Deyuan Meng · Yuxin Wu

Distributed control is a hot topic in many research f ields.Thanks to the development of communication and computation technologies, plenty of complex engineering problems can resort to the mechanism of distributed control. Of note is that most of the distributed control design only leverages the real-time information of agents, but as a consequence, only some specif ic steady-state objectives and/or system performances can be realized. Nonetheless, in many real scenarios,of particular importance are the transient behaviors of control systems. To deal with such problems in conventional situations, iterative learning control (ILC) may provide a good alternative method that has been widely studied. ILC is one of the most popular intelligent control methodologies,which is particularly applicable for control systems running in a repetitive process over some f inite time horizon of interest (see, e.g., [ 1, 2] and references therein). As a result, a classic ILC system generally evolves in the presence of two axes, i.e., the f inite time axis and the inf inite iteration axis.The working mechanism of ILC is the learning from the past experiences, which enables it to improve the transient performances of control systems (along the time axis). Because the implementation of ILC never needs the accurate model information but only the measurement and control data, it belongs to the framework of the model-free control methods in some sense, and thus, is eff ective in many practical applications, such as robotics, batch processes, and transportation systems.

In recent years, there emerge new challenging practical problems with transient concerns. Take, for example, the smart factory. In addition to the fact that the data fusion of wireless sensor networks may have some transient requirements, the smart shop-f loor objects, such as machines, conveyers, and products, can all be regarded as agents such that some transient system-wide goals may need to be achieved through the collaborations of all smart objects [ 3]. Other typical examples include the f light control of unmanned aerial vehicles and the formation of a group of satellites[ 4]. This motivates the integration of ILC to the distributed control. Through the synthesis of ILC and distributed control, all agents are capable of learning from not only the information of other agents, but also the experiences in the past iterations [ 5]. Therefore, it may be a feasible way to enhance the performances of multi-agent systems such that some signif icant transient behaviors can be achieved.

The design of distributed ILC algorithms mainly follows the nearest neighbor rule, under which the control input of each agent updates only based on its control input information and its neighbors’ output information in the previous iterations. Through agents learning from each other, all agents can remain in a preset formation at every instant over the time horizon of interest such that the transient formation is realized for multi-agent systems after several iterations.The transient formation problem for multi-agent systems with respect to distributed ILC algorithms can generally be classif ied into two categories, namely, the leaderless framework [ 6, 7] and the leader-follower framework [ 8, 9]. The former one only attempts to keep all agents in the desired formation without any requirements of the formation center,which depends on both the initial conditions of agents and the topology structures, whereas the latter one aims to drive all agents to form a specif ic formation centered on a reference trajectory given by a (virtual) leader agent. The convergence analysis of multi-agent systems along the iteration axis is commonly transformed to the equivalent stability analysis of some related auxiliary error systems. For the leaderless case, the error systems are generally developed based on the error of the relative outputs (the agent’s output minus its formation position) between one specif ied agent and other agents, and for the leader-follower case, the error systems can be simply constructed by the error between the agent’s relative output and the reference formation center.To perform the stability analysis of the associated error systems, some feasible analysis methods have been investigated to cope with diff erent agent dynamics thanks to the unique characteristics of ILC, mainly including the contraction mapping-based method and the composite energy functionbased method (see, e.g., [ 10]).

For leaderless multi-agent systems, a distributed ILC algorithm was f irst presented in [ 6] concerning agents with unknown nonlinear dynamics to accomplish the prescribed transient formation under a topology-dependent description,where the f ixed topology of a special structure was involved.Then, a relatively thorough transient formation result was provided in [ 7], in which the 2-D Roesser system-based method was employed to disclose the necessity and suffi -ciency of the joint spanning tree condition along the iteration axis for the realization of specif ic transient formation. Note that due to the iteration-variation of topologies, the repetitiveness of multi-agent systems along the iteration axis cannot be guaranteed, which is regarded as a basic requirement for the implementation of ILC (see, e.g., [ 1, 2]). It clearly indicates that thanks to the agents’ learning from each other,ILC can eff ectively work without the repetitiveness condition. As an improvement, the robust transient formation problem was further discussed in [ 11], which revealed that both the iteration-varying initial shifts and the external disturbances have no essential impacts on realizing the transient formation objective of non-identical time-varying nonlinear multi-agent systems if their uncertainties exponentially converge to zero along the iteration axis.

The transient formation problem for leader-follower multi-agent systems is generally easier to be dealt with to some extent compared with that under the leaderless framework. To be specif ic, for the leader-follower multi-agent systems, the stability analysis of the constructed error systems can be directly implemented by taking advantage of the topology structures among agents together with some common ILC analysis methods. This attributes to the fact that the simple construction of their error systems preserves the complete structure of Laplacian matrix in the system dynamics, and thus, the good properties of Laplacian matrix can be utilized. By contrast,the dynamics of error systems for leaderless multi-agent systems are much more complicated, as a consequence of which the convergence analysis methods for them can be similarly applied to the leader-follower case, but not vice versa. Although the transient formation problem of leader-follower multi-agent systems may be less challenging from the technical perspective, it is more popular in some practical applications, such as the quadrotors formation [ 12], the mobile robots formation [ 13], and the large-scale building temperature regulation [ 14], since it can ensure all agents to move along the targeted reference center as well as forming the desired formation. It is worth mentioning that the leader-follower framework has its own limitations. For example, it may not be applicable for multi-agent systems with antagonistic interactions [ 15]to reach some desired transient behaviors, which, however,is possible under the leaderless framework.

Here we only provide a brief review on the studies of distributed ILC for multi-agent systems. By benef iting from the learning mechanism of ILC, the transient behaviors of multi-agent systems can be prominently improved. However, how to use data to realize the transient formation of multi-agent systems without any model information remains unsolved. In addition, how to better overcome the inf luences of iteration-varying uncertainties is still an open question.These problems may be possible future directions to enhance the distributed ILC algorithms for multi-agent systems.

AcknowledgementThis work was supported by the National Natural Science Foundation of China (Nos. 61922007, 61873013).