diff --git a/paper/paper.org b/paper/paper.org index 2957695..757e95d 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -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 <> -** 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 diff --git a/paper/paper.pdf b/paper/paper.pdf index eeacf96..bd9b677 100644 Binary files a/paper/paper.pdf and b/paper/paper.pdf differ diff --git a/paper/paper.tex b/paper/paper.tex index 31ce588..b2ce3d4 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -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}. @@ -269,19 +269,29 @@ 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 \\ - 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} @@ -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}