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 performance 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.