Time-varying group formation tracking control for second-order multi-agent systems with communication delays and multiple leaders

Abstract

Time-varying group formation tracking control problems for second-order multi-agent systems with communication delays and multiple leaders are addressed in this paper. The followers are divided into multiple subgroups, and each subgroup is driven to complete the time-varying sub-formation and to track the average state of entire leaders simultaneously. A distributed protocol that considers communication delays and uses only local interaction information is presented. According to Lyapunov-like stability approach, sufficient conditions for the multi-agent systems to realize time-varying group formation tracking with multiple leaders are given. A variable-substitution based method is developed to get the unknown feedback matrix of the protocol and an algorithm is given to summarize the steps to construct the protocol. The theoretical results can be used in the problem of multiple targets enclosed by several subgroups of unmanned aerial vehicles. Finally, numerical examples are applied to verify the feasibility of the proposed results.

Introduction

In recent years, cooperative control technology of multi-agent system has rapidly developed due to its widespread applications in military and civilian areas, including underwater robots [1], unmanned ground vehicles (UGVs) [2], unmanned aerial vehicles (UAVs) [3], satellites [4], etc. The cooperative control technology has developed several branches so far, where consensus control [5], [6], formation control [7], [8], and tracking control [9], [10] are three of the most representative branches. Researchers have already made a lot of progress on cooperative control techniques and now increased attention is being devoted to more comprehensive and more effective approaches for multi-agent systems.

Formation control approaches have been extensively applied to multi-agent systems as mainstream cooperative control techniques, where all agents are required to reach the specified formation. Based on the consensus strategy, a formation control method that uses partial neighboring state information was proposed for multi-agent systems with classical second-order dynamics in [11]. The formation control methods, which considered finite-time constraint, were developed for multi-agent systems with first-order dynamics and second-order dynamics in [12] and [13], respectively. In [14], the decentralized information feedback based formation approach was presented for first-order multi-agent systems under switching undirected topology to accomplish the time-invariable formation. The work in [15] discussed the feasible formation methods for second-order multi-agent systems with different transmission delays to fulfill the desired formation. A formation method based leader-follower strategy for second-order multi-agent systems with communication delays and nonlinear dynamics was presented in [16]. In contrast with the fixed formation cases, the time-varying formation problems in industrial applications including target tracking, securing, and enclosing, have more research significance. A formation control method, which considered collective circular behavior, was developed for multi-agent systems under jointly connected topologies to complete the obstacle avoidance while preserving the formation in [17]. In [18], based on direction information and neighboring relative position, a decentralized formation control method was introduced for heteroid multi-agent systems to realize the circular formation. The work in [19] discussed the three-dimensional circular formation problem and gave a layered formation control approach to perform the desired formation. The aforementioned works only study the specified formation problem that all agents complete the single formation together. However, in the complex applications, such as multiple targets enclosing and cooperative searching, the agents of multi-agent systems are required to build several groups to perform different tasks simultaneously. According to the practical demands, the group formation control, which splits the agents into multiple sub-groups while achieving the sub-formations, are presented in [20], [21]. The time-varying group formation approach was developed for time-invariant multi-agent systems with second-order dynamics under directed topologies in [20]. Furthermore, using the Riccati equation based stability analysis method, sufficient conditions for the multi-agent system with general linear dynamics to complete the time-varying group formation were analyzed in [21]. These research results gave theoretical supports for the applications of multi-agent systems. In [22], the distributed estimator and formation controller were presented for multiple nonholonomic mobile robots to form the formation while tracking the leader robot, and experiment examples were given to evaluate the feasibility of the results. In [23], the binary-tree network based cooperative control approach was proposed for mutli-UAV system under switching topology to complete the formation flight. Based on the consensus strategy, outdoor experiments with switching topology were applied to test the feasibility of multi-UAV formation control approach in [24]. The work in [25] presented a distributed robust method for nonlinear multi-UAV system with disturbances to attain the desired formation and the robustness property of the system was analyzed.

The above researches study the fixed formation, time-varying formation, and group formation control problems. These results only take the stability or stabilization into consideration. However, in many engineering applications, the agents are driven to track the trajectory provided by the other cooperative/non-cooperative agents while forming the effective formation, which has important research value. In [26], the agents were required to realize the fixed formation while following the virtual leader. A differential formation tracking method considering collision avoidance was presented for multi-agent systems in [27]. Utilizing complex Laplacian theory, the formation tracking strategies were proposed for multi-agent systems in [28], respectively. The work in [29] studied the finite-time formation control problem for multi-agent systems and presented the fast terminal sliding mode based control approach to complete the predefined formation tracking. The formation tracking control problems for general multi-agent systems under switching topology were addressed in [30], where the formation is time-varying and a local neighboring state based nonlinear formation tracking protocol was given. In [31], a formation tracking method with state observer was provided to multi-UAV system to complete the output-feedback formation tracking. In [32], an adaptive control scheme was presented for multi-UAV system under uncertainties and nonholonomic constraints to attain the time-varying formation tracking. The works in [26], [27], [28], [29], [30], [31], [32], [33], [34] only assumed that there only exists single leader/target in multi-agent systems. However, if multiple leaders/targets exist in multi-agent systems, the problem will become more complicated since the followers of multi-agent systems are required to track the leaders/targets and to preserve the feasible formation simultaneously, ensuring that each leader/target can lie inside it. The formation tracking problem with multiple leaders was addressed in [35], where the relative state based protocol was constructed for performing the time-varying formation tracking. In [36], an adaptive fault-tolerant formation tracking control protocol, which considered input saturation and actuator failure, was designed for time-invariant multi-agent systems with multiple leaders, and sufficient conditions for the closed-loop multi-agent system to complete the given time-varying formation were presented. The works in [37] proposed a fully-distributed control protocol for the high-order linear multi-agent system with multiple leaders to address the time-varying group formation tracking problem. The above mentioned formation tracking problems considered the influences from switching topology, input saturation, and actuator failure, etc. However, as one of the main influencing factors, communication delays are inevitable in applications of multi-agent systems, which are not considered in the above researches. It can influence the performance of multi-agent systems and even cause the divergence. Therefore, the study on communication delays in multi-agent systems is valuable. According to the reviews on the existing researches, the time-varying group formation tracking problem with communication delays and multiple leaders is still open.

This paper discusses group formation tracking problems for second-order multi-agent systems with communication delays and multiple leaders, where the followers formed several subgroups to attain the time-varying sub-formations while tracking multiple leaders in groups. First, the distributed protocol that utilizes the local interaction information is presented to attain the time-varying group formation tracking with multiple leaders. Second, applying the Lyapunov-like stability method and matrix decomposition method, sufficient conditions for the closed-loop multi-agent systems with communication delays and multiple leaders to preform the time-varying group formation tracking are presented. Moreover, based on the variable-substitution strategy, an approach to obtain the undetermined feedback matrices of the time-varying group formation tracking protocol is presented. In addition, an algorithm is provided to summarize the steps of constructing the given protocol. Finally, numerical examples on multiple targets enclosed by several subgroups of UAVs are given.

Compared with the prior researches, the contributions of this paper are as followings. First, the works in [15], [16], [17] only took the fixed formation into consideration. However, this paper discusses the distributed time-varying group formation tracking problem, where all the followers are divided into multiple subgroups and each subgroup is controlled to track the average state of leaders while forming the desired sub-formation. Note that preserving time-varying sub-formations and tracking leaders concurrently bring challenges to multi-agent system. Second, in contrast with the previous results in [26], [27], [28], [29], [30], [33], [34], which only considered multi-agent systems with a single leader, the cases with multiple leaders are investigated in this paper. The proposed results can also be used for consensus tracking and targets enclosing problems. Third, communication delays that constantly exist in practical multi-agent systems were not considered in [35], [36], [37], while the distributed protocol considering time-varying delays are presented for the multi-agent system to attain the time-varying group formation tracking with multiple leaders.

The structure of this work is given as followings. Necessary theories and definitions related with the time-varying group formation tracking problem are given in Section 2. In Section 3, sufficient conditions for the closed-loop multi-agent system with communication delay and multiple leaders to realize time-varying group formation tracking are proposed. In addition, based on the variable-substitution strategy, a feasible approach to acquire the undetermined feedback matrix of the protocol is presented. In Section 4, numerical examples on multiple UAVs enclosing problem are developed. Finally, Section 5 concludes the whole works.