Chaotic Quantized Feedback Stabilizers: The Scalar Case

Abstract

In this paper we consider practical stabilization strategies of scalar linear systems by means of quantized feedback maps which use a minimal number of quantization levels. These stabilization schemes are based on the chaotic properties of piecewise affine maps and their perfor- mance can be analyzed in terms of the mean time needed to shrink the system from an initial interval into a fixed target interval. We show here that this entrance time grows linearly with respect to the contraction rate defined as the quotient of the length of the initial and target interval respectively. Estimations are obtained using denumerable Markov chains arguments. 1. Introduction and problem statement. Control problems where the in- formation flow between the plant and the controller is an important feature to be considered in the design, have become very popular in the last few years. See (1, 2, 5, 6, 7, 12, 13, 14) and the reference therein. Indeed, information flow be- comes important in situations where a channel with limited information rate has to be used between the plant and the controller or when the controller needs to be simple and is only allowed to process a limited number of information per time. We expect that an information flow constraint will in general degrade the performances of the feedback loop scheme and we also expect in general a trade-off between performances and the amount of information exchange allowed in the loop. The specific problem we consider in this paper is the stabilization problem for a scalar linear system