Stability and Stabilizability of Discrete-time Structured Linear Systems

Abstract

This work presents a graph theoretic approach to the investigation of stability and stabilizability of discrete-time structured linear systems – i.e., discrete-time dynamical systems defined by linear maps whose entries are only known to be either zero or nonzero (unknown) values. The main result consists in a necessary and sufficient condition for each element of the family of systems represented by a given discrete-time structured linear system to be asymptotically stable. In particular, under the stated condition, convergence to zero of the free state evolution of each system of the family is shown to be achieved in a finite number of steps, through what will be referred to as a dead-beat behavior. The notions of essential state feedback and essential output injection are then introduced and a sufficient condition for stabilizability by essential state feedback and by essential output injection, respectively, is given. An obstruction to stabilizability by essential state feedback or by essential output injection, respectively, is also pointed out.