Stabilization of linear systems with limited information

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.