A realization approach to stochastic model reduction and balanced stochastic realizations

This paper considers the model reduction problem for discrete-time stochastic systems. Two approaches are presented. The first approach is based on viewing the model reduction problem as a reduced order stochastic realization problem. In this approach the state vector for the realization is picked form the canonical decomposition of the Hankel matrix which is obtained from the cross-covariance of the future with the past. Furthermore this choice provides a special ordering for the state vector. Using this ordering and a measure for the mutual information between the past and the future an approximation scheme is developed which leads to the new reduced order realization algorithm. Next the concept of balanced stochastic realization is developed. Using this notion the second approach for model reduction is obtained. In this approach a transformation is derived by appropriately factoring the solutions of algebraic Riccati equations. Use of this transformation then leads to the balanced stochastic realization Whose subsystem gives essentially the same reduced order model as that given by the first approach. Uniqueness and symmetry results for the balanced realization are given.