Abstract
An consensus problem of multiagent systems is studied by introducing disturbances into the systems. Based on control theory and consensus theory, a condition is derived to guarantee the systems both reach consensus and have a certain property. Finally, an example is worked out to demonstrate the effectiveness of the theoretical results.
1. Introduction
The research booms for consensus problems of multiagent systems following DeGroot’s literature [1]. Consensus problem of multiagent systems has attracted considerable attention in recent years. Among various studies of linear-consensus algorithms, a noticeable phenomenon is the fact that algebraic graph theory and linear matrix inequalities (LMIs) play an important role in dealing with consensus problems.
There has been a tremendous amount of interest in consensus problems of multiagent systems. In 2003, Jadbabaie et al. explained the phenomenon mentioned in [2] by theoretical techniques like undirected graph theory, algebraic theory, and the special properties of stochastic matrices in [3]. The distributed complete consensus problems for continuous-time have been intensively investigated [4–15], since the theoretical framework of consensus problems for first-order multiagent networks was proposed and solved by Olfati-Saber and Murray in [16]. They presented the conditions on consensus in terms of graphs for three cases. Furthermore, Ren and Beard presented looser conditions to guarantee the complete consensus based on the graph theory in [17] than the results in [16]. The consensus problem for discrete-time multiagent systems has attracted numerous researchers from mathematics, physics, biology, sociology, control science, and computer science. Xiao and Wang investigated the consensus problems for discrete-time multiagent system with time-varying delays; it is worth mentioning that the augmentation system was first proposed to deal with the consensus problem in [18]. In [19], the cluster consensus problem for first-order discrete-time multiagent system based on the stochastic matrix theory was solved.
In recent years, many researchers analysed the external disturbances effects on stability and the convergence performance of networks, which is called consensus. In [20], the consensus problems were investigated for both discrete-time and continuous-time multiagent systems. In [21], the sufficient conditions to guarantee all agents achieve consensus were obtained by algebraic graph theory firstly. After that, Guo et al. extended general consensus problem to cluster synchronization in [22] and obtained some sufficient conditions to realize cluster synchronization of the Lurie dynamical networks both without time delay and with time delay. In [23], Liu and Chen addressed consensus control for second-order multiagent systems; a sufficient condition is derived to guarantee consensus for the systems by Lyapunov-Krasovkii functional.
Inspired by the above analysis, for discrete-time multiagent systems, we discussed a distributed consensus problem of which the objective is to design appropriate protocols to reach consensus while satisfying the described performance for multiagent system with external disturbances. The paper is organized as follows. Section 2 presents the mathematical model as well as relevant graph theory. Some analysis results on are derived in Section 3. A simulation result is presented in Section 4. The conclusion is given in Section 5.
Notations. Throughout this paper, the following notations are used: denotes the identity matrix with an appropriate dimension; denotes the space of square integrable vector functions over ; represents an ellipsis for the term introduced by symmetry.
2. Preliminaries
2.1. Graph Theory
Denote by an undirected graph with nodes, where and indicate the set of vertices and edges, respectively. Here, if agent can communicate with agent . In this paper, we assume that there is no self-loop in the graph; that is, . Let be the neighborhood set of vertex . For any edge of the undirected graph , if and only if edge . The term of path refers to a sequence of distinct vertices. A path between two vertices and is the sequence , where for . Graph is called connected if there exists a path between any two vertices in . Let be the adjacency matrix of , where if and only if ; otherwise . Define as the Laplacian matrix of the undirected graph , where is diagonal with . The degree of node is defined as .
2.2. Mathematical Models
We suppose that a multiagent system under consideration consists of agents with discrete-time dynamics, and the dynamics of th agent is given bywhere is the state, is the control input to be designed, and and are the external disturbance and the output, respectively. Here, we assume parameters , , and are all known constants.
Here, for (1), we design the following consensus protocol:where is the entry of weighted matrix .
To end this section we give two definitions.
Definition 1. Given the dynamic system (1), one says that protocol (2) asymptotically solves an consensus problem if the states of agents satisfy , .
Definition 2. Let be the initial state of agent . Then protocol (2) is said to solve an average consensus problem asymptotically if the states of agents satisfy , .
3. Analysis Results
In this section, we discuss an consensus problem of the proposed protocol for the discrete-time multiagent system. We first transfer system (1) into one in the form of matrix for completing theory analysis conveniently.
Letand it is easy to know that when .
Based on (1) and (3), one gets
Define , , and , and then (1) can be rewritten into a vector form aswhere And , , and .
LetIt is obvious that is a full row rank matrix. We introduce a linear transformation for (5) and note that . Thus, (5) is transformed to Furthermore, (4) can be written aswhere , , , and .
Definition 3. System (9) is said to realize consensus with performance index , if(1)system (9) with zero disturbance can reach consensus;(2)system (9) satisfies , where is the 2-norm of a vector in and is a constant.
Lemma 4 (Schur Complement). is a symmetric matrix, where with ; then the following three conditions are equivalent:(a);(b), ;(c), .
Based on the above analysis, we can derive the following main result.
Theorem 5. For given constants , system (1) can achieve consensus if there exists a symmetric positive definite matrix , such that
Proof. We first consider the system without disturbances as follows:and we prove that system (9) without disturbances is stable.
Define a Lyapunov function as where is a symmetric positive definite matrix.
The difference of Lyapunov function yields The difference is negative if and only if which is equivalent by Schur Complement to Under (10), it is easy to know that the inequality above holds by denoting .
Hence, system (11) is stable; that is, system (9) without disturbances is stable.
In order to prove that the system has the consensus property, that is, satisfying the performance , we define Since the system is stable and , for , we haveSince , we have Furthermore, letting , we can obtain where and , , and . It is easy to know that if . That indicates .
By Schur Complement Lemma, we know that if and only if Furthermore, by using Schur Complement Lemma again, we can derive equivalently which exactly is (10) by letting . According to Definition 3, we obtain that system (1) can achieve consensus. The proof is completed.
4. Simulation
In this section, we provide some simulations of two protocols for system (1) under the effect of different positive parameter with agents. The state trajectories of all agents are described in all the figures. In particular, the solid curve and the dotted curve describe the state trajectories of agent and agent , respectively; the remaining curve describes the state trajectory of agent . The horizontal axis and vertical axis describe time and state trajectories of all agents, respectively. In order to verify the effect of positive parameter , we present some simulations for Example 7 under different .
Example 6. Consider a multiagent system with agents and the weighted adjacency matrix is We suppose the disturbance is ; the positive parameter which is defined in Definition 3. Let the initial values be , , and . The state trajectories of all agents are described in Figure 1. By simple matrix computation, we know that the model satisfies the condition in Theorem 5. It is easy to see that the system reaches consensus (three curves reach coincident) at about time in Figure 1.
Example 7. Consider a multiagent system with agents and the weighted adjacency matrix isWe suppose the disturbance is ; we provide some simulations with different . Let the initial state values be .
In Figure 2(a), the consensus for system (1) is under topology (2) and ; in particular, all the agents converge to a constant consensus state at about time . In Figure 2(b), the consensus for system (1) is under topology (2) and ; in particular, all the agents converge to a constant consensus state at about time . In Figure 3(a), the consensus for system (1) is under topology (2) and ; in particular, all the agents converge to a time-varying consensus state at about time . In Figure 3(b), the consensus for system (1) is under topology (2) and ; in particular, all the agents converge to a time-varying consensus state at about time . In Figure 4(a), the consensus for system (1) is under topology (2) and ; in particular, all the agents converge to a time-varying consensus state at about time . In Figure 4(b), the consensus for system (1) is under topology (2) and ; in particular, all the agents converge to a time-varying consensus state at about time .
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5. Conclusions
An consensus problem for discrete-time multiagent system with external disturbances is investigated. Based on system transformation as well as Lyapunov stability theory, the sufficient condition in form of LMIs is obtained to guarantee consensus of discrete-time multiagent with disturbances. A simulation result is finally provided to verify the effectiveness of our theoretical results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the reviewers for their valuable comments and suggestions. This work was supported by NSFC (61403064) and the Fundamental Research Funds for the Central Universities (ZYGX2013Z005).