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.
Highlights
Piecewise linear (PWL) models can represent a wide class of practical systems and can exhibit interesting nonlinear behaviours such as periodic steady-state oscillations
We consider the problem of the computation of periodic solutions for linear timeinvariant dynamical systems whose inputs are related to the outputs through a static multi input–multi output PWL relation, possibly set-valued
That class of dynamical systems has attracted a considerable interest in the literature, and it allows to capture even complex behaviours arising in many practical systems
Summary
Piecewise linear (PWL) models can represent a wide class of practical systems and can exhibit interesting nonlinear behaviours such as periodic steady-state oscillations. A nonlinear oscillator is analysed by linearizing the system along the solution predicted by the harmonic balance technique and by computing the Floquet’s multipliers by using a time-domain numerical algorithm. In [18], the harmonic balance method is implemented together with the envelope following method in time domain Such approach is used to compute the steady-state behaviour and the associated period of nonlinear circuits forced by two input signals with different oscillation frequencies. We propose the use of mixed quadratic complementarity problems (MQCPs) for the computation of periodic solutions exhibited by PWL systems.
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