On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

  • Authors:

  • P. Pepe;I. Karafyllis;Z. -P. Jiang

  • Affiliations:

  • Dipartimento di Ingegneria Elettrica e dell'Informazione, Universití degli Studi dell'Aquila, Monteluco di Roio, 67040 L'Aquila, Italy;Department of Environmental Engineering, Technical University of Crete, 73100 Chania, Greece;Department of Electrical and Computer Engineering, Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2008

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Abstract

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale's form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale's form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.