Control of chaotic dynamical systems using OGY
Gert Witvoet · TU/e Research Portal · 2005
Chaos is the general name for non-linear dynamical systems which behave noise-like.Chaos is indecomposable, is highly dependent on the initial condition and consists of a large number of periodic points and orbits.Because of this, the solution of a chaotic system is difficult to predict, which calls for a way to control it.The control algorithm of Ott, Grebogi and Yorke (OGY, [13]) manages to do this.The basic observation behind OGY is that a chaotic attractor has a large number of unstable periodic solutions embedded within itself.Furthermore, by slightly perturbing an accessible parameter of the system, it is possible to push the system towards one of these orbits.Since chaos behaves ergodically, at some point in time the solution will come into the vicinity of a certain point of the orbit where a linearization is valid.With this linearization a simple pole placement method can be used to calculate a control effort to direct the system towards this point and thus the orbit.OGY is a discrete control algorithm, perturbing the system at discrete moments in time.OGY can therefore easily be used on discrete systems like the Hénon map.By first discretizing a system using the Poincaré map it can also be used on continuous systems like the Duffing oscillator.Periodic orbits will become a simple sequence of points on the Poincaré map (m points for a period m orbit), around which the OGY algorithm finds a linearization.The accuracy of this linearization is very important in the implementation of OGY.In cases where no model is present the values of the periodic points should be estimated using the recurrence method and the matrices A and B in vii CONTENTS 5.2.2Control without delay coordinates . . . . . . . . . . . . . . . . . . . . .5.2.3Control with delay coordinates . . . . . . . . . . . . . . . . . . . . . . .5.2.4 Alternative method for delay coordinates . . . . . . . . . . . . . . . . .5.3 Implementation in Simulink R . . . . . . . .