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2020-10-05 15:42:08 +02:00
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@@ -221,13 +221,46 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g
** Example
#+name: fig:figure_name
#+caption: Figure caption
#+name: fig:sensors_nominal_dynamics
#+caption: Sensor nominal dynamics from the velocity of the object to the output voltage
#+attr_latex: :scale 1
[[file:figs/sensors_nominal_dynamics.pdf]]
#+name: fig:sensors_noise
#+caption: Amplitude spectral density of the sensors $\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|$
#+attr_latex: :scale 1
[[file:figs/sensors_noise.pdf]]
#+name: fig:htwo_comp_filters
#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
#+attr_latex: :scale 1
[[file:figs/htwo_comp_filters.pdf]]
#+name: fig:psd_sensors_htwo_synthesis
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
#+attr_latex: :scale 1
[[file:figs/psd_sensors_htwo_synthesis.pdf]]
#+name: fig:super_sensor_time_domain_h2
#+caption: Noise of individual sensors and noise of the super sensor
#+attr_latex: :scale 1
[[file:figs/super_sensor_time_domain_h2.pdf]]
** Robustness Problem
#+name: fig:sensors_nominal_dynamics_and_uncertainty
#+caption: Nominal Sensor Dynamics $\hat{G}_i$ (solid lines) as well as the spread of the dynamical uncertainty (background color)
#+attr_latex: :scale 1
[[file:figs/sensors_nominal_dynamics_and_uncertainty.pdf]]
#+name: fig:super_sensor_dynamical_uncertainty_H2
#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
#+attr_latex: :scale 1
[[file:figs/super_sensor_dynamical_uncertainty_H2.pdf]]
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
<<sec:robust_fusion>>
@@ -334,6 +367,33 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
** Example
#+name: fig:sensors_uncertainty_weights
#+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
#+attr_latex: :scale 1
[[file:figs/sensors_uncertainty_weights.pdf]]
#+name: fig:weight_uncertainty_bounds_Wu
#+caption: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)
#+attr_latex: :scale 1
[[file:figs/weight_uncertainty_bounds_Wu.pdf]]
#+name: fig:hinf_comp_filters
#+caption: Obtained complementary filters using the $\mathcal{H}_\infty$ Synthesis
#+attr_latex: :scale 1
[[file:figs/hinf_comp_filters.pdf]]
#+name: fig:super_sensor_dynamical_uncertainty_Hinf
#+caption: Super sensor dynamical uncertainty (solid curve) when using the $\mathcal{H}_\infty$ Synthesis
#+attr_latex: :scale 1
[[file:figs/super_sensor_dynamical_uncertainty_Hinf.pdf]]
#+name: fig:psd_sensors_hinf_synthesis
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/psd_sensors_hinf_synthesis.pdf]]
* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
<<sec:optimal_robust_fusion>>
@@ -414,6 +474,26 @@ The synthesis objective is to:
** Example
#+name: fig:htwo_hinf_comp_filters
#+caption: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/htwo_hinf_comp_filters.pdf]]
#+name: fig:psd_sensors_htwo_hinf_synthesis
#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/psd_sensors_htwo_hinf_synthesis.pdf]]
#+name: fig:super_sensor_time_domain_h2_hinf
#+caption: Noise of individual sensors and noise of the super sensor
#+attr_latex: :scale 1
[[file:figs/super_sensor_time_domain_h2_hinf.pdf]]
#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf
#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
#+attr_latex: :scale 1
[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf]]
* Experimental Validation
<<sec:experimental_validation>>

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paper/paper.tex
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@@ -1,4 +1,4 @@
% Created 2020-09-23 mer. 14:15
% Created 2020-10-05 lun. 15:33
% 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{2020-09-23}
\date{2020-10-05}
\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:org88afd51}
\label{sec:org26a7400}
\label{sec:introduction}
\begin{itemize}
@@ -61,12 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{itemize}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:org5853545}
\label{sec:org49e80fd}
\label{sec:optimal_fusion}
\subsection{Sensor Model}
\label{sec:org565ea86}
\label{sec:org9555932}
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 \(G_i(s)\).
@@ -102,8 +101,7 @@ 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:org1ae73e8}
\label{sec:orga12ae12}
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.
@@ -140,7 +138,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
\end{equation}
\subsection{Super Sensor Noise}
\label{sec:orgb2e8dd6}
\label{sec:org924b750}
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 +152,7 @@ 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:orga4cf5f1}
\label{sec:org042a601}
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 \(n\):
@@ -170,8 +168,8 @@ This can be cast into an \(\mathcal{H}_2\) synthesis problem by considering the
\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
@@ -198,17 +196,62 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
\end{figure}
\subsection{Example}
\label{sec:org74634c9}
\label{sec:org98c54c2}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_nominal_dynamics.pdf}
\caption{\label{fig:sensors_nominal_dynamics}Sensor nominal dynamics from the velocity of the object to the output voltage}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_noise.pdf}
\caption{\label{fig:sensors_noise}Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/htwo_comp_filters.pdf}
\caption{\label{fig:htwo_comp_filters}Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/psd_sensors_htwo_synthesis.pdf}
\caption{\label{fig:psd_sensors_htwo_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2.pdf}
\caption{\label{fig:super_sensor_time_domain_h2}Noise of individual sensors and noise of the super sensor}
\end{figure}
\subsection{Robustness Problem}
\label{sec:org5fda5c1}
\label{sec:org81a0772}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_nominal_dynamics_and_uncertainty.pdf}
\caption{\label{fig:sensors_nominal_dynamics_and_uncertainty}Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_H2.pdf}
\caption{\label{fig:super_sensor_dynamical_uncertainty_H2}Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis}
\end{figure}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgc88050f}
\label{sec:org78ced60}
\label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty}
\label{sec:orgb09aa5a}
\label{sec:org9df3b01}
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 +271,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:org1d92a74}
\label{sec:orgf4531ff}
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:
@@ -253,8 +296,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
\end{figure}
\subsection{Super Sensor Dynamical Uncertainty}
\label{sec:org81db1d8}
\label{sec:orgf5bb33e}
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}.
@@ -269,7 +311,7 @@ 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:org0e2a7a8}
\label{sec:orgf07efa7}
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.
@@ -279,7 +321,7 @@ To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such t
\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:
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}
@@ -289,9 +331,9 @@ This problem can thus be dealt with an \(\mathcal{H}_\infty\) synthesis problem
\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}
@@ -315,15 +357,58 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:org0122000}
\label{sec:org0ca6ef9}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensors_uncertainty_weights.pdf}
\caption{\label{fig:sensors_uncertainty_weights}Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/weight_uncertainty_bounds_Wu.pdf}
\caption{\label{fig:weight_uncertainty_bounds_Wu}Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/hinf_comp_filters.pdf}
\caption{\label{fig:hinf_comp_filters}Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Hinf.pdf}
\caption{\label{fig:super_sensor_dynamical_uncertainty_Hinf}Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/psd_sensors_hinf_synthesis.pdf}
\caption{\label{fig:psd_sensors_hinf_synthesis}Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the \(\mathcal{H}_\infty\) synthesis}
\end{figure}
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgdf5a196}
\label{sec:orgf642e73}
\label{sec:optimal_robust_fusion}
\subsection{Sensor Fusion Architecture}
\label{sec:orge16b510}
\subsection{Sensor with noise and model uncertainty}
\label{sec:org8949812}
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}
\begin{figure}[htbp]
\centering
@@ -331,6 +416,26 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty}
\end{figure}
\subsection{Sensor Fusion Architecture}
\label{sec:orgcbc3d54}
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 \(x\)
\begin{figure}[htbp]
\centering
@@ -338,11 +443,34 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\caption{\label{fig:sensor_fusion_arch_full}Super Sensor Fusion with both sensor noise and sensor model uncertainty}
\end{figure}
\subsection{Synthesis Objective}
\label{sec:orgb4b43b3}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgb9b52ad}
\label{sec:org9d3f160}
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:
\begin{itemize}
\item Keep the \(\mathcal{H}_\infty\) norm from \(w\) to \((z_{\infty,1}, z_{\infty,2})\) below \(1\)
\item Minimize the \(\mathcal{H}_2\) norm from \(w\) to \((z_{2,1}, z_{2,2})\)
\end{itemize}
\begin{figure}[htbp]
\centering
@@ -351,30 +479,54 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
\end{figure}
\subsection{Example}
\label{sec:orgc881f20}
\label{sec:org85f304b}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/htwo_hinf_comp_filters.pdf}
\caption{\label{fig:htwo_hinf_comp_filters}Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/psd_sensors_htwo_hinf_synthesis.pdf}
\caption{\label{fig:psd_sensors_htwo_hinf_synthesis}Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_time_domain_h2_hinf.pdf}
\caption{\label{fig:super_sensor_time_domain_h2_hinf}Noise of individual sensors and noise of the super sensor}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf}
\caption{\label{fig:super_sensor_dynamical_uncertainty_Htwo_Hinf}Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\end{figure}
\section{Experimental Validation}
\label{sec:org05b79a0}
\label{sec:org49bf34a}
\label{sec:experimental_validation}
\subsection{Experimental Setup}
\label{sec:orgc3daf35}
\label{sec:orgdd8fce6}
\subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:org26fedf6}
\label{sec:org21add72}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:org72f2969}
\label{sec:org30521a3}
\subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgf66f78b}
\label{sec:org86cde79}
\section{Conclusion}
\label{sec:orge0f0a43}
\label{sec:org16245b7}
\label{sec:conclusion}
\section{Acknowledgment}
\label{sec:orgb16559e}
\label{sec:orgd992049}
\bibliography{ref}
\end{document}