Add weight on the uncertainty

paper/paper.org

@@ -289,14 +289,24 @@ 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.
 
-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{pmatrix}
   z_1 \\ z_2 \\ v
 \end{pmatrix} = \underbrace{\begin{bmatrix}
-  W_1 & W_1 \\
-  0   & 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}
@@ -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:
 #+NAME: eq:Hinf_norm
 \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}
 
 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)
 \end{equation}
 
-
 #+name: fig:h_infinity_robust_fusion
 #+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
 #+attr_latex: :scale 1