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A complementarity approach for the computation of periodic oscillations in piecewise linear systems

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Abstract

Piecewise linear (PWL) systems can exhibit quite complex behaviours. In this paper, the complementarity framework is used for computing periodic steady-state trajectories belonging to linear time-invariant systems with PWL, possibly set-valued, feedback relations. The computation of the periodic solutions is formulated in terms of a mixed quadratic complementarity problem. Suitable anchor equations are used as problem constraints in order to determine the unknown period and to fix the phase of the steady-state oscillation. The accuracy of the complementarity problem solution is shown through numerical investigations of stable and unstable oscillations exhibited by practical PWL systems: a neural oscillator, a deadzone feedback system, a stick–slip system, a repressilator and a relay feedback system.

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Notes

  1. \(\text {vec}\{A\}\) denotes the column vector obtained by stacking in one column all the columns of the matrix A. By \(\text {vec}(c_1, c_2, \ldots ,c_n)\) we will denote the column obtained by stacking all the \(c_i\) columns, even when they have a different number of components.

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Authors and Affiliations

  1. Department of Engineering, State University of Rio de Janeiro (UERJ), R. Sao Francisco Xavier, 524, Maracana, Rio de Janeiro, Brazil

    Valentina Sessa

  2. Department of Engineering, University of Sannio, Piazza Roma 21, 82100, Benevento, Italy

    Luigi Iannelli & Francesco Vasca

  3. INRIA, Bipop Team-Project, Inovallee de Montbonnot, 655 Avenue, Europe, 38334, Saint Ismier Cedex, France

    Vincent Acary

Authors
  1. Valentina Sessa
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  2. Luigi Iannelli
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  3. Francesco Vasca
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  4. Vincent Acary
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Correspondence to Valentina Sessa.

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Sessa, V., Iannelli, L., Vasca, F. et al. A complementarity approach for the computation of periodic oscillations in piecewise linear systems. Nonlinear Dyn 85, 1255–1273 (2016). https://doi.org/10.1007/s11071-016-2758-5

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  • DOI: https://doi.org/10.1007/s11071-016-2758-5

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