Reduction for Structured Aggregated Markov Models Based on Reachable Space

The order of an aggregated Markov model (AMM) is an index of complexity and is closely related to the reachable subspace of a model. The AMM is called reachable-space reducible when the reachable subspace is not the whole space. Previous results demonstrate that there exists a reduced-order quasi-realization, which may not satisfy the nonnegative constraints, equivalent to a given reachable-space reducible AMM. This article focuses on the structured AMM where the transition and observation matrices have certain structured patterns. Sufficient conditions are derived for a structured AMM to be reachable-space reducible. Moreover, in this case, we show that a real reduced-order realization, instead of a quasi-realization, is obtained by choosing suitable bases for supersets of the reachable space. Finally, examples are given to support our results.