Moment stabilization over Markov channels

Abstract

The problem of stabilization of a discrete-time linear dynamical system over a Markov time-varying digital feedback channel is studied. We extend previous results for mean-square stabilization to m-th moment stabilization in the general case of systems with unbounded disturbances. Since the index m gives an estimate of the quality of the stability attainable, in the sense that large stabilization errors occur more rarely as m increases, one interpretation of our results is that in order to achieve stronger stability one needs to assume stricter conditions on the disturbances and on the quality of the communication channel. On the technical side, we provide a general lower bound on the norm of a random continuum vector using the differential entropy function and a tight condition for the m-th moment stability of an inhomogeneous Markov jump linear system. These tools could be useful to prove stabilization results for other system models.

Anytime Capacity of a Class of Markov Channels

“…In contrast, the subsampling technique used in the direct part of Theorem 1 is not suitable to prove strong moment stability because it cannot prevent undesirable fluctuations in the system dynamics between any two sampling periods. In [49] we presented a different proof technique based on the binomial expansion of (a k z k + w k ) m which can be used to prove that (6) is a sufficient condition for the strong moment stability of (1). This approach however requires m to be an integer number, as opposed to the subsampling technique which can be applied to any m > 0.…”