Application of fractional-order calculus in a class of multi-agent systems

Abstract

This chapter is concerned with fractional-order consensus problem in multi-agent systems. A brief introduction of fractional-order calculus is given in Sect. 9.1.The design of observer for consensus of a linear fractional-order multi-agent system is discussed in Sect. 9.2. Section 9.3 considers a multi-agent system consisting of second-order leader and fractional-order followers where a necessary and sufficient condition of tracking consensus is derived by using only the relative local position information of neighboring agents. The stabilization consensus problem of uncertain fractional-order multi-agent system is investigated in Sect. 9.4. Coordination ofmulti-agent systems has numerous applications.Examples include flocking, swarming, formation control, and sensor networks [1-4]. The study within this field focuses on analyzing how globally coordinative group behavior emerges as a result of local interactions among the individuals. In many cooperative multi-agent systems, a group of agents only share information with their neighbors locally and simultaneously try to agree on certain global criteria of common interest. Recently, consensus, one of the most typical collective behaviors, plays an important role in the distributed coordination which usually refers to the problem of how to reach an agreement, such as the positions, velocities, and attitudes, among a group of autonomous mobile agents in a dynamical system [5-9]. Recently, second-order consensus problem has come to be an important topic [10-15], where each agent is governed by second-order dynamics. In general, the second-order consensus problem refers to the problem of reaching an agreement among a group of autonomous agents governed by second-order dynamics. Some sufficient conditions were derived for reaching second-order consensus in linear models [12, 14]. In [14], some necessary and sufficient conditions were obtained for second-order consensus in a linear multi-agent system containing a directed spanning tree with or without delay. It was found that both the real and imaginary parts of the eigenvalues of Laplacian matrix of the network play key roles in reaching second-order consensus in general. In the leader-following case, each agent should exchange both position and velocity information with its neighbors in order to reach second-order tracking consensus. Fractional calculus can be dated back to the seventeenth century. Different from the integer orders of derivatives and integrals in the conventional calculus, the orders of derivatives and integrals in fractional calculus can be any given positive real numbers. It should be noted that, to date, most papers have studied integer-order multi-agent systems. However, it has been pointed out by many researchers that many physical systems are more suitable to be described by fractional-order dynamic equations rather than by the classic integer-order ones, such as, vehicles moving on top of viscoelastic materials (e.g., sand, muddy road) or high-speed aircrafts traveling in an environment with the influence of particles (e.g., rain, snow) [16]. Moreover, many phenomena can be explained naturally by the collective group behavior of agents with fractional-order dynamics. For example, the synchronized motion of agents in fractional circumstances, such as macromolecule fluids and porous media. Fractional-order derivatives provide an excellent instrument for the description of memories which are neglected in the classical integer-order models. In addition, fractional-order systems include traditional integer-order systems as special cases. Most of the real systems can be described by factional-order systems essentially [17-21]. In the past, the integer-order systems were used to describe the nature. But in recent years, itwas found that the traditional integer-order differential equation can not be used to accurately describe many phenomena in nature, while the fractionalorder system is able to accumulate a certain range of all the information with good memories and thus has wider applications. This provides an excellent tool for the description of memories which are often neglected in the classical integer-order models. Thus it is more reasonable to model the practical systems by fractionalorder systems. The calculation of fractional order in many areas of science and engineering plays an important role, which has also attracted increasing attention of many researchers from different fields. Sometimes, the state information of agent cannot be measured directly in multiagent systems, so the observer-based control laws depending on the output measurements of the agents are needed [22-24]. The investigation on consensus of fractional-order systems meets certain difficulties for lacking the similar stability theory of integer-order ones. The typically used methods are Laplace transforms and numerical simulations,which all have some limitations in dealingwith the large-scale systems. In addition, the system’s evolution will be inevitably affected by external disturbance and uncertain parameters in most practical systems. Thus, it is important to include the uncertainties or disturbances into the model of considered system. Based on the above discussions, this chapter mainly investigates the applications of fractional-order calculus in multi-agent cooperative control problem by using the properties of fractional-order calculus. The main contribution is that some new control protocols are derived to reach consensus in a nominal multi-agent system or stabilization consensus in a class of uncertain fractional-order multi-agent systems.