Dynamical Analysis And Control For Several Classes Of Fractional-order Systems

In recent years, fractional calculus has attracted tremendous attention of many researchers and become a hot topic because it was widely applied in many fields such as fluid mechan-ics, viscoelasticity, material science, quantum mechanics, bioengineering, biological models, medicine, etc. The fractional calculus is a generalization of the classical differentiation and integration to arbitrary non-integer order. The most advantage of fractional-order models in comparison with classical integer-order calculus is that fractional-order systems have infi-nite memory and hereditary properties which made it more effective and accurate to describe some physical phenomenon. This paper focuses on several typical fractional-order systems, including fractional-order(memristor-based) neural network, fractional-order multi-agent sys-tem, fractional-order chaotic system and fractional-order differential inclusions. Based on the fractional calculus theory, fixed pointed theory, inequality theory, differential inclusions, graph theory and control theory, a series of sufficient conditions for ensuring existence, uniqueness stability and the method for realizing synchronization and consensus controller design are presented. The main contents can be summarized as follows:(1) The stability of two kinds of fractional-order neural networks are discussed. First, for fractional-order neural networks without delays, by using of fractional calculus theory, fixed pointed theory and inequality theory, sufficient conditions for ensuring the existence and u-niqueness of nontrivial solution are obtained. Based on the obtained results, the nontrivial solution is global existence and finite time stability. Second, for fractional-order neural net-works with delays, sufficient conditions for the uniform stability of such network are proposed. Moreover, the existence, uniqueness and stability of the nontrivial solution are also proved. The validity of the results is illustrated by numerical examples. The obtained results are provided theory basis for the design and application of fractional-order neural network.(2) The synchronization of drive-response memristor-based neural networks with fractional-order derivative is considered. First, based on the differential inclusions theory, the equation with discontinuous right-hand hands is transformed into differential inclusions problems, then the sufficient condition of synchronization for drive-response fractional-order memristor-based neural networks with uncertainty is presented by means of the equivalent representation the-orem of fractional-order differential equation, Lyapunov function and matrix inequality tech-nique. The obtained result is represented by linear matrix inequality, therefore it is simple and easy to implement. The results laid a foundation for the engineering application of fractional-order memristor-based neural networks.(3) The consensus control for fractional-order multi-agent systems is considered. Based on the information of the second neighbors, by designing a new distributed protocol, some con-sensus criteria for a class of fractional-order multi-agent systems with positive real uncertainty in form of linear matrix inequality with fractional order a (0<a<1 and 1<a<2) are proposed respectively. The matrix to determine feedback gains is very simple and easily solved by MATLAB. Numerical examples and simulation are presented to demonstrate the effectiveness of the proposed approach.(4) The synchronization control for two kinds of fractional-order chaotic system are con-sidered. First, based on the stability theory of incommensurate fractional-order linear system theory, some criteria of generalized projective synchronization for a class of fractional-order nonlinear chaotic systems are presented by designing the error feedback controller. The pro-posed controllers are simple and easy to implement. Second, for a class of fractional-order chaotic systems with parameter uncertainties and external disturbances, using of the stability theory of fractional-order linear system and the theory of fractional-order sliding-mode con-trol, a sufficient condition of robustness synchronization for the fractional order drive-response chaotic systems is obtained by designing the output feedback sliding mode controller. Nu-merical examples are given to show the effectiveness of the conclusions. The obtained results provided theory basis for the application of secure communication.(5) The existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions is investigated. Utilizing the theory of fractional calculus, differen- tial inclusions, fixed point theory for multivalued maps and the technique of inequality, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. An illustrative example shows the validity of the theoretical results.