King, Christopher K. and Griggs, Wynita M. and Shorten, Robert N.
A Kalman-Yakubovich-Popov-type lemma for systems with
certain state-dependent constraints.
Automatica, 47 (9).
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.
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