On achievable behavior of stochastic descriptor systems

A theme widely studied in the control systems and discrete-event systems communities concerns the characterization of the behavior that is achievable by a plant system via interconnection with a controller system. Many results have been found for several classes of discrete, continuous and hybrid systems. However, to the best of our knowledge, current literature does not yet consider stochastic systems. In this paper we aim at filling this gap and address this control problem for the class of Stochastic Descriptor Systems (SDS): given a plant SDS and a desired stochastic behavior, expressed as a SDS, we study conditions under which there exists a SDS controller such that the interconnection between the plant and the controller yields a system having the same stochastic external behavior of the desired SDS. Necessary and sufficient conditions are derived in terms of stochastic behavioral inclusions. When such conditions are satisfied we show that the so-called canonical controller, obtained as the interconnection of the plant SDS and the desired SDS, provides a solution to the stochastic control problem considered.