Update generalized plant
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@ -96,7 +96,6 @@
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<<sec:optimal_fusion>>
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** Sensor Model
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Let's consider a sensor measuring a physical quantity $x$ (Figure ref:fig:sensor_model_noise).
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The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function $G_i(s)$.
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@ -131,7 +130,6 @@ In order to obtain an estimate $\hat{x}_i$ of $x$, a model $\hat{G}_i$ of the (t
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[[file:figs/sensor_model_noise.pdf]]
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** Sensor Fusion Architecture
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Let's now consider two sensors measuring the same physical quantity $x$ but with different dynamics $(G_1, G_2)$ and noise characteristics $(N_1, N_2)$ (Figure ref:fig:sensor_fusion_noise_arch).
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The noise sources $\tilde{n}_1$ and $\tilde{n}_2$ are considered to be uncorrelated.
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@ -195,8 +193,8 @@ This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the fo
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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N_1 & N_1 \\
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0 & N_2 \\
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N_1 & -N_1 \\
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0 & N_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
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w \\ u
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@ -223,6 +221,11 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
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** Example
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#+name: fig:figure_name
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#+caption: Figure caption
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#+attr_latex: :scale 1
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[[file:figs/sensors_nominal_dynamics.pdf]]
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** Robustness Problem
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* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
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@ -270,7 +273,6 @@ As $H_1$ and $H_2$ are complementary filters, we finally have:
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[[file:figs/sensor_fusion_arch_uncertainty.pdf]]
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** Super Sensor Dynamical Uncertainty
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The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency $\omega$ is bounded in the complex plane by a circle centered on 1 and with a radius equal to $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$ as shown in Figure ref:fig:uncertainty_set_super_sensor.
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@ -306,9 +308,9 @@ This problem can thus be dealt with an $\mathcal{H}_\infty$ synthesis problem by
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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W_u W_1 & W_u W_1 \\
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0 & W_u W_2 \\
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1 & 0
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W_u W_1 & -W_u W_1 \\
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0 & W_u W_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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