Stabilizing quantized feedback with minimal information flow: the scalar case

A state feedback with finitely many quantization levels yields only the so called practical stabilization, namely the convergence of any initial state belonging to a bigger bounded region into another smaller target region of the state space. The ratio between the measure of the starting region and the target region is called contraction of the closed loop system. In the analysis of the performance of a stabilization strategy based on a quantized state feedback two parameters play a central role: the number of quantization levels used by the feedback and the convergence time of the closed loop system. In this paper we propose a definition of optimality for a quantized stabilization strategy. This definition is based on how the number of quantization levels and the convergence time grow with the contraction. Then, we analyze the performance and prove the optimality of three different stabilizing quantized feedbacks strategy for scalar linear systems.