Balanced Truncation for a Class of Stochastic Jump Linear Systems and Model Reduction for Hidden Markov Models

This paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete-time stochastic jump linear systems. The approximation error, which is captured by means of the stochastic L ₂ gain, is bounded from above by twice the sum of singular numbers associated to the truncated states, similar to the case of linear time-invariant systems. A two step model reduction algorithm for hidden Markov models is also developed. The first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden Markov models and a subclass of stochastic jump linear systems. In a second step the positivity constraints, which reflect the hidden Markov model structure, are enforced by solving a low dimensional optimization problem.