Stabilization of LTI Systems with Quantized State-Quantized Input Static Feedback

This paper is concerned with the stabilizability problem for discrete-time linear systems subject to a uniform quantization of the control set and to a regular state quantization, both fixed a priori. As it is well known, for quantized systems only weak (practical) stability properties can be achieved. Therefore, we focus on the existence and construction of quantized controllers capable of steering a system to within invariant neighborhoods of the equilibrium. We first consider uniformly quantized, unbounded input sets for which an increasing family of invariant sets is constructed and quantized controllers realizing invariance are characterized. The family contains a minimal set depending only on the quantization resolution. The analysis is then extended to cases where the control set is bounded: for any given state-space set of the family above, the minimal diameter of the control set which ensures its invariance is found. The finite control set so determined also guarantees that all the states of the set can be controlled in finite time to within the family's minimal set. It is noteworthy that the same property holds for systems without state quantization: hence, to ensure invariance and attractivity properties, the necessary control set diameter is invariant with state quantization; yet the minimal invariant set is larger. An example is finally reported to illustrate the above results.