Robust Stability of Uncertain Discrete-time Linear Systems with Input and Output Quantization

This paper investigates the robust stability of uncertain discrete-time linear systems with both input and output quantization. Specifically, the output of the plant and the output of the dynamic controller are quantized via two independent static logarithmic quantizers. In fact, there are three blocks of uncertainties under consideration due to the double quantization and uncertain plant. First, a necessary and sufficient condition in terms of LMIs is proposed for the quadratic stability of the closed-loop system with double quantization and norm bounded uncertainty in the plant. Moreover, it is shown that the proposed condition can be exploited to derive the coarsest logarithmic quantization density under which the uncertain plant can be quadratically stabilized via quantized state feedback. Lastly, a new class of Lyapunov function which depends on the quantization errors in a multilinear way is developed to obtain less conservative results.