In this paper, we show that the coarsest, or least dense, quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special linear quadratic regulator (LQR) problem. We provide a closed form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback controllers, and quantized state estimators. This leads to the design of hybrid output feedback controllers. The theory is then extended to sampling and quantization of continuous time linear systems sampled at constant time intervals. We generalize the definition of density of quantization to the density of sampling and quantization in a natural way, and search for the coarsest sampling and quantization scheme that ensures stability. We show that the resulting optimal sampling time is only function of the sum of the unstable eigenvalues of the continuous time system, and that the associated optimal quantizer is logarithmic with the logarithmic base being a universal constant independent of the system. The coarsest sampling and quantization scheme so obtained is related to the concept of minimal attention control recently introduced by Brockett. Finally, by relaxing the definition of quadratic stability, we show how to construct logarithmic quantizers with only finite number of quantization levels and still achieve practical stability of the closed-loop system. This final result provides a way to practically implement the theory developed in this paper.

No abstract

In this paper, we show a general equivalence between feedback stabilization through an analog communication channel, and a communication scheme based on feedback which is a generalization of that of Schalkwijk and Kailath. We also show that the achievable transmission rate of the scheme is given by the Bode's sensitivity integral formula, which characterizes a fundamental limitation of causal feedback. Therefore, we can now use the many results and design tools from control theory to design feedback communication schemes providing desired communication rates, and to generate lower bounds on the channel feedback capacity. We consider single user Gaussian channels with memory and memory-less multiuser broadcast, multiple access, and interference channels. In all cases, the results we obtain either achieve the feedback capacity, when this is known, recover known best rates, or provide new best achievable rates.Index Terms-Feedback capacity, fundamental limits on communication and control, networked systems.

No abstract

No abstract