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

Christopher K. King, Wynita M. Griggs, Robert N. Shorten

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)


In this note, a result is presented that may be considered an extension of the classical KalmanYakubovichPopov (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.

Original languageEnglish
Pages (from-to)2107-2111
Number of pages5
Issue number9
Publication statusPublished - Sep 2011
Externally publishedYes


  • Convex cone
  • Frequency domain inequality
  • Kalman-Yakubovich-Popov lemma
  • Linear matrix inequality
  • Lyapunov function
  • Nonlinear systems
  • State space
  • State-dependent constraints
  • Switched systems

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