Simultaneous balancing and model reduction of switched linear systems

In this paper, first, balanced truncation of linear systems is revisited. Then, simultaneous balancing of multiple linear systems is investigated. Necessary and sufficient conditions are introduced to identify the case where simultaneous balancing is possible. The validity of these conditions is not limited to a certain type of balancing, and they are applicable for different types of balancing corresponding to different equations, like Lyapunov or Riccati equations. The results obtained are used for model order reduction of switched linear systems (SLS) by simultaneous balanced truncation. Finally, we give conditions under which global uniform exponential stability is preserved after simultaneous balanced truncation of the original switched linear system.