Stabilization of periodic orbits via time-delay feedback control

Ferroresonance is a special case of jump resonance in which the nonlinearity is iron-core magnetization. Jump resonance refers to a condition in a sinusoidally excited system. If an incremental change in the amplitude or frequency of the input to the system, or in the magnitude of parameters of the system, causes a sudden jump in signal amplitude somewhere in the system, jump resonance is said to have occurred. Thus, the jump can occur in the voltage, current, flux linkages, or in all three. In electric networks, ferroresonance oscillations can seriously damage the equipment. They are caused by the interaction of the system's capacitance with the magnetic characteristics of the voltage transformers. Due to the nonlinear nature of these phenomena, the system's behaviour is extremely sensitive to initial conditions. One of the most exciting and interesting ideas developed in nonlinear dynamics concerns chaotic behaviour of a complex system. A deterministic system is said to be chaotic whenever its evolution sensitively depends on the initial conditions. The theory of chaos gained an important role in understanding complex behaviour of the dynamic system. Ferroresonant oscillations can be successfully described as chaotic. Chaotic dynamics has two additional characteristics. First, there is an infinite number of unstable periodic orbits embedded in the chaotic set. Secondly, during the temporal evolution, the system visits a small neighbourhood of every point in each one of the periodic orbits embedded within the chaotic attractor. Therefore, a chaotic dynamics can also be seen as shadowing some periodic behaviour at a given time, and erratically jumping rom one periodic orbit to another. The idea of controlling chaotic dynamics is to apply small perturbations to stabilise a trajectory when it approaches a desired periodic orbit embedded in the attractor. Indeed, if a small perturbation can give rise to a very large response in the course of time, then it should be possible to direct the trajectory to any desired part of the attractor and produce a series of desired dynamical states by an appropriate perturbation. The problem of controlling chaos has gained particular attention in recent years. Two general methods were proposed: the Ott- Grebogi-Yorke (OGY) method [1] and the Pyragas, or adaptive method [2]. We apply the second method, appropriate for periodically forced chaotic systems like the power system that we control. We describe the design of the feedback controller and present the results of the stabilisation procedure.

کلیدواژه ها:

Ferroresonance ، chaos ، controlling chaos ، stability