c" class="figure">
 <p><img src="figs/hinf_comp_H1_H2_syn.png" alt="hinf_comp_H1_H2_syn.png" />
 </p>
 <p><span class="figure-number">Figure 16: </span>Comparison of \(H_1(s)\) with \(1-H_2(s)\), and \(H_2(s)\) with \(1-H_1(s)\)</p>
 </div>
 </div>
 </div>
+
+<div id="outline-container-orgc9d9779" class="outline-3">
+<h3 id="orgc9d9779"><span class="section-number-3">4.3</span> Using Feedback architecture</h3>
+<div class="outline-text-3" id="text-4-3">
+<div class="org-src-container">
+<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>11; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2;
+W1 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+
+n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; G0 = 1000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2;
+W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+</pre>
 </div>
 
-<div id="outline-container-orgf082879" class="outline-2">
+<p>
-<h2 id="orgf082879"><span class="section-number-2">5</span> Impose a positive slope at DC or a negative slope at infinite frequency</h2>
+Let&rsquo;s first synthesize \(H_1(s)\):
+</p>
+<div class="org-src-container">
+<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1;
+     0   W2;
+     1  <span class="org-type">-</span>1];
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">[K, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'lmi'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">H1 = inv(1 <span class="org-type">+</span> K);
+H2 = 1 <span class="org-type">-</span> H1;
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org0aa4270" class="outline-3">
+<h3 id="org0aa4270"><span class="section-number-3">4.4</span> Adding feature in the filters</h3>
+<div class="outline-text-3" id="text-4-4">
+<div class="org-src-container">
+<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>11; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2;
+W1 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+
+n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; G0 = 1000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2;
+W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">Wf = (1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1)<span class="org-type">/</span>s;
+Wf = s<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1e2);
+
+<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">W2 </span></span><span class="org-comment">= W2/Wf/(1 + s/2/pi/1e3);</span>
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>Wf<span class="org-type">*</span>W1;
+     0   Wf<span class="org-type">*</span>W2;
+     1  <span class="org-type">-</span>Wf];
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">[Ka, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'lmi'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">K = Ka<span class="org-type">*</span>Wf;
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">H1 = inv(1 <span class="org-type">+</span> K);
+H2 = 1 <span class="org-type">-</span> H1;
+</pre>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgb026d30" class="outline-2">
+<h2 id="orgb026d30"><span class="section-number-2">5</span> Impose a positive slope at DC or a negative slope at infinite frequency</h2>
 <div class="outline-text-2" id="text-5">
 </div>
-<div id="outline-container-org96df1d1" class="outline-3">
+<div id="outline-container-org4d4e2ad" class="outline-3">
-<h3 id="org96df1d1"><span class="section-number-3">5.1</span> Manually shift zeros to the origin after synthesis</h3>
+<h3 id="org4d4e2ad"><span class="section-number-3">5.1</span> Manually shift zeros to the origin after synthesis</h3>
 <div class="outline-text-3" id="text-5-1">
 <p>
 Suppose we want \(H_2(s)\) to be an high pass filter with a slope of +2 at low frequency (from 0Hz).
@@ -1084,7 +1191,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> co
 <p>
 Looking at the zeros, we see two low frequency complex conjugate zeros.
 </p>
-<pre class="example" id="orge6a4f22">
+<pre class="example" id="org45e17b6">
 z{1}
 ans =
           -4690930.24283199 +                     0i
@@ -1119,12 +1226,12 @@ And as usual, \(H_{1z}(s)\) is defined as the complementary of \(H_{2z}(s)\):
 </div>
 
 <p>
-The bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\) are shown in Figure <a href="#org4631977">17</a>.
+The bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\) are shown in Figure <a href="#orga18e536">17</a>.
 And we see that \(H_{1z}(s)\) is slightly modified when setting the zeros at the origin for \(H_{2z}(s)\).
 </p>
 
 
-<div id="org4631977" class="figure">
+<div id="orga18e536" class="figure">
 <p><img src="figs/comp_filters_shift_zero.png" alt="comp_filters_shift_zero.png" />
 </p>
 <p><span class="figure-number">Figure 17: </span>Bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\)</p>
@@ -1132,16 +1239,16 @@ And we see that \(H_{1z}(s)\) is slightly modified when setting the zeros at the
 </div>
 </div>
 
-<div id="outline-container-org6b92ce0" class="outline-3">
+<div id="outline-container-orgf1693db" class="outline-3">
-<h3 id="org6b92ce0"><span class="section-number-3">5.2</span> Imposing a positive slope at DC during the synthesis phase</h3>
+<h3 id="orgf1693db"><span class="section-number-3">5.2</span> Imposing a positive slope at DC during the synthesis phase</h3>
 <div class="outline-text-3" id="text-5-2">
 <p>
 Suppose we want to synthesize \(H_2(s)\) such that it has a slope of +2 from DC.
-We can include this &ldquo;feature&rdquo; in the generalized plant as shown in Figure <a href="#org00fe83c">18</a>.
+We can include this &ldquo;feature&rdquo; in the generalized plant as shown in Figure <a href="#orgf2b673c">18</a>.
 </p>
 
 
-<div id="org00fe83c" class="figure">
+<div id="orgf2b673c" class="figure">
 <p><img src="figs/h_infinity_arch_H2_feature.png" alt="h_infinity_arch_H2_feature.png" />
 </p>
 <p><span class="figure-number">Figure 18: </span>Generalized plant with included wanted feature represented by \(H_{2w}(s)\)</p>
@@ -1184,7 +1291,7 @@ H2w = (s<span class="org-type">/</span>w0<span class="org-type">/</span>(s<span
 </div>
 
 <p>
-We define the generalized plant as shown in Figure <a href="#org00fe83c">18</a>.
+We define the generalized plant as shown in Figure <a href="#orgf2b673c">18</a>.
 </p>
 <div class="org-src-container">
 <pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1;
@@ -1218,11 +1325,11 @@ And we define \(H_1(s)\) to be the complementary of \(H_2(s)\):
 </div>
 
 <p>
-The obtained complementary filters are shown in Figure <a href="#org551ae15">19</a>.
+The obtained complementary filters are shown in Figure <a href="#orgf11ca9c">19</a>.
 </p>
 
 
-<div id="org551ae15" class="figure">
+<div id="orgf11ca9c" class="figure">
 <p><img src="figs/comp_filters_H2_feature.png" alt="comp_filters_H2_feature.png" />
 </p>
 <p><span class="figure-number">Figure 19: </span>Obtained complementary fitlers</p>
@@ -1230,15 +1337,15 @@ The obtained complementary filters are shown in Figure <a href="#org551ae15">19<
 </div>
 </div>
 
-<div id="outline-container-org71e3235" class="outline-3">
+<div id="outline-container-org52c02ef" class="outline-3">
-<h3 id="org71e3235"><span class="section-number-3">5.3</span> Imposing a negative slope at infinity frequency during the synthesis phase</h3>
+<h3 id="org52c02ef"><span class="section-number-3">5.3</span> Imposing a negative slope at infinity frequency during the synthesis phase</h3>
 <div class="outline-text-3" id="text-5-3">
 <p>
 Let&rsquo;s suppose we now want to shape a low pass filter that as a negative slope until infinite frequency.
 </p>
 
 <p>
-The used technique is the same as in the previous section, and the generalized plant is shown in Figure <a href="#org00fe83c">18</a>.
+The used technique is the same as in the previous section, and the generalized plant is shown in Figure <a href="#orgf2b673c">18</a>.
 </p>
 
 <p>
@@ -1296,11 +1403,11 @@ And \(H_1(s)\) is defined as follows:
 </div>
 
 <p>
-The obtained complementary filters are shown in Figure <a href="#org7f33e5d">20</a>.
+The obtained complementary filters are shown in Figure <a href="#orgab78578">20</a>.
 </p>
 
 
-<div id="org7f33e5d" class="figure">
+<div id="orgab78578" class="figure">
 <p><img src="figs/comp_filters_H2_feature_neg_slope.png" alt="comp_filters_H2_feature_neg_slope.png" />
 </p>
 <p><span class="figure-number">Figure 20: </span>Obtained complementary fitlers</p>
@@ -1318,10 +1425,179 @@ The obtained complementary filters are shown in Figure <a href="#org7f33e5d">20<
 <div class="csl-bib-body">
   <div class="csl-entry"><a name="citeproc_bib_item_1"></a>Hua, Wensheng. 2005. “Low Frequency Vibration Isolation and Alignment System for Advanced LIGO.” stanford university.</div>
 </div>
+
+
+<div id="outline-container-org29e79e7" class="outline-2">
+<h2 id="org29e79e7"><span class="section-number-2">6</span> Functions</h2>
+<div class="outline-text-2" id="text-6">
+</div>
+<div id="outline-container-orgfbca8a7" class="outline-3">
+<h3 id="orgfbca8a7"><span class="section-number-3">6.1</span> <code>generateWF</code>: Generate Weighting Functions</h3>
+<div class="outline-text-3" id="text-6-1">
+<p>
+<a id="org0d68f63"></a>
+</p>
+
+<p>
+This Matlab function is accessible <a href="matlab/src/generateWF.m">here</a>.
+</p>
+</div>
+
+<div id="outline-container-org95965a4" class="outline-4">
+<h4 id="org95965a4">Function description</h4>
+<div class="outline-text-4" id="text-org95965a4">
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[W]</span> = <span class="org-function-name">generateWF</span>(<span class="org-variable-name">args</span>)
+<span class="org-comment">% createWeight -</span>
+<span class="org-comment">%</span>
+<span class="org-comment">% Syntax: [W] = generateWeight(args)</span>
+<span class="org-comment">%</span>
+<span class="org-comment">% Inputs:</span>
+<span class="org-comment">%    - n  - Weight Order (integer)</span>
+<span class="org-comment">%    - G0 - Low frequency Gain</span>
+<span class="org-comment">%    - G1 - High frequency Gain</span>
+<span class="org-comment">%    - Gc - Gain of the weight at frequency w0</span>
+<span class="org-comment">%    - w0 - Frequency at which |W(j w0)| = Gc [rad/s]</span>
+<span class="org-comment">%</span>
+<span class="org-comment">% Outputs:</span>
+<span class="org-comment">%    - W - Generated Weight</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgaeb61de" class="outline-4">
+<h4 id="orgaeb61de">Optional Parameters</h4>
+<div class="outline-text-4" id="text-orgaeb61de">
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-keyword">arguments</span>
+    <span class="org-variable-name">args</span>.n    (1,1) double {mustBeInteger, mustBePositive} = 1
+    <span class="org-variable-name">args</span>.G0   (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+    <span class="org-variable-name">args</span>.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
+    <span class="org-variable-name">args</span>.Gc   (1,1) double {mustBeNumeric, mustBePositive} = 1
+    <span class="org-variable-name">args</span>.w0   (1,1) double {mustBeNumeric, mustBePositive} = 1
+<span class="org-keyword">end</span>
+
+mustBeBetween(args.G0, args.Gc, args.Ginf);
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-orgd91e493" class="outline-4">
+<h4 id="orgd91e493">Generate the Weighting function</h4>
+<div class="outline-text-4" id="text-orgd91e493">
+<div class="org-src-container">
+<pre class="src src-matlab">s = zpk(<span class="org-string">'s'</span>);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">W = (((1<span class="org-type">/</span>args.w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n))<span class="org-type">/</span>(1<span class="org-type">-</span>(args.Gc<span class="org-type">/</span>args.Ginf)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n)))<span class="org-type">*</span>s <span class="org-type">+</span> ...
+      (args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n))<span class="org-type">/</span>...
+     ((1<span class="org-type">/</span>args.Ginf)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n)<span class="org-type">*</span>(1<span class="org-type">/</span>args.w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n))<span class="org-type">/</span>(1<span class="org-type">-</span>(args.Gc<span class="org-type">/</span>args.Ginf)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n)))<span class="org-type">*</span>s <span class="org-type">+</span> ...
+      (1<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n)))<span class="org-type">^</span>args.n;
+</pre>
+</div>
+</div>
+</div>
+
+
+<div id="outline-container-org84eb536" class="outline-4">
+<h4 id="org84eb536">Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains</h4>
+<div class="outline-text-4" id="text-org84eb536">
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-comment">% Custom validation function</span>
+<span class="org-keyword">function</span> <span class="org-function-name">mustBeBetween</span>(<span class="org-variable-name">a</span>,<span class="org-variable-name">b</span>,<span class="org-variable-name">c</span>)
+    <span class="org-keyword">if</span> <span class="org-type">~</span>((a <span class="org-type">&gt;</span> b <span class="org-type">&amp;&amp;</span> b <span class="org-type">&gt;</span> c) <span class="org-type">||</span> (c <span class="org-type">&gt;</span> b <span class="org-type">&amp;&amp;</span> b <span class="org-type">&gt;</span> a))
+        eid = <span class="org-string">'createWeight:inputError'</span>;
+        msg = <span class="org-string">'Gc should be between G0 and Ginf.'</span>;
+        throwAsCaller(MException(eid,msg))
+    <span class="org-keyword">end</span>
+</pre>
+</div>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org2f9ac50" class="outline-3">
+<h3 id="org2f9ac50"><span class="section-number-3">6.2</span> <code>generateCF</code>: Generate Complementary Filters</h3>
+<div class="outline-text-3" id="text-6-2">
+<p>
+<a id="orgf94493e"></a>
+</p>
+
+<p>
+This Matlab function is accessible <a href="matlab/src/generateCF.m">here</a>.
+</p>
+</div>
+
+<div id="outline-container-org7c373f8" class="outline-4">
+<h4 id="org7c373f8">Function description</h4>
+<div class="outline-text-4" id="text-org7c373f8">
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[H1, H2]</span> = <span class="org-function-name">generateCF</span>(<span class="org-variable-name">W1</span>, <span class="org-variable-name">W2</span>, <span class="org-variable-name">args</span>)
+<span class="org-comment">% createWeight -</span>
+<span class="org-comment">%</span>
+<span class="org-comment">% Syntax: [W] = generateCF(args)</span>
+<span class="org-comment">%</span>
+<span class="org-comment">% Inputs:</span>
+<span class="org-comment">%    - W1 - Weighting Function for H1</span>
+<span class="org-comment">%    - W2 - Weighting Function for H2</span>
+<span class="org-comment">%    - args:</span>
+<span class="org-comment">%      - method  - H-Infinity solver ('lmi' or 'ric')</span>
+<span class="org-comment">%      - display - Display synthesis results ('on' or 'off')</span>
+<span class="org-comment">%</span>
+<span class="org-comment">% Outputs:</span>
+<span class="org-comment">%    - H1 - Generated H1 Filter</span>
+<span class="org-comment">%    - H2 - Generated H2 Filter</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org00c9d5d" class="outline-4">
+<h4 id="org00c9d5d">Optional Parameters</h4>
+<div class="outline-text-4" id="text-org00c9d5d">
+<div class="org-src-container">
+<pre class="src src-matlab"><span class="org-keyword">arguments</span>
+    <span class="org-variable-name">W1</span>
+    <span class="org-variable-name">W2</span>
+    <span class="org-variable-name">args</span>.method  char {mustBeMember(args.method,{<span class="org-string">'lmi'</span>, <span class="org-string">'ric'</span>})} = <span class="org-string">'ric'</span>
+    <span class="org-variable-name">args</span>.display char {mustBeMember(args.display,{<span class="org-string">'on'</span>, <span class="org-string">'off'</span>})} = <span class="org-string">'on'</span>
+<span class="org-keyword">end</span>
+</pre>
+</div>
+</div>
+</div>
+
+<div id="outline-container-org92a7c04" class="outline-4">
+<h4 id="org92a7c04">H-Infinity Synthesis</h4>
+<div class="outline-text-4" id="text-org92a7c04">
+<div class="org-src-container">
+<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1;
+     0   W2;
+     1   0];
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">[H2, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, args.method, <span class="org-string">'DISPLAY'</span>, args.display);
+</pre>
+</div>
+
+<div class="org-src-container">
+<pre class="src src-matlab">H1 = 1 <span class="org-type">-</span> H2;
+</pre>
+</div>
+</div>
+</div>
+</div>
+</div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Thomas Dehaeze</p>
-<p class="date">Created: 2020-12-11 ven. 14:05</p>
+<p class="date">Created: 2021-04-29 jeu. 17:25</p>
 </div>
 </body>
 </html>

matlab/index.org

@@ -43,7 +43,7 @@ One use of complementary filter is described below:
 #+end_quote
 
 This document is divided into several sections:
-- in section [[sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters
+- in section [[#sec: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
 
@@ -55,8 +55,8 @@ This document is divided into several sections:
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m
 :header-args:matlab+: :comments org :mkdirp yes
+:CUSTOM_ID: sec:h_inf_synthesis_complementary_filters
 :END:
-<<sec:h_inf_synthesis_complementary_filters>>
 
 ** Introduction                                                      :ignore:
 #+begin_note
@@ -93,7 +93,7 @@ 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-tikz/h_infinity_robust_fusion.png]]
+[[file:figs/h_infinity_robust_fusion.png]]
 
 The $\mathcal{H}_\infty$ synthesis applied on this generalized plant will give a transfer function $H_2$ (figure [[fig:h_infinity_robst_fusion]]) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2]$ is less than one:
 \[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
@@ -352,7 +352,7 @@ For that, we use the $\mathcal{H}_\infty$ synthesis with the architecture shown
 
 #+name: fig:comp_filter_three_hinf
 #+caption: Generalized architecture for generating 3 complementary filters
-[[file:figs-tikz/comp_filter_three_hinf.png]]
+[[file:figs/comp_filter_three_hinf.png]]
 
 The $\mathcal{H}_\infty$ objective is:
 \begin{align*}
@@ -585,7 +585,7 @@ exportFig('figs/ligo_specifications.pdf', 'width', 'wide', 'height', 'normal');
 #+RESULTS:
 [[file:figs/ligo_specifications.png]]
 
-** TODO FIR Filter
+** FIR Filter
 We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
 For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]].
 
@@ -937,7 +937,7 @@ exportFig('figs/hinf_synthesis_ligo_results.pdf', 'width', 'wide', 'height', 'no
 #+RESULTS:
 [[file:figs/hinf_synthesis_ligo_results.png]]
 
-** TODO Compare FIR and H-Infinity Filters
+** Compare FIR and H-Infinity Filters
 Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
 
 #+begin_src matlab :exports none
@@ -965,25 +965,25 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Magnitude');
 set(gca, 'XTickLabel',[]);
 ylim([5e-3, 10]);
-legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+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*angle(squeeze(freqresp(Hh, freqs, 'Hz'))), '-');
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
 set(gca,'ColorOrderIndex',2);
-plot(freqs, 180/pi*angle(squeeze(freqresp(Hl, freqs, 'Hz'))), '-');
+plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
 
 set(gca,'ColorOrderIndex',1);
-plot(w, 180/pi*angle(H), '--');
+plot(w, 180/pi*unwrap(angle(H)), '--');
-set(gca,'   H2ColorOrderIndex',2);
+set(gca,'ColorOrderIndex',2);
-plot(w, 180/pi*angle(1-H), '--');
+plot(w, 180/pi*unwrap(angle(1-H)), '--');
 set(gca, 'XScale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
-yticks([-180:90:180]); ylim([-180, 180]);
+yticks([-450:90:180]); ylim([-450, 200]);
 
 linkaxes([ax1,ax2],'x');
 xlim([freqs(1), freqs(end)]);
@@ -1897,7 +1897,9 @@ function [H1, H2] = generateCF(W1, W2, args)
 % Inputs:
 %    - W1 - Weighting Function for H1
 %    - W2 - Weighting Function for H2
-%    - args -
+%    - args:
+%      - method  - H-Infinity solver ('lmi' or 'ric')
+%      - display - Display synthesis results ('on' or 'off')
 %
 % Outputs:
 %    - H1 - Generated H1 Filter

matlab/matlab/src/generateCF.m

@@ -6,7 +6,9 @@ function [H1, H2] = generateCF(W1, W2, args)
 % Inputs:
 %    - W1 - Weighting Function for H1
 %    - W2 - Weighting Function for H2
-%    - args -
+%    - args:
+%      - method  - H-Infinity solver ('lmi' or 'ric')
+%      - display - Display synthesis results ('on' or 'off')
 %
 % Outputs:
 %    - H1 - Generated H1 Filter

tikz/index.org

@@ -6,54 +6,182 @@
 #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
 #+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
 
-#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/tikz/}{config.tex}")
+#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/papers/dehaeze21_desig_compl_filte/tikz/}{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 raw replace :buffer no
+#+PROPERTY: header-args:latex+ :results file raw replace
+#+PROPERTY: header-args:latex+ :buffer no
 #+PROPERTY: header-args:latex+ :eval no-export
 #+PROPERTY: header-args:latex+ :exports both
 #+PROPERTY: header-args:latex+ :mkdirp yes
-#+PROPERTY: header-args:latex+ :noweb yes
 #+PROPERTY: header-args:latex+ :output-dir figs
 #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
 :END:
 
 Configuration file is accessible [[file:config.org][here]].
 
-* Fig 1: Sensor Fusion Architecture
+* Sensor Model
+#+begin_src latex :file sensor_model.pdf
+\begin{tikzpicture}
+  \node[addb](add1){};
+  \node[block, right=0.8 of add1](G1){$G_1(s)$};
+
+  \draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
+  \draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
+  \draw[->] (add1.east) -- (G1.west);
+  \draw[->] (G1.east) -- ++(0.7, 0) node[above left]{$\tilde{x}_1$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
+    \node[below left] at (sensor1.north east) {Sensor 1};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:sensor_model
+#+caption: Basic Sensor Model
+#+RESULTS:
+[[file:figs/sensor_model.png]]
+
+* Sensor Model with calibration
+#+begin_src latex :file sensor_model_calibrated.pdf
+\begin{tikzpicture}
+  \node[addb](add1){};
+  \node[block, right=0.8 of add1](G1){$G_1(s)$};
+  \node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
+
+  \draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
+  \draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
+  \draw[->] (add1.east) -- (G1.west);
+  \draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
+  \draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_1$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
+    \node[below left] at (sensor1cal.north east) {Calibration};
+
+    \node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
+    \node[below left] at (sensor1.north east) {Sensor 1};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:sensor_model_calibrated
+#+caption: Calibrated Sensor
+#+RESULTS:
+[[file:figs/sensor_model_calibrated.png]]
+
+* Sensor Model with Uncertainty
+#+begin_src latex :file sensor_model_uncertainty.pdf
+\begin{tikzpicture}
+  \node[branch] (input) at (0,0) {};
+  \node[block, above right= 0.4 and 0.4 of input](W1){$w_1(s)$};
+  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
+  \node[addb] (addu) at ($(delta1.east|-input) + (0.4, 0)$) {};
+  \node[addb, right=0.4 of addu] (addn) {};
+  \node[block, right=0.4 of addn] (G1) {$\hat{G}_1(s)$};
+  \node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
+
+  \draw[->] ($(input)+(-0.7, 0)$) node[above right]{$x$} -- (addu);
+  \draw[->] (input.center) |- (W1.west);
+  \draw[->] (W1.east) -- (delta1.west);
+  \draw[->] (delta1.east) -| (addu.north);
+  \draw[->] (addu.east) -- (addn.west);
+  \draw[->] (addn.east) -- (G1.west);
+  \draw[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
+  \draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
+  \draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_1$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(input.west |- G1inv.south) (delta1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
+    \node[below left] at (sensor1cal.north east) {Calibration};
+
+    \node[fit={(input.west |- G1.south) (delta1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
+    \node[below left] at (sensor1.north east) {Sensor 1};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:sensor_model_uncertainty
+#+caption: Input Uncertainty
+#+RESULTS:
+[[file:figs/sensor_model_uncertainty.png]]
+
+* Sensor Model with Uncertainty - Simplified
+#+begin_src latex :file sensor_model_uncertainty_simplified.pdf
+\begin{tikzpicture}
+  \node[branch] (input) at (0,0) {};
+  \node[block, above right= 0.4 and 0.4 of input](W1){$w_1(s)$};
+  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
+  \node[addb] (addu) at ($(delta1.east|-input) + (0.4, 0)$) {};
+  \node[addb, right=0.4 of addu] (addn) {};
+
+  \draw[->] ($(input)+(-0.7, 0)$) node[above right]{$x$} -- (addu);
+  \draw[->] (input.center) |- (W1.west);
+  \draw[->] (W1.east) -- (delta1.west);
+  \draw[->] (delta1.east) -| (addu.north);
+  \draw[->] (addu.east) -- (addn.west);
+  \draw[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
+  \draw[->] (addn.east) -- ++(0.9, 0) node[above left]{$\hat{x}_1$};
+
+  \begin{scope}[on background layer]
+    \node[fit={(input.west |- addu.south) ($(delta1.north -| addn.east) + (0.1, 0.3)$)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
+    \node[below left] at (sensor1cal.north east) {Calibrated Sensor};
+  \end{scope}
+\end{tikzpicture}
+#+end_src
+
+#+name: fig:sensor_model_uncertainty_simplified
+#+caption: Input Uncertainty
+#+RESULTS:
+[[file:figs/sensor_model_uncertainty_simplified.png]]
+
+* Sensor Fusion Architecture
 #+begin_src latex :file fusion_super_sensor.pdf :tangle figs/fusion_super_sensor.tex
+\definecolor{myblue}{rgb}{0, 0.447, 0.741}
+\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
+
 \begin{tikzpicture}
   \node[branch] (x) at (0, 0);
-    \node[block, above right=0.5 and 0.5 of x](G1){$G_1(s)$};
+  \node[addb, above right=0.8 and 0.5 of x](add1){};
-    \node[block, below right=0.5 and 0.5 of x](G2){$G_2(s)$};
+  \node[addb, below right=0.8 and 0.5 of x](add2){};
-    \node[addb, right=0.8 of G1](add1){};
+  \node[block, right=0.8 of add1](G1){$G_1(s)$};
-    \node[addb, right=0.8 of G2](add2){};
+  \node[block, right=0.8 of add2](G2){$G_2(s)$};
-    \node[block, right=0.8 of add1](H1){$H_1(s)$};
+  \node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
-    \node[block, right=0.8 of add2](H2){$H_2(s)$};
+  \node[block, right=0.8 of G2](G2inv){$\hat{G}_2^{-2}(s)$};
-    \node[addb, right=5 of x](add){};
+  \node[block, right=0.8 of G1inv](H1){$H_1(s)$};
+  \node[block, right=0.8 of G2inv](H2){$H_2(s)$};
+  \node[addb, right=7 of x](add){};
 
   \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
-    \draw[->] (x.center) |- (G1.west);
+  \draw[->] (x.center) |- (add1.west);
-    \draw[->] (x.center) |- (G2.west);
+  \draw[->] (x.center) |- (add2.west);
-    \draw[->] (G1.east) -- (add1.west);
+  \draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
-    \draw[->] (G2.east) -- (add2.west);
+  \draw[->] (add1.east) -- (G1.west);
-    \draw[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){$n_1$};
+  \draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
-    \draw[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){$n_2$};
+  \draw[->] (G1inv.east) -- (H1.west) node[above left]{$\hat{x}_1$};
-    \draw[->] (add1.east) -- (H1.west);
+  \draw[<-] (add2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};
-    \draw[->] (add2.east) -- (H2.west);
+  \draw[->] (add2.east) -- (G2.west);
+  \draw[->] (G2.east) -- (G2inv.west) node[above left]{$\tilde{x}_2$};
+  \draw[->] (G2inv.east) -- (H2.west) node[above left]{$\hat{x}_2$};
   \draw[->] (H1) -| (add.north);
   \draw[->] (H2) -| (add.south);
   \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
 
   \begin{scope}[on background layer]
-      \node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
+    \node[fit={(G2.south-|x) (n1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
     \node[below left] at (supersensor.north east) {Super Sensor};
 
-      \node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
+    \node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
-      \node[below right] at (sensor1.north west) {Sensor 1};
+    \node[below left] at (sensor1cal.north east) {Calibration};
-      \node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
+    \node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=myblue!30!white, draw, inner sep=3pt] (sensor1) {};
-      \node[below right] at (sensor2.north west) {Sensor 2};
+    \node[below left] at (sensor1.north east) {Sensor 1};
+
+    \node[fit={(add2.west |- G2inv.south) (n2.north -| G2inv.east)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
+    \node[below left] at (sensor2cal.north east) {Calibration};
+    \node[fit={(add2.west |- G2.south) (n2.north -| G2.east)}, fill=myred!30!white, draw, inner sep=3pt] (sensor2) {};
+    \node[below left] at (sensor2.north east) {Sensor 2};
   \end{scope}
 \end{tikzpicture}
 #+end_src
@@ -63,40 +191,59 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/fusion_super_sensor.png]]
 
-* Fig 2: Sensor fusion architecture with sensor dynamics uncertainty
+* Sensor fusion architecture with sensor dynamics uncertainty
-#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/sensor_fusion_dynamic_uncertainty.tex
+#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/fusion_super_sensor.tex
+\definecolor{myblue}{rgb}{0, 0.447, 0.741}
+\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
+
 \begin{tikzpicture}
   \node[branch] (x) at (0, 0);
-    \node[addb, above right=0.8 and 4 of x](add1){};
+
-    \node[addb, below right=0.8 and 4 of x](add2){};
+  \node[branch, above right=0.9 and 0.3 of x] (input1) {};
-    \node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$};
+  \node[branch, below right=0.9 and 0.3 of x] (input2) {};
-    \node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$};
+  \node[block, above right= 0.4 and 0.4 of input1](W1){$w_1(s)$};
-    \node[block, left=0.5 of delta1](W1){$w_1(s)$};
+  \node[block, above right= 0.4 and 0.4 of input2](W2){$w_2(s)$};
-    \node[block, left=0.5 of delta2](W2){$w_2(s)$};
+  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
-    \node[block, right=0.5 of add1](H1){$H_1(s)$};
+  \node[block, right=0.4 of W2](delta2){$\Delta_2(s)$};
-    \node[block, right=0.5 of add2](H2){$H_2(s)$};
+  \node[addb] (addu1) at ($(delta1.east|-input1) + (0.4, 0)$) {};
-    \node[addb, right=6 of x](add){};
+  \node[addb] (addu2) at ($(delta2.east|-input2) + (0.4, 0)$) {};
+  \node[addb, right=0.4 of addu1] (addn1) {};
+  \node[addb, right=0.4 of addu2] (addn2) {};
+  \node[block, right=0.9 of addn1](H1){$H_1(s)$};
+  \node[block, right=0.9 of addn2](H2){$H_2(s)$};
+
+  \node[addb, right=7 of x](add){};
+
 
   \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
-    \draw[->] (x.center) |- (add1.west);
+  \draw[->] (x.center) |- (addu1.west);
-    \draw[->] (x.center) |- (add2.west);
+  \draw[->] (x.center) |- (addu2.west);
-    \draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west);
+  \draw[->] (input1.center) |- (W1.west);
-    \draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S1){} |- (W2.west);
   \draw[->] (W1.east) -- (delta1.west);
+  \draw[->] (delta1.east) -| (addu1.north);
+  \draw[->] (addu1.east) -- (addn1.west);
+  \draw[<-] (addn1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
+  \draw[->] (input2.center) |- (W2.west);
   \draw[->] (W2.east) -- (delta2.west);
-    \draw[->] (delta1.east) -| (add1.north);
+  \draw[->] (delta2.east) -| (addu2.north);
-    \draw[->] (delta2.east) -| (add2.north);
+  \draw[->] (addu2.east) -- (addn2.west);
-    \draw[->] (add1.east) -- (H1.west);
+  \draw[<-] (addn2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};
-    \draw[->] (add2.east) -- (H2.west);
+
-    \draw[->] (H1.east) -| (add.north);
+  \draw[->] (addn1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
-    \draw[->] (H2.east) -| (add.south);
+  \draw[->] (addn2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
+  \draw[->] (H1) -| (add.north);
+  \draw[->] (H2) -| (add.south);
   \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
 
   \begin{scope}[on background layer]
-      \node[block, fit={($(W1.north-|S1)+(-0.2, 0.2)$) ($(add1.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
+    \node[fit={(addn2.south-|x) (delta1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
-      \node[above right] at (sensor1.south west) {Sensor 1};
+    \node[below left] at (supersensor.north east) {Super Sensor};
-      \node[block, fit={($(W2.north-|S1)+(-0.2, 0.2)$) ($(add2.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
+
-      \node[above right] at (sensor2.south west) {Sensor 2};
+    \node[fit={(input1.west |- addu1.south) ($(delta1.north -| addn1.east) + (0.1, 0.0)$)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
+    \node[below left] at (sensor1cal.north east) {Sensor 1};
+
+    \node[fit={(input2.west |- addu2.south) ($(delta2.north -| addn1.east) + (0.1, 0.0)$)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
+    \node[below left] at (sensor2cal.north east) {Sensor 2};
   \end{scope}
 \end{tikzpicture}
 #+end_src
@@ -106,31 +253,35 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/sensor_fusion_dynamic_uncertainty.png]]
 
-* Fig 3: Uncertainty set of the super sensor dynamics
+* Uncertainty set of the super sensor dynamics
 #+begin_src latex :file uncertainty_set_super_sensor.pdf :tangle figs/uncertainty_set_super_sensor.tex :exports both
+\definecolor{myblue}{rgb}{0, 0.447, 0.741}
+\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
+
 \begin{tikzpicture}
   \begin{scope}[shift={(4, 0)}]
 
     % Uncertainty Circle
     \node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
-      \path[draw, dotted] (0, 0) circle [radius=1.0];
+    \path[draw, fill=myblue!20!white] (0, 0) circle [radius=1.0];
-      \path[draw, dashed] (135:1.0) circle [radius=0.8];
+    \path[draw, fill=myred!20!white] (135:1.0) circle [radius=0.8];
+    \path[draw, dashed] (0, 0) circle [radius=1.0];
 
     % Center of Circle
     \node[below] at (0, 0){$1$};
 
-      \draw[<->, dashed] (0, 0)   node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
+    \draw[<->] (0, 0)   node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
-      \draw[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
+    \draw[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);
 
     \node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
-      \draw[->, dashed, out=-90, in=0] (l1.south) to (r1);
+    \draw[->, out=-90, in=0] (l1.south) to (r1);
 
-      \node[] (l2) at (-2.5, 1.5) {$|w_2 H_2|$};
+    \node[] (l2) at (-3.2, 1.2) {$|w_2 H_2|$};
-      \draw[->, dashed, out=0, in=-180] (l2.east) to (r2);
+    \draw[->, out=0, in=-180] (l2.east) to (r2);
 
-      \draw[<->, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
+    \draw[<->] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
     \node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
-      \draw[->, dashed, out=90, in=-90] (l3.north) to (r3);
+    \draw[->, out=90, in=-90] (l3.north) to (r3);
   \end{scope}
 
   % Real and Imaginary Axis
@@ -147,10 +298,10 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/uncertainty_set_super_sensor.png]]
 
-* Fig 4: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
+* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
 #+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
 \begin{tikzpicture}
-     \node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
+  \node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
   \node[above] at (P.north) {$P(s)$};
 
   \coordinate[] (inputw)  at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
@@ -181,98 +332,7 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/h_infinity_robust_fusion.png]]
 
-* Fig 5: Magnitude of a weighting function generated using the proposed formula
+* Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
-#+begin_src matlab :exports none :results none
-  s = zpk('s');
-
-  freqs = logspace(-1, 2, 500);
-
-  n = 2;
-  w0 = 2*pi*10;
-  G0 = 1e-3;
-  G1 = 10;
-  Gc = 2;
-
-  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;
-
-  T = table(freqs', ...
-            abs(squeeze(freqresp(W, freqs, 'Hz'))), ...
-            'VariableNames', {'freqs', 'ampl'});
-  writetable(T, '../matlab/mat/weight_formula.csv');
-#+end_src
-
-
-#+begin_src latex :file weight_formula.pdf :tangle figs/weight_formula.tex :exports both
-  \setlength\fwidth{6.5cm}
-  \setlength\fheight{3.5cm}
-
-  \begin{tikzpicture}
-    \begin{axis}[%
-      width=1.0\fwidth,
-      height=1.0\fheight,
-      at={(0.0\fwidth, 0.0\fheight)},
-      scale only axis,
-      xmode=log,
-      xmin=0.1,
-      xmax=100,
-      xtick={0.1,1,10, 100},
-      xminorticks=true,
-      ymode=log,
-      ymin=0.0005,
-      ymax=20,
-      ytick={0.001, 0.01, 0.1, 1, 10},
-      yminorticks=true,
-      ylabel={Magnitude},
-      xlabel={Frequency [Hz]},
-      xminorgrids,
-      yminorgrids,
-      ]
-
-      \addplot [color=black, line width=1.5pt, forget plot]
-      table [x=freqs, y=ampl, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matweight_formula.csv};
-
-      \addplot [color=black, dashed, line width=1.5pt]
-      table[row sep=crcr]{%
-        1     10\\
-        100   10\\
-      };
-      \addplot [color=black, dashed, line width=1.5pt]
-      table[row sep=crcr]{%
-        0.1  0.001\\
-        3    0.001\\
-      };
-
-      \addplot [color=black, line width=1.5pt]
-      table[row sep=crcr]{%
-        0.1  1\\
-        100  1\\
-      };
-
-      \addplot [color=black, dashed, line width=1.5pt]
-      table[row sep=crcr]{%
-        10  2\\
-        10  1\\
-      };
-
-      \node[below] at (2, 10) {$G_\infty$};
-      \node[above] at (2, 0.001) {$G_0$};
-
-      \node[branch] at (10, 2){};
-      \draw[dashed, line cap=round] (7, 2) -- (20, 2) node[right]{$G_c$};
-      \draw[dashed, line cap=round] (10, 2) -- (10, 1) node[below]{$\omega_c$};
-
-      \node[right] at (3, 0.1) {$+n$};
-
-    \end{axis}
-  \end{tikzpicture}
-#+end_src
-
-#+name: fig:weight_formula
-#+caption: Magnitude of a weighting function generated using the proposed formula ([[./figs/weight_formula.png][png]], [[./figs/weight_formula.pdf][pdf]], [[./figs/weight_formula.tex][tex]]).
-#+RESULTS:
-[[file:figs/weight_formula.png]]
-
-* Fig 6: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
 #+begin_src latex :file hinf_synthesis_results.pdf :tangle figs/hinf_synthesis_results.tex :exports both
   \setlength\fwidth{6.5cm}
   \setlength\fheight{6cm}
@@ -348,7 +408,7 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/hinf_synthesis_results.png]]
 
-* Fig 7: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
+* Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
 #+begin_src latex :file comp_filter_three_hinf.pdf :tangle figs/comp_filter_three_hinf.tex
   \begin{tikzpicture}
      \node[block={5.0cm}{3.5cm}, fill=black!20!white, dashed] (P) {};
@@ -393,7 +453,7 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/comp_filter_three_hinf.png]]
 
-* Fig 8: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
+* Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
 #+begin_src latex :file hinf_three_synthesis_results.pdf :tangle figs/hinf_three_synthesis_results.tex :exports both
   \setlength\fwidth{6.5cm}
   \setlength\fheight{6cm}
@@ -482,7 +542,7 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/hinf_three_synthesis_results.png]]
 
-* Fig 9: Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis
+* Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis
 #+begin_src latex :file ligo_weights.pdf :tangle figs/ligo_weights.tex :exports both
   \setlength\fwidth{6.5cm}
   \setlength\fheight{3.2cm}
@@ -557,7 +617,7 @@ Configuration file is accessible [[file:config.org][here]].
 #+RESULTS:
 [[file:figs/ligo_weights.png]]
 
-* Fig 10: Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
+*  Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
 #+begin_src latex :file comp_fir_ligo_hinf.pdf :tangle figs/comp_fir_ligo_hinf.tex :exports both
   \setlength\fwidth{6.5cm}
   \setlength\fheight{6.8cm}