Stabilizing a linear system with quantized state feedback

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 is addressed. It is found that there is no control strategy that stabilizes the system in the traditional sense of making all closed-loop trajectories asymptotic to zero. If the system is not excessively unstable, feedback strategies that bring closed-loop trajectories arbitrarily close to zero for a long time can be implemented. It is also found that when the ordinary linear feedback of quantized state measurements is applied, the resulting closed-loop system behaves chaotically. The asymptotic pseudorandom closed-loop system dynamics differ substantially from what would be predicted by a conventional signal-with-noise analysis of the quantization´s effects. Probabilistic reformulations of the stability problem in terms of the invariant measure are considered