Bifurcation analysis for T system with delayed feedback and its application to control of chaos

In this paper, a three-dimensional autonomous nonlinear system called the T system which has potential application in secure communications is considered. Regarding the delay as parameter, we investigate the effect of delay on the dynamics of T system with delayed feedback. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associated characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.     

A new chaos synchronization scheme and its application to secure communications

Under the framework of drive-response systems, a new method of complete dislocated general hybrid projective synchronization (CDGHPS) is proposed. In this design, every state variable of drive system does not equal the corresponding state variable of response system, but equal other ones of response system while evolving in time. Especially, complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization can be considered all as the special cases of the proposed method. In addition, this method is applied to secure communication through chaotic masking, the unpredictability of the scaling factor in projective synchronization can additionally enhance the security of communication. In consideration of random white noise, we study the random white noise perturbing for the transmission of an information signal. Finally, eliminate noise using wavelet transform. Numerical simulations are given to show the effectiveness of these methods.     

Robust output feedback control of time-delay nonlinear systems with dead-zone input and application to chemical reactor system

Nonlinear Dynamics - This paper investigates the problem of robust output feedback control for a class of time-delay nonlinear systems with unknown continuous time varying output function. Unlike...     

A data-driven quadratic stability condition and its application for stabilizing unknown nonlinear systems

This paper proposes a data-driven stability criterion for quadratic stabilization of unknown nonlinear discrete-time systems. The novelty of this quadratic stability criterion lies in the direct use of the time series of system states, instead of using mathematical models. The data-driven stability criterion is utilized to design a control for stabilizing unknown nonlinear systems using online black-box system identification. The effectiveness and the adaptability of the proposed approach are compared with those of adaptive feedback linearization method with an example of stabilizing a nonlinear aeroelastic system.     

A stability result for the relativistic Vlasov-Maxwell system

We consider a space-periodic version of the relativistic Vlasov-Maxwell system describing a collisionless plasma consisting of electrons and positively charged ions. As our main result, we prove that certain spacially homogeneous stationary solutions are nonlinearly stable. To this end we also establish global existence of weak solutions to the corresponding initial value problem. Our investigation is motivated by a corresponding result for the Vlasov-Poisson system, cf. [1, 14].     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

Control and synchronization of chaos systems using time-delay estimation and supervising switching control

In this paper, we present a new technique, developed using time-delay estimation (TDE) and supervising switching control (SSC), for the control and synchronization of chaos systems. The proposed technique consists of three units: a time-delay estimation unit that cancels system dynamics, a pole placement control unit that shapes error dynamics, and an SSC unit that is activated when the system dynamics are rapidly changing. We prove the stability of the closed-loop system using the Lyapunov analysis method. To verify the control and synchronization performance of the proposed technique (TDE-SSC), we compare it with TDC using numerical simulation. Our results indicate that the proposed scheme is an easily understood, numerically efficient, robust, and accurate solution for the control and synchronization of chaos systems.     

Noise-to-state finite-time practical stability for random nonlinear systems and its application in random Lagrange systems

The noise-to-state finite-time practical stability for random nonlinear systems and its application is studied in this paper. The definition of noise-to-state finite-time practical stability is firstly introduced in probability sense for random nonlinear systems. Next, the related stability criterion is also given by Lyapunov approach. For random benchmark system, the finite-time adaptive tracking control problem is investigated by the vectorial backstepping method and the obtained stability theorem. Simulation example illustrates that the constructed controller design scheme is effective and feasible.

     

A note on polynomial chaos expansions for designing a linear feedback control for nonlinear systems

This paper presents a polynomial chaos-based framework for designing optimal linear feedback control laws for nonlinear systems with stochastic parametric uncertainty. The spectral decomposition of the original stochastic dynamical model in an orthogonal polynomial basis, prescribed by the Wiener–Askey scheme, provides a deterministic model from which the optimal linear control law is designed. Optimality of the proposed control law is proved by solving the Hamilton–Jacobi–Bellman equation, and asymptotic stability of the controlled nonlinear systems is guaranteed in the Lyapunov sense. We are especially interested in synchronization of chaotic systems. For this reason, the control strategy is applied in the trajectory tracking of periodic orbits for the Duffing oscillator and the Rössler system with uncertain stochastic parameters and initial conditions. The results are verified with Monte Carlo simulations.     

Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators

This study aims to show that a multiple delay feedback control method can stabilize unstable fixed points of time-delay nonlinear oscillators. The boundary curves of stability in a control parameter space are derived using linear stability analysis. A simple procedure for designing a feedback gain is provided. The main advantage of this procedure is that the designed controller can stabilize a system even if the controller delay times are long. These analytical results are experimentally verified using electronic circuits.     

A criterion for the stability of motion of nonlinear system

As to an autonomous nonlinear system, the stability of the equilibrium state in a sufficiently small neighborhood of the equilibrium state can be determined by eigenvalues of the linear part of the nonlinear system provided that the eigenvalues are not in a critical case. Many methods may be used to detect the stability for a linear system. A lot of researches for determining the stability of a nonlinear system are completed by mathematicians and mechanicians but most of them are methods for the special forms of nonlinear systems. Till now, none of these methods can be conveniently applied to all nonlinear systems. The method introduced by this paper gives the necessary and sufficient conditions of the stability of a nonlinear system. The familiar Krasovski's method is a special case of this method.     

A criterion of robustness intelligent nonlinear control for multiple time-delay systems based on fuzzy Lyapunov methods

A control system with state feedback controllers, in which the fuzzy Lyapunov approach is developed for the stability criterion, is studied. The proposed intelligent design provides a systematic and effective framework for the control systems. The global nonlinear controller is constructed based on T–S (Takagi–Sugeno) fuzzy controller design techniques, blending all such local state feedback controllers. Based on this design, the stability conditions of a multiple time-delay system are derived in terms of the fuzzy Lyapunov theory. The effectiveness and the feasibility of the proposed controller design method are demonstrated through numerical simulations.     

A graphic stability criterion for non-commensurate fractional-order time-delay systems

This study focuses on a graphical approach to determine the stability of non-commensurate fractional-order systems with time-delay. By the iterative decomposition technique, the fractional-order systems described by the transfer functions are represented as a series of closed loop systems. The innermost closed loop system is straightforward to test the stability by its coefficient and the fractional-order. A function with respect to each open loop system is defined, and the stability of a non-commensurate fractional-order system is determined by the unstable poles of the innermost system and the times of the curve depending on the defined function encircling the origin in the clockwise direction. Finally, three illustrative examples are provided to demonstrate the effectiveness of the proposed criterion.     

A discrete nonlinear stepping optimization method and its application

Most of the practical design variables should always be discrete quantity within engineering optimization design problems. To obtain the true optimization solution, a discrete optimization method must be used. In this paper, a new method called step optimization search method is presented to solve the discrete quantity mathematic programming problems. The basic idea of this method is to find out an initial feasible point and then to search the optimum point step by step in the neighbouring region of this point so as to obtain an improved new discrete point. Respectively, the new point can be taken as initial one, and the whole process can be carried out once more until the optimum solution of the problem is obtained. Some results of numerical examples of practical problems show that this new method can solve problems quickly and simply and can be applied in a lot of engineering design problems.     

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.