Absolute stability of Lur´e systems: A complementarity and passivity approach

The stability problem of Lur´e systems composed by a dynamical linear time invariant system closed in feedback through a static piecewise linear mapping is considered. By exploiting complementarity representations of piecewise linear characteristics, generalized sector conditions expressed in terms of constrained relations are established. The proposed form is shown to be suitable for getting sector conditions of multi-input multi-output coupled characteristics as well as for representing classical conic sectors. Then, starting from the passivity assumption of the complementarity system representation, an operative result is obtained which allows to verify absolute stability of Lur´e systems with feedback constrained relations and possibly nonzero equilibrium. Some examples illustrate the effectiveness of the proposed approach.