tdehaeze/dehaeze20_optim_robus_compl_filte
1
0
Fork 0
Code Issues Pull Requests Releases Activity

Add test about Hinf synthesis

Browse Source
This commit is contained in:
Thomas Dehaeze 2020-09-22 21:58:37 +02:00
parent f091ca5411
commit 9a1379d4eb
3 changed files with 204 additions and 91 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

110
paper/paper.org
View File

@ -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}
\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
&= \left| N_i(j\omega) \right|^2
\end{aligned}
\begin{aligned}
\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
&= \left| N_i(j\omega) \right|^2
\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
\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}
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
\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}
\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
\end{split}
\begin{aligned}
\hat{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
& \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
&= \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
\end{equation}
\begin{equation}
\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
\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}
#+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

BIN
paper/paper.pdf
View File

Binary file not shown.

185
paper/paper.tex
View File

@ -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}
Section \ref{sec:robust_fusion}
Section \ref{sec:optimal_robust_fusion}
Section \ref{sec:experimental_validation}
\begin{itemize}
\item Section \ref{sec:optimal_fusion}
\item Section \ref{sec:robust_fusion}
\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}
\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
&= \left| N_i(j\omega) \right|^2
\end{aligned}
\begin{aligned}
\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
&= \left| N_i(j\omega) \right|^2
\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}
\caption{\label{fig:sensor_model}Sensor Model}
\includegraphics[scale=1]{figs/sensor_model_noise.pdf}
\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}
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 \(\hat{x}\).
\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 \(\hat{x}\).
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}
\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
\end{split}
\begin{aligned}
\hat{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
& \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
&= \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
\end{equation}
\begin{equation}
\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
\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{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}
\label{sec:org41ccb1e}
At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
\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}