Stability of discrete-time systems with quantized input and state measurements

This note focuses on linear discrete-time systems controlled using a quantized input computed from quantized measurements. Nominally stabilizing, but otherwise arbitrary, state feedback gains could result in limit cycling or nonzero equilibrium points. Although a single quantizer is a sector nonlinearity, the presence of a quantizer at each state measurement channel makes traditional absolute stability theory not applicable in a direct way. A global asymptotic stability condition is obtained by means of a result which allows us to apply discrete positive real theory to systems with a sector nonlinearity which is multiplicatively perturbed by a bounded function of the state. The stability result is readily applicable by evaluating the location of the polar plot of a system transfer function relative to a vertical line whose abcissa depends on the one-norm of the feedback gain. A graphical method is also described that can be used to determine the equilibrium points of the closed-loop system for any given feedback gain.