Continuous control of chaos by self-controlling feedback
TL;DR: In this paper, the stabilization of unstable periodic orbits of a chaotic system is achieved either by combined feedback with the use of a specially designed external oscillator, or by delayed self-controlling feedback without using of any external force.
About: This article is published in Physics Letters A.The article was published on 1992-11-23. It has received 2957 citations till now. The article focuses on the topics: Control of chaos & System dynamics.
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26 Jun 2003
TL;DR: Preface, Notations 1.Introduction to Time-Delay Systems I.Robust Stability Analysis II.Input-output stability A.LMI and Quadratic Integral Inequalities Bibliography Index
Abstract: Preface, Notations 1.Introduction to Time-Delay Systems I.Frequency-Domain Approach 2.Systems with Commensurate Delays 3.Systems withIncommensurate Delays 4.Robust Stability Analysis II.Time Domain Approach 5.Systems with Single Delay 6.Robust Stability Analysis 7.Systems with Multiple and Distributed Delays III.Input-Output Approach 8.Input-output stability A.Matrix Facts B.LMI and Quadratic Integral Inequalities Bibliography Index
4,200 citations
Book•
02 May 2008
TL;DR: Fuzzy Control Systems Design and Analysis offers an advanced treatment of fuzzy control that makes a useful reference for researchers and a reliable text for advanced graduate students in the field.
Abstract: From the Publisher:
A comprehensive treatment of model-based fuzzy control systems
This volume offers full coverage of the systematic framework for the stability and design of nonlinear fuzzy control systems. Building on the Takagi-Sugeno fuzzy model, authors Tanaka and Wang address a number of important issues in fuzzy control systems, including stability analysis, systematic design procedures, incorporation of performance specifications, numerical implementations, and practical applications.
Issues that have not been fully treated in existing texts, such as stability analysis, systematic design, and performance analysis, are crucial to the validity and applicability of fuzzy control methodology. Fuzzy Control Systems Design and Analysis addresses these issues in the framework of parallel distributed compensation, a controller structure devised in accordance with the fuzzy model.
This balanced treatment features an overview of fuzzy control, modeling, and stability analysis, as well as a section on the use of linear matrix inequalities (LMI) as an approach to fuzzy design and control. It also covers advanced topics in model-based fuzzy control systems, including modeling and control of chaotic systems. Later sections offer practical examples in the form of detailed theoretical and experimental studies of fuzzy control in robotic systems and a discussion of future directions in the field.
Fuzzy Control Systems Design and Analysis offers an advanced treatment of fuzzy control that makes a useful reference for researchers and a reliable text for advanced graduate students in the field.
3,183 citations
TL;DR: The extreme sensitivity of chaotic systems to tiny perturbations (the ‘butterfly effect’) can be used both to stabilize regular dynamic behaviours and to direct chaotic trajectories rapidly to a desired state.
Abstract: The extreme sensitivity of chaotic systems to tiny perturbations (the ‘butterfly effect’) can be used both to stabilize regular dynamic behaviours and to direct chaotic trajectories rapidly to a desired state. Incorporating chaos deliberately into practical systems therefore offers the possibility of achieving greater flexibility in their performance.
719 citations
TL;DR: In this article, several theorems on the stability of impulsive control systems are presented, which are then used to find the conditions under which the chaotic systems can be asymptotically controlled to the origin.
Abstract: Impulsive control of a chaotic system is ideal for designing digital control schemes where the control laws are generated by digital devices which are discrete in time. In this paper, several theorems on the stability of impulsive control systems are presented. These theorems are then used to find the conditions under which the chaotic systems can be asymptotically controlled to the origin by using impulsive control. Given the parameters of the chaotic system and the impulsive control law, an estimation of the upper bound of the impulse interval is given. We also present a theory of impulsive synchronization of two chaotic systems. A promising application of impulsive synchronization of chaotic systems to a secure communication scheme is presented. In this secure communication scheme, the transmitted signals are divided into small time frames. In each time frame, the synchronization impulses and the scrambled message signal are embedded. Conventional cryptographic methods are used to scramble the message signal. Simulation results based on a typical chaotic system; namely, Chua's oscillator, are provided.
715 citations
References
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 citations
TL;DR: This chapter describes the linking of two chaotic systems with a common signal or signals and highlights that when the signs of the Lyapunov exponents for the subsystems are all negative the systems are synchronized.
Abstract: Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.
9,201 citations
TL;DR: A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable as mentioned in this paper, which allows for a "folded" Poincare map (horseshoe map).
3,334 citations
TL;DR: It is shown that driving with chaotic signals can be done in a robust fashion, rather insensitive to changes in system parameters, and the calculation of the stability criteria leads naturally to an estimate for the convergence of the driven system to its stable state.
Abstract: We generalize the idea of driving a stable system to the situation when the drive signal is chaotic. This leads to the concept of conditional Lyapunov exponents and also generalizes the usual criteria of the linear stability theorem. We show that driving with chaotic signals can be done in a robust fashion, rather insensitive to changes in system parameters. The calculation of the stability criteria leads naturally to an estimate for the convergence of the driven system to its stable state. We focus on a homogeneous driving situation that leads to the construction of synchronized chaotic subsystems. We apply these ideas to the Lorenz and R\"ossler systems, as well as to an electronic circuit and its numerical model.
888 citations
TL;DR: It was demonstrated that one can convert the motion of a chaotic dynamical system to periodic motion by controlling the system about one of the many unstable periodic orbits embedded in the chaotic attractor, through only small time dependent perturbations in an accessible system parameter.
Abstract: It was demonstrated that one can convert the motion of a chaotic dynamical system to periodic motion by controlling the system about one of the many unstable periodic orbits embedded in the chaotic attractor, through only small time dependent perturbations in an accessible system parameter. They demonstrated their method numerically by controlling the Henon map. Far from being a numerical curiosity that requires experimentally unattainable precision, it was believed that this method can be widely implemented in a variety of systems including chemical, biological, optical, electronic, and mechanical systems. The method is based on the observation that unstable periodic orbits are dense in a typical chaotic attractor. Their method assumes only the following four points. First, the dynamics of the system can be represented as arising from an n-dimensional non-linear map. Second, there is a specific periodic orbit of the map which lies in the attractor and around which one wishes to stabilize the dynamics, and so on. To control the chaos, one attempts to confine the iterates of the map to a small neighborhood of the desired orbit.
545 citations