Update generalized plant

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Thomas Dehaeze 2020-10-05 11:47:57 +02:00
parent dcd65832dc
commit b53bee9e37

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