Balanced Approximation of Stochastic Systems

Abstract

The state of a linear system is an information interface between past inputs and future outputs, and system approximation (even identification) is essentially a problem of approximating a large-dimensional interface by a low-order partial state. Balanced Model Reduction [IEEE Trans. Automat. Control , 26 (1981), pp. 17–31], the Fujishige–Nagai–Sawaragi Model Reduction Algorithm [Internal. J. Control, 22 (1975), pp. 807–819], and the Principal Hankel Components Algorithm for system identification [Proc. 12th Asilomar Conference on Circuits Systems and Computers, Pacific Grove, CA, November 1978] approximate this input-output interface by its principal components. First generalizations of balanced model reduction to the stochastic system approximation problem are presented. Then the ideas of principal components to the problem of approximating the information interface between two random vectors are generalized; this leads to three approximate stochastic realization methods based on singular value decomposition. These methods and their relationship to the different kinds of balanced stochastic model reduction are discussed.