Research On Leader-following Consensus Of Fractional-order Nonlinear Multi-agent Systems

In recent years,with the development and widespread use of distributed artificial intelligence technology,the coordination control of multi-agent systems(MASs)has attracted a large number attention of researchers.Most of the existing mathematical modeling methods of MASs are based on integral differential equations,which means the model assumption is the physical structure of MASs is completely rigid.However,in the real physical system,there exist some flexible structural characteristics more or less,so the dynamic characteristics of MASs are described only by integral differential equations will make the system miss part of the information in most cases.Fractional-order is more accurate than integral-order in modeling the actual physical system and more suitable for the specific physical characteristics.Furthermore,considering the uncertainty of the system and the limited network bandwidth,the application of adaptive control and event triggered control can improve the impact of these two problems on the dynamic performance of the system to a certain extent.The main works of the paper are as follows,which focuses on the leader-following consensus of fractional-order MASs:(1)The consensus problems of double-integral fractional-order MASs with linearity and nonlinearity are considered.In the ideal case,a new centralized control strategy based on state errors is designed by using the inverse matrix of the sum of Laplacian matrix and leader adjacency matrix as the gain matrix of state error variables.Two Lyapunov functions are selected respectively to prove the position state and velocity state are finally synchronized.With nonlinear disturbance,a novel distributed control strategy is designed based on topology structure,which only need the state information from the adjacency agents.Then,a special matrix is defined to construct the state error system of MASs.Furthermore,the consensus can be proved by Lyapunov stability to be achieved when the lower bound of the eigenvalue of the matrix is greater than Lipschitz constant of the nonlinear function.(2)The consensus problem of leader-following fractional-order multi-agent systems is considered.Consider the unnecessary waste of network resources,a novel distributed event-triggered control method is designed with defining a inner coupling matrix of state errors.The triggering condition is whether the norm of tracking error exceeds the corresponding exponential function value.The Lyapunov function is selected based on the tracking errors,and the sufficient conditions can be reached by the stability criterion.Furthermore,by using the Kronecker product characteristics,Shure complement lemma and fractional Lyapunov stability criteria,the effectiveness of the event-triggered controller are verified and the Zero behavior does not exist under triggering conditions,which means the smallest triggering interval is not zero.Finally,two simulation examples under different triggering conditions are given,which not only proves the correctness and effectiveness of the above derivation,but also gives the difference of dynamic performance of the system under different triggering conditions.(3)The adaptive event-triggered consensus of leader-following nonlinear fractional-order MASs is considered.Considering uncertainty and control cost saving of MASs,a novel adaptive event-triggered control protocol composed of gain matrix,adaptive parameters and state errors is designed,and the corresponding trigger conditions and adaptive parameter updating law are presented.The sufficient condition of achieving consensus is derived by using Lyapunov stability and linear matrix inequality.Furthermore,that the system will not behave Zeno phenomenon under the control of the designed adaptive event triggered controller is proved using by matrix analysis,fractional integral and limit transformation.Finally,a specific simulation example is designed to show the curves of state error,control input,event trigger time and adaptive parameters respectively,which makes the above theoretical results more convincing.