A New Insight on State-Trimness and Minimal State-Space Realizations of Input-Output Behaviors

In this paper we study the property of state-trimness introduced in [4], which, together with observability, is a necessary and sufficient condition for the minimality of the state-space realizations of a given input-output behavior. More concretely, we show how to compute the trim subspace T of a non state-trim state-space system Sigma and prove that the restriction of Sigma to T is a state-space system with the same input-output behavior as Sigma. Combined with Kalman's observability decomposition, this allows obtaining minimal state-space realizations from non minimal ones.