Asynchronous impulsive containment control in switched multi-agent systems☆
Introduction
Consensus of multi-agent systems has regained popularity due to its wide applications in distributed computing. When the computational issues are sophisticated for an individual agent, the collective computation of multi-agent systems is an alternative to solve these challenging problems in a networked environment, e.g., surveillance and monitoring of power networks [18], [28], [38], collaborative information processing [1], and distributed sensor fusion scheme from spatially distributed sensor networks [41]. Indeed, consensus based algorithms can be considered as an effective method for performing distributed computation problems over a network environment. Based on consensus of multi-agent systems, a large number of control schemes have been investigated for various scenarios, e.g., synchronization of networks [10], [36], distributed optimization [33], [44], distributed filtering [4], [43], and formation control [22].
The consensus problem is derived from distributed and parallel computation, where all of agents are required to reach the agreement of states. Inspired by biological collective behaviors [32], Jadbabaie [14] provides a theoretical explanation for the collective behavior. After this study, insightful researches on the consensus [8], [19], [20], [34] and the synchronization problems [9], [45] are published. Regarding the different distributed protocols, analytic approaches via eigenvalue analysis [42], nonnegative matrix [26] and Lyapunov functions [13], [40] have been successfully employed. Using the eigenvalue analysis, some necessary and sufficient conditions are obtained for the second-order consensus in [42]. Moreover, high-order consensus problem of the multi-agent dynamical network with switching topology is studied in [35] by Lyapunov functions. By consideration of the limited communication channel, distributed event-triggered based strategies [7] are designed by Lyapunov functions for consensus of multi-agent systems. In addition, finite-time control algorithms [3] are developed for the second-order multi-agent system via nonsmooth analysis.
As known, the switching topology widely exists in the multi-agent systems. Concerning the synchronization problem of the multi-agent systems with switching topology, a multiple-Lyapunov function approach [37] is used for pinning the network into a synchronous state. Leader-following consensus of multi-agent system is studied in [2], [31] and [29]. Recently, discontinuous control method has attracted much attentions, i.e. impulsive control [5], [15], [39] and [24]. Impulsive control based consensus problems have been studied for the multi-agent system with sampled information in [6]. Impulsive synchronization problems of the network have been investigated in [11]. A further study on the impulsive synchronization schemes of complex network with switching topology can be found in [17].
However, the impulsive containment control algorithm has merely been developed for multi-agent systems with ordinary nonlinearities, such as nonlinear dynamics and time delays. Especially, the multi-agent system involving nonlinearity with topology switching behaviors brings about several challenges for deriving a containment control strategy. Furthermore, for impulsive control in a switched system, the existing algorithms rigorously require each time instant of the impulsive control coinciding with the time instant when the switching behavior is triggered [21], [25], [27]. If the impulsive controller fails to perform precisely at the switching instant, the system may not be stabilized. Meanwhile, the time delays from a networked environment may also jeopardize the effectiveness of an impulsive controller through which there is a time difference between the impulsive control and the switching signal. These issues makes it impossible to efficiently implement the impulsive control scheme for the multi-agent systems. To better handle these problems, we propose an impulsive containment control for multi-agent systems with time delay and switching behavior, which asynchronously performs impulsive control for the whole time horizon. Undoubtedly, this strategy will enhance the flexibility of impulsive control for a switched system. In this work, the main contributions are twofold. First, an impulsive containment control algorithm is built for a multi-agent system with switching behavior and time delays. To ensure stability of the containment control scheme, we use an asynchronous impulsive control formulation derived from a distributed synthesis, where local information exchange is only allowed. Second, multiple Lyapunov functions and the Razumikhin technique are employed to derive impulsive containment control criterion. Both analytical and numerical conditions of impulsive control algorithms are proposed. In fact, the relationship between impulsive control and the average dwell time is established for the multi-agent system.
The rest of the paper is organized as follows. First, basic mathematical descriptions of the switched multi-agent system with the time delay are defined in Section 2. Then, impulsive containment control is designed for the multiple leader-following based multi-agent system. Following on, the stability analysis of the impulsive controlled system is considered via the Lyapunov–Razumikhin technique. Moreover, the discussion on the difference between the asynchronous and the synchronous impulsive control is presented. Section 4 is devoted verifying the effectiveness of the impulsive containment control algorithm by a numerical simulation. The last section concludes the paper and briefly discusses future research.
Let Rn denote the n-dimensional Euclidean space and ‖•‖ be the Euclidean norm in Rn. . Let In be the n × n identity matrix (or simply I if no confusion arises), and 0n × m is the n × m matrix with all elements zero (or simply 0 if no confusion arises). Define the finite set with a finite positive integer S, . Given a matrix X, X > 0(<, ≥, ≤) means that X is a symmetric positive definite matrix (negative definite, positive semi-definite, negative semi-definite, respectively). Denote the maximum and minimum eigenvalue of the matrix by λmax (•) and λmin (•), respectively. The superscript T stands for matrix transposition. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The notation ⌈x⌉ stands for the minimal integer not less than x.
Consider a switched multi-agent system consists of N agents with nonlinear dynamics. Assume that there are m leaders and followers. The interaction for all agents can be naturally modeled by a directed graph where is a set of agents, is a set of edges, is a weighted adjacency matrix. If then aij > 0, otherwise . A directed path from node i to node j is a sequence of edges where . Let denote the set of followers, and denote the set of leaders. The Laplacian matrix where and for i ≠ j. The Laplacian matrix L can be partitioned as:
where L11 ∈ Rm × m and denote the communication among the followers and the communication among the followers and leaders, respectively.
Assumption 1 In the directed graph for each follower, there exists at least one leader that has a directed path to that follower.
From [23], the following lemma can be obtained.
Lemma 1 If Assumption 1 holds, then all the eigenvalues of L11 have positive real parts and all the elements of are nonnegative and where is the number of leaders.
Generally, the dynamics of the multi-agent system is described by
where xi(t) ∈ Rn is the position state of the ith agent. ui(t) ∈ Rn is its control input. The time delay d(t) may be unknown (constant or time-varying) but bounded by a known constant where is upper bound of the time delay. σ(t) is switching signal. with a constant s being the total number of the switching modes. For each time instant the switching signal σ establishes the state change among different subsystems with respect to the multi-agent system. The logical rules that generate the switching signals constitute the switching logic, and the index σ(t) is called the active mode at the time instant t. Let be the starting mode.
Definition 1 see [12] For the switching signal σ and any ta > tb > t0, let Nσ(ta, tb) be the switching numbers of σ over the interval [tb, ta). We have
where and N0 are called average dwell time and the chatter bound, respectively.
gi(·, ·, ·): R × Rn × Rn → Rn, are the nonlinear functions which satisfy the following assumption.
Assumption 2 The nonlinear function satisfies the convex Lipschitz condition. That is, for any given nonnegative constant there exist two positive numbers ks and ls, such that for y, z, yi, zi ∈ Rn,
Definition 2 [30] A subset C of Rn is said to be convex if whenever x ∈ C, y ∈ C and 0 < λ < 1. The convex hull of a finite set of points (q is a positive integer) is the minimal convex set containing all points in . We use to denote it, i.e.
Definition 3 The containment control is achieved in the multi-agent system (2) if for any initial values xi(0) as . That is, each state of the followers will converge into the convex hull formed by the states of the leaders as .
To achieve the containment control, the impulsive control is designed as where α is the control gain to be determined, is the adjacency matrix associated with the directed graph and δ( · ) is the Dirac impulsive function. tk, are the impulsive instants and . That is, t1 < t2 < ⋅⋅⋅tk < ⋅⋅⋅ are the impulsive instant sequences, where t1 > t0, and is the initial time.
The multi-agent system (2) under impulsive control (4) can be reformulated by the following impulsive differential equation. where and .
We assume that xi(t), are right-hand continuous at that is, where .
Let
Then, according to (1), the system (5) can be written in a vector form:
By defining the error then we can derive the following error system
Remark 1 It is noticeable that the impulsive instants of the proposed control strategy are able to force the system reach the containment control in an arbitrary way as long as the convergence criterion can be satisfied. In other words, the time instants of impulsive control unnecessarily coincide with the triggered switching signal, for details please refer to the section of the comparison.
Section snippets
Asynchronous impulsive containment control
In this section, we consider the global exponential stability of the aforementioned error system (7) by employing the multiple Lyapunov functions and the Razumikhin technique. Precisely, the multiple Lyapunov functions of the error system are used to estimate the destabilizing effect of each subsystem which is usually associated with the switched multi-agent system. The time delays are handled by using the Razumikhin technique. By mathematical induction method, the criterion of global
Simulation
In this section, we will give an illustrative example to demonstrate the validity of the previous theoretical results. Suppose that the interaction graph with two leaders and ten followers are chosen as illustrated in Fig. 1, where the graph connection matrix is given as follow
Note that, by the network topology L, there is a
Conclusion
This paper introduced a novel impulsive containment control for the multi-agent system with the state switching and time delay, where an asynchronous distributed strategy is utilized. In terms of Lyapunov–Razumikhin technique, the global exponential stability criterion of the impulsive controlled system is established, which indicates the multi-agent system is able to achieve the containment control. More specifically, the proposed impulsive control algorithm takes an asynchronous strategy that
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This paper was not presented at any IFAC meeting. This work was supported by the Australian Research Council (Nos. DP130104765, DP140100544), the National Natural Science Foundation of China under grants (No. 61304152) and the open fund of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education (No. MCCSE2015A02).