On the Kalman-Yakubovich-Popov Lemma for Positive Systems

The classical Kalman-Yakubovich-Popov lemma gives conditions for solvability of a certain inequality in terms of a symmetric matrix. The lemma has numerous applications in systems theory and control. Recently, it has been shown that for positive systems, important versions of the lemma can equivalently be stated in terms of a diagonal matrix rather than a general symmetric one. This paper generalizes these results and a new proof is given

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 IRn. 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 IRn, with the transfer function condition reducing to the condition of Strict Positive Realness

A stability criterion for systems with neutrally stable modes and deadzone nonlinearities

Stability analysis is considered for feedback interconnections of deadzone nonlinearities with linear systems that has a neutrally stable mode. Such systems do not have a unique equilibrium point and the standard techniques from passivity and Lyapunov theory cannot be applied. A stability criterion that generalizes the Popov criterion for this class of systems is derived in this report and several examples will prove its applicability