diff --git a/paper/paper.org b/paper/paper.org index 4d82d0d..c649de2 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -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 <> @@ -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 <> @@ -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 <> diff --git a/paper/paper.pdf b/paper/paper.pdf index bd9b677..6889ae1 100644 Binary files a/paper/paper.pdf and b/paper/paper.pdf differ diff --git a/paper/paper.tex b/paper/paper.tex index b2ce3d4..9e766a2 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -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}