#+TITLE: Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Tikz Figures
:DRAWER:
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP:   ../index.html

#+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/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 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+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:

Configuration file is accessible [[file:config.org][here]].

* 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[addb, above right=0.8 and 0.5 of x](add1){};
  \node[addb, below right=0.8 and 0.5 of x](add2){};
  \node[block, right=0.8 of add1](G1){$G_1(s)$};
  \node[block, right=0.8 of add2](G2){$G_2(s)$};
  \node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
  \node[block, right=0.8 of G2](G2inv){$\hat{G}_2^{-2}(s)$};
  \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) |- (add1.west);
  \draw[->] (x.center) |- (add2.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) -- (H1.west) node[above left]{$\hat{x}_1$};
  \draw[<-] (add2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};
  \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) (n1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
    \node[below left] at (supersensor.north east) {Super Sensor};

    \node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=myblue!20!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=myblue!30!white, draw, inner sep=3pt] (sensor1) {};
    \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

#+name: fig:fusion_super_sensor
#+caption: Sensor Fusion Architecture ([[./figs/fusion_super_sensor.png][png]], [[./figs/fusion_super_sensor.pdf][pdf]], [[./figs/fusion_super_sensor.tex][tex]]).
#+RESULTS:
[[file:figs/fusion_super_sensor.png]]

* Sensor fusion architecture with sensor dynamics uncertainty
#+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[branch, above right=0.9 and 0.3 of x] (input1) {};
  \node[branch, below right=0.9 and 0.3 of x] (input2) {};
  \node[block, above right= 0.4 and 0.4 of input1](W1){$w_1(s)$};
  \node[block, above right= 0.4 and 0.4 of input2](W2){$w_2(s)$};
  \node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
  \node[block, right=0.4 of W2](delta2){$\Delta_2(s)$};
  \node[addb] (addu1) at ($(delta1.east|-input1) + (0.4, 0)$) {};
  \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) |- (addu1.west);
  \draw[->] (x.center) |- (addu2.west);
  \draw[->] (input1.center) |- (W1.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[->] (delta2.east) -| (addu2.north);
  \draw[->] (addu2.east) -- (addn2.west);
  \draw[<-] (addn2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};

  \draw[->] (addn1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
  \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[fit={(addn2.south-|x) (delta1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
    \node[below left] at (supersensor.north east) {Super Sensor};

    \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

#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_dynamic_uncertainty.png][png]], [[./figs/sensor_fusion_dynamic_uncertainty.pdf][pdf]], [[./figs/sensor_fusion_dynamic_uncertainty.tex][tex]]).
#+RESULTS:
[[file:figs/sensor_fusion_dynamic_uncertainty.png]]

* 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, fill=myblue!20!white] (0, 0) circle [radius=1.0];
    \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[<->] (0, 0)   node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
    \draw[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);

    \node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
    \draw[->, out=-90, in=0] (l1.south) to (r1);

    \node[] (l2) at (-3.2, 1.2) {$|w_2 H_2|$};
    \draw[->, out=0, in=-180] (l2.east) to (r2);

    \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[->, out=90, in=-90] (l3.north) to (r3);
  \end{scope}

  % Real and Imaginary Axis
  \draw[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
  \draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};

  \draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
  \draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
\end{tikzpicture}
#+end_src

#+name: fig:uncertainty_set_super_sensor
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle), of the sensor 1 (dotted circle) and of the sensor 2 (dashed circle) ([[./figs/uncertainty_set_super_sensor.png][png]], [[./figs/uncertainty_set_super_sensor.pdf][pdf]], [[./figs/uncertainty_set_super_sensor.tex][tex]]).
#+RESULTS:
[[file:figs/uncertainty_set_super_sensor.png]]

* 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}{3.0cm}, fill=black!10!white] (P) {};
  \node[above] at (P.north) {$P(s)$};

  \coordinate[] (inputw)  at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
  \coordinate[] (inputu)  at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);

  \coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
  \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);

  \node[block, left=1.4 of output1] (W1){$W_1(s)$};
  \node[block, left=1.4 of output2] (W2){$W_2(s)$};
  \node[addb={+}{}{}{}{-}, left=of W1] (sub) {};

  \node[block, below=0.3 of P] (H2) {$H_2(s)$};

  \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
  \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
  \draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
  \draw[->] (sub.east) -- (W1.west);
  \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
  \draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
  \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
\end{tikzpicture}
#+end_src

#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters ([[./figs/h_infinity_robust_fusion.png][png]], [[./figs/h_infinity_robust_fusion.pdf][pdf]], [[./figs/h_infinity_robust_fusion.tex][tex]]).
#+RESULTS:
[[file:figs/h_infinity_robust_fusion.png]]

* 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}

  \begin{tikzpicture}
    \begin{axis}[%
      width=1.0\fwidth,
      height=0.5\fheight,
      at={(0.0\fwidth, 0.47\fheight)},
      scale only axis,
      xmode=log,
      xmin=0.1,
      xmax=1000,
      xtick={0.1, 1, 10, 100, 1000},
      xticklabels={{}},
      xminorticks=true,
      ymode=log,
      ymin=0.0005,
      ymax=20,
      ytick={0.001, 0.01, 0.1, 1, 10},
      yminorticks=true,
      ylabel={Magnitude},
      xminorgrids,
      yminorgrids,
      ]
      \addplot [color=mycolor1, line width=1.5pt, forget plot]
      table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};

      \addplot [color=mycolor2, line width=1.5pt, forget plot]
      table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};

      \addplot [color=mycolor1, dashed, line width=1.5pt, forget plot]
      table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};

      \addplot [color=mycolor2, dashed, line width=1.5pt, forget plot]
      table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_weights.csv};
    \end{axis}

    \begin{axis}[%
      width=1.0\fwidth,
      height=0.45\fheight,
      at={(0.0\fwidth, 0.0\fheight)},
      scale only axis,
      xmode=log,
      xmin=0.1,
      xmax=1000,
      xtick={0.1, 1, 10, 100, 1000},
      xminorticks=true,
      xlabel={Frequency [Hz]},
      ymin=-200,
      ymax=200,
      ytick={-180,  -90,    0,   90,  180},
      ylabel={Phase [deg]},
      xminorgrids,
      legend style={at={(1,1.1)}, outer sep=2pt , anchor=north east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}},
      ]
      \addlegendimage{color=mycolor1, dashed, line width=1.5pt}
      \addlegendentry{$W_1^{-1}$};
      \addlegendimage{color=mycolor2, dashed, line width=1.5pt}
      \addlegendentry{$W_2^{-1}$};
      \addplot [color=mycolor1, line width=1.5pt]
      table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
      \addlegendentry{$H_1$};
      \addplot [color=mycolor2, line width=1.5pt]
      table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_filters_results.csv};
      \addlegendentry{$H_2$};
    \end{axis}
  \end{tikzpicture}
#+end_src

#+name: fig:hinf_synthesis_results
#+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis ([[./figs/hinf_synthesis_results.png][png]], [[./figs/hinf_synthesis_results.pdf][pdf]], [[./figs/hinf_synthesis_results.tex][tex]]).
#+RESULTS:
[[file:figs/hinf_synthesis_results.png]]

* 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) {};
     \node[above] at (P.north) {$P(s)$};

     \coordinate[] (inputw)  at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
     \coordinate[] (inputu)  at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);

     \coordinate[] (output1) at ($(P.south east)!0.8!(P.north east)  + (0.7, 0)$);
     \coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
     \coordinate[] (output3) at ($(P.south east)!0.3!(P.north east)  + (0.7, 0)$);
     \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east)  + (0.7, 0)$);

     \node[block, left=1.4 of output1] (W1){$W_1(s)$};
     \node[block, left=1.4 of output2] (W2){$W_2(s)$};
     \node[block, left=1.4 of output3] (W3){$W_3(s)$};
     \node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
     \node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};

     \node[block, below=0.3 of P] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};

     \draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
     \draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
     \draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
     \draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv) |- (H.east);
     \draw[->] ($(H.south west)!0.7!(H.north west)$) -| (inputu|-W2) -- (W2.west);
     \draw[->] ($(H.south west)!0.3!(H.north west)$) -| ($(inputu|-W3)+(0.4, 0)$) -- (W3.west);

     \draw[->] (sub2.east) -- (sub1.west);
     \draw[->] (sub1.east) -- (W1.west);
     \draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
     \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
     \draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
     \node[above] at (W2-|w){$u_1$};
     \node[above] at (W3-|w){$u_2$};
     \node[above] at (outputv-|z){$v$};
  \end{tikzpicture}
#+end_src

#+name: fig:comp_filter_three_hinf
#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters ([[./figs/comp_filter_three_hinf.png][png]], [[./figs/comp_filter_three_hinf.pdf][pdf]], [[./figs/comp_filter_three_hinf.tex][tex]]).
#+RESULTS:
[[file:figs/comp_filter_three_hinf.png]]

* 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}

  \begin{tikzpicture}
    \begin{axis}[%
      width=1.0\fwidth,
      height=0.55\fheight,
      at={(0.0\fwidth, 0.42\fheight)},
      scale only axis,
      xmode=log,
      xmin=0.1,
      xmax=100,
      xticklabels={{}},
      xminorticks=true,
      ymode=log,
      ymin=0.0005,
      ymax=20,
      ytick={0.001, 0.01, 0.1, 1, 10},
      yminorticks=true,
      ylabel={Magnitude},
      xminorgrids,
      yminorgrids,
      legend columns=2,
      legend style={
        /tikz/column 2/.style={
          column sep=5pt,
        },
        at={(1,0)}, outer sep=2pt , anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}
      },
      ]
      \addplot [color=mycolor1, dashed, line width=1.5pt]
      table [x=freqs, y=W1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
      \addlegendentry{${W_1}^{-1}$};
      \addplot [color=mycolor1, line width=1.5pt]
      table [x=freqs, y=H1, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
      \addlegendentry{$H_1$};


      \addplot [color=mycolor2, dashed, line width=1.5pt]
      table [x=freqs, y=W2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
      \addlegendentry{${W_2}^{-1}$};
      \addplot [color=mycolor2, line width=1.5pt]
      table [x=freqs, y=H2, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
      \addlegendentry{$H_2$};

      \addplot [color=mycolor3, dashed, line width=1.5pt]
      table [x=freqs, y=W3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_weights.csv};
      \addlegendentry{${W_3}^{-1}$};
      \addplot [color=mycolor3, line width=1.5pt]
      table [x=freqs, y=H3, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
      \addlegendentry{$H_3$};
    \end{axis}

    \begin{axis}[%
      width=1.0\fwidth,
      height=0.4\fheight,
      at={(0.0\fwidth, 0.0\fheight)},
      scale only axis,
      xmode=log,
      xmin=0.1,
      xmax=100,
      xminorticks=true,
      xlabel={Frequency [Hz]},
      ymin=-240,
      ymax=240,
      ytick={-180,  -90,    0,   90,  180},
      ylabel={Phase [deg]},
      xminorgrids,
      ]

      \addplot [color=mycolor1, line width=1.5pt]
      table [x=freqs, y=H1p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};

      \addplot [color=mycolor2, line width=1.5pt]
      table [x=freqs, y=H2p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};

      \addplot [color=mycolor3, line width=1.5pt]
      table [x=freqs, y=H3p, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/mathinf_three_results.csv};
    \end{axis}
  \end{tikzpicture}
#+end_src

#+name: fig:hinf_three_synthesis_results
#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis ([[./figs/hinf_three_synthesis_results.png][png]], [[./figs/hinf_three_synthesis_results.pdf][pdf]], [[./figs/hinf_three_synthesis_results.tex][tex]]).
#+RESULTS:
[[file:figs/hinf_three_synthesis_results.png]]

* 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}

  \begin{tikzpicture}
    \begin{axis}[%
      width=1.0\fwidth,
      height=1.0\fheight,
      at={(0.0\fwidth, 0.0\fheight)},
      scale only axis,
      separate axis lines,
      every outer x axis line/.append style={black},
      every x tick label/.append style={font=\color{black}},
      every x tick/.append style={black},
      xmode=log,
      xmin=0.001,
      xmax=1,
      xminorticks=true,
      xlabel={Frequency [Hz]},
      every outer y axis line/.append style={black},
      every y tick label/.append style={font=\color{black}},
      every y tick/.append style={black},
      ymode=log,
      ymin=0.005,
      ymax=20,
      yminorticks=true,
      ylabel={Magnitude},
      axis background/.style={fill=white},
      xmajorgrids,
      xminorgrids,
      ymajorgrids,
      yminorgrids,
      legend style={at={(0,1)}, outer sep=2pt, anchor=north west, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
      ]

      \addplot [color=mycolor1, line width=1.5pt]
        table [x=freqs, y=wHm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
      \addlegendentry{$|w_H|^{-1}$}

      \addplot [color=mycolor2, line width=1.5pt]
        table [x=freqs, y=wLm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matligo_weights.csv};
      \addlegendentry{$|w_L|^{-1}$}

      \addplot [color=black, dotted, line width=1.5pt]
      table[row sep=crcr]{%
        0.0005	0.008\\
        0.008   0.008\\
      };
      \addlegendentry{Specifications}

      \addplot [color=black, dotted, line width=1.5pt, forget plot]
      table[row sep=crcr]{%
        0.008	0.008\\
        0.04	1\\
      };
      \addplot [color=black, dotted, line width=1.5pt, forget plot]
      table[row sep=crcr]{%
        0.04  3\\
        0.1   3\\
      };
      \addplot [color=black, dotted, line width=1.5pt]
      table[row sep=crcr]{%
        0.1	0.045\\
        2   0.045\\
      };
    \end{axis}
  \end{tikzpicture}
#+end_src

#+name: fig:ligo_weights
#+caption: Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis ([[./figs/ligo_weights.png][png]], [[./figs/ligo_weights.pdf][pdf]], [[./figs/ligo_weights.tex][tex]]).
#+RESULTS:
[[file:figs/ligo_weights.png]]

*  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}

  \begin{tikzpicture}
    \begin{axis}[%
      width=1.0\fwidth,
      height=0.60\fheight,
      at={(0.0\fwidth, 0.32\fheight)},
      scale only axis,
      xmode=log,
      xmin=0.001,
      xmax=1,
      xtick={0.001,0.01,0.1,1},
      xticklabels={{}},
      xminorticks=true,
      ymode=log,
      ymin=0.002,
      ymax=5,
      ytick={0.001, 0.01, 0.1, 1, 10},
      yminorticks=true,
      ylabel={Magnitude},
      xminorgrids,
      yminorgrids,
      legend style={at={(1,0)}, outer sep=2pt, anchor=south east, legend cell align=left, align=left, draw=black, nodes={scale=0.7, transform shape}}
      ]
      \addplot [color=mycolor1, line width=1.5pt]
        table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
      \addlegendentry{$H_H(s)$ - $\mathcal{H}_\infty$}
      \addplot [color=mycolor1, dashed, line width=1.5pt]
        table [x=freqs, y=Hhm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
      \addlegendentry{$H_H(s)$ - FIR}
      \addplot [color=mycolor2, line width=1.5pt]
        table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
      \addlegendentry{$H_L(s)$ - $\mathcal{H}_\infty$}
      \addplot [color=mycolor2, dashed, line width=1.5pt]
        table [x=freqs, y=Hlm, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
      \addlegendentry{$H_L(s)$ - FIR}
    \end{axis}

    \begin{axis}[%
      width=1.0\fwidth,
      height=0.3\fheight,
      at={(0.0\fwidth, 0.0\fheight)},
      scale only axis,
      xmode=log,
      xmin=0.001,
      xmax=1,
      xtick={0.001,  0.01,   0.1,     1},
      xminorticks=true,
      xlabel={Frequency [Hz]},
      ymin=-180,
      ymax=180,
      ytick={-180,  -90,    0,   90,  180},
      ylabel={Phase [deg]},
      xminorgrids,
      ]
      \addplot [color=mycolor1, line width=1.5pt, forget plot]
        table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
      \addplot [color=mycolor1, dashed, line width=1.5pt, forget plot]
        table [x=freqs, y=Hhp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
      \addplot [color=mycolor2, line width=1.5pt, forget plot]
        table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_hinf.csv};
      \addplot [color=mycolor2, dashed, line width=1.5pt, forget plot]
        table [x=freqs, y=Hlp, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matcomp_ligo_fir.csv};
    \end{axis}
  \end{tikzpicture}
#+end_src

#+name: fig:comp_fir_ligo_hinf
#+caption: Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed) ([[./figs/comp_fir_ligo_hinf.png][png]], [[./figs/comp_fir_ligo_hinf.pdf][pdf]], [[./figs/comp_fir_ligo_hinf.tex][tex]]).
#+RESULTS:
[[file:figs/comp_fir_ligo_hinf.png]]