Application Of Fractional-order Calculus In A Class Of Multi-agent System

The real world can be described by factional-order systems essentially. In the past, the integer-order systems were used to describe the nature. But in recent years,it was found that the traditional integer-order differential equation can not be used to accurately describe many phenomena in nature, while the fractional-order system is be able to accumulate a certain range of all information with good memories. This provide an excellent tool for the description of memories which are often neglected in the classical integer-order models. Thus it is more realistic to model the actual world by fractional-order calculus. Meanwhile, as an extension of the integer-order system, the fractional-order system contains the case of the classical integer ones. The calculation of fractional-order in many areas of science and engineering plays an increasingly important role based on the properties of fractional-order derivatives and integrals which has also attracted increasing attention of many researchers from different fields.In the fields of automatic control, signal processing and some others, the problems of stability, synchronization, and consensus have been a hot issue of concern to researchers. Con-sensus is a typical behavior of cooperative control of multi-agent system which means the final state of all agents can reach the same value. In addition, cooperative control of the multi-agent system which consists of a group of agents by information communication has been widely in-vestigated recently. The state information of agent cannot be measured directly sometimes in multi-agent system, so the observer design is needed to based on the measurement of outputs in order to reach the control of whole system. The consensus investigation of fractional-order systems exists certain difficulties for lacking of the similar stability theory of integer-order ones. The typical methods used are Laplace transform、Linear Matrix Inequality and nu-merical simulations, which all have some limitation when dealing with large-scale systems. In addition, the system will be affected by the influence of external disturbance and uncertain parameters inevitably in most practical systems which can affect the stability and performance of the system. Based on the above discussions, this paper mainly investigate the application of fractional-order calculus in multi-agent cooperative control problem by using the excellent properties of fractional-order calculus.The main work of this paper is listed as follows:In the first chapter, a brief introduction of fractional-order system,background and mean-ing of consensus behaviors in multi-agent systems as well as some preliminary knowledge which contain graph theory, basic functions in fractional-order calculus and definition of fractional-order calculus are given. The main results of this thesis are based on the above basic knowledge.The second chapter studies the design of observer for the consensus of a class of linear fractional-order multi-agent system. An observer-type control protocol of related output mea-surement of neighbors is derived, so the consensus problem can be casted into the asymptotic stability problem of a series of matrices with lower dimension which can simplify the computa-tional complexity. A sufficient condition and a necessary and sufficient condition are derived. Finally, numerical simulations are presented to verify the theoretical analysis.The third chapter investigates the leader-following consensus problem of second-order multi-agent system. In most of the previous studies, to reach the general second-order leader-following consensus, one should use both position and velocity information of the neighbors. As we know, the velocity information is more difficult to be measured in practice. Based on this, this chapter considers a system which consists of second-order leader and fractional-order followers, a necessary and sufficient condition of tracking consensus is derived only by using the local position information of the neighbors and the leader if they are informed. The numerical simulation is presented to verify the effectiveness of the result.The forth chapter considers the consensus problem of uncertain fractional-order multi-agent system. In order to reach consensus with external disturbance and uncertain parameters, a control protocol based on the related information of neighbors as well as its absolute infor-mation is given. The consensus problem when 0< a< 1 and 1<a<2 are discussed, respectively under the control protocol. Two sufficient conditions are then derived. A simple numerical example is given to verify the validity of the results finally.The last chapter summarizes the research work of this dissertation. Additionally, the possible improved theories are proposed and the prospect for the further work is illustrated.