文档介绍：Mathematical Control and System Theory
of Stochastic Systems in Discrete-Time
Jan H. van Schuppen
CWI (Centre for Mathematics puter Science)
. Box 94079, 1090 GB Amsterdam, herlands
Version July 15, 2009
c . van Schuppen, Amsterdam, March 2008
Please do not quote these notes without written permission of the author.
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Contents
1 Introduction to control of stochastic systems 11
2 Stochastic Systems 13
Modelling of signals as outputs of stochastic systems . . . . . 13
Definition of a stochastic system . . . . . . . . . . . . . . . . 16
Gaussian system representations in state space form . . . . . . 24
Stochastic systems with invariant measures . . . . . . . . . . 29
Forward and backward representations of
Gaussian systems . . . . . . . . . . . . . . . . . . . . . . . . 34
Other models of stationary Gaussian processes . . . . . . . . . 39
Finite stochastic systems . . . . . . . . . . . . . . . . . . . . 44
Other classes of stochastic systems . . . . . . . . . . . . . . . 47
σ-Algebraic stochastic systems . . . . . . . . . . . . . . . . . 49
Factor models and dynamic factor systems . . . . . . . . . . . 49
Stochastic multiparameter systems . . . . . . . . . . . . . . . 52
Stochastic stability . . . . . . . . . . . . . . . . . . . . . . . 53
Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Stochastic Realization 59
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Problem formulation . . . . . . . . . . . . . . . . . . . . . . 62
Stochastic observability and stochastic reconstructibility . . . 64
Weak Gaussian stochastic realization problem . . . . . . . . . 69
Realization algorithms . . . . . . . . . . . . . . . . . . . . . 77
Special Gaussian stochastic realizations . . . . . . . . . . . . 80
Strong Gaus