A symbolic dynamics approach to performance analysis of quantized feedback systems: the scalar case

When dealing with the control of a large number of interacting systems, the fact that the flow of information has to be limited becomes an essential feature of the control design. The first consequence of the limited information flow constraint is that the signals that the controllers and the systems exchange have to be quantized. Though quantization has already been extensively considered in the control literature, its analysis from the point of view of the information flow demand has been considered only recently. Limiting the information flow between a plant and a controller will necessarily lead to a performance degradation of the feedback loop, and we expect a trade-off between the achievable performance and the amount of information exchange allowed in the loop. Most of the success of modern digital communication theory in the last 50 years is due to the contributions of information theory, which proposed a symbolically based analysis of the communication channel performance. The same goal is much more difficult to reach in digital control theory. This paper proposes an attempt toward this direction. The main contribution of this paper is to provide a complete analysis of the trade-off between performance and information flow in the simple case of the stabilization of a scalar linear system by means of a memoryless quantized feedback map.