A new insight on state-trimness and minimal state-space realizations of input-output behaviors

Abstract

In this paper we study the property of state-trimness introduced in~\cite{RapiWill97}, 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 $\mathcal T$ of a non state-trim state-space system $\Sigma$ and prove that the restriction of $\Sigma$ to $\mathcal T$ 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.