A Kalman–Yakubovich–Popov-type lemma for systems with certain state-dependent constraints

In this note, a result is presented that may be considered an extension of the classical Kalman–Yakubovich–Popov (KYP) lemma. Motivated by problems in the design of switched systems, we wish to infer the existence of a quadratic Lyapunov function (QLF) for a nonlinear system in the case where a matrix defining one system is a rank-1 perturbation of the other and where switching between the systems is orchestrated according to a conic partitioning of the state space Rⁿ. We show that a necessary and sufficient condition for the existence of a QLF reduces to checking a single constraint on a sum of transfer functions irrespective of problem dimension. Furthermore, we demonstrate that our conditions reduce to the classical KYP lemma when the conic partition of the state space is Rⁿ, with the transfer function condition reducing to the condition of Strict Positive Realness.