Add test about Hinf synthesis

paper/paper.org

@@ -42,11 +42,11 @@
 \bibliographystyle{IEEEtran}
 :END:
 
-* LaTeX Config                                                     :noexport:
+* LaTeX Config                                                      :noexport:
 #+begin_src latex :tangle config.tex
 #+end_src
 
-* Build                                                            :noexport:
+* Build                                                             :noexport:
 #+NAME: startblock
 #+BEGIN_SRC emacs-lisp :results none
   (add-to-list 'org-latex-classes
@@ -74,7 +74,7 @@
   (setq org-latex-with-hyperref nil)
 #+END_SRC
 
-* Abstract                                                           :ignore:
+* Abstract                                                            :ignore:
 #+begin_abstract
   Abstract text to be done
 #+end_abstract
@@ -97,17 +97,17 @@
 
 ** Sensor Model
 
-Let's consider a sensor measuring a physical quantity $x$ (Figure [[fig: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 $G_i(s)$.
 
 The noise of sensor can be described by the Power Spectral Density (PSD) $\Phi_{n_i}(\omega)$.
 
 This is approximated by shaping a white noise with unitary PSD $\tilde{n}_i$ eqref:eq:unitary_sensor_noise_psd with a LTI transfer function $N_i(s)$:
 \begin{equation}
-\begin{aligned}
+  \begin{aligned}
-  \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
+    \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
-                     &= \left| N_i(j\omega) \right|^2
+    &= \left| N_i(j\omega) \right|^2
-\end{aligned}
+  \end{aligned}
 \end{equation}
 
 #+name: eq:unitary_sensor_noise_psd
@@ -120,19 +120,19 @@ The output of the sensor $v_i$:
   v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i
 \end{equation}
 
-In order to obtain an estimate $\hat{x}_i$ of $x$, a model $\hat{G}_i$ of the (true) sensor dynamics $G_i$ is inverted and applied at the output (Figure [[fig:sensor_model]]):
+In order to obtain an estimate $\hat{x}_i$ of $x$, a model $\hat{G}_i$ of the (true) sensor dynamics $G_i$ is inverted and applied at the output (Figure ref:fig:sensor_model_noise):
 \begin{equation}
   \hat{x}_i = \left( \hat{G}_i^{-1} G_i \right) x + \left( \hat{G}_i^{-1} G_i N_i \right) \tilde{n}_i
 \end{equation}
 
-#+name: fig:sensor_model
+#+name: fig:sensor_model_noise
 #+caption: Sensor Model
 #+attr_latex: :scale 1
-[[file:figs/sensor_model.pdf]]
+[[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 [[fig:sensor_fusion_noise_arch]]).
+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.
 
@@ -190,7 +190,7 @@ And the goal is the minimize the Root Mean Square (RMS) value of $n$:
 
 Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\sigma_n$ is minimized.
 
-This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the following generalized plant (also represented in Figure [[fig:h_two_optimal_fusion]]):
+This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the following generalized plant (also represented in Figure ref:fig:h_two_optimal_fusion):
 \begin{equation}
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
@@ -230,28 +230,38 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
 
 ** Representation of Sensor Dynamical Uncertainty
 
-Suppose that the sensor dynamics $G_i(s)$ can be modelled by a nominal d
+In Section ref:sec:optimal_fusion, the model $\hat{G}_i(s)$ of the sensor was considered to be perfect.
-\begin{equation}
+In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
-  G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
-\end{equation}
 
+The Uncertainty on the sensor dynamics $G_i(s)$ is here modelled by (input) multiplicative uncertainty:
+\begin{equation}
+  G_i(s) = \hat{G}_i(s) \left( 1 + W_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
+\end{equation}
+where $\hat{G}_i(s)$ is the nominal model, $W_i$ a weight representing the size of the uncertainty at each frequency, and $\Delta_i$ is any complex perturbation such that $\left\| \Delta_i \right\|_\infty < 1$.
+
+The sensor can then be represented as shown in Figure ref:fig:sensor_model_uncertainty.
+
+#+name: fig:sensor_model_uncertainty
+#+caption: Sensor Model including Dynamical Uncertainty
+#+attr_latex: :scale 1
+[[file:figs/sensor_model_uncertainty.pdf]]
 
 ** Sensor Fusion Architecture
+Let's consider the sensor fusion architecture shown in Figure ref:fig:sensor_fusion_arch_uncertainty where the dynamical uncertainties of both sensors are included.
+
+The super sensor estimate is then:
 \begin{equation}
-\begin{split}
+\begin{aligned}
-  \hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\
+  \hat{x} &= \Big(  H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
-             + & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x
+          & \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
-\end{split}
+          &= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x
+\end{aligned}
 \end{equation}
 with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$.
 
-Suppose the model inversion is equal to the nominal model:
+As $H_1$ and $H_2$ are complementary filters, we finally have:
 \begin{equation}
-  \hat{G}_i = G_i
+  \hat{x} = \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right) x, \quad \|\Delta_i\|_\infty<1
-\end{equation}
-
-\begin{equation}
-  \hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
 \end{equation}
 
 #+name: fig:sensor_fusion_arch_uncertainty
@@ -261,16 +271,48 @@ Suppose the model inversion is equal to the nominal model:
 
 ** 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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$.
+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.
+
+
+And we can see that the dynamical uncertainty of the super sensor is equal to the sum of the individual sensor uncertainties filtered out by the complementary filters.
 
 #+name: fig:uncertainty_set_super_sensor
 #+caption: Super Sensor model uncertainty displayed in the complex plane
 #+attr_latex: :scale 1
 [[file:figs/uncertainty_set_super_sensor.pdf]]
 
-** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
+# Some comments on the weights
+At frequencies where $\left|W_i(j\omega)\right| > 1$ the uncertainty exceeds $100\%$ and sensor fusion is impossible.
+
+** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
+In order for the fusion to be "robust", meaning no phase drop will be induced in the super sensor dynamics,
+
+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:
+\begin{equation}
+\begin{pmatrix}
+  z_1 \\ z_2 \\ v
+\end{pmatrix} = \underbrace{\begin{bmatrix}
+  W_1 & W_1 \\
+  0   & W_2 \\
+  1   &  0
+\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
+  w \\ u
+\end{pmatrix}
+\end{equation}
+
+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
+\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 $H_2(s)$:
+\begin{equation}
+  H_1(s) = 1 - H_2(s)
+\end{equation}
 
-In order to minimize the super sensor dynamical uncertainty
 
 #+name: fig:h_infinity_robust_fusion
 #+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
@@ -279,11 +321,19 @@ In order to minimize the super sensor dynamical uncertainty
 
 ** Example
 
+# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors
+
 * Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
 <<sec:optimal_robust_fusion>>
 
 ** Sensor Fusion Architecture
 
+#+name: fig:sensor_model_noise_uncertainty
+#+caption: Sensor Model including Noise and Dynamical Uncertainty
+#+attr_latex: :scale 1
+[[file:figs/sensor_model_noise_uncertainty.pdf]]
+
+
 #+name: fig:sensor_fusion_arch_full
 #+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
 #+attr_latex: :scale 1

paper/paper.tex

@@ -1,4 +1,4 @@
-% Created 2020-09-22 mar. 11:10
+% Created 2020-09-22 mar. 21:58
 % Intended LaTeX compiler: pdflatex
 \documentclass[conference]{IEEEtran}
 \usepackage[utf8]{inputenc}
@@ -50,32 +50,34 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
 \end{IEEEkeywords}
 
 \section{Introduction}
-\label{sec:orgfaa194e}
+\label{sec:orgc2fc7e2}
 \label{sec:introduction}
 
-Section \ref{sec:optimal_fusion}
+\begin{itemize}
-Section \ref{sec:robust_fusion}
+\item Section \ref{sec:optimal_fusion}
-Section \ref{sec:optimal_robust_fusion}
+\item Section \ref{sec:robust_fusion}
-Section \ref{sec:experimental_validation}
+\item Section \ref{sec:optimal_robust_fusion}
+\item Section \ref{sec:experimental_validation}
+\end{itemize}
 
 \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
-\label{sec:org08f9f0e}
+\label{sec:org2031a7c}
 \label{sec:optimal_fusion}
 
 \subsection{Sensor Model}
-\label{sec:orgaa5ec56}
+\label{sec:org32da471}
 
-Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig: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 \(G_i(s)\).
 
 The noise of sensor can be described by the Power Spectral Density (PSD) \(\Phi_{n_i}(\omega)\).
 
 This is approximated by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_sensor_noise_psd} with a LTI transfer function \(N_i(s)\):
 \begin{equation}
-\begin{aligned}
+  \begin{aligned}
-  \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
+    \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
-                     &= \left| N_i(j\omega) \right|^2
+    &= \left| N_i(j\omega) \right|^2
-\end{aligned}
+  \end{aligned}
 \end{equation}
 
 \begin{equation}
@@ -88,19 +90,19 @@ The output of the sensor \(v_i\):
   v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i
 \end{equation}
 
-In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of the (true) sensor dynamics \(G_i\) is inverted and applied at the output (Figure \ref{fig:sensor_model}):
+In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of the (true) sensor dynamics \(G_i\) is inverted and applied at the output (Figure \ref{fig:sensor_model_noise}):
 \begin{equation}
   \hat{x}_i = \left( \hat{G}_i^{-1} G_i \right) x + \left( \hat{G}_i^{-1} G_i N_i \right) \tilde{n}_i
 \end{equation}
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/sensor_model.pdf}
+\includegraphics[scale=1]{figs/sensor_model_noise.pdf}
-\caption{\label{fig:sensor_model}Sensor Model}
+\caption{\label{fig:sensor_model_noise}Sensor Model}
 \end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:org17e7387}
+\label{sec:orgf3af62a}
 
 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}).
 
@@ -138,7 +140,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
 \end{equation}
 
 \subsection{Super Sensor Noise}
-\label{sec:orgb010f68}
+\label{sec:orga39f54c}
 Let's note \(n\) the super sensor noise.
 \begin{equation}
   n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
@@ -152,31 +154,42 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
 It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
 
 \subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
-\label{sec:orgf1d735c}
+\label{sec:org536193f}
-
+The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise \(n\) of the estimation \(\hat{x}\).
-The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise  \(n\) of the estimation \(\hat{x}\).
 
 And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
 \begin{equation}
 \label{eq:rms_value_estimation}
-  \sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2
+  \sigma_{n} = \sqrt{\int_0^\infty \Phi_{n}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2
 \end{equation}
 
-Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized.
+Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\sigma_n\) is minimized.
 
+This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the following generalized plant (also represented in Figure \ref{fig:h_two_optimal_fusion}):
 \begin{equation}
 \begin{pmatrix}
   z_1 \\ z_2 \\ v
-\end{pmatrix} = \begin{bmatrix}
+\end{pmatrix} = \underbrace{\begin{bmatrix}
   N_1 & N_1 \\
   0   & N_2 \\
   1   &  0
-\end{bmatrix} \begin{pmatrix}
+\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
   w \\ u
 \end{pmatrix}
 \end{equation}
 
-The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
+Applying the \(\mathcal{H}_2\) synthesis on \(P_{\mathcal{H}_2}\) will generate a filter \(H_2(s)\) such that the \(\mathcal{H}_2\) norm from \(w\) to \((z_1,z_2)\) is minimized:
+\begin{equation}
+\label{eq:H2_norm}
+   \left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_2 = \left\| \begin{matrix} N_1 (1 - H_2) \\ N_2 H_2 \end{matrix} \right\|_2
+\end{equation}
+
+The \(\mathcal{H}_2\) norm of Eq. \eqref{eq:H2_norm} is equals to \(\sigma_n\) by defining \(H_1(s)\) to be the complementary filter of \(H_2(s)\):
+\begin{equation}
+  H_1(s) = 1 - H_2(s)
+\end{equation}
+
+We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}\) generates two complementary filters \(H_1(s)\) and \(H_2(s)\) such that the RMS value of super sensor noise is minimized.
 
 \begin{figure}[htbp]
 \centering
@@ -185,41 +198,52 @@ The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the
 \end{figure}
 
 \subsection{Example}
-\label{sec:org074433c}
+\label{sec:orgd689dc3}
 
 \subsection{Robustness Problem}
-\label{sec:org21dc09f}
+\label{sec:orgc57d2ad}
 
 \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
-\label{sec:org2041184}
+\label{sec:orgeed5209}
 \label{sec:robust_fusion}
 
 \subsection{Representation of Sensor Dynamical Uncertainty}
-\label{sec:orgfd12a50}
+\label{sec:org7f4d435}
 
-Suppose that the sensor dynamics \(G_i(s)\) can be modelled by a nominal d
+In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
+In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
+
+The Uncertainty on the sensor dynamics \(G_i(s)\) is here modelled by (input) multiplicative uncertainty:
 \begin{equation}
-  G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
+  G_i(s) = \hat{G}_i(s) \left( 1 + W_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
 \end{equation}
+where \(\hat{G}_i(s)\) is the nominal model, \(W_i\) a weight representing the size of the uncertainty at each frequency, and \(\Delta_i\) is any complex perturbation such that \(\left\| \Delta_i \right\|_\infty < 1\).
 
+The sensor can then be represented as shown in Figure \ref{fig:sensor_model_uncertainty}.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_uncertainty.pdf}
+\caption{\label{fig:sensor_model_uncertainty}Sensor Model including Dynamical Uncertainty}
+\end{figure}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:org11c9d00}
+\label{sec:orgd4a5727}
+Let's consider the sensor fusion architecture shown in Figure \ref{fig:sensor_fusion_arch_uncertainty} where the dynamical uncertainties of both sensors are included.
+
+The super sensor estimate is then:
 \begin{equation}
-\begin{split}
+\begin{aligned}
-  \hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\
+  \hat{x} &= \Big(  H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
-             + & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x
+          & \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
-\end{split}
+          &= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x
+\end{aligned}
 \end{equation}
 with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\).
 
-Suppose the model inversion is equal to the nominal model:
+As \(H_1\) and \(H_2\) are complementary filters, we finally have:
 \begin{equation}
-  \hat{G}_i = G_i
+  \hat{x} = \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right) x, \quad \|\Delta_i\|_\infty<1
-\end{equation}
-
-\begin{equation}
-  \hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
 \end{equation}
 
 \begin{figure}[htbp]
@@ -229,9 +253,12 @@ Suppose the model inversion is equal to the nominal model:
 \end{figure}
 
 \subsection{Super Sensor Dynamical Uncertainty}
-\label{sec:org6673a25}
+\label{sec:org7eede13}
 
-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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\).
+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}.
+
+
+And we can see that the dynamical uncertainty of the super sensor is equal to the sum of the individual sensor uncertainties filtered out by the complementary filters.
 
 \begin{figure}[htbp]
 \centering
@@ -239,10 +266,38 @@ The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at freque
 \caption{\label{fig:uncertainty_set_super_sensor}Super Sensor model uncertainty displayed in the complex plane}
 \end{figure}
 
-\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
+At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
-\label{sec:org41ccb1e}
+
+\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
+\label{sec:org0b02610}
+In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
+
+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:
+\begin{equation}
+\begin{pmatrix}
+  z_1 \\ z_2 \\ v
+\end{pmatrix} = \underbrace{\begin{bmatrix}
+  W_1 & W_1 \\
+  0   & W_2 \\
+  1   &  0
+\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
+  w \\ u
+\end{pmatrix}
+\end{equation}
+
+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:
+\begin{equation}
+\label{eq:Hinf_norm}
+   \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
+\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 \(H_2(s)\):
+\begin{equation}
+  H_1(s) = 1 - H_2(s)
+\end{equation}
 
-In order to minimize the super sensor dynamical uncertainty
 
 \begin{figure}[htbp]
 \centering
@@ -251,14 +306,22 @@ In order to minimize the super sensor dynamical uncertainty
 \end{figure}
 
 \subsection{Example}
-\label{sec:orgba594da}
+\label{sec:orgfe98b6f}
+
 
 \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:orgc07eeab}
+\label{sec:org9114fff}
 \label{sec:optimal_robust_fusion}
 
 \subsection{Sensor Fusion Architecture}
-\label{sec:orgddd6d33}
+\label{sec:org7816cc1}
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/sensor_model_noise_uncertainty.pdf}
+\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
+\end{figure}
+
 
 \begin{figure}[htbp]
 \centering
@@ -267,10 +330,10 @@ In order to minimize the super sensor dynamical uncertainty
 \end{figure}
 
 \subsection{Synthesis Objective}
-\label{sec:org79824da}
+\label{sec:org39451fc}
 
 \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:org247ac1c}
+\label{sec:orga8ff805}
 
 \begin{figure}[htbp]
 \centering
@@ -279,30 +342,30 @@ In order to minimize the super sensor dynamical uncertainty
 \end{figure}
 
 \subsection{Example}
-\label{sec:org7af2158}
+\label{sec:orga353d87}
 
 \section{Experimental Validation}
-\label{sec:orgb54c59b}
+\label{sec:orgb00dce4}
 \label{sec:experimental_validation}
 
 \subsection{Experimental Setup}
-\label{sec:org40eadad}
+\label{sec:orgc725d26}
 
 \subsection{Sensor Noise and Dynamical Uncertainty}
-\label{sec:orgb81743f}
+\label{sec:org0b05001}
 
 \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
-\label{sec:orgb2232ac}
+\label{sec:org9c0559a}
 
 \subsection{Super Sensor Noise and Dynamical Uncertainty}
-\label{sec:orgd80a558}
+\label{sec:orgc629276}
 
 \section{Conclusion}
-\label{sec:org0da5eb6}
+\label{sec:orgdd3a6b6}
 \label{sec:conclusion}
 
 \section{Acknowledgment}
-\label{sec:orge5b9b80}
+\label{sec:orge958f77}
 
 \bibliography{ref}
 \end{document}