Computing period and shape of oscillations in piecewise linear Lur'e systems: A complementarity approach

Abstract

e n tA c a r y ♭ ,B e r n a r dB r o g l i a t o ♭ and Francesco Vasca Abstract-Autonomous piecewise linear systems in the Lur'e form may exhibit periodic steady-state oscillations. For many practical systems belonging to this class the period and the shape of the oscillation is difficult to be predicted a priori. In this paper the complementarity approach is used to tackle the issue. The complementarity formalism is used to represent the closed-loop system and a phase condition acting as an anchor equation for the periodic solution. By discretizing the dynamics a mixed complementarity problem is formulated. The corresponding solution provides an accurate prediction of the steady-state oscillation and its period. Numerical results show the effectiveness of the proposed technique for the computation of stable and sliding periodic solutions. The analysis of the steady-state solution of a Colpitts oscillator is considered as an illustration.

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

“…18 Repressilator network. Error in the steady-state period computation obtained with different discretization techniques and with N ∈ [30,1080] The relay characteristic can be modelled as the MLCP (4) with M R = 0, β R = 0, = −1 and u = 1. We choose H u = 1 and H y = 1 in (3c) and (3d), respectively, and we write the dynamical system in form (7) with the matrices given by (6).…”

“…In this paper, we propose the use of mixed quadratic complementarity problems (MQCPs) for the computation of periodic solutions exhibited by PWL systems. The analysis here proposed extends the approach presented in [30] by allowing nonsmooth solutions and by considering a more general class of PWL systems which includes those analysed in [27][28][29][30]. In particular, the PWL system is characterized by a nonlinearity which is representable as a linear combination of saturations and relays characteristics.…”

“…In [29], the periodicity condition was included as a constraint of the complementarity problem, but an a priori estimation of the period was still required. Instead, in [30], the period is considered as a further unknown and it is computed together with the periodic solution by constructing a suitable mixed complementarity problem. However, such approach fails if the solution does not satisfy the assumption of being sufficiently smooth.…”