Research On Cooperative Control Of Fractional Multi-agent Systems With Delays On Signed Networks

During the past decades,the cooperative control of fractional multi-agent systems is favored by scholars because of its extensive application in various fields.But most of the early research results focus on the fractional multi-agent systems which only include cooperative relations among agents.However,there are both cooperative and competitive relationships among agents in some practical multi-agent systems,such as social networks,multi-robot systems,biological systems,etc.Such networks with both cooperative and competitive relationships are generally called signed networks,in recent years,the cooperative control issue on signed network has gradually become a research hotspot and been concerned and discussed by many scholars.In addition,time delay is often inevitable in practical systems,and will affect the performance of system,and even make the system lose stability.Therefore,it is of great theoretical and practical significance to study the cooperative control of fractional multi-agent systems with time delays on signed networks.This paper aims at two kinds of multi-agent systems based on Caputo and Riemann-Liouville’s fractional delay equations,on the basis of signed graph theory,Lyapunov function approach and fractional Razumikhin way,we systematically investigate the bipartite consensus and bipartite containment control problems of fractional multi-agent systems with delays.The main contents are as below:1.The bipartite consensus of Caputo’s fractional multi-agent systems with delay on signed directed networks is addressed.First,a delayed control protocol is developed to resolve bipartite consensus on fixed signed directed network,by means of signed graph theory and fractional-order Razumikhin theorem,a valid approach is presented to get over the difficulty resulting from fractional derivative and time delay,and several feasible conditions are obtained.Secondly,the communication interaction of multi-agent systems is usually time-varying in practical applications,which may be modeled by switching signed networks.Therefore,the bipartite consensus of systems on switching networks is further investigated by the aid of common Lyapunov function method.2.The bipartite consensus of Riemann-Liouville’s fractional multi-agent systems with delay on signed directed networks is addressed.First,two delay control protocols are designed to study the leader-following and leaderless bipartite consensus of RiemannLiouville’s fractional multi-agent systems with discrete delay,several feasible criteria are provided by applying classical Lyapunov direct method and signed graph theory.Secondly,the sufficient conditions on attractiveness of Riemann-Liouville’s fractional systems with mixed delays are derived by utilizing classical Lyapunov direct approach,properties of fractional calculus and analytical technique.Based on attractiveness conclusions and signed graph theory,the bipartite consensus of Riemann-Liouville’s fractional systems with distributed and input delays is further studied,and several valid and practicable criteria are presented.The main merit of proposed approach is that it allows us to calculate the first-order derivative of the corresponding Lyapunov function instead of its fractional derivative,avoiding computational complexity.3.The bipartite containment control of Caputo’s fractional multi-agent systems with input delay on fixed and switching signed directed networks is addressed.Delayed control protocol and switching control protocols are put forward,several sufficient conditions are provided to realize the bipartite containment control of systems by means of using signed graph theory,fractional Razumikhin technique and common Lyapunov function approach.The presented method can well overcome the difficulty resulting from fractional calculus,time delay and switchings.4.The observer-based bipartite containment control of Caputo’s fractional multiagent systems with distributed and input delays is addressed.First,a follower-based observer is designed by the local output information.On the basis of the state information observed,a delayed control protocol is proposed to achieve bipartite containment control on fixed signed directed network,in order to overcome the difficulty caused by distributed and input delays,we first employ fractional Razumikhin method to deal with the distributed delay,and utilize the matrix transformation and the property of similarity matrix to solve the coupling terms generated by the simultaneous design of observer and delayed controller,then several feasible conditions are obtained to realize the bipartite containment control of systems with the help of signed graph theory,the conditions can also ensure that the observation error to be asymptotically stable.Secondly,the observerbased bipartite containment control on switching networks is further realized by utilizing common Lyapunov function approach.