Synchronization of nonlinear networked agents under event-triggered control☆
Introduction
In many practical applications, e.g., complex network [22], [27], [42], formation control [8], [14], [16], attitude alignment [7], [10], [11], cooperative control multiple missiles [24], flocking [31], rendezvous [6], the consensus theory plays a significant role. The aim of the consensus theory is to construct a control algorithm based on the local information such that all the states will reach a common value [12], [30], [32], [47]. From the viewpoint of control, the difficulty and challenging for multi-agent system lies that not only single agent but also the cooperative control among multiple agents should be addressed.
In the literature, there have been many results about the consensus algorithms for different kinds of multiple dynamical systems, for example for single-integrator and second-order multi-agent dynamical systems in [18], [20], [23], [28], [35]. For the nonlinear cases, the consensus control protocols were given in [13], [25], [26], [36], [38], [39], the distributed optimization consensus control laws was presented in [29]. Considering many controllers are realized through digital computers [3], some works about consensus for different nonlinear multi-agent systems through discrete-time control have been reported. In [4], the consensus problem via sampled-data control for double-integrator multi-agent systems was investigated. If there is time-varying communication topology, the work [17] presented the sampled-data control algorithm. If there are only the position information, the consensus control law via output feedback was designed in [43]. If there is communication delay between many agents, the work [44] considered the delayed consensus control algorithms. If the agent has second-order nonlinear dynamical structure, the corresponding sampled-data consensus algorithm was given in the work [9].
Note that the previous mentioned results are based on traditional time-triggered control mechanism, i.e., periodic sample-data control method. That is to say that the controller’s update is periodic no matter whether the system needs it or not. Recently, the event-triggered control method, i.e., the controller only need to be updated in some discrete triggered time instants, which is determined by a predefined state-dependent criterion, has been gained increasing attentions [1], [2], [33], [34], [37], [45], [46]. The main superiority of this control method is that the need for communication can be reduced while the system’s performance can be guaranteed. In [5], an overview about consensus control for multi-agent systems via event-triggered was provided and many different kinds of event-triggered control strategies are discussed. The work [21] investigated the problem of decentralized networked control systems via an asynchronous event-triggered. Combining the technique of output feedback control, the observer-based event-triggered control method was studied in the work [15]. The attitude synchronization problem and the cooperative control algorithms for multiple rigid spacecraft was investigated in [40] where the communication pressure can be reduced by using event-triggered mechanism. Considering the limited communication resources and unknown-but-bounded process and measurement noise, the work [19] proposed a distributed event-based consensus control algorithm.
Different from the previous event-triggered control results, this paper mainly considered more general systems, i.e., a class of second-order multi-agent systems involving nonlinear dynamic, where the nonlinear term is allowed to contain unknown parameters. How to design an event-triggered synchronization algorithm for this kind of nonlinear multi-agent systems is very challenging. To conquer this difficulty, the idea of feedback domination is developed from the continuous-time controller design to the event-triggered controller design. Specifically, based on the consensus theory, a distributed synchronization control algorithm and an event-triggered mechanism are first proposed. Based on the Lyapunov method and feedback domination technique, it is shown that the stability of closed-loop system is guaranteed under the appropriate gains condition. In addition, the rigorous proof shows that there is a lower bound for any twice controls’ update, which implies that the Zeno behavior can be avoided. In a word, under the proposed control scheme, not only the global states synchronization can be achieved but also controller’s update frequency can be greatly reduced.
The paper is organized as follows. In Section 2, the dynamics model of multi-agents system and some necessary assumptions and lemmas are introduced. Section 3 proposes an event-triggered state feedback controller and an event-trigged mechanism. The stability analysis of the closed-loop system is given. In Section 4, an example is presented. At last, the whole work is concluded in Section 5.
Notations: Throughout this paper, some necessary notations are listed. R denotes the set of all real numbers and Rn stands for the n-dimensional real vector space. Rn × n is the set of n × n matrices. The superscript T stands for transposition. is a matrix with aij be the entry of ith row and jth column. represents a diagonal matrix with diagonal elements . The minimum eigenvalue and maximum eigenvalue of A are denoted, respectively, as λmax(A) and λmin(A).
Section snippets
Problem formulation
The second-order synchronization problem via event-triggered control for a class of nonlinear multi-agent systems is discussed. Specifically speaking, the dynamics of each agent is described as that in [36]:where (xi, vi) are the system’s state, ui is the control signal, f( · ) is continuous function used to describe the nonlinear dynamics which is allowed to contain unknown parameter. Assumption 2.1 It is assumed that there is a known constant ρ > 0 such that
Main results
Next, we will present our main results. That is to say that under Assumptions 2.1 and 2.2, it will be shown that the distributed synchronization problem for nonlinear multi-agent systems (1) via event-triggered control is solvable. Theorem 3.1 Under Assumptions 2.1and 2.2, for the considered second-order nonlinear multi-agent systems (1), if ui i chosen astogether with the event-trigged mechanism
Numerical examples and simulations
The distributed synchronization control problem for second-order nonlinear multi-agent systems by using event-triggered control is investigated, which includes four agents labeled from 1 to 4. The corresponding communication topology graph G is shown in Fig. 1, and the weight of adjacency agent is given by the numbers along the edges. It is not difficult to find that the graph G is detail-balanced with and strongly connected.
Like the simulation example as that [36], the
Conclusion
The states synchronization problem via distributed event-triggered control for second-order nonlinear multi-agent systems has been investigated in this paper. Based on graph theory, a distributed event-triggered synchronization control algorithm has been explicitly constructed. Using the feedback domination idea, the rigorous proof has been shown that the global states synchronization can be achieved under the proposed controller and some sufficient conditions. Simulation results has been shown
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This work was supported by the National Natural Science Foundation of China under grant nos. 61673153, 61673104, 61773216, in part by the Australian Research Council under grant DE180101268, and the Fundamental Research Funds for the Central Universities of China under grants no. JZ2016HGXJ0023, JZ2017HGPA0163.