Quadratic optimal synchronization of uncertain chaotic systems

This paper investigates the quadratic optimal synchronization of uncertain chaotic systems with parameter mismatch, parametric perturbations and external disturbances on both master and slave systems. A robust control scheme based on Lyapunov stability theory and quadratic optimal control approach is derived to realize chaotic synchronization. The sufficient criterion for stability condition is formulated in a linear matrix inequality (LMI) form. The effect of uncertain parameters and external disturbance is suppressed to an H_∞ norm constraint. An adaptive algorithm is proposed to adjust the uncertain bound in the robust controller avoiding the chattering phenomena. The simulation results for synchronization of the Chua’s circuit system and the Lorenz system demonstrate the effectiveness of the proposed scheme.     

Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory

In this paper stabilizing unstable periodic orbits (UPO) in a chaotic fractional order system is studied. Firstly, a technique for finding unstable periodic orbits in chaotic fractional order systems is stated. Then by applying this technique to the fractional van der Pol and fractional Duffing systems as two demonstrative examples, their unstable periodic orbits are found. After that, a method is presented for stabilization of the discovered UPOs based on the theories of stability of linear integer order and fractional order systems. Finally, based on the proposed idea a linear feedback controller is applied to the systems and simulations are done for demonstration of controller performance.     

Hybrid chaos control of continuous unified chaotic systems using discrete rippling sliding mode control

In this paper, a new systematic design procedure to stabilize continuous unified chaotic systems based on discrete sliding mode control (DSMC) is presented. In contrast to the previous works, the concept of rippling control is newly introduced such that the design of DSMC can be simplified and only a single controller is needed to realize chaos suppression. As expected, under the proposed DSMC law, the unified system can be stabilized in a manner of ripple effect, even when the external uncertainty is present. Last, two examples are included to illustrate the effectiveness of the proposed rippling DSMC developed in this paper.     

Implementation of dynamic programming for chaos control in discrete systems

In this paper the control of discrete chaotic systems by designing linear feedback controllers is presented. The linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming. For suppressing chaos with minimum control effort, the system is stabilized on its first order unstable fixed point (UFP). The presented method also could be employed to make any desired nth order fixed point of the system, stable. Two different methods for higher order UFPs stabilization are suggested. Afterwards, these methods are applied to two well-known chaotic discrete systems: the Logistic and the Henon Maps. For each of them, the first and second UFPs in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance.     

Robust H∞ control for linear Markovian jump systems with unknown nonlinearities

This paper studies the problem of stochastic stability and disturbance attenuation for a class of linear continuous-time uncertain systems with Markovian jumping parameters. The uncertainties are assumed to be nonlinear and state, control and external disturbance dependent. A sufficient condition is provided to solve the above problem. An H_∞ controller is designed such that the resulting closed-loop system is stochastically stable and has a disturbance attenuation γ for all admissible uncertainties. It is shown that the control law is in terms of the solutions of a set of coupled Riccati inequalities. A numerical example is included to demonstrate the potential of the proposed technique.     

Finite-time stabilization of a class of chaotic systems via adaptive control method

This paper investigates the stabilization of three dimensional chaotic systems in a finite time by extending our previous method for chaos stabilization. Based on the finite-time stability theory, a control law is proposed to realize finite-time stabilization of three dimensional chaotic systems. In comparison with the previous methods, the controller obtained by our method is simpler than those. Moreover, the method obtained in this paper is suitable for a class of three dimensional chaotic systems. The efficiency of the control scheme is revealed by some illustrative simulations.     

Chaos synchronization between two different chaotic systems via nonlinear feedback control

This work presents chaos synchronization between two different chaotic systems via nonlinear feedback control. On the basis of a converse Lyapunov theorem and balanced gain scheme, control gains of controller are derived to achieve chaos synchronization for the unified chaotic systems. Numerical simulations are shown to verify the results.     

Sliding mode control of uncertain unified chaotic systems

This paper investigates the chaos control of the uncertain unified chaotic systems by means of sliding mode control. A proportional plus integral sliding surface is introduced to obtain a sliding mode control law. To confirm the validity of the proposed method, numerical simulations are presented graphically.     

Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique

In this paper, the problem of finite-time chaos synchronization between two different chaotic systems with fully unknown parameters is investigated. First, a new nonsingular terminal sliding surface is introduced and its finite-time convergence to the zero equilibrium is proved. Then, appropriate adaptive laws are derived to tackle the unknown parameters of the systems. Afterwards, based on the adaptive laws and finite-time control idea, an adaptive sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time. It is mathematically proved that the introduced sliding mode technique has finite-time convergence and stability in both reaching and sliding mode phases. Finally, some numerical simulations are presented to demonstrate the applicability and effectiveness of the proposed technique.     

Nonlinear discrete systems with global boundary conditions

This paper is devoted to the study of boundary value problems on infinite time intervals for nonlinear, discrete-time systems. For problems of the form

     

Recovering unknown model parameters contained in a class of time-variant chaotic dynamical systems

This work deals mainly with the problem of recovering all unknown parameters for a class of time-variant chaotic dynamical systems from given time sequence. Based on synchronization between a chaotic sender system and an additional receiver system, a procedure, which combines a linear feedback technique with updated feedback gain and an adapted control strategy associated with the law of estimated parameters, is developed to dynamically determine the values of unknown parameters contained in the sender system. To promote widespread applications, the structure of the receiver system can be independent of that of the sender system. The effectiveness of this procedure is guaranteed by the periodic version of the classical LaSalle invariance principle of differential equations. Illustrations are presented for a harmonically excited Duffing oscillator and a four dimensional chaotic oscillator. The numerical results reveal the present procedure not only can precisely recover unknown model parameters, but also can rapidly response to sudden changes in unknown parameters. In addition, it has great robustness against the disturbance of noise.     

Robust H∞ control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach

This paper addresses the problem of robust H_∞ control for a class of switched nonlinear cascade systems with parameter uncertainty using the multiple Lyapunov functions (MLFs) approach. Each subsystem under consideration is composed of two cascade-connected parts. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi equations, sufficient conditions for the solvability of the robust H_∞ control problem are presented. As an application, the hybrid robust H_∞ control problem for a class of uncertain non-switched nonlinear cascade systems is solved when no single continuous controller is effective. Finally, a numerical example is provided to demonstrate the feasibility of the proposed method.     

Feedback control of bursting and multistability in chaotic systems

With a modulated CO₂ laser as a standard model of periodically driven multistable systems, we experimentally demonstrate that a small-amplitude optoelectronic feedback perturbation can efficiently transform a bursting chaotic system to a nonchaotic one. Numerical simulations are in excellent agreement with the experimental results. The control has also been equally effective in the case of a driven FitzHugh-Nagumo model of Neuroscience.     

Connections among several chaos feedback control approaches and chaotic vibration control of mechanical systems

This study reveals the essential connections among several popular chaos feedback control approaches, such as delayed feedback control (DFC), stability transformation method (STM), adaptive adjustment method (AAM), parameter adjustment method, relaxed Newton method, and speed feedback control method (SFCM), etc. Meanwhile, the generality and practical applicability of these approaches are evaluated and compared. It is shown that for discrete chaotic maps, STM can be regarded as a kind of predictive feedback control, and AAM is actually a special case of STM which is merely effective for a particular dynamical system. The parameter adjustment method is only a different expression of the relaxed Newton method, and both of them represent just one search direction of STM, i.e., the gradient direction. Moreover, the intrinsic relation between the STM and SFCM for controlling the equilibrium of continuous autonomous systems is investigated, indicating that STM can be viewed as a special form of the SFCM. Finally, both the STM and SFCM are extended to control the chaotic vibrations of non-autonomous mechanical systems effectively.     

Robust finite-time H∞ control for a class of uncertain switched neutral systems

This paper investigates the robust finite-time H_∞ control problem for a class of uncertain switched neutral systems with unknown time-varying disturbance. The uncertainties under consideration are norm bounded. By using the average dwell time approach, a sufficient condition for finite-time boundedness of switched neutral systems is derived. Then, finite-time H_∞ performance analysis for switched neutral systems is developed, and a robust finite-time H_∞ state feedback controller is proposed to guarantee that the closed-loop system is finite-time bounded with H_∞ disturbance attenuation level γ. All the results are given in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed method.     

Optimal estimation and control of discrete multiplicative systems with unknown second-order statistics

The problem of estimation and control for discrete-time systems with multiplicative noise is examined. Such systems occur naturally in the modeling of stochastic systems with random or unknown coefficients and appear to be robust in contrast to LQG regulators which are sensitive to errors in the coefficients.The statistics of the white sequences of the system are unknown. The problem of stochastic estimation and control of such a system is difficult not only because of the unknown statistics but also because the state is not Gaussian.The approach of this work is to convert the stochastic problem to a deterministic game-theoretic one. We find the estimator and controller so as to minimize a suitable performance measure assuming the worst behavior of nature.A set of necessary and sufficient conditions is developed for the existence of a saddle-point estimator. When both estimation and control are considered, two difficulties appear: the optimality conditions are only necessary and the separation principle collapses. As a result, the saddle-point conditions are only necessary. If the covariances belong to sets with maximal points, then the necessary conditions are satisfied at these points. If, on the other hand, they belong to convex and compact sets and the system has a steady state, then the estimation problem alone has always a saddle-point solution.     

Space-time discontinuous galerkin method for maxwell equations in dispersive media

In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L²-stability and error estimate of order O(τ^r+1+h^k+1/2) are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r+1 in temporal variable t.     

L 1-optimization in linear systems with bounded controls

In a linear control system over a fixed time interval, we treat steering to the origin by controls (i) having realistic constraints on their values, while minimizing theL ₁-cost of control, (ii) without such constraints, and (iii) minimizing the coordinate bound, with or without a givenL ₁-cost restriction. Existence (also, nonexistence) results are obtained, together with further necessary conditions for control optimality.The author was introduced to these problems by Professors P. Hagedorn and W. Krabs and profited much by discussions with them and with other workers at the TH Darmstadt. The incisive idea that feedback controls still exist if state space is augmented by one dimension (see Example, Section 1) is due to Professor Hagedorn and obviously merits further treatment. Professor E. N. Chukwu pointed out an error in a preprint version of this paper, and the present formulation of Theorem 4.6 is the result of discussions with him. In a colloquium at Würzburg University, Professor P. C. Parks made the point that a portion ofmodern control theory is implicit in the work of Achiezer: in connection with the present material, the so-called M-moment problem, and our Theorem 4.6. The referee kindly pointed out the reference for the example in Section 1. This paper is a revised version of a report, Preprint No. 279, prepared while the author held a Humboldt Award at the TH Darmstadt.     

Suboptimal feedback control of linear periodic systems

A method is developed for the approximate design of an optimal state regulator for a linear periodically varying system with quadratic performance index. The periodic term is taken to be a perturbation to the system. By making use of a power-series expansion in a small parameter, associated with periodic terms, a set of matrix equations is derived for determining successively a feedback gain. Given periodic terms of a Fourier-series form, explicit solutions are obtained for those matrix equations. A sufficient condition for existence and periodicity of the solution is also shown. Further, the performance degradation resulting from a truncation of the power-series solution is investigated. The method may effectively be used in a computer-programmed computation.