Topological Entropy and Data Rate for Practical Stability: a Scalar Case

A special class of nonlinear systems, i.e. expanding piecewise affine discrete-time scalar systems with limited data rate, is used to investigate the role of topological entropy and date rate in a practical stability problem. This special class of nonlinear system as an abstract model is an extension of discrete-time unstable scalar systems, a well-known model for quantized feedback design. For such systems with finite quantization levels, how to design a quantized feedback controller to achieve practical stability is considered as a boundability problem. Unlike the existing results about topological entropy for nonlinear stabilization and optimal control for linear systems, for the boundability problem under consideration, the feedback topological entropy defined in this paper is not equal to the minimum number of the quantization interval (i.e. the minimal information rate) and only provides a necessary condition for the boundability of our system under some condition. A necessary and sufficient condition for the boundability is also presented in terms of an inequality related to data rate and the minimum number of quantization intervals.