dehaeze20_optim_robus_compl_filte/tikz/index.org

Robust and Optimal Sensor Fusion - Tikz Figures

• Sensor Model - Noise
• Sensor Fusion Architecture
• Architecture used for $\mathcal{H}_2$ synthesis of complementary filters
• Sensor Model - Uncertainty
• Sensor fusion architecture with sensor dynamics uncertainty
• Uncertainty set of the super sensor dynamics
• Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
• Sensor Model - Noise and Uncertainty
• Sensor fusion architecture with sensor dynamics uncertainty and noise
• Mixed H2/H-Infinity Synthesis

Configuration file is accessible here.

Sensor Model - Noise

  \begin{tikzpicture}
    \node[addb](add1){};
    \node[block, right=0.5 of add1](G1){$G_i$};
    \node[block, right=0.8 of G1](Ginv1){$\hat{G}_i^{-1}$};
    \node[block, above=0.5 of add1](N1){$N_i$};

    \draw[<-] (add1.west) -- ++(-1.0, 0) node[above right]{$x$};
    \draw[->] (add1.east) -- (G1.west);
    \draw[->] (N1.south) -- (add1.north)node[above left]{$n_i$};
    \draw[<-] (N1.north)node[above left](n1){$\tilde{n}_i$} -- ++(0, 0.4);
    \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_i$};
    \draw[->] (Ginv1.east) -- ++(0.7, 0) node[above left]{$\hat{x}_i$};

    \begin{scope}[on background layer]
      \node[fit={($(G1.south east)+(0.15, -0.15)$) ($(n1.north west)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
      \node[below left, align=right] at (sensor1.north east) {Sensor};
    \end{scope}
  \end{tikzpicture}

Sensor Model (png, pdf).

Sensor Fusion Architecture

  \begin{tikzpicture}
    \node[branch] (x) at (0, 0);
    \node[addb, above right=1.05 and 0.6 of x](add1){};
    \node[addb, below right=1.05 and 0.6 of x](add2){};
    \node[block, right=0.4 of add1](G1){$G_1$};
    \node[block, right=0.4 of add2](G2){$G_2$};
    \node[block, right=0.7 of G1](Ginv1){$\hat{G}_1^{-1}$};
    \node[block, right=0.7 of G2](Ginv2){$\hat{G}_2^{-1}$};
    \node[block, right=0.6 of Ginv1](H1){$H_1$};
    \node[block, right=0.6 of Ginv2](H2){$H_2$};
    \node[block, above=0.5 of add1](N1){$N_1$};
    \node[block, above=0.5 of add2](N2){$N_2$};
    \node[addb, right=5.3 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.east) -- (G1.west);
    \draw[->] (add2.east) -- (G2.west);
    \draw[->] (N1.south) -- (add1.north)node[above left]{$n_1$};
    \draw[->] (N2.south) -- (add2.north)node[above left]{$n_2$};
    \draw[<-] (N1.north)node[above left](n1){$\tilde{n}_1$} -- ++(0, 0.4);
    \draw[<-] (N2.north)node[above left](n2){$\tilde{n}_2$} -- ++(0, 0.4);
    \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_1$};
    \draw[->] (G2.east) -- (Ginv2.west)node[above left]{$v_2$};
    \draw[->] (Ginv1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
    \draw[->] (Ginv2.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={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.2)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
      \node[below left] at (supersensor.north east) {Super Sensor};

      \node[fit={($(G1.south east)+(0.15, -0.15)$) ($(n1.north west)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
      \node[below left] at (sensor1.north east) {Sensor 1};

      \node[fit={($(G2.south east)+(0.15, -0.15)$) ($(n2.north west)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
      \node[below left] at (sensor2.north east) {Sensor 2};
    \end{scope}
  \end{tikzpicture}

Sensor Fusion Architecture (png, pdf).

Architecture used for $\mathcal{H}_2$ synthesis of complementary filters

  \begin{tikzpicture}
     \node[block={3.5cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
     \node[above] at (P.north) {$P_{\mathcal{H}_2}$};

     \coordinate[] (inputw)  at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
     \coordinate[] (inputu)  at ($(P.south west)!0.5!(P.north west) + (-0.7, 0)$);

     \coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (output2) at ($(P.south east)!0.5!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (outputv) at ($(P.south east)!0.2!(P.north east) + ( 0.7, 0)$);

     \node[block, left=0.9 of output1] (N1){$N_1$};
     \node[block, left=0.9 of output2] (N2){$N_2$};
     \node[addb={+}{}{}{}{-}, left=of N1] (sub) {};

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

     \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
     \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west);
     \draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
     \draw[->] (sub.east) -- (N1.west);
     \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
     \draw[->] (N1.east) -- (output1)node[above left]{$z_1$};
     \draw[->] (N2.east) -- (output2)node[above left]{$z_2$};
  \end{tikzpicture}

Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters (png, pdf).

Sensor Model - Uncertainty

  \begin{tikzpicture}
    \node[branch](b1){};
    \node[block, above right=0.3 and 0.25 of b1](W1){$W_i$};
    \node[block, right=0.3 of W1](delta1){$\Delta_i$};
    \node[addb, right=0.3 of b1-|delta1](add1){};
    \node[block, right=0.3 of add1](G1){$\hat{G}_i$};
    \node[block, right=0.7 of G1](Ginv1){$\hat{G}_i^{-1}$};

    \draw[->] ($(g1)+(-0.6,0)$) node[above right]{$x$} -- (add1.west);
    \draw[->] (g1) |- (W1.west);
    \draw[->] (W1.east) -- (delta1.west);
    \draw[->] (delta1.east) -| (add1.north);
    \draw[->] (add1.east) -- (G1.west);
    \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_i$};
    \draw[->] (Ginv1.east) -- ++(0.7, 0) node[above left]{$\hat{x}_i$};

    \begin{scope}[on background layer]
      \node[block, fit={($(b1|-W1.north) + (-0.15, 0.15)$) ($(G1.south east)+(0.15, -0.15)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
      \node[below left] at (sensor1.north east) {Sensor};
    \end{scope}
  \end{tikzpicture}

Sensor Model including dynamical uncertainty (png, pdf).

Sensor fusion architecture with sensor dynamics uncertainty

  \begin{tikzpicture}
    \node[branch] (x) at (0, 0);
    \node[branch, above right=0.8 and 0.2 of x](b1){};
    \node[branch, below right=0.8 and 0.2 of x](b2){};
    \node[block, above right=0.3 and 0.25 of b1](W1){$W_1$};
    \node[block, above right=0.3 and 0.25 of b2](W2){$W_2$};
    \node[block, right=0.3 of W1](delta1){$\Delta_1$};
    \node[block, right=0.3 of W2](delta2){$\Delta_2$};
    \node[addb, right=0.3 of b1-|delta1](add1){};
    \node[addb, right=0.3 of b2-|delta2](add2){};
    \node[block, right=0.3 of add1](G1){$\hat{G}_1$};
    \node[block, right=0.3 of add2](G2){$\hat{G}_2$};
    \node[block, right=0.7 of G1](Ginv1){$\hat{G}_1^{-1}$};
    \node[block, right=0.7 of G2](Ginv2){$\hat{G}_2^{-1}$};
    \node[block, right=0.35 of Ginv1](H1){$H_1$};
    \node[block, right=0.35 of Ginv2](H2){$H_2$};
    \node[addb, right=6.8 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.east) -- (G1.west);
    \draw[->] (add2.east) -- (G2.west);
    \draw[->] (b1) |- (W1.west);
    \draw[->] (b2) |- (W2.west);
    \draw[->] (W1.east) -- (delta1.west);
    \draw[->] (W2.east) -- (delta2.west);
    \draw[->] (delta1.east) -| (add1.north);
    \draw[->] (delta2.east) -| (add2.north);
    \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_1$};
    \draw[->] (G2.east) -- (Ginv2.west)node[above left]{$v_2$};
    \draw[->] (Ginv1.east) -- (H1.west);
    \draw[->] (Ginv2.east) -- (H2.west);
    \draw[->] (H1.east) -| (add.north);
    \draw[->] (H2.east) -| (add.south);
    \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};

    \begin{scope}[on background layer]
      \node[fit={($(H2.south-|x)+(-0.2, -0.3)$) ($(delta1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
      \node[below left] at (supersensor.north east) {Super Sensor};

      \node[block, fit={($(b1|-W1.north) + (-0.15, 0.15)$) ($(G1.south east)+(0.15, -0.15)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
      \node[below left] at (sensor1.north east) {Sensor 1};
      \node[block, fit={($(b2|-W2.north) + (-0.15, 0.15)$) ($(G2.south east)+(0.15, -0.15)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
      \node[below left] at (sensor2.north east) {Sensor 2};
    \end{scope}
  \end{tikzpicture}

Sensor fusion architecture with sensor dynamics uncertainty (png, pdf).

Uncertainty set of the super sensor dynamics

  \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, dashed] (135:1.0) circle [radius=0.8];

      % Center of Circle
      \node[below] at (0, 0){$1$};

      \draw[<->, dashed] (0, 0)   node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
      \draw[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);

      \node[] (l1) at (2, 1.5) {$|W_1 H_1|$};
      \draw[->, dashed, out=-90, in=0] (l1.south) to (r1);

      \node[] (l2) at (-2.5, 1.5) {$|W_2 H_2|$};
      \draw[->, dashed, out=0, in=-180] (l2.east) to (r2);

      \draw[<->, dashed] (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);
    \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}

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) (png, pdf).

Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters

  \begin{tikzpicture}
     \node[block={4.2cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
     \node[above] at (P.north) {$P_{\mathcal{H}_\infty}$};

     \coordinate[] (inputw)  at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
     \coordinate[] (inputu)  at ($(P.south west)!0.5!(P.north west) + (-0.7, 0)$);

     \coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (output2) at ($(P.south east)!0.5!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (outputv) at ($(P.south east)!0.2!(P.north east) + ( 0.7, 0)$);

     \node[block, left=0.9 of output1] (W1){$W_1$};
     \node[block, left=0.9 of output2] (W2){$W_2$};
     \node[block, left=0.4 of W1] (Wu1){$W_u$};
     \node[block, left=0.4 of W2] (Wu2){$W_u$};
     \node[addb={+}{}{}{}{-}, left=of Wu1] (sub) {};

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

     \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
     \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (Wu2.west);
     \draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
     \draw[->] (sub.east) -- (Wu1.west);
     \draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
     \draw[->] (Wu1.east) -- (W1.west);
     \draw[->] (Wu2.east) -- (W2.west);
     \draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
     \draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
  \end{tikzpicture}

Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters (png, pdf).

Sensor Model - Noise and Uncertainty

  \begin{tikzpicture}
    \node[addb](add1){};

    \node[block, above=0.5 of add1](N1){$N_i$};
    \node[branch, right=0.4 of add1](b1){};
    \node[block, above right=0.3 and 0.25 of b1](W1){$W_i$};
    \node[block, right=0.3 of W1](delta1){$\Delta_i$};
    \node[addb, right=0.3 of b1-|delta1](addu){};
    \node[block, right=0.3 of addu](G1){$\hat{G}_i$};
    \node[block, right=0.7 of G1](Ginv1){$\hat{G}_i^{-1}$};

    \draw[<-] (add1.west) -- ++(-1.0, 0) node[above right]{$x$};
    \draw[->] (add1.east) -- (addu.west);
    \draw[->] (b1) |- (W1.west);
    \draw[->] (W1.east) -- (delta1.west);
    \draw[->] (delta1.east) -| (addu.north);
    \draw[->] (addu.east) -- (G1.west);

    \draw[->] (N1.south) -- (add1.north)node[above left]{$n_i$};
    \draw[<-] (N1.north)node[above left](n1){$\tilde{n}_i$} -- ++(0, 0.4);
    \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_i$};
    \draw[->] (Ginv1.east) -- ++(0.7, 0) node[above left]{$\hat{x}_i$};

    \begin{scope}[on background layer]
      \node[fit={($(G1.south east)+(0.15, -0.15)$) ($(n1.north west)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
      \node[below left, align=right] at (sensor1.north east) {Sensor};
    \end{scope}
  \end{tikzpicture}

Sensor Model (png, pdf).

Sensor fusion architecture with sensor dynamics uncertainty and noise

  \begin{tikzpicture}
    \node[branch] (x) at (0, 0);
    \node[addb, above right=1.0 and 0.6 of x](addn1){};
    \node[addb, below right=1.0 and 0.6 of x](addn2){};
    \node[addb, right=2.6 of addn1](add1){};
    \node[addb, right=2.6 of addn2](add2){};
    \node[block, above left=0.2 and 0 of add1](delta1){$\Delta_1$};
    \node[block, above left=0.2 and 0 of add2](delta2){$\Delta_2$};
    \node[block, left=0.4 of delta1](W1){$W_1$};
    \node[block, left=0.4 of delta2](W2){$W_2$};
    \node[block, above=0.5 of addn1](N1) {$N_1$};
    \node[block, above=0.5 of addn2](N2) {$N_2$};
    \node[block, right=0.7 of add1](H1){$H_1$};
    \node[block, right=0.7 of add2](H2){$H_2$};
    \node[addb, right=5.8 of x](add){};

    \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
    \draw[->] (x.center) |- (addn1.west);
    \draw[->] (x.center) |- (addn2.west);
    \draw[->] ($(addn1-|W1.west)+(-0.3, 0)$)node[branch](S1){} |- (W1.west);
    \draw[->] ($(addn2-|W2.west)+(-0.3, 0)$)node[branch](S2){} |- (W2.west);
    \draw[->] (W1.east) -- (delta1.west);
    \draw[->] (W2.east) -- (delta2.west);
    \draw[->] (delta1.east) -| (add1.north);
    \draw[->] (delta2.east) -| (add2.north);
    \draw[->] (addn1.east) -- (add1.west);
    \draw[->] (addn2.east) -- (add2.west);
    \draw[->] (add1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
    \draw[->] (add2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
    \draw[<-] (N1.north)node[above left](n1){$\tilde{n}_1$} -- ++(0, 0.4);
    \draw[<-] (N2.north)node[above left](n2){$\tilde{n}_2$} -- ++(0, 0.4);
    \draw[->] (N1.south) -- (addn1.north)node[above left]{$n_1$};
    \draw[->] (N2.south) -- (addn2.north)node[above left]{$n_2$};
    \draw[->] (H1.east) -| (add.north);
    \draw[->] (H2.east) -| (add.south);
    \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};

    \begin{scope}[on background layer]
      \node[fit={($(H2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
      \node[below left] at (supersensor.north east) {Super Sensor};

      \node[fit={(n1.north west) ($(add1.south -| add1.east) + (0.1, -0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
      \node[below left] at (sensor1.north east) {Sensor 1};

      \node[fit={(n2.north west) ($(add2.south -| add2.east) + (0.1, -0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
      \node[below left] at (sensor2.north east) {Sensor 2};
    \end{scope}
  \end{tikzpicture}

Sensor fusion architecture with sensor dynamics uncertainty (png, pdf).

Mixed H2/H-Infinity Synthesis

  \begin{tikzpicture}
     \node[block={4.2cm}{4.0cm}, fill=black!20!white, dashed] (P) {};
     \node[above] at (P.north) {$P_{\mathcal{H}_2/\mathcal{H}_\infty}$};

     \coordinate[] (inputw)  at ($(P.south west)!0.85!(P.north west) + (-0.7, 0)$);
     \coordinate[] (inputu)  at ($(P.south west)!0.25!(P.north west) + (-0.7, 0)$);

     \coordinate[] (output1) at ($(P.south east)!0.85!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (output2) at ($(P.south east)!0.65!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (output3) at ($(P.south east)!0.45!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (output4) at ($(P.south east)!0.25!(P.north east) + ( 0.7, 0)$);
     \coordinate[] (outputv) at ($(P.south east)!0.10!(P.north east) + ( 0.7, 0)$);

     \node[block, left=1.0 of output1] (W1){$W_1$};
     \node[block, left=1.0 of output2] (W2){$W_2$};
     \node[block, left=0.4 of W1] (Wu1){$W_u$};
     \node[block, left=0.4 of W2] (Wu2){$W_u$};
     \node[addb={+}{}{}{}{-}, left=0.4 of Wu1] (sub1) {};

     \node[block, left=1.0 of output3] (N1){$N_1$};
     \node[block, left=1.0 of output4] (N2){$N_2$};
     \node[addb={+}{}{}{}{-}, left=0.6 of N1] (sub2) {};

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

     \draw[->] (inputw) node[above right]{$w$} -- (sub1.west);
     \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west);
     \draw[->] (inputu-|sub1) node[branch]{} -- (sub1.south);
     \draw[->] (inputu-|sub2) node[branch]{} -- (sub2.south);
     \draw[->] (sub1|-W2) node[branch]{} -- (Wu2.west);
     \draw[->] (sub1.east) -- (Wu1.west);
     \draw[->] (sub2.east) -- (N1.west);
     \draw[->] ($(sub1.west)+(-0.6, 0)$) node[branch](w_branch){} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
     \draw[->] (w_branch|-sub2) node[branch]{} -- (sub2.west);
     \draw[->] (Wu1.east) -- (W1.west);
     \draw[->] (Wu2.east) -- (W2.west);
     \draw[->] (W1.east) -- (output1)node[above left](z1){$z_1$};
     \draw[->] (W2.east) -- (output2)node[above left](z2){$z_2$};
     \draw[->] (N1.east) -- (output3)node[above left](z3){$z_3$};
     \draw[->] (N2.east) -- (output4)node[above left](z4){$z_4$};


    \draw [decoration={brace, raise=5pt}, decorate] (z1.north east) -- node[right=6pt]{$z_{\mathcal{H}_\infty}$} (z2.south east);
    \draw [decoration={brace, raise=5pt}, decorate] (z3.north east) -- node[right=6pt]{$z_{\mathcal{H}_2}$} (z4.south east);
  \end{tikzpicture}

Mixed H2/H-Infinity Synthesis (png, pdf).