Cooperative control with designated convergence rate for high-order integrators under heterogeneous couplings☆
Introduction
In recent decades, cooperative control is of considerable interest for its widely use in multi-agent systems (MASs), ranging from consensus [1], [2], [3], [4], flocking [5], swarming [6], formation [7], etc. The consensus control problem is one of the fundamental topics in cooperative control. For simple models, first-order and second-order MASs, the asymptotic consensus can be achieved by using the conventional distributed control algorithms, see for example, in [8], [9], [10], [11], [12]. In practical applications, since the MAS models can be much more complex in many aspects, such as system uncertainties [13], network topologies [14], and requirements of dynamic performance [15], there are pending problems to be solved, i.e., the convergence problem.
Fast convergence which implies better performance is of great importance for MASs. However, it is worth mentioning that only the asymptotic stability was considered in most of the existing literature, which means the consensus can be achieved with time asymptotic trends to infinity. The authors in [16] studied the cooperative robust regulation problem for a class of MASs subject to uncertainties, where the internal model design method was adopted to achieve asymptotic tracking. An output formation-containment protocol was designed in [17] to reach asymptotic convergence. Some results on finite-time and fixed-time consensus have been reported in recent years. In [18], the authors exploited the finite-time consensus problem for networks of dynamic agents. They proposed a distributed control algorithm under the bidirectional and unidirectional topologies. A fixed-time consensus tracking problem for perturbed high-order integrator MASs was investigated in [19]. The authors presented a fixed-time distributed observer, based on which the leader–follower tracking consensus algorithm was proposed in a cascade control structure. Although they took the convergence time into consideration, the convergence rate was not specified. Consequently, it is significant to study how to guarantee the designated convergence rate (DCR) in the cooperative control problem. A robust distributed model predictive control strategy was introduced in [20] for a class of MASs with dynamically decoupled subsystems. The contraction theory was utilized to provide the capability of exponential convergence. In [21] and [22], the designated convergence rate problem was investigated in the context of consensus problem for coupled multiple second-order agents. The authors proposed the conditions to guarantee the convergence rate under non-equal velocity and position couplings. It should be pointed out that the condition of solving the DCR problem for high-order MASs has been rarely studied in existing works.
For the cooperative control problem, either the individual agent dynamic or the exchange of information among the agents was mainly adopted in the existing literature. Recently, more efforts have been spared on developing the communication protocol to satisfy the practical requirements, such as limited resources [23], and convergence rate [21]. Therefore, the research on the communication network has attracted increasingly attention in this field. In [24], the authors investigated the consensus tracking problem under heterogeneous position and velocity topologies. Some sufficient conditions for solving the consensus problem were illustrated. The adaptive state couplings of the complex network were taken into account in [25], where the finite-time and synchronization conditions were presented. For the intermittent communication networks, the consensus algorithms have been proposed in the works of [26], [27]. In practical applications, the considering systems are usually high-order, and the problem of cooperative control is required to be solved for high-order dynamics. In [28], a linear consensus algorithm which includes a feedback controller and interactions from the neighbors, is developed for high-order MASs. Tian [29] made a significant progress and developed consensus algorithm for high-order heterogeneous MASs with unknown communication delays, where a necessary and sufficient condition for the existence of consensus was proposed in this paper. Furthermore, the distributed formation control problem [30] and the containment control problem [31] of high-order MASs were investigated in recent literature.
In this paper, we investigate the cooperative control of high-order integrators under heterogeneous couplings. Inspired by the consensus control algorithms based on DCR problems proposed in [21] and [22], we extend their work from double-integrators to high-order integrators. Furthermore, we investigate the output leaderless consensus of heterogeneous high-order integrators. Although the proposed distributed control algorithms require full state of heterogeneous couplings among agents, the main contribution of this paper is summarized as follows:
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The distributed control algorithms for high-order integrators under heterogeneous couplings are developed based on the DCR problem, which has been rarely investigated in literature. Furthermore, necessary and sufficient conditions for state consensus of high-order integrators are proposed by theoretical analysis.
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The DCR based consensus control algorithms can explicitly show the convergence margin, which is more realistic and robust than most conventional consensus control algorithms. The proposed method considers the output consensus of heterogeneous high-order integrators under heterogeneous couplings, which is a more general control approach.
The rest of the paper is organized as follows. In Section 2, we first give the basic concept about graph theory, and then we introduce the DCR problem of this study. In Section 3, the state and output consensus problems are investigated with theoretical analysis. In Section 4, simulation examples are provided to verify the effectiveness of the proposed algorithms. Finally, the conclusion is drawn in Section 5.
Section snippets
Preliminaries
We first revisit basic definitions on graph theory. A weighted directed graph is denoted by . Let be the set of nodes, and denotes the edges. Let be the adjacency matrix, in which represents coupling strength. If is an edge, we have while if is not an edge. Throughout we assume that each is a measurable function. Let the neighbors of node be . Denote
Main results
Before stating the main results of this section, the following lemma is important for the analysis. Lemma 5 If the bounded variable is defined aswith and where are constant values such that the associated polynomial is Hurwitz stable. Then, all functions are bounded. Furthermore, if as then the function and its derivatives [32,33]
Simulation examples
In this section, we give two examples to illustrate the effectiveness of the proposed algorithms. Example 1 In this example, we consider a group of four 6-order agents in the form of Eq. (15). Let the Laplacian matrices in control algorithms (19) bewhere the eigenvalues of Laplacian matrices of and are and respectively. Furthermore, we define and where which is the DCR
Conclusion
In this paper, the DCR problem of high-order integrators with heterogeneous couplings has been first investigated, and then the output leaderless consensus of heterogeneous high-order integrators with heterogeneous couplings has been solved. The developed control algorithms are more general approaches that can deal with a broad spectrum of real systems, including high-order integrators and heterogeneous couplings. Both theoretical analysis and numerical simulations have validated the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work was supported in part by the National Natural Science Foundation of China (Nos. 61973074, 62111530149, 61921004, and U1713209).