Forced Solutions of Disturbed Pendulum-Like Lur'e Systems

The mathematical model of a viscously damped pendulum is an example of a Lur'e system with periodic nonlinearity. Systems of this type arise in many applications, describing e.g. phase-locked loops and other synchronization systems arising in communication engineering, networks of oscillators and power generators. Periodic nonlinearities usually imply multistability of the system and the existence of multiple stable and unstable equilibria. This makes inapplicable many tools of classical nonlinear control, developed for systems with globally stable equilibria. To study asymptotic properties of such systems, special techniques have been developed stemming from Popov's method of “a priori integral indices”, or integral quadratic constraints. These tools lead to efficient frequency-domain criteria, providing convergence of any solution to one of equilibria. In this paper, we further develop Popov's method, addressing the problem of robustness of the convergence property against external disturbances that do not oscillate at infinity (more precisely, decomposable into the sum of a constant excitation and decaying $L_{1}$ or $L_{2}$ signal). Will the forced solutions of the disturbed system also converge to one of the equilibria points (in general, the set of equilibria depends on the disturbance)? In this paper, we find a sufficient frequency-domain condition ensuring such a robust convergence, showing also that a relaxed form of this condition guarantees absence of high-frequency periodic oscillations in the system.