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2015 | OriginalPaper | Chapter

6. Linear Finite-Dimensional Stochastic Systems

Authors : Anders Lindquist, Giorgio Picci

Published in: Linear Stochastic Systems

Publisher: Springer Berlin Heidelberg

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Abstract

This chapter is an introduction to linear state-space modeling of second-order, wide-sense stationary, stochastic vector processes. In particular, we shall discuss modeling of discrete-time purely non deterministic processes with a rational spectral density matrix. These processes turn out to admit representations as the output y of a finite-dimensional linear system
$$\displaystyle{ \left \{\begin{array}{lcl} x(t + 1) &=& Ax(t) + Bw(t)\quad \\ y(t) &=& Cx(t) + Dw(t)\quad \end{array} \right. }$$
driven by a white noise input {w(t)}, where A, B, C and D are constant matrices of appropriate dimensions. These state-space descriptions provide a natural and useful class of parametrized stochastic models widely used in control and signal processing, leading to simple recursive estimation algorithms. Stochastic realization theory consists in characterizing and determining any such representation. This is in turn related to spectral factorization. The structure of these stochastic models is described in geometric terms based on coordinate-free representations and on elementary Hilbert space concepts.

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previous chapter Spectral Factorization in Continuous Time
next chapter The Geometry of Splitting Subspaces
Footnotes
1
Some background on deterministic state-space modeling is presented in Appendix A.
 
2
The direction of the arrows reflects anticausality; i.e., the fact that the future of \(\bar{w}\) is mapped into the past of y.
 
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Metadata
Title
Linear Finite-Dimensional Stochastic Systems
Authors
Anders Lindquist
Giorgio Picci
Copyright Year
2015
Publisher
Springer Berlin Heidelberg
Book
Linear Stochastic Systems

Print ISBN: 978-3-662-45749-8

Electronic ISBN: 978-3-662-45750-4

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-662-45750-4_6