Guaranteed absolute stability and robustness of a class of delay systems with local sector nonlinearities via piecewise linear Lyapunov function

The problem of absolute stability with guaranteed domain of attraction is investigated for a class of delay systems with local sector constraints on their nonlinear parts, using piecewise linear Lyapunov functions along with mixed monotone decomposition of the systems. First, a Razumikhin type lemma is applied to obtain a necessary and sufficient condition for positive invariance of the domain of sector constraints with respect to the system under consideration. Then, by Lyapunov-Razumikhin method, this condition is further strengthened to ensure the attractiveness of the origin with respect to the domain so as to achieve the largest absolute stability domain. In general, the existence of a rectangular type of guaranteed stability domain is related to the well known M-matrix conditions. Robustness issues are also examined for system parameter uncertainties described by matrix polytopes and interval matrices, and vertex and extreme point results are obtained. Further extension of the study to general delayed MIMO Lur´e systems is briefly discussed. Finally, an illustrative example is given