LMI-based stabilization of a class of fractional-order chaotic systems

Based on the theory of stabilization of fractional-order LTI interval systems, a simple controller for stabilization of a class of fractional-order chaotic systems is proposed in this paper. We consider the structure of the chaotic systems as fractional-order LTI interval systems due to the limited amplitude of chaotic trajectories. We introduce a simple feedback controller for the interval system and then, based on a recently established theorem for stabilization of interval systems, we reach to a linear matrix inequality (LMI) problem. Solving the LMI yields an appropriate decoupling feedback control law which suffices to bring the chaotic trajectories to the origin. Several illustrative examples are given which show the effectiveness of the method.     

Robust adaptive fractional-order observer for a class of fractional-order nonlinear systems with unknown parameters

Nonlinear Dynamics - This paper investigates the parameter and state estimation problems for a class of fractional-order nonlinear systems subject to the perturbation on the observer gain. The...     

Feedback stabilization of a class of nonlinear second-order systems

The goal of this paper is to study stabilization techniques for a system described by nonlinear second-order differential equations. The problem is to determine the feedback control as a function of the state variables. It is shown that the following controllers can asymptotically stabilize the system: linear position feedback, linear velocity feedback and a group of nonlinear feedbacks. The stability of the corresponding closed-loop system is proved by imposing a suitable Lyapunov function and then using LaSalle’s invariance principle. The results of numerical computations are included to verify theoretical analysis and mathematical formulation. Some application examples from robotics, mechanics and electronics are presented.     

Asymptotic stabilization of general nonlinear fractional-order systems with multiple time delays

Nonlinear Dynamics - In this paper, two new control methods based on a Lyapunov-like function and a vector Lyapunov function separately were put forward to solve the asymptotic stabilization...     

Studying on the stability of fractional-order nonlinear system

In this paper, we study the properties of the Mittag?CLeffler function and propose an approach for calculating the maximum norm of eigenvalue of Jacobian matrix of nonlinear system. Then a simple approach is proposed for judging the stability of fractional nonlinear system. Based on the approach, the maximum decay rate of fractional nonlinear system can be calculated. Finally, some examples are provided to illustrate the approach.     

Impulsive stabilization of chaos in fractional-order systems

Nonlinear Dynamics - This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos...     

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lü system, and fractional-order Duffing system.     

Parametric identification of fractional-order nonlinear systems

Nonlinear Dynamics - This work presents a new method for the identification of fractional-order nonlinear systems from time domain data. A parametric identification technique for integer-order...     

Asymptotic stabilization of a class of nonlinear SDEs

Nonlinear Dynamics - In this paper, we investigate the asymptotic stability of the probability density function (pdf) of the states of a class of nonlinear SDEs. We use the Detailed-balance...     

Output feedback stabilization based on dynamic surface control for a class of uncertain stochastic nonlinear systems

In this paper, the dynamic surface control (DSC) algorithm is proposed for a class of stochastic nonlinear systems with nonminimum phase and the standard output-feedback form. The proposed algorithm is a stochastic vision by combining the traditional back-stepping together with the DSC technique, which can overcome the problem of ‘explosion of complexity’ in the back-stepping designing procedure for the stochastic nonlinear systems. Thus, it can reduce the computation complexity and is easy to be used in the actual implementation. It is shown that all the signals of the resulting closed-loop system are uniformly ultimately bounded.     

Stability analysis of Caputo fractional-order nonlinear systems revisited

In this paper stability analysis of fractional-order nonlinear systems is studied. An extension of Lyapunov direct method for fractional-order systems using Bihari’s and Bellman–Gronwall’s inequality and a proof of comparison theorem for fractional-order systems are proposed.     

Passive decomposition and gradient control of fractional-order nonlinear systems

Nonlinear Dynamics - Vector field decomposition is essential in physics and engineering in order to analyze several dynamical systems that appear in these areas. This paper proposes some novel...     

External stability of Caputo fractional-order nonlinear control systems: advances in the Lyapunov function method

Nonlinear Dynamics - Nonlinear behavior of bubbles, and most importantly $$\frac{1}{2}$$ -order subharmonics (SH), are used to increase the contrast-to-tissue ratio (CTR) in diagnostic ultrasound...