Absolute stability and guaranteed domain of attraction for MIMO discrete-time Lur´e systems

The problem of absolute stability with a guaranteed domain of attraction is studied for a class of MIMO discrete-time Lur´e systems with local sector nonlinearities by means of piecewise-linear Lyapunov functions along with mixed monotone decomposition of the systems. A sufficient condition is presented which ensures that the domain of attraction coincides with the domain of the sector constraints, so as to achieve the largest absolute stability domain. Particular cases where necessary and sufficient results are available are also examined. Moreover, the existence of positively invariant rectangular sets for the system under consideration is related to well-known M-matrix conditions. Such sets may provide invariant estimates of the guaranteed domain of attraction. The robustness issue is further discussed for system parameter uncertainties described by matrix polytopes, and vertex results are obtained. The main result is illustrated with an example.