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Add some text about H2/Hinf synthesis

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Thomas Dehaeze 2020-09-23 15:33:27 +02:00
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@ -294,7 +294,9 @@ To define what by "small" we mean, we use a weighting filter $W_u(s)$ such that
\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:
# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors
This is actually almost equivalent as to have (within a factor $\sqrt{2}$):
\begin{equation}
\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
\end{equation}
@ -330,28 +332,79 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
** 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
** Sensor with noise and model uncertainty
We wish now to combine the two previous synthesis, that is to say
The sensors are now modelled by a white noise with unitary PSD $\tilde{n}_i$ shaped by a LTI transfer function $N_i(s)$.
The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
\begin{equation}
v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
\end{equation}
Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate $\hat{x}_i$ of $x$:
\begin{equation}
\hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
\end{equation}
#+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]]
** Sensor Fusion Architecture
For reason of space, the blocks $\hat{G}_i$ and $\hat{G}_i^{-1}$ are omitted.
\begin{equation}
\begin{aligned}
\hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
\end{aligned}
\end{equation}
The estimate $\hat{x}$ of $x$
#+name: fig:sensor_fusion_arch_full
#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
#+attr_latex: :scale 1
[[file:figs/sensor_fusion_arch_full.pdf]]
** Synthesis Objective
** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
The synthesis objective is to generate two complementary filters $H_1(s)$ and $H_2(s)$ such that the uncertainty associated with the super sensor is kept reasonably small and such that the RMS value of super sensors noise is minimized.
To specify how small we want the super sensor dynamic spread, we use a weighting filter $W_u(s)$ as was done in Section ref:sec:robust_fusion.
This synthesis problem can be solved using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis on the following generalized plant:
\begin{equation}
\begin{pmatrix}
z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
\end{pmatrix} = \underbrace{\begin{bmatrix}
W_u W_1 & W_u W_1 \\
0 & W_u W_2 \\
N_1 & N_1 \\
0 & N_2 \\
1 & 0
\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
w \\ u
\end{pmatrix}
\end{equation}
The synthesis objective is to:
- Keep the $\mathcal{H}_\infty$ norm from $w$ to $(z_{\infty,1}, z_{\infty,2})$ below $1$
- Minimize the $\mathcal{H}_2$ norm from $w$ to $(z_{2,1}, z_{2,2})$
#+name: fig:mixed_h2_hinf_synthesis
#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :scale 1

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@ -1,4 +1,4 @@
% Created 2020-09-22 mar. 21:58
% Created 2020-09-23 mer. 14:15
% Intended LaTeX compiler: pdflatex
\documentclass[conference]{IEEEtran}
\usepackage[utf8]{inputenc}
@ -35,7 +35,7 @@
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\usepackage{showframe}
\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
\date{2020-09-22}
\date{2020-09-23}
\title{Optimal and Robust Sensor Fusion}
\begin{document}
@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{IEEEkeywords}
\section{Introduction}
\label{sec:orgc2fc7e2}
\label{sec:org88afd51}
\label{sec:introduction}
\begin{itemize}
@ -61,11 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{itemize}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:org2031a7c}
\label{sec:org5853545}
\label{sec:optimal_fusion}
\subsection{Sensor Model}
\label{sec:org32da471}
\label{sec:org565ea86}
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 \(G_i(s)\).
@ -102,7 +102,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:orgf3af62a}
\label{sec:org1ae73e8}
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}).
@ -140,7 +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:orga39f54c}
\label{sec:orgb2e8dd6}
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
@ -154,7 +154,7 @@ 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:org536193f}
\label{sec:orga4cf5f1}
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 \(n\):
@ -198,17 +198,17 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure}
\subsection{Example}
\label{sec:orgd689dc3}
\label{sec:org74634c9}
\subsection{Robustness Problem}
\label{sec:orgc57d2ad}
\label{sec:org5fda5c1}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgeed5209}
\label{sec:orgc88050f}
\label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty}
\label{sec:org7f4d435}
\label{sec:orgb09aa5a}
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.
@ -228,7 +228,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:orgd4a5727}
\label{sec:org1d92a74}
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:
@ -253,7 +253,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
\end{figure}
\subsection{Super Sensor Dynamical Uncertainty}
\label{sec:org7eede13}
\label{sec:org81db1d8}
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}.
@ -269,18 +269,28 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
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}
\label{sec:org0e2a7a8}
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:
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 \\
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
@ -290,7 +300,7 @@ This problem can be dealt with an \(\mathcal{H}_\infty\) synthesis problem by co
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
\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 \(H_2(s)\):
@ -298,7 +308,6 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
H_1(s) = 1 - H_2(s)
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf}
@ -306,15 +315,15 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:orgfe98b6f}
\label{sec:org0122000}
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org9114fff}
\label{sec:orgdf5a196}
\label{sec:optimal_robust_fusion}
\subsection{Sensor Fusion Architecture}
\label{sec:org7816cc1}
\label{sec:orge16b510}
\begin{figure}[htbp]
\centering
@ -330,10 +339,10 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Synthesis Objective}
\label{sec:org39451fc}
\label{sec:orgb4b43b3}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orga8ff805}
\label{sec:orgb9b52ad}
\begin{figure}[htbp]
\centering
@ -342,30 +351,30 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:orga353d87}
\label{sec:orgc881f20}
\section{Experimental Validation}
\label{sec:orgb00dce4}
\label{sec:org05b79a0}
\label{sec:experimental_validation}
\subsection{Experimental Setup}
\label{sec:orgc725d26}
\label{sec:orgc3daf35}
\subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:org0b05001}
\label{sec:org26fedf6}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org9c0559a}
\label{sec:org72f2969}
\subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgc629276}
\label{sec:orgf66f78b}
\section{Conclusion}
\label{sec:orgdd3a6b6}
\label{sec:orge0f0a43}
\label{sec:conclusion}
\section{Acknowledgment}
\label{sec:orge958f77}
\label{sec:orgb16559e}
\bibliography{ref}
\end{document}