Stability And Synchronization Control Of Fractional-order Nonlinear Systems

In recent years, with the leap forward development of computer science and technology, andfractional calculus theories are becoming more mature, fractional-order calculus as extensionof integer order calculus in order have attracted the attention of many scholars both at homeand abroad. Due to its powerful advantage and broad application prospects in many aspectssuch as physics, chemistry, biology and engineering, fractional calculus has become immedi-ate research focus. Especially, stability and control of the fractional-order nonlinear systemare the hot and difficult topic. This paper focuses on several typical fractional-order nonlin-ear systems, including fractional-order semilinear system, fractional-order complex network,fractional-order chaotic system and fractional-order(delayed) neural network, mainly concernson stability, stabilization and synchronous controller design, a series of conditions for ensuringstability and the method for stabilization and synchronization controller design are presented.The main contents include:1Stability and stabilization problems for a class of semilinear fractional-order systemsare considered. Based on the stability theorem of fractional-order linear system, Mittag-Lefflerfunction, Laplace transform, Gronwall inequality as well as inequality scaling skills, by esti-mating analytical solutions expression of system, several sufficient conditions for ensuring thelocal asymptotic stability and global asymptotic stability of the fractional-order nonlinear sys-tems for fractional order α belong to0<α≤1and1<α <2are presented. On this basis, aappropriate linear feedback controller is designed to achieve the stabilization of such systems.Only discussing and controlling the linear parts, without making any changes on the nonlin-ear parts, which has good theory meaning and project application value. Theoretical proof andexperimental results show that the feasibility and effectiveness of the conclusions.2Synchronization control for two kinds of fractional-order complex network are discussed.Based on the stability theorem of fractional-order system and pinning control scheme, by de-signing two suitable feedback controller, some sufficient conditions to realize cluster synchro-nization and adaptive synchronization are obtained respectively. For cluster synchronization,only the nodes in one community which have direct connections to the nodes in other commu-nities are needed to be controlled, resulting in reduced control cost. Numerical simulations il-lustrate that cluster synchronization performance is influenced by inner-coupling matrix, controlgain, coupling strength and topological structures of the networks. For adaptive synchroniza-tion, by designing a suitable adaptive pinning controller and a feedback gain updating law, somesufficient local asymptotical synchronization criteria and global asymptotical ones are derivedrespectively, which succeed in solving the problem about how many nodes are need to be con- trolled and how much coupling strength should be applied to ensure the synchronization of theentire fractional-order networks. Moreover, the coupling configuration matrices and the innercoupling matrices are not assumed to be symmetric and irreducible and the considered modelis more general. Theoretical proof and numerical simulations are presented to demonstrate thevalidity and feasibility of the proposed synchronization criteria.3Stabilization and synchronization control for two kinds of fractional-order chaotic sys-tem are considered. Firstly, based on the stability theory of fractional-order differential equa-tions and Routh-Hurwitz stability condition, by designing the linear state and error feedbackcontroller, some criteria for control and synchronization of a class of fractional-order nonlinearsystems are proposed, respectively. The proposed controllers are simple and easy to implement,which eliminate chaotic behavior and achieve synchronization effectively, and avoid the short-comings in the existing relevant results. Secondly, for a class of fractional-order chaotic systemswith parameter disturbance, on basis of the stability theory of fractional-order linear system andby constructing a suitable observer, a analysis method for synchronization control of the frac-tional order uncertain chaotic systems is obtained, and necessary and sufficient conditions inform of LMI for synchronizing such systems with fractional order α belong to0<α <1and1≤α <2are presented respectively. The matrix to determine feedback gains is very simpleand easily solved by MATLAB. Theoretical analysis and numerical simulation are proved thevalidity of the conclusions.4Stability and synchronization controller design for two kinds of fractional-order neuralnetworks are discussed. Firstly, for fractional-order neural networks without delay, by using offractional calculus theorem and Mittag-Leffler function, a simple and effective sufficient condi-tion in form of M matrix is established for the Mittag-Leffler stability and asymptotic stability.Based on the obtained results, a linear feedback error controller is designed for synchronizingsuch system. Secondly, for fractional-order neural networks with delay, a sufficient conditionfor the uniform stability of such network is proposed. Moreover, the existence, uniqueness andstability of its equilibrium point are also proved. A linear feedback error controller is designedfor synchronizing such system. The obtained results are provided theory basis for the designand application of fractional-order neural network. The correctness of the obtained results areillustrated by theoretical analysis and numerical simulation examples.