Chaotic Behavior of Digital Control Systems and its Continuous Dependence on Parameters

It has recently been demonstrated that controlling an unstable discrete-time linear system by means of a feedback control strategy that employs quantized measurements of the system´s state leads in many cases to chaotic asymptotic behavior on the part of the state. The chaotic asymptotics of such a system admit an economical description in terms of a probability density on the system´s state space that is preserved by the closed-loop dynamics and with respect to which the closed-loop system is ergodic. In the present paper, we extend these earlier results to apply to a broader class of systems than those we have previously considered; we find in particular that a suitably generalized invariant measure analysis goes through for essentially any scalar discrete-time unstable nonlinear system. We also present a preliminary investigation of the manner in which the quantitative features of the invariant density depend on the parameters of the system and the feedback strategy one employs. We discover that in many cases the invariant density is robust in the sense that it depends continuously on the parameters of the system and the feedback controller. In addition, we introduce a notion of statistical control in terms of which we are able to phrase and answer some important prescriptive questions about the systems under consideration.