Multi-consensus of multi-agent systems with various intelligences using switched impulsive protocols
Introduction
In recent years, distributed consensus of multi-agent systems has attracted increasing interest from many fields, such as biology, control engineering, and physics. The objective is to design distributed protocols such that a team of agents can reach an agreement on certain quantities of common interest [17], [18], [19], [20], [22]. Ding et al. [4] considered leader-following consensus for a multi-agent system. A novel distributed event-triggered transmission strategy was proposed for sampled-data consensus of multi-agent systems in [9]. When multiple agents cooperatively accomplish a complex task, the states of agents may converge to several consistent values at some stages due to different task distributions or changing situations. This complex phenomenon widely exists in animal foraging [3] and social networks [11].
Yu et al. [25], [26] proposed a group consensus problem for multi-agent systems. In [25], group consensus was addressed on a communication network with a switching topology, where a double-tree-form transformation was introduced under which dynamic equations were transformed into reduced-order systems. Under the condition of balanced in-degrees, some necessary and/or sufficient conditions were presented for the multi-agent system to achieve group consensus. In [26], a novel consensus protocol was designed to solve the group consensus problem when the information exchange was directed. Hu et al. [12] investigated group consensus with discontinuous information transmissions among groups. A hybrid protocol was proposed, including a continuous-time signal that depicts the information exchange in the same group and a discrete-time signal that describes the information exchange among different groups. Feng et al. [5], [6] considered group consensus of a second-order multi-agent network with a fixed topology. In [5], a distributed protocol was designed for achieving group consensus with discrete-time dynamics, and an algebraic condition was established for selecting proper control parameters. The group consensus problem with continuous-time dynamics was considered in [6], where two kinds of protocols were proposed for achieving static group consensus and dynamic group consensus, respectively.
In the aforementioned papers, multiple agents were grouped artificially, and all results were established based on a common assumption that the interactions among groups were balanced. Also, only algebraic conditions were obtained for group consensus. Qin et al. [23] explored the topology under which group consensus can be reached regardless of the magnitudes of the coupling strengths among agents. Assuming that the groups interact with each other in an acyclic mode, it was shown that group consensus of generic linear multi-agent systems is related only to the network topology; while group consensus with a switching topology could be maintained under arbitrary switching if the underlying topology satisfies certain connectivity conditions. Ji et al. [14] investigated the group consensus problem for linearly coupled first-order and second-order multi-agent systems, respectively. However, if the grouping schemes or weights among agents are changing, the group consensus problem could not be solved using the protocols proposed in [5], [6], [12], [23], [25], [26]. That is, group consensus depends on a special way of grouping the agents.
A multi-consensus problem for multi-agent systems was proposed in [10], [13]. Compared with the existing group consensus results, multi-consensus has the following differences:
- (1)
In many real-world applications, the agents are intelligent with collective behavior. So far, research on multi-agent system has focused on collective behavior, ignoring the agent intelligence characteristic. In the multi-consensus problem, on one hand, the interaction among agents may include competition, abstention, and cooperation [13]; on the other hand, agents have distinct levels of intelligence to perform a cooperative task or to deal with a sophisticated situation. Intelligent coordination thus has potential applications not only in engineering and technology but also in economics and social studies.
- (2)
Several consistent states can occur for a grouping or non-grouping multi-agent network by designing appropriate distributed control protocols.
- (3)
Multi-consensus is a collective behavior existing in some stages of a dynamic evolution process of multi-agent systems. That is, the collective behavior evolves over time, tasks, and environment. Multi-agent systems reach consensus at some stages, and reach multi-consensus in other stages. Therefore, the communication outage among subnetworks is not considered due to the movement of agents in the multi-consensus problem.
In summary, group consensus can be recognized as one special case of multi-consensus. The impulsive strategy enjoys many advantages, such as robustness, fast transient, less energy, and simpler implementation [8]. In this paper, the multi-consensus with non-grouping of the agents is investigated using switched impulsive protocols. The contributions of this paper include:
- (1)
The vertices of a multi-agent network are individuals with certain intelligence. The concept of intelligence degree is introduced to describe the level of agent intelligence.
- (2)
Compared with the group consensus [5], [6], [12], [23], [25], [26] and the multi-consensus proposed in [10], [13], the multi-agent network here is not grouped artificially and the assumption that each block matrix of the control matrix has an identical row-sum is no longer needed.
- (3)
The control protocols proposed in [5], [6], [12], [13], [14], [23], [25], [26] used the sampled position and velocity data simultaneously. Distributed switched impulsive protocols are proposed here to reduce the burden of sensors in a multi-agent network, which use sampled position data and sampled velocity data alternately at sampling instants.
- (4)
Only algebraic conditions are obtained in [5], [6], [10], [12], [13], [23], [25], [26]. Graph-theoretic conditions are derived in this paper. Specifically, with the proposed control protocols, multi-consensus can be asymptotically achieved if and only if the directed network has a spanning tree and the feedback gains and sampling periods of the control protocols satisfy some conditions.
The rest of this paper is organized as follows. In Section 2, the algebraic graph theory is briefly reviewed and the multi-consensus problem is formulated. Three switched impulsive protocols are proposed in Section 3. The main analytic results of this paper are given in Section 4. Numerical examples are illustrated in Section 5, followed by conclusions in Section 6.
The following notations will be used throughout this paper. Let and and |z|, and represent the modulus, real part, and imaginary part of z, respectively. Let 0n and 1n denote, respectively, the n × 1 column vector of all zeros and all ones. In denotes the n × n identity matrix and Om × n denotes the m × n all-zero matrix. is called a nonnegative matrix, denoted by A ≥ 0, if all its entries are nonnegative. Let be the diagonal matrix with diagonal entries . The symbol ⊗ is the Kronecker product operator.
Section snippets
Preliminaries
A directed graph of order n consists of a vertex set and a link set with a nonnegative adjacency matrix . indicates a directed link from vertex j to vertex i. wij > 0 if and only if and otherwise. Moreover, assume that there are no self-loops, i.e., for all . The in-neighbor set of vertex i is denoted by . Call the in-degree of vertex i and the in-degree matrix of the directed
Distributed switched impulsive protocols
The sampling time sequence satisfies 0 ≤ t0 < t1 < ⋅⋅⋅ < tl < ⋅⋅⋅, and the sampling period is constant. In order to achieve stationary multi-consensus of the multi-agent system (1), a distributed switched impulsive protocol is proposed as
where α1 > 0 and β1 > 0 are feedback gains to be determined, and δ(·) is the Dirac δ function.
Remark 1 In engineering problems, the impulsive control can be implemented
Analysis of multi-consensus
Let be the ith eigenvalue of and . has a right and a left eigenvector and associated with the eigenvalue 0, respectively, where .
A simulation example
In this section, a simulation example is provided to illustrate the effectiveness of the proposed switched impulsive protocols. Consider a multi-agent network with nine agents. There are four agents with intelligence degree and five agents with . The adjacency matrix is given as
In what follows, three types of multi-consensus are verified with random initial states, see Figs. 1–3. The red solid
Conclusions
Multi-consensus of multi-agent systems with distinct levels of intelligence is investigated in this paper. A novel framework is proposed for solving the multi-consensus problem, which does not require grouping of the agents. Three switched impulsive protocols are proposed, by which the stationary multi-consensus, the first dynamic multi-consensus, and the second dynamic multi-consensus are solved without the common assumption that each block matrix of the control matrix has an identical
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants 61370093, 61473128, 61572084, 61503133, and 61503129. The authors are grateful to Professor Guanrong Chen of the City University of Hong Kong for his suggustion and modification.
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