Consensus control of higher-order nonlinear multi-agent systems with unknown control directions☆
Introduction
Due to its various potential applications, distributed control of multi-agent systems has become a focal topic in a diversity of research communities over the last fifteen years. Distributed consensus control, as a fundamental problem of multi-agents, intends to design control protocols only using local neighborhood information such that all agents reach a same final state [1], [2]. In this respect, considerable research efforts have been made to deal with the design of consensus controllers for different classes of agent dynamics. These include first-order systems [3], second-order systems [2], higher-order systems [4], and even Lagrangian systems [5]. The interested readers are invited to consult [6], [7] for a more comprehensive exposition because the literature on the subject is extensive. It is worth noting that the control direction of each agent is assumed to be known in these eminent works [2], [3], [4], [5]. However, in real-life applications, the control direction may not be known a priori, e.g., autopilot design of uncertain ships [8] and adaptive visual servoing control [9]. Therefore, research in a multi-agent system with unknown control directions has more realistic significance.
Recently, a series of seminal results has emerged in consensus of multi-agents with unknown control directions. Nussbaum functions are introduced for handling unknown control directions of first- and second-order multi-agents in [10], [11]. The control directions are assumed to be unknown but identical. Such assumptions may render the application of the proposed control methods to a comparatively small kind of systems. In [12], [13], Nussbaum-type adaptive controllers are proposed for first- and second-order multi-agents with unknown control directions by constructing a sub-Lyapunov function. The control directions of these works are allowed to be unknown and nonidentical. An alternative solution to first- and second-order linear multi-agents was provided in [14], by introducing nonlinear PI control laws. But, in practice, there exist a lot of systems that cannot be modeled by first- and second-order systems and possess higher-order dynamics. This gap in knowledge has been underlined in many articles; see, for instance, [15], [16]. In [17], the consensus of higher-order nonlinear multi-agents with unknown control directions is studied under undirected graphs. For undirected communication topologies, it is possible to leverage the property of symmetry in the graph to design Lyapunov functions as in [3], [10], [17]. But for a nonsymmetric directed graph, the design of a suitable positive definite Lyapunov function is quite challenging. The problem will become even more difficult when there further exist unknown control directions and higher-order dynamics since the agent dynamics gets more involved with the graph topology. More recently, using the backstepping technique and the knowledge of the Laplacian matrix, a distributed adaptive controller is deduced for higher-order multi-agents with unknown control directions [18]. However, backstepping is a recursive design procedure in nature, of which complexity increases extraordinarily with the order of the systems.
Motivated by the above facts, this paper deals with the consensus control of higher-order nonlinear multi-agent systems with uncertain control directions under directed graphs. Specifically, we consider two types of multi-agent systems: (M1) the multi-agents with uncertain dynamics and unknown identical control directions and (M2) the multi-agents with known dynamics and unknown nonidentical control directions. We propose a distributed adaptive control protocol for M1 type of multi-agents based on NN and Nussbaum functions. Besides, we give a distributed nonlinear PI control protocol for M2 type of multi-agents. It is proved via Lyapunov stability analysis that our developed protocols can ensure the asymptotical convergence of the consensus errors and global uniform boundedness of all the closed-loop signals. We make the following specific contributions. (1). We construct auxiliary surface variables for both types of multi-agents, which generalizes the existing results [10], [11], [12], [14]. And these surface variables avoid us resorting to the backstepping technique as in [18] and therefore facilitate the derivation and implementation of the distributed control protocol. (2). For the consensus of M1 type of multi-agents, we propose parameters adaptation schemes to compensate for unknown information related to the communication topology which allows us to derive the consensus protocol without requiring the knowledge of the global information or assuming undirected graph conditions.
Compared with the existing works, the presented protocols in this paper have the following advantages.
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In this work, we provide a solution for the consensus problem of higher-order nonlinear multi-agent systems with unknown and nonidentical control directions. Such a problem, as stated in the relevant literature [10], [11], has not been addressed and is still an interesting open issue for further research.
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Though [12], [13], [14] investigated the distributed consensus problem of multi-agent systems with unknown nonidentical control directions, the results are restricted to first- and second-order multi-agents. Moreover, only linear systems can be dealt with by the proposed control methods in [12], [14]. On the contrary, the considered multi-agent system model in this paper has higher-order nonlinear dynamics and is more general than those in [12], [13], [14].
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Different from [17], [18], the backstepping design procedure is not employed for deducing the control protocol for each agent. Additionally, the requirement that some control directions of agents are known in [18] is relaxed to that the control direction of agents can be completely unknown and nonidentical provided their dynamics is known.
Notation: Throughout this article, we use σ( · ) to denote the minimum singular value of a matrix, and max {ρi} to denote the maximum value of ρi, ; Matrix A > 0 (A ≥ 0) means that A is positive definite (positive semidefinite).
Section snippets
Graph theory
A directed graph without self-loops is used to characterize the communication topology among the N agents, where is the node set and is the edge set. An edge stands for that node j can obtain information from node i, and node i is a neighbor of node j. The set of all neighbors of node i is denoted by . A directed path from node i1 to node im is a sequence of ordered edges in the form of . A directed graph is said to contain a directed spanning
Consensus protocol for unknown identical control directions
Assumption 1 The nonlinear dynamics fi(xi) in (1) is unknown, and all agents have unknown but identical control directions, i.e., for 1 ≤ i ≠ j ≤ N. Assumption 2 The directed graph is strongly connected. Lemma 4 The following three statements hold. If is strongly connected, there exists a vector with and θi > 0, such that . If is undirected connected, then the eigenvalues of can be arranged as . If is undirected connected, then or if[19]
Illustrative example
To illustrate the performance of our distributed consensus control protocols, we consider a group of four third-order agents described by (1), in which and .
Conclusion
In this paper, we have studied the distributed consensus control problem of nonlinear multi-agent systems under directed graphs. The considered multi-agent system model is more general than those in most existing results on distributed consensus control in the following twofold: (i) the agents have high-order dynamics, and (ii) the control directions of agents can be unknown and nonidentical. Our main tools are NN, Nussbaum functions and nonlinear PI functions. It has been proved that our
Conflict of interest
There are no conflicts of interest.
Zhihua Zhang was born in Nantong, China, in 1983.He received the M.E. degree from University of Shanghai for Science and Technology, Shanghai, China in 2010 and is currently pursuing the Ph.D. degree in control science and engineering at University of Shanghai for Science and Technology, Shanghai, China. He is currently with Nantong Shipping College as a lecturer. His current research interests include nonlinear control theory, distributed control of nonlinear systems and adaptive control, and
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Zhihua Zhang was born in Nantong, China, in 1983.He received the M.E. degree from University of Shanghai for Science and Technology, Shanghai, China in 2010 and is currently pursuing the Ph.D. degree in control science and engineering at University of Shanghai for Science and Technology, Shanghai, China. He is currently with Nantong Shipping College as a lecturer. His current research interests include nonlinear control theory, distributed control of nonlinear systems and adaptive control, and robotics.
Chaoli Wang (M’09) received the B.S. and M.Sc. degrees from Mathematics Department, Lanzhou University, Lanzhou, China, in 1986 and 1992, respectively, and the Ph.D. degree in control theory and engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1999. He is a Professor with the School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, China. From 1999 to 2000, he was a Post-Doctoral Research Fellow with the Robotics Laboratory of Chinese Academy of Sciences, Shenyang, China. From 2001 to 2002, he was a Research Associate with the Department of Automation and Computer-Aided Engineering, the Chinese University of Hong Kong, Hong Kong. Since 2003, he has been with the Department of Electrical Engineering, University of Shanghai for Science and Technology, Shanghai, China. His current research interests include nonlinear control, robust control, robot dynamic and control, visual servoing feedback control, and pattern identification.
Xuan Cai received the M.E. degree from University of Shanghai for Science and Technology, Shanghai, China in 2015 and is currently pursuing the Ph.D. degree in control science and engineering at University of Shanghai for Science and Technology, Shanghai, China. His current research interests include nonlinear control theory, distributed control of nonlinear systems and adaptive control.
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This paper was partially supported by the National Natural Science Foundation of China (61374040, 61673277, 61503262) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (18KJB580013).