Balanced realizations of regime-switching linear systems

In this work, we establish a framework for balanced realization of linear systems subject to regime switching modulated by a continuous-time Markov chain with a finite state space. First, a definition of balanced realization is given. Then a rho-balanced realization is developed to approximate the system of balancing equations, which is a system of time-varying algebraic equations. When the state space of the Markov chain is large, the computational effort becomes a real concern. To resolve this problem, we introduce a two-time-scale formulation and use decomposition/aggregation and averaging techniques to reduce the computational complexity. Based on the two-time-scale formulation, further approximation procedures are developed. Numerical examples are also presented for demonstration.