OGY Method Research Articles

Control of Three Synchronous Generators Infinite-bus System by OGY Method

Control of Three Synchronous Generators Infinite-bus System by OGY Method

338 モデルベーストOGY法によるダフィング系のカオス振動制御について

338 モデルベーストOGY法によるダフィング系のカオス振動制御について

"MIDDLE" PERIODIC ORBIT AND ITS APPLICATION TO CHAOS CONTROL

In this paper, a new method, by which any point in a chaotic attractor can be guided to any target periodic orbit, is proposed. The "Middle" periodic orbit is used to lead an initial point in a chaotic attractor to a neighborhood of the target orbit, and then controlling chaos can be achieved by the improved OGY method. The time needed in the method using "Middle" periodic orbit is less than that of the OGY method, and is inversely proportional to the square of the topological entropy of the given map. An example is used to illustrate the results.

2211 OGY 法による連続力学系のカオス制御について

2211 OGY 法による連続力学系のカオス制御について

CONTROL OF CHAOS: SURVEY 1997-2000

A brief survey of the emerging field termed “Control of Chaos” is given based on about 200 publications in peer reviewed journals. Three major branches of research are discussed in detail: “nonfeedback control” (based on periodic excitation of the system); “OGY method” (based on linearization of Poincaré map) and “Pyragas method” (based on a time-delay feedback). Some unsolved problems concerning the justification of chaos control methods are presented.

An invariant-manifold-based method for chaos control

In this paper, we extend the OGY chaos-control method to be one based on the invariant manifold theory and the sliding mode control concept. This extended-control method not only can deal with higher order chaotic systems in the same spirit of the OGY method, but also can remove the reliance of the control on eigenvalues and eigenvectors of the system Jacobians, resulting in an even simpler but more effective controller. The novelty of the new design lies in the construction of suitable invariant manifolds according to the desired dynamic properties. The controller is then forcing the system state to lie on the intersection of the selected invariant manifolds, so that once the invariant manifolds are reached,the chaotic system will be guided toward a desired fixed point that corresponds to an originally targeted unstable periodic orbit of the given system. Such an idea is directly relevant to the sliding mode control approach. This new method is particularly useful for controlling higher order chaotic systems, especially in the case where some of the eigenvalues of the system Jacobian are complex conjugates. The effectiveness of the proposed method is tested by numerical examples of the third-order continuous-time Lorenz system and the fourth-order discrete-time double rotor map.

ON THE CONTROL OF CHAOTIC DYNAMICAL SYSTEMS USING NONLINEAR APPROXIMATIONS

The technique of using nonlinear approximations to increase the effectiveness of the OGY method of controlling chaotic dynamical systems is investigated and a way of further increasing the effectiveness of this general approach is presented.

Chaos and complexity in a simple model of production dynamics

We consider complex dynamical behavior in a simple model of production dynamics, based on the Wiendahl’s funnel approach. In the case of continuous order flow a model of three parallel funnels reduces to the one‐dimensional Bernoulli‐type map, and demonstrates strong chaotic properties. The optimization of production costs is possible with the OGY method of chaos control. The dynamics changes drastically in the case of discrete order flow. We discuss different dynamical behaviors, the complexity and the stability of this discrete system.

USING CONTROL OF CHAOS TO REFINE APPROXIMATIONS TO PERIODIC POINTS

The OGY method for controlling chaos using parameter perturbations is analyzed assuming that the regression approximation to the fixed point is not exact. This leads to a method for obtaining a new approximation to the fixed point which can be applied iteratively and which gives a fixed point of high accuracy. The method is extended to systems with noise and so is appropriate for use in experimental systems.

Prediction-based control of chaos

Pyragas proposes a practically useful control method, called delayed feedback control, for control of chaos. Conditions for (local) stabilization by the delayed feedback control, however, are more restricted than those by the OGY method. In order to overcome this problem, we propose a novel control method, called a prediction-based feedback control, for discrete-time chaotic systems. Moreover, we give necessary and sufficient conditions for exponential stabilization of fixed points by the proposed method.

The control of higher dimensional chaos: comparative results for the chaotic satellite attitude control problem

Using simulations this paper reports the success and failure of three experiments each using a different technique to control the same higher dimensional chaotic system. The three techniques are: a simple delayed feedback control method originally suggested by Pyragas, the Otani–Jones technique, and a higher dimensional variation of the OGY method. The three methods are applied to a six-dimensional system which describes the attitude dynamics of a satellite (rigid body) subjected to deterministic external perturbations which induce chaotic motion when no control is effected. The attitude of the satellite is controlled by three orthogonal pairs of thrusters aligned with the principal axes and the system can be described by a modified set of Euler equations. The three control methods are compared in terms of the prior calculations required, the real-time computational requirements, and the effectiveness of the method in stabilizing the system. The results show that the method of Pyragas compares very favourably with the other two techniques, requiring no prior calculation and having a very low real-time computational overhead. Pyragas’ method also provides the most satisfactory control solution. The problems in presenting a theoretical justification of Pyragas’ method are briefly discussed and the connection between the method and a recent technique for the location of unstable fixed points in chaotic systems is highlighted.

CONTROLLING CHAOS USING NONLINEAR APPROXIMATIONS FOR A PENDULUM WITH FEEDFORWARD AND FEEDBACK CONTROL

The OGY method is one of the most powerful techniques for controlling chaotic dynamical systems, but often requires very long time for stabilization of the targets. Recently, it was shown that the problem can be improved by using nonlinear approximations for the chaotic dynamical systems and the stable manifolds of the targets. We consider a pendulum with feedforward and feedback control and apply the improved method to the associated Poincaré map. The derivatives of the Poincaré map, which are necessary for application of the method, are computed by solving certain differential equations. Numerical examples are given and the effectiveness of the control method is demonstrated.

Control of a Chaotic Relay System Using the OGY Method

Abstract The control method of Ott, Grebogi, and Yorke (OGY) is used to stabilize the unstable periodic orbits of a chaotic relay system. Small variations in the height of the relay output are used as control input. The influence of the control activation bound is studied in detail via the one-dimensional Poincare map of the controlled system. The reduced sensitivity of the multi-step OGY method for higher period orbits can thus directly be verified.

Delayed feedback control with nonlinear estimation in chaotic discrete-time systems

The delayed feedback control method proposed by Pyragas is practically useful for control of chaos. Conditions for (local) stabilization by the delayed feedback, however, are more restricted than those by the OGY method. In order to overcome this problem, we proposed a delayed feedback control method using nonlinear estimation of stabilized orbits. Moreover, we give several conditions for stabilization by the proposed method, using robust control theory and linear matrix inequalities.

Experimental Switchings in Bistability Domains Induced by Resonant Perturbations

We show that a combination of a resonant perturbation which induces bistability in the system with the targeting technique based on the action of a short-lived pulse perturbation makes the nonfeedback control of nonlinear systems (not only in a chaotic state) more flexible and even competitive with OGY's method in the sense of fast switching between controlled orbits belonging to coexisting attractors. For different initial states we present several experimental and numerical examples of such a type of control, which does not require feedback system.

Suppression and Excitation of Chaos: The Example of the Glow Discharge

I report on the experimental observation of excitation and suppression of chaos through time dependent perturbations in the dynamical variable of a glow discharge. The interaction of the external signal with the dynamical system is explained in terms of the 1D map associated to the glow discharge. Numerical simulations are also performed with the logistic map. The proposed mechanism of exciting and/or suppressing chaos is in accordance with the OGY method of controlling chaos.

A new approach for controlling chaotic dynamical systems

The chaos control method proposed by Ott and his coworkers and now called the OGY method has attracted much attention. However, in some applications this technique requires a very long time until the control applies and it is not so effective. In this Letter, we present a new chaos control method in which this problem is improved. The main difference from the OGY method is the use of nonlinear approximations for chaotic dynamical systems and stable manifolds of the target points. We give an example for the Hénon map to demonstrate the effectiveness of the present method. Our method is also shown to be robust to modeling errors like the OGY method.

An improvement and proof of OGY method

OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and gives a mathematics proof of it.

Stabilizing unstable orbits

For the purpose of controlling chaos, we describe a method to eliminate the deviations of the trajectories from the desired orbit in the fastest way, independent of the orbit being periodic or not. This is especially useful for cases where the OGY method does not work, namely when the desired orbit has complex eigenvalues, the eigenvalues in the stable eigendirections are near unity, and the unstable manifold is multi-dimensional. Applications for many aspects concerning chaos control are discussed. We demonstrate the the method by an application to the control of the kicked double rotor map in the presence of noise.

Adaptive neuro-genetic control of chaos applied to the attitude control problem

Conventional adaptive control techniques have, for the most part, been based on methods for linear or weakly non-linear systems. More recently, neural network and genetic algorithm controllers have started to be applied to complex, non-linear dynamic systems. The control of chaotic dynamic systems poses a series of especially challenging problems. In this paper, an adaptive control architecture using neural networks and genetic algorithms is applied to a complex, highly nonlinear, chaotic dynamic system: the adaptive attitude control problem (for a satellite), in the presence of large, external forces (which left to themselves led the system into a chaotic motion). In contrast to the OGY method, which uses small control adjustments to stabilize a chaotic system in an otherwise unstable but natural periodic orbit of the system, the neuro-genetic controller may use large control adjustments and proves capable of effectively attaining any specified system state, with no a prioriknowledge of the dynamics, even in the presence of significant noise.