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Stabilizing a linear system with quantized state feedback. (English) Zbl 0719.93067
Summary: This paper 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 admissible control strategy that stabilizes the system in the traditional sense of making all closed-loop trajectories tend asymptotically to zero. Still, if the system is not excessively unstable, one can implement feedback strategies that bring closed-loop trajectories arbitrarily close to zero for an arbitrarily long time. It is found that when ordinary “linear” feedback of quantized state measurements is applied, the resulting closed-loop system behaves chaotically. When the state is one- dimensional, a quantitative statistical analysis of the resulting closed- loop dynamics reveals the existence of an invariant probability measure on the state space that is absolutely continuous with respect to Lebesgue measure and with respect to which the closed-loop system is ergodic. The asymptotically pseudorandom closed-loop system dynamics differ substantially from what would be predicted by a conventional signal-plus- noise analysis of the quantization’s effect. Probabilistic reformulations of the stabilization problem in terms of the invariant measure are then considered.

93D15Stabilization of systems by feedback
93C55Discrete-time control systems
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