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

  • Authors:

  • Christopher K. King;Wynita M. Griggs;Robert N. Shorten

  • Affiliations:

  • Department of Mathematics, Northeastern University, Boston, MA 02115, USA;Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland;Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2011

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Abstract

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 R^n. 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 R^n, with the transfer function condition reducing to the condition of Strict Positive Realness.