Construction of Periodic Counterexamples to the Discrete-Time Kalman Conjecture

Peter Seiler, Joaquin Carrasco

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This letter considers the Lurye system of a discrete-time, linear time-invariant plant in negative feedback with a nonlinearity. Both monotone and slope-restricted nonlinearities are considered. The main result is a procedure to construct destabilizing nonlinearities for the Lurye system. If the plant satisfies a certain phase condition then a monotone nonlinearity can be constructed so that the Lurye system has a non-trivial periodic cycle. Several examples are provided to demonstrate the construction. This represents a contribution for absolute stability analysis since the constructed nonlinearity provides a less conservative upper bound than existing bounds in the literature.
Original languageEnglish
Pages (from-to)1291-1296
Number of pages6
JournalIEEE Control Systems Letters
Issue number4
Publication statusPublished - 23 Oct 2021


  • Stability of nonlinear systems
  • absolute stability


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