Cooperative control of multi-agent systems has attracted considerable attention due to its wide applications in many areas such as mobile robots, unmanned air vehicles, satellites, and air-/spacecrafts. One of the basic problems in these applications is the consensus of multi-agent systems [1], [2], [3], [4] and [5]. Until very recently, there have been numerous new developments devoted to proposing protocols and solutions to consensus problems; see, e.g., [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and [17]. Among these studies, an important issue discussed in the protocol design is how to achieve consensus within a finite time [14], [15], [16] and [17]. It is mainly because the convergence rate is often considered as an important index for the performance evaluation of consensus protocols. As a consequence, the finite-time consensus is usually required in practice. Besides, the finite-time consensus is desirable in many situations such as the case requiring crucial control accuracies, and it can also show better performance in disturbance rejection and robustness against uncertainty [15].
With regard to effective finite-time techniques, iterative learning control (ILC, [18]) has been well known for its ability to achieve a perfect tracking of the desired target. Recently, ILC has been developed to improve the control performances of complex multiple systems, e.g., see [19] for satellites, [20] for multi-agent systems, and [21] for mobile robots. It has been shown that the introduction of ILC can help to provide some desired performances in achieving control objectives of multiple systems such as trajectory-keeping, reference-tracking, and leader–follower formation. Motivated by the developments of [19], [20] and [21] and the important applications of consensus theory claimed in [14], [15], [16] and [17], this paper attempts to solve the consensus problems for multi-agent systems through designing some distributed protocols in the two-dimensional system framework of ILC1. To our knowledge, there have been no studies on the consensus theory from the viewpoint of two-dimensional systems, and all existing consensus results are proposed in the one-dimensional system framework evolving along the time axis.
In this paper, effective distributed protocols are presented by incorporating the idea of ILC, and thus are termed as iterative learning protocols. On one hand, an iterative learning protocol improves its own performances gradually via an iteration process, which results in fundamentally a two-dimensional system with evolution along two independent axes, like the general ILC [18]. On the other hand, an iterative learning protocol is constructed for each agent based on communications with its neighbors, rather than the fixed targets, like the general protocols [4]. With these two features, it shows that iterative learning protocols can be used to effectively enable all agents to achieve the finite-time consensus, and design conditions can be obtained for the selections of learning gains. If, for any prescribed desired terminal output, it can be detected by some of the agents, then the iterative learning protocols can be modified to solve the finite-time consensus problem just at the desired terminal output, which coincides with the perfect tracking objective of ILC. The main contributions of the proposed results are as follows: (i) new methods to design consensus algorithms are provided by using the iterative learning approach, which can determine a protocol to enable the multi-agent system in a directed graph to achieve consensus in a finite time, and (ii) new methods to analyze iterative learning algorithms are developed by combining the graph theory, which can address the convergence problem of ILC processes not satisfying the contraction mapping principle. To verify the effectiveness of our iterative learning protocols, simulation results are finally given by considering a system of six agents which have relative degrees of different orders.
This paper is organized as follows. In Section 2, some preliminaries are introduced for the graph theory, and two finite-time consensus objectives are formulated under the two-dimensional system framework of ILC. Two classes of iterative learning protocols are presented in Section 3, together with their consensus analysis. Simulation results, and then conclusions, are given in Sections 4 and 5, respectively.
A finite-time output consensus problem for multi-agent systems has been discussed in this paper. By combining the effective ILC approaches, some distributed iterative learning protocols have been given. On one hand, this has provided alternative ways to address the consensus problems from the viewpoint of two-dimensional systems, which is challenging for the distributed control of multi-agent systems. On the other hand, the ILC approach has been developed greatly to address the control problems of complex distributed systems, which contributes to the ILC applications. Through making each agent learn from its dynamic neighbors, an iterative learning protocol has been shown with the ability to enable all agents to achieve the output consensus within a finite time. Based on this fact, the iterative learning protocol can be further modified to derive the finite-time consensus at any desired terminal output provided the desired terminal output can be detected by some of the agents. The effectiveness of our proposed iterative learning protocols has been illustrated finally through the numerical simulations.
