Minimal Realization Problems in Jump Linear System: Some Preliminaries

We address two fundamental problems in the context of jump linear systems (JLS). The first problem is concerned with characterizing the minimal state space dimension solely from input-output pairs and without any knowledge of the number of mode switches. The second problem is concerned with characterizing the number of discrete modes of the JLS. For the first problem, we develop a linear system theory-based approach and construct an appropriate Hankel--like matrix. The rank of this matrix gives us the state space dimension. For the second problem, we show that the minimal number of modes corresponds to the minimal rank of a positive semi-definite matrix obtained via a non--convex formulation.