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Unformatted text preview: OptimizationBased Control Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT v2.1a, January 3, 2010 c circlecopyrt California Institute of Technology All rights reserved. This manuscript is for review purposes only and may not be reproduced, in whole or in part, without written consent from the author. Chapter 6 Sensor Fusion In this chapter we consider the problem of combining the data from different sensors to obtain an estimate of a (common) dynamical system. Unlike the previous chap ters, we focus here on discretetime processes, leaving the continuoustime case to the exercises. We begin with a summary of the input/output properties of discrete time systems with stochastic inputs, then present the discretetime Kalman filter, and use that formalism to formulate and present solutions for the sensor fusion problem. Some advanced methods of estimation and fusion are also summarized at the end of the chapter that demonstrate how to move beyond the linear, Gaussian process assumptions. Prerequisites. The material in this chapter is designed to be reasonably selfcontained, so that it can be used without covering Sections ?? – ?? or Chapter ?? of this sup plement. We assume rudimentary familiarity with discretetime linear systems,, at the level of the brief descriptions in Chapters 2 and 5 of ˚ AM08, and discretetime random processes as described in Section ?? of these notes. 6.1 DiscreteTime Stochastic Systems We begin with a concise overview of stochastic system in discrete time, echoing our development of continuoustime random systems described in Chapter ?? . We consider systems of the form X [ k + 1] = AX [ k ] + Bu [ k ] + FW [ k ] , Y [ k ] = CX [ k ] + V [ k ] , (6.1) where X ∈ R n represents the state, u ∈ R m represents the (deterministic) input, W ∈ R q represents process disturbances, Y ∈ R p represents the system output and W ∈ R p represents measurement noise. As in the case of continuoustime systems, we are interested in the response of the system to the random input W [ k ]. We will assume that W is a Gaussian process with zero mean and correlation function ρ W ( k,k + d ) (or correlation matrix R W ( k,k + d ) if W is vector valued). As in the continuous case, we say that a random process is white noise if R W ( k,k + d ) = R W δ ( d ) with δ ( d ) = 1 if d = 0 and 0 otherwise. (Note that in the discretetime case, white noise has finite covariance.) To compute the response Y [ k ] of the system, we look at the properties of the state vector X [ k ]. For simplicity, we take u = 0 (since the system is linear, we can always add it back in by superposition). Note first that the state at time k + l can 6.2. KALMAN FILTERS IN DISCRETE TIME (AM08) 62 be written as X [ k + l ] = AX [ k + l − 1] + FW [ x + l − 1] = A ( AX [ k + l − 2] + FW [ x + l − 2]) + FW [ x + l − 1] = A l X [ k ] + l summationdisplay j =1 A j 1 FW [ k + l − j ] ....
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This note was uploaded on 01/04/2012 for the course CDS 110b taught by Professor Murray,r during the Fall '08 term at Caltech.
 Fall '08
 Murray,R

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