Stability analysis and synthesis for scalar linear systems with a quantized feedback

It is quite well-known that control systems in which the feedback is quantized may exhibit the typical wild behavior of nonlinear systems. In the classical literature devoted to this problem usually the flow of information in the feedback is not considered as a critical constraint. Consequently, in this case it was natural in control synthesis to simply approximate the non-quantized feedback that the classical methods provided with the quantized one. If, on the other hand, the flow of information has to be limited, for instance because of the use of a limited capacity transmission channel, then some specific considerations are in order. The aim of the paper is to obtain a detailed analysis of linear systems with quantized feedback in the simple scalar case. In the scalar case a rather complete analysis is possible through a geometric characterization of asymptotically stable closed loop maps. Moreover a lower bound on the number of quantization levels is obtained in this case, showing that the logarithmic connection between the number of quantization levels and the rate of convergence is quite intrinsic in this approach