Average Reachability of Continuous-time Markov Jump Linear Systems and the Linear Minimum Mean Square Estimator

Abstract

In this paper we study the average reachability gramian for continuous-time linear systems with additive noise and jump parameters driven by a general Markov chain. We define a rather natural reachability concept by requiring that the average reachability gramian be positive definite. Aiming at a testable condition, we introduce a set of reachability matrices for this class of systems and employ invariance properties of the null space of the noise coefficient matrices to show that the system is reachable if and only if these matrices are of full rank. We also show for reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear minimum mean square state estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, the average boundedness of the error covariance matrix is invariant to a type of perturbation in the noise model, meaning that the estimates are not overly sensitive, which consists in a property that is desirable in applications and sometimes referred to as stability of the estimator.