Tracking Consensus For Fractional-order Multi-agent Systems With Adaptive Control

The tracking consensus is an important part of multi-agent cooperative control for multiagent systems(MASs),and has broad application prospects in engineering practice.Therefore,it has been written them into papers in recent decades.The existing integer-order control protocols often lose some important information when describing complex systems by its inherent shortcomings.They may be resulted in poor consistency and work unstably after running for a period of time.Compared with these integer-order shortcomings,fractionalorder protocols have natural advantages in systems that require memory and heritability,and are more suitable for describing complex control systems.The fractional-order protocols have stronger application functions for the integer models in MASs.Therefore,MASs in this paper focuses on considering the tracking consensus with fractional-order control protocols.The main tasks are as follows:(1)The tracking consensus are considered without the relative-state of the agent for fractional-order multi-agent systems(FOMASs).Two fractional-order adaptive tracking edgebased protocols are designed by the relative output of agents(including linear and non-linear protocols).In the undirected and connected communication topology graph,two distributed adaptive control protocols are proposed by the relative-output and protocol-state in FOMASs.The main contents include the time-varying control input,time-varying coupling weight and the fractional protocol-state.To design the agent’s protocol state and control input,the system assigns a time-varying adaptive coupling weight to each communication edge.The error function is designed by the relative-output and the protocol-state.Through the fractional Lyapunov stability theorem and the Kronecker product property of matrices,it is proved that the error function would tend to 0,and the system will be asymptotically stable and achieve consensus under the fractional-order protocol.Finally,two example simulations in MATLAB verify the accuracy of the theoretical results.(2)Two node-based tracking adaptive protocols are studied to achieve consensus for FOASs(two types of obtainable relative-state and only obtainable relative-output).In an undirected and connected communication topology graph,the relative-states are used to design the coupling weight and control input when the relative-states can be obtained.The error function is the common state difference between the follower and the leader.When the error function tends to be stable and is 0,the system achieves tracking consensus.Without the relative-state of the agent,the design process is similar to the above(1).But the designed adaptive control protocol is also different,especially the coupling weight and control input.The node-based time-varying coupling weights are considered for complex systems.Finally,the fractional Lyapunov stability theorem and the Kronecker product property of matrices are used to prove asymptotically stable and achieve consensus under this protocol.The accuracy of the theoretical results is verified by two MATLAB simulations.(3)The paper consider fractional Lyapunov functions and several variants of LMI.Several fractional-order adaptive control protocols use graph theory knowledge,fractionalorder Barbalat lemma,Kronecker product and Schur complement lemma.Complete the theoretical proof in the designed Lyapunov function stability proof,and are used in the.System tracking consensus is obtained by the theorems.Finally,the examples of MATLAB are used to simulate the accuracy of the theoretical results.