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Rhodes_paper

# Rhodes_paper - 688 ~ E E E RANSACTIONS T O N A TJTON.~IC...

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688 ~ E E E TRANSACTIONS ON ATJTON.~IC CONTROL, VOL. AC-16, NO. 6, DECEMBER 1971 A Tutorial Introduction to Estimation and Filtering Absiracf-In this tutorial paper the basic principles of least squares estimation are introduced and applied to the solution of some filtering, prediction, and smoothing problems involving stochastic linear dynamic systems. In particular, the paper includes derivations of the discrete-time and Continuous-time Kalman aters and their prediction and smoothing counterparts, with remarks on the modiii- cations that are necessary if the noise processes are colored and cor- related. The examination of these state estimation problems is pre- ceded by a derivation of both the unconstrained and the linear least squares estimator of one random vector in terms of another, and an examination of the properties of each, with particular attention to the case of jointly Gaussian vectors. The paper concludes with a discus- sion of the duality between least squares estimation problems and least squares optimal control problems. T I. INTRODUCTION HIS paper contains a tutorial introduction to t,he basic principles of least squares estimat.ion and their application to the solution of some stat.e-est.imation prob- lems associated wit.h finite-dimensional linear dynamic systemsoperatingina stochast.ic environment.. The ex- position begins n-it11 t,he problem of cstimat.ing on(: random variable or vector in t,erms of anot>her, and proceeds throughaderivation of the discrete-time Kalman filter a.nd predict.or to a derivat.ion of the continuous-t.ime Ihl- man-Bucy filter and predictor, and an examination of t.he continuous-t,ime smoothing problem. The paper concludes vit.11 a discussion of t,he duality bet.wen least squares esti- mation and least squares opt.ima1 control problems. The development, in each section dram nontrivially on that in preceding semions and on the following a.ssumed prerequisites. 1) Familiarity v,-it,h the elements of probability t.heory through t,he concept of jointly distributed random vari- ables and random vectors described by their joint. prob- . ability density function, and t,he associated means, co- . variances, and condhional expectations. For a. discussion of these topics, see, for example, [l]. 2) Beginning xvith Section VII, a.n exposure t.o t.he state-space description of linear dynamic systems, t.he dual concepts of controllability and observability, and t,he least. squares (“linear-quadrat.ic”) regulator problem in both finite and infinite t.ime. For an exposition of these t.opics the reader is referred t.o the appropriate papers in this issue or, for example, t.o [2]. In Section 11, we esanune t.he least. squares estimation of one random vector in t.erms of anot.her. Some important, 3Iannscript received July 19, 19il. Paper recommended by D. G.

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