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Add weight on the uncertainty

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Thomas Dehaeze 2020-09-23 14:10:55 +02:00
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@ -289,13 +289,23 @@ In order for the fusion to be "robust", meaning no phase drop will be induced in
The goal is to design two complementary filters $H_1(s)$ and $H_2(s)$ such that the super sensor noise uncertainty is kept reasonably small. The goal is to design two complementary filters $H_1(s)$ and $H_2(s)$ such that the super sensor noise uncertainty is kept reasonably small.
This problem can be dealt with an $\mathcal{H}_\infty$ synthesis problem by considering the following generalized plant: To define what by "small" we mean, we use a weighting filter $W_u(s)$ such that the synthesis objective is:
\begin{equation}
\left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
\end{equation}
This is actually almost equivalent (to within a factor $\sqrt{2}$) equivalent as to have:
\begin{equation}
\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
\end{equation}
This problem can thus be dealt with an $\mathcal{H}_\infty$ synthesis problem by considering the following generalized plant (Figure ref:fig:h_infinity_robust_fusion):
\begin{equation} \begin{equation}
\begin{pmatrix} \begin{pmatrix}
z_1 \\ z_2 \\ v z_1 \\ z_2 \\ v
\end{pmatrix} = \underbrace{\begin{bmatrix} \end{pmatrix} = \underbrace{\begin{bmatrix}
W_1 & W_1 \\ W_u W_1 & W_u W_1 \\
0 & W_2 \\ 0 & W_u W_2 \\
1 & 0 1 & 0
\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix} \end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
w \\ u w \\ u
@ -305,7 +315,7 @@ This problem can be dealt with an $\mathcal{H}_\infty$ synthesis problem by cons
Applying the $\mathcal{H}_\infty$ synthesis on $P_{\mathcal{H}_\infty}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_\infty$ norm from $w$ to $(z_1,z_2)$ is minimized: Applying the $\mathcal{H}_\infty$ synthesis on $P_{\mathcal{H}_\infty}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_\infty$ norm from $w$ to $(z_1,z_2)$ is minimized:
#+NAME: eq:Hinf_norm #+NAME: eq:Hinf_norm
\begin{equation} \begin{equation}
\left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_1 (1 - H_2) \\ W_2 H_2 \end{matrix} \right\|_\infty \left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_u W_1 (1 - H_2) \\ W_u W_2 H_2 \end{matrix} \right\|_\infty
\end{equation} \end{equation}
The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$ by defining $H_1(s)$ to be the complementary filter of $H_2(s)$: The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$ by defining $H_1(s)$ to be the complementary filter of $H_2(s)$:
@ -313,7 +323,6 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
H_1(s) = 1 - H_2(s) H_1(s) = 1 - H_2(s)
\end{equation} \end{equation}
#+name: fig:h_infinity_robust_fusion #+name: fig:h_infinity_robust_fusion
#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters #+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :scale 1 #+attr_latex: :scale 1