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

    King, Christopher K. and Griggs, Wynita M. and Shorten, Robert N. (2011) A Kalman-Yakubovich-Popov-type lemma for systems with certain state-dependent constraints. Automatica, 47 (9). pp. 2107-2111. ISSN 0005-1098

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    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.

    Item Type: Article
    Additional Information: Preprint version of original published article. The definitive version of this article is available at
    Keywords: Kalman-Yakubovich-Popov lemma; nonlinear systems; switched systems; Lyapunov function; state space; state-dependent constraints; convex cone; frequency domain inequality; linear matrix inequality;
    Academic Unit: Faculty of Science and Engineering > Research Institutes > Hamilton Institute
    Item ID: 3602
    Depositing User: Dr. Robert Shorten
    Date Deposited: 25 Apr 2012 15:22
    Journal or Publication Title: Automatica
    Publisher: Elsevier
    Refereed: No

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