LettersAdaptive NN consensus tracking control of a class of nonlinear multi-agent systems
Introduction
In recent decades, cooperative control of multi-agent systems has received significant attention, because of its wide applications in diverse areas, such as multiple robots [1], unmanned air vehicles [2], autonomous underwater vehicles [3], spacecraft [4], and so on. A fundamental problem for coordinated control is consensus control, which means designing an appropriate network protocol so that all agents asymptotically reach an agreement. The consensus problem of multi-agent systems as a fresh research topic has become an important research direction of control theory due to its broad applications [5], [6].
A particularly interesting topic of consensus control is the leader-following consensus problem of multi-agent systems. The leader is an independent agent and it is followed by all the other ones. In fact, the leader-following strategy is an energy saving mechanism in many biological systems [7], and it can also enhance the communication and orientation of the flock [8].
Recently, based on matrix theory, algebraic graph theory and control theory, many meaningful research results of the leader-following and leaderless consensus problems of linear multi-agent systems have been reported [9], [10], [11], [12], [13], [14], [15]. It is well known that nonlinearity, which is an inherent quality in most practical control systems, is more complex and challenging than linear nature. For nonlinear multi-agent systems, several valuable results have also emerged and received widespread attention [16], [17], [18], [19], [20]. In [16], Lu et al. proposed a signal impulsive control approach for a class of directed dynamical network with impulsive coupling. By several simple criteria, it is concluded that the single impulsive control can stabilize the class of impulsive dynamical networks. For a class of multi-agent systems with non-identical unknown nonlinear dynamics, a decentralized adaptive leader-following consensus algorithm is developed in [17] by using both relative position feedback and local consensus error feedback of neighboring agents. Su et al. [18] proposed a second-order consensus algorithm for a class of multiple nonlinear dynamical mobile agents with a virtual leader. A primary contribution of the research work is that nonlinear multi-agent with directed topology is developed. In [19], [20], pinning-controlled consensus control method is studied for nonlinear multi-agent systems. Because a coupling term is contained in each agent׳s dynamic, the consensus objective can be realized by only controlling a small fraction of agents. A main advantage of the consensus strategy is that the control cost is greatly reduced. However, all these eminent consensus control approaches do not consider the improvement of multi-agent system robustness. Although adaptive robust control has been well-developed in tracking control, for examples [23], [24], [25], there are few research results to be reported about the adaptive robust consensus control.
Since it is proven that the neural networks (NNs) have the universal approximation property [21], the neural networks become an attractive and powerful tool for stabilizing complex nonlinear dynamic systems. In addition, the fuzzy logic systems (FLS) are also the universal approximator [22]. Active research has been carried out in adaptive tracking control of nonlinear systems by using the fact that neural networks or fuzzy logic systems can approximate a wide range of nonlinear functions to any desired accuracy under certain conditions [23], [24], [25], [26], [27], [28]. Recently, a neural network-based leader–follower consensus control for a class of nonlinear multi-agent system with modeling uncertainties has been proposed in [29], and received widespread attentions.
However, for the most of the existing adaptive NN consensus control of nonlinear multi-agent systems, such as [29], [30], [31], [32], the quantity of the adjusted parameters depends on the number of input neurons of neural networks (or hidden layer nodes in the fuzzy systems). In general, the number of the input neurons or the hidden layer nodes must be sufficiently large in order to improve approximation accuracy. Therefore, if the schemes proposed in [29], [30], [31], [32] are employed to stabilize the corresponding nonlinear multi-agent system, the online computation burden would be very heavy. From the viewpoint of the engineering application, the consensus control performance is affected, and the running cost is increased.
In this paper, a robust adaptive consensus tracking control for a class of nonlinear multi-agent systems with modeling uncertainties and external disturbances is researched. In the consensus control design, RBFNN is used to construct the adaptive approximators to counteract uncertainties which derive from the unknown nonlinear function of system dynamic. The main contributions of the research work are (1) Compared with existing consensus algorithms, the number of adaptive parameters is greatly decreased because only a scalar adjustable parameter, which is the estimation of the norm of the optimal neural weight vector, is adaptively updated for each agent; (2) By adaptive adjusting the estimations of unknown bounds of neural approximation errors, the adaptive consensus control method can enhance the system robustness against modeling uncertainties. Finally, it is proven that all the signals of the multi-agent system are uniformly bounded and the consensus tracking errors converge to a small neighborhood of zero.
The paper is organized as follows. In Section 2, a description of nonlinear multi-agent systems is given, and the relevant notation, assumption, lemma and preliminaries are stated. In Section 3, the robust adaptive consensus control is designed, and stability result is proven. In Section 4, a numerical simulation example is given to demonstrate the effectiveness of the proposed consensus control approach. In Section 5, the conclusion is made for this paper.
Section snippets
Problem statement and preliminaries
Consider a class of nonlinear multi-agent systems consisting of following agents and a leader. The dynamic of each following agent is described as follows:where is the agent׳s state, , are unknown smooth nonlinear functions. is the control input. is the unknown external disturbance.
The leader of the nonlinear multi-agent system (1) is an independent agent and it is described by the
Consensus protocol
Let denote the tracking error between the following agent and leader (2), and then we define consensus tracking errors aswhere is the row and the column element of the graph adjacency matrix . Eq. (7) can be rewritten as Lemma 3 [39]: Let the undirected graph be a connected graph and , where . Then if and only if ,
Simulations
In order to demonstrate the effectiveness of the proposed control method, consider a nonlinear multi-agent system with six agents and every agent׳s dynamic is described as follows:where , and are the system function , the control gain function and the external disturbance of the differential equation (1), , , are shown in Table 1. Obviously, by choosing
Conclusion
In this paper, a robust adaptive consensus tracking control scheme for a class of perturbed nonlinear multi-agent systems with modeling uncertainties is proposed. The neural networks are used to approximate the nonlinear function. Because the NN weight vector is transformed to a scalar parameter, only a small number of the adjustable parameters are online updated in each agent׳s dynamic. So compared with the existing consensus control methods, the approach can alleviate the online computation
Acknowledgments
This work is supported by the Research Foundation Project of Binzhou University (No. BZXYG1113).
Jun Feng received the M.S. degree in Computer Applied Technology from Qufu Normal University, Qufu, China, in 2009 and she is currently a Ph.D. student of Nanjing Aeronautics and Astronautics University. She is also a lecturer with the department of information engineering, Binzhou University. Her research interests include consensus control of nonlinear multi-agent systems and artificial intelligence.
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Jun Feng received the M.S. degree in Computer Applied Technology from Qufu Normal University, Qufu, China, in 2009 and she is currently a Ph.D. student of Nanjing Aeronautics and Astronautics University. She is also a lecturer with the department of information engineering, Binzhou University. Her research interests include consensus control of nonlinear multi-agent systems and artificial intelligence.
Guo-Xing Wen received the M.S. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China, in 2011 and he is a Ph.D. student of Macau University. His research interests include adaptive nonlinear control of discrete systems, consensus control of nonlinear multi-agent systems.