Update generalized plant

paper/paper.org

@@ -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}