Control Of Chaotic Dynamical Systems Research Articles

Control simulation experiments of extreme events with the Lorenz-96 model

Abstract. The control simulation experiment (CSE) is a recently developed approach to investigate the controllability of dynamical systems, extending the well-known observing system simulation experiment (OSSE) in meteorology. For effective control of chaotic dynamical systems, it is essential to exploit the high sensitivity to initial conditions for dragging a system away from an undesired regime by applying minimal perturbations. In this study, we design a CSE for reducing the number of extreme events in the Lorenz-96 model. The 40 variables of this model represent idealized meteorological quantities evenly distributed on a latitude circle. The reduction of occurrence of extreme events over 100-year runs of the model is discussed as a function of the parameters of the CSE: the ensemble forecast length for detecting extreme events in advance, the magnitude and localization of the perturbations, and the quality and coverage of the observations. The design of the CSE is aimed at reducing weather extremes when applied to more realistic weather prediction models.

Nonlinear Time Series Analysis in Unstable Periodic Orbits Identification-Control Methods of Nonlinear Systems

The main purpose of this paper is to present a solution to the well-known problems generated by classical control methods through the analysis of nonlinear time series. Among the problems analyzed, for which an explanation has been sought for a long time, we list the significant reduction in control power and the identification of unstable periodic orbits (UPOs) in chaotic time series. To accurately identify the type of behavior of complex systems, a new solution is presented that involves a method of two-dimensional representation specific to the graphical point of view, and in particular the recurrence plot (RP). An example of the issue studied is presented by applying the recurrence graph to identify the UPO in a chaotic attractor. To identify a certain type of behavior in the numerical data of chaotic systems, nonlinear time series will be used, as a novelty element, to locate unstable periodic orbits. Another area of use for the theories presented above, following the application of these methods, is related to the control of chaotic dynamical systems by using RP in control techniques. Thus, the authors’ contributions are outlined by using the recurrence graph, which is used to identify the UPO from a chaotic attractor, in the control techniques that modify a system variable. These control techniques are part of the closed loop or feedback strategies that describe control as a function of the current state of the UPO stabilization system. To exemplify the advantages of the methods presented above, the use of the recurrence graph in the control of a buck converter through the application of a phase difference signal was analyzed. The study on the command of a direct current motor using a buck converter shows, through a final concrete application, the advantages of using these analysis methods in controlling dynamic systems.

Flatness-Based Adaptive Neurofuzzy Control of Chaotic Dynamical Systems

The paper proposes a solution to the problem of control of nonlinear chaotic dynamical systems, which is based on differential flatness theory and on adaptive fuzzy control. An adaptive fuzzy controller is designed for chaotic dynamical systems, under the constraint that the system’s model is unknown. The control algorithm aims at satisfying the $$H_\infty $$ tracking performance criterion, which means that the influence of the modeling errors and the external disturbances on the tracking error is attenuated to an arbitrary desirable level. After transforming the chaotic system’s model into a linear form, the resulting control inputs are shown to contain nonlinear elements which depend on the system’s parameters. The nonlinear terms which appear in the control inputs are approximated with the use of neuro-fuzzy networks. It is shown that a suitable learning law can be defined for the aforementioned neuro-fuzzy approximators so as to preserve the closed-loop system stability. With the use of Lyapunov stability analysis it is proven that the proposed adaptive fuzzy control scheme results in $$H_{\infty }$$ tracking performance. The efficiency of the adaptive fuzzy control method is checked through simulation experiments, using as case study the Lorenz chaotic oscillator.

LQR based optimal control of chaotic dynamical systems

This article studies the linear quadratic regulator (LQR)-based optimal control law for chaotic dynamical system. First, we show a systematic and effective framework for modelling of chaotic system by considering the dynamics of bouncing ball on sinusoidally oscillating surface and explore the similarity between dynamics of the standard chaotic map and bouncing ball dynamics. Then we investigated the discrete LQR optimal control technique to the control of chaotic system. The investigated design method is conceptually simple and computationally efficient. The effectiveness of the design method is demonstrated by simulation results.

Adaptive sliding mode control of chaotic dynamical systems with application to synchronization

We address the problem of control and synchronization of a class of uncertain chaotic systems. Our approach follows techniques of sliding mode control and adaptive estimation law. The adaptive algorithm is constructed based on the sliding mode control to ensure perfect tracking and synchronization in presence of system uncertainty and external disturbance. Stability of the closed-loop system is proved using Lyapunov stability theory. Our theoretical findings are supported by simulation results.

Stabilization of parameters perturbation chaotic system via adaptive backstepping technique

The work of Yassen [M.T. Yassen, Chaos control of chaotic dynamical systems using backstepping design, Chaos Soliton Fract. 27 (2006) 537–548] which mainly investigated the stabilization problem for a class of chaotic systems without the parameters perturbation. This paper is concerned with stabilization problem for a class of parameters perturbation chaotic systems via both backstepping design method and adaptive technique. The proposed controllers can guarantee that the parameters perturbation systems will be stabilized at a fixed bounded point. Furthermore, the paper also proposes controllers to stabilize the uncertain chaotic system at equilibrium point with only backstepping design method. Finally, numerical simulations are given to illustrate the effectiveness of the proposed controllers.

Chaos control of chaotic dynamical systems using backstepping design

Abstract This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lu systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.

Comments on: “Control of chaotic dynamical systems using support vector machines” [Phys. Lett. A 317 (2003) 429

In the above Letter, the authors proposed an interesting approach to design the feedback controller for chaotic systems. However, since the problem how to produce the nonlinear compensation term was not considered, the control scheme present by them is not reasonable and cannot be implemented in general cases.

QUALITATIVE RESONANCE OF SHIL'NIKOV-LIKE STRANGE ATTRACTORS, PART I: EXPERIMENTAL EVIDENCE

This is the first of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, the results of an experimental analysis are presented together with an intuitive geometrical qualitative model of the phenomenon.

QUALITATIVE RESONANCE OF SHIL'NIKOV-LIKE STRANGE ATTRACTORS, PART II: MATHEMATICAL ANALYSIS

This is the second of two papers introducing a new dynamical phenomenon, strongly related to the problems of synchronization and control of chaotic dynamical systems, and presenting the corresponding mathematical analysis, conducted both experimentally and theoretically. In particular, it is shown that different dynamical models (ordinary differential equations) admitting chaotic behavior organized by a homoclinic bifurcation to a saddle-focus (Shil'nikov-like chaos) tend to have a particular selective property when externally perturbed. Namely, these systems settle on a very narrow chaotic behavior, which is strongly correlated to the forcing signal, when they are slightly perturbed with an external signal which is similar to their corresponding generating cycle. Here, the "generating cycle" is understood to be the saddle cycle colliding with the equilibrium at the homoclinic bifurcation. On the other hand, when they are slightly perturbed with a generic signal, which has no particular correlation with their generating cycle, their chaotic behavior is reinforced. This peculiar behavior has been called qualitative resonance underlining the fact that such chaotic systems tend to resonate with signals that are qualitatively similar to an observable of their corresponding generating cycle. Here, a detailed mathematical analysis of the qualitative resonance phenomenon is presented, confirming the intuitions given by the geometrical model discussed in Part I.

Hybrid internal model control and proportional control of chaotic dynamical systems

Hybrid internal model control and proportional control of chaotic dynamical systems

Hybrid internal model control and proportional control of chaotic dynamical systems

Hybrid internal model control and proportional control of chaotic dynamical systems

Control of chaotic dynamical systems using support vector machines

This work provides a methodology for the control of complex non-linear systems by compensating for the non-linear part using support vector machines (SVM) and subsequently developing simple linear feedback law for control. The method tested for the benchmark Rössler and Lorenz chaotic oscillators shows excellent performance.

Control of chaotic dynamical systems using radial basis function network approximators

This paper presents a general control method based on radial basis function networks (RBFNs) for chaotic dynamical systems . For many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part , under some mild conditions the RBFN can be used to well approximate the nonlinear part of the system dynamics. The resulting system is then dominated by the linear part, with some small or weak residual nonlinearities due to the RBFN approximation errors. Thus, a simple linear state-feedback controller can be devised, to drive the system response to a desirable set-point. In addition to some theoretical analysis, computer simulations on two representative continuous-time chaotic systems (the Duffing and the Lorenz systems) are presented to demonstrate the effectiveness of the proposed method.

Adaptive control of chaotic dynamical systems using invariant manifold approach

In this brief, an adaptive chaos control method is developed for stabilizing chaotic systems at their unknown equilibrium(s) using the invariant manifold theory. The developed method overcomes the problem that the equilibrium(s) of the chaotic systems are dependent on the unknown system parameters, which makes direct application of the conventional adaptive control difficult. Further development of the adaptive chaos control is undertaken for the situation where the parameter estimates are only allowed to vary within a bounded set due to the sensitivity of chaotic systems to parameter variations. A sufficient condition for convergence of system states and parameter estimates is obtained. The design method developed then is applied to stabilizing the Lorenz chaotic system at an unknown equilibrium. Both mathematical and computational results have demonstrated the effectiveness of this method.

Feedback controlling against chaos

A P–D (Proportional plus Derivative) feedback controller is proposed for the control of chaotic dynamic systems. Using this feedback law the design requirement of command matching is satisfied independently of the perturbations of the initial conditions of the dynamic plant. The problem is solved for all nonlinear systems having independent inputs and independent performance variables. The controller is implemented without knowledge of the system’s initial conditions and thus it is independent of their perturbations.

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.

Neurocontrol: Recent advances and links with the human brain

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.

CONTROLLING CHAOS OF AN UNCERTAIN LOZI SYSTEM VIA ADAPTIVE TECHNIQUES

A case on the control of chaotic dynamical systems is presented. The chaotic Lozi system is taken with the aim to drive its trajectories to certain given reference points even when there are uncertainties on the parameters of the system. The control objective is reached by applying adaptive control laws.

Progress in the control of chaos

Abstract In this review, we summarize several of the key contributions made over the past 5 years to the control of chaotic dynamical systems. The idea that chaotic systems can in fact be controlled may be counter-intuitive; after all they are unpredictable in the long term. Nevertheless, numerous theorists have independently developed methods which can be applied to chaotic systems, and many experimentalists have demonstrated that physical chaotic systems respond well to both simple and sophisticated control strategies. The great bulk of these researchers have restricted their study to low-dimensional systems, and correspondingly we critique this work at length. Most recently, a few researchers have proposed control techniques for application to high- or infinite-dimensional systems, and we describe this work in some detail as well.