Cone-copositivity for absolute stability of Lur´e systems

The absolute stability problem of single-input single-output Lur´e systems with uncertain feedback belonging to a sector bounded by possibly asymmetric piecewise linear characteristics is considered. These characteristics determine a state space partition into slabs. Sign conditions of quadratic functions and quadratic forms constrained to the slabs are reformulated in terms of cone-constrained linear matrix inequalities. It is shown that these conditions can be solved by defining a suitable copositive programming. Copositivity is then exploited in order to determine a sufficient condition for the existence of a piecewise quadratic Lyapunov function which is used to get an absolute stability result for the Lur´e system. An example with asymmetric sector bounds shows the effectiveness of the the proposed approach.