Hypercubes are minimal controlled invariants for discrete-time linear systems with quantized scalar input

Quantized linear systems are a widely studied class of nonlinear dynamics resulting from the control of a linear system through finite inputs. The stabilization problem for these models shall be studied in terms of the so called practical stability notion that essentially consists in confining the trajectories into sufficiently small neighborhoods of the equilibrium (ultimate boundedness) . In this paper, we study the problem of describing the smallest sets into which any feedback can ultimately confine the state, for a given linear single-input system with an assigned finite set of admissible input values (quantization). We show that a controlled invariant set of minimal size is contained in a family of sets (namely, hypercubes in controller-canonical form), previously introduced in [14, 15] . A comparison is presented which quantifies the improvement in tightness of the proposed analysis technique with respect to classical results using quadratic Lyapunov functions.