Add some text about H2/Hinf synthesis
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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
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\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
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\end{equation}
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This is actually almost equivalent (to within a factor $\sqrt{2}$) equivalent as to have:
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# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors
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This is actually almost equivalent as to have (within a factor $\sqrt{2}$):
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\begin{equation}
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\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
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\end{equation}
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@ -330,28 +332,79 @@ The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$
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** Example
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# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors
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* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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<<sec:optimal_robust_fusion>>
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** Sensor Fusion Architecture
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** Sensor with noise and model uncertainty
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We wish now to combine the two previous synthesis, that is to say
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The sensors are now modelled by a white noise with unitary PSD $\tilde{n}_i$ shaped by a LTI transfer function $N_i(s)$.
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The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
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\begin{equation}
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v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
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\end{equation}
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Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate $\hat{x}_i$ of $x$:
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\begin{equation}
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\hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
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\end{equation}
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#+name: fig:sensor_model_noise_uncertainty
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#+caption: Sensor Model including Noise and Dynamical Uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_model_noise_uncertainty.pdf]]
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** Sensor Fusion Architecture
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For reason of space, the blocks $\hat{G}_i$ and $\hat{G}_i^{-1}$ are omitted.
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\begin{equation}
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\begin{aligned}
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\hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
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&+ \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
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
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&+ \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
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\end{aligned}
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\end{equation}
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The estimate $\hat{x}$ of $x$
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#+name: fig:sensor_fusion_arch_full
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#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_full.pdf]]
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** Synthesis Objective
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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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.
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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.
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This synthesis problem can be solved using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis on the following generalized plant:
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\begin{equation}
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\begin{pmatrix}
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z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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W_u W_1 & W_u W_1 \\
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0 & W_u W_2 \\
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N_1 & N_1 \\
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0 & N_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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The synthesis objective is to:
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- Keep the $\mathcal{H}_\infty$ norm from $w$ to $(z_{\infty,1}, z_{\infty,2})$ below $1$
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- Minimize the $\mathcal{H}_2$ norm from $w$ to $(z_{2,1}, z_{2,2})$
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#+name: fig:mixed_h2_hinf_synthesis
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#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
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#+attr_latex: :scale 1
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BIN
paper/paper.pdf
BIN
paper/paper.pdf
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@ -1,4 +1,4 @@
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% Created 2020-09-22 mar. 21:58
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% Created 2020-09-23 mer. 14:15
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% Intended LaTeX compiler: pdflatex
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\documentclass[conference]{IEEEtran}
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\usepackage[utf8]{inputenc}
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@ -35,7 +35,7 @@
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\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
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\usepackage{showframe}
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\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 }}
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\date{2020-09-22}
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\date{2020-09-23}
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\title{Optimal and Robust Sensor Fusion}
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\begin{document}
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@ -50,7 +50,7 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{IEEEkeywords}
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\section{Introduction}
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\label{sec:orgc2fc7e2}
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\label{sec:org88afd51}
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\label{sec:introduction}
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\begin{itemize}
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@ -61,11 +61,11 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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\end{itemize}
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\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
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\label{sec:org2031a7c}
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\label{sec:org5853545}
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\label{sec:optimal_fusion}
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\subsection{Sensor Model}
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\label{sec:org32da471}
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\label{sec:org565ea86}
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Let's consider a sensor measuring a physical quantity \(x\) (Figure \ref{fig:sensor_model_noise}).
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The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function \(G_i(s)\).
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@ -102,7 +102,7 @@ In order to obtain an estimate \(\hat{x}_i\) of \(x\), a model \(\hat{G}_i\) of
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgf3af62a}
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\label{sec:org1ae73e8}
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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}).
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@ -140,7 +140,7 @@ In such case, the super sensor estimate \(\hat{x}\) is equal to \(x\) plus the n
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\end{equation}
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\subsection{Super Sensor Noise}
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\label{sec:orga39f54c}
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\label{sec:orgb2e8dd6}
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Let's note \(n\) the super sensor noise.
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\begin{equation}
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n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
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@ -154,7 +154,7 @@ As the noise of both sensors are considered to be uncorrelated, the PSD of the s
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It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
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\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
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\label{sec:org536193f}
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\label{sec:orga4cf5f1}
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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 \(\hat{x}\).
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And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
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@ -198,17 +198,17 @@ We then have that the \(\mathcal{H}_2\) synthesis applied on \(P_{\mathcal{H}_2}
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\end{figure}
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\subsection{Example}
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\label{sec:orgd689dc3}
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\label{sec:org74634c9}
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\subsection{Robustness Problem}
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\label{sec:orgc57d2ad}
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\label{sec:org5fda5c1}
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\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
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\label{sec:orgeed5209}
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\label{sec:orgc88050f}
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\label{sec:robust_fusion}
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\subsection{Representation of Sensor Dynamical Uncertainty}
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\label{sec:org7f4d435}
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\label{sec:orgb09aa5a}
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In Section \ref{sec:optimal_fusion}, the model \(\hat{G}_i(s)\) of the sensor was considered to be perfect.
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In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
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@ -228,7 +228,7 @@ The sensor can then be represented as shown in Figure \ref{fig:sensor_model_unce
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgd4a5727}
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\label{sec:org1d92a74}
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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.
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The super sensor estimate is then:
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@ -253,7 +253,7 @@ As \(H_1\) and \(H_2\) are complementary filters, we finally have:
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\end{figure}
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\subsection{Super Sensor Dynamical Uncertainty}
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\label{sec:org7eede13}
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\label{sec:org81db1d8}
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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}.
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@ -269,19 +269,29 @@ And we can see that the dynamical uncertainty of the super sensor is equal to th
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At frequencies where \(\left|W_i(j\omega)\right| > 1\) the uncertainty exceeds \(100\%\) and sensor fusion is impossible.
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\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
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\label{sec:org0b02610}
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\label{sec:org0e2a7a8}
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In order for the fusion to be ``robust'', meaning no phase drop will be induced in the super sensor dynamics,
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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.
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This problem can be dealt with an \(\mathcal{H}_\infty\) synthesis problem by considering the following generalized plant:
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To define what by ``small'' we mean, we use a weighting filter \(W_u(s)\) such that the synthesis objective is:
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\begin{equation}
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\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
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\end{equation}
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This is actually almost equivalent (to within a factor \(\sqrt{2}\)) equivalent as to have:
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\begin{equation}
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\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
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\end{equation}
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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}):
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\begin{equation}
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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W_1 & W_1 \\
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0 & W_2 \\
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1 & 0
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W_u W_1 & W_u W_1 \\
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0 & W_u W_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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@ -290,7 +300,7 @@ This problem can be dealt with an \(\mathcal{H}_\infty\) synthesis problem by co
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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:
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\begin{equation}
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\label{eq:Hinf_norm}
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\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
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\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
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\end{equation}
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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 \(H_2(s)\):
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@ -298,7 +308,6 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
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H_1(s) = 1 - H_2(s)
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\end{equation}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/h_infinity_robust_fusion.pdf}
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@ -306,15 +315,15 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
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\end{figure}
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\subsection{Example}
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\label{sec:orgfe98b6f}
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\label{sec:org0122000}
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\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:org9114fff}
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\label{sec:orgdf5a196}
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\label{sec:optimal_robust_fusion}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org7816cc1}
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\label{sec:orge16b510}
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\begin{figure}[htbp]
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\centering
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@ -330,10 +339,10 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
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\end{figure}
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\subsection{Synthesis Objective}
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\label{sec:org39451fc}
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\label{sec:orgb4b43b3}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:orga8ff805}
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\label{sec:orgb9b52ad}
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\begin{figure}[htbp]
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\centering
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@ -342,30 +351,30 @@ The \(\mathcal{H}_\infty\) norm of Eq. \eqref{eq:Hinf_norm} is equals to \(\sigm
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\end{figure}
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\subsection{Example}
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\label{sec:orga353d87}
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\label{sec:orgc881f20}
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\section{Experimental Validation}
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\label{sec:orgb00dce4}
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\label{sec:org05b79a0}
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\label{sec:experimental_validation}
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\subsection{Experimental Setup}
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\label{sec:orgc725d26}
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\label{sec:orgc3daf35}
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\subsection{Sensor Noise and Dynamical Uncertainty}
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\label{sec:org0b05001}
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\label{sec:org26fedf6}
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\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
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\label{sec:org9c0559a}
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\label{sec:org72f2969}
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\subsection{Super Sensor Noise and Dynamical Uncertainty}
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\label{sec:orgc629276}
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\label{sec:orgf66f78b}
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\section{Conclusion}
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\label{sec:orgdd3a6b6}
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\label{sec:orge0f0a43}
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\label{sec:conclusion}
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\section{Acknowledgment}
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\label{sec:orge958f77}
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\label{sec:orgb16559e}
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\bibliography{ref}
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\end{document}
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