The `stabilization´ of linear systems with quantized feedback

The author addresses the problem of stabilizing an unstable time-invariant discrete-time linear system by means of state feedback when the measurements of the state are quantized. It is found that there is no finite-memory control strategy which stabilizes the system in the traditional sense of making all closed-loop trajectories tend asymptotically to zero. If the system is not excessively unstable, however, one can implement feedback strategies which bring closed-loop trajectories arbitrarily close to zero for an arbitrarily long time. If the system is too unstable to apply these control laws, it is found that when ordinary linear feedback of quantized state measurements is applied, the resulting closed-loop system exhibits qualitative behavior which is chaotic. When the state is one-dimensional, a quantitative statistical analysis of the resulting closed-loop dynamics reveals the existence of an invariant measure on the state space which is absolutely continuous with respect to Lebesgue measure and with respect to which the closed-loop system is ergodic