Stabilization of linear systems with limited information

In this paper, we show that the coarsest, or least dense, quantizer that qu adratically 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 logarithm ic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback controllers, and quantized stat e estimators. This leads to the design of hybrid output feedback controller s. The theory is then extended to sampling and quantization of continuous t ime linear systems sampled at constant time intervals. We generalize the de finition of density of quantization to the density of sampling and quantiza tion in a natural way, and search for the coarsest sampling and quantizatio n scheme that ensures stability. We show that the resulting optimal samplin g time is only function of the sum of the unstable eigenvalues of the conti nuous time system, and that the associated optimal quantizer is logarithmic with the logarithmic base being a universal constant independent of the sy stem. 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 t o construct logarithmic quantizers with only finite number of quantization levels and still achieve practical stability of the closed-loop system. Thi s final result provides a way to practically implement the theory developed in this paper.