Generally, a multi-agent system is composed of a number of intelligent agents that communicate and interact with each other, in which each agent corresponds to an autonomous dynamic systems. Multi-agent systems can be employed to handle problems that are always difficult or even infeasible for an individual agent to figure out. Therefore, attention has increasingly been focused on the field of analysis and control design of multi-agent systems during the past few decades. In particular, considerable research activity has been dedicated to consensus study for multi-agent systems owing to its great application potential in various areas, including smart grids,^[1] mobile robots,^[2] sensor networks,^[3] as well as multiple underwater vehicles.^[4] In Ref. [5], Hong et al. investigated the consensus of a class of multi-agent systems with an active leader, whose state keeps changing and can be unmeasurable. It is shown in Ref. [5] that, by introducing a local control scheme, each agent is able to follow the leader asymptotically on connected undirected graphs. For directed networks, Ren considered multi-vehicle systems with bounded control inputs in Ref. [6], where both consensus algorithms and experiment platforms were developed; Olfati-Saber and Murray discussed continuous- and discrete-time multi-agent systems in Ref. [7], where several consensus conditions were proposed with the help of graph theory and Lyapunov theory for different cases, including the fixed topology without communication delays, the fixed topology with communication time delays, and the dynamical switching topologies. For recent results on the consensus of multi-agent systems, one can refer to Refs. [8]–[15] and the references therein.

To ensure the consensus of multi-agent systems, in most studies the communication topology is required to have a spanning tree. Sometimes, however, the whole topology does not meet the condition while it can be partitioned into several parts where each subgraph contains a spanning tree. In the light of this, many researchers have carried out studies on group consensus. For instance, Lu et al. discussed the group consensus of a family of interacting second-order integrators via pinning control in Ref. [16], where it is found that the system is able to attain group consensus by choosing pinned nodes according to the topological structure. The idea of pinning control was extended to multi-agent systems with nonlinear dynamics or general connected topology in Refs. [17] and [18]. Feng and Zheng considered a family of discrete-time agents with heterogeneous dynamics in Ref. [19], in which two algorithms were presented for ensuring the group consensus by means of matrix and graph theory. Note that references [16]–[19] are only concerned with the fixed topology case. In some applications, the communication topology among the interacting agents could change considerably over time and it is often the case that the next topology only depends on the current one. It is thus not surprising that many efforts have been put into multi-agent systems with Markovian switching communication topologies in recent years and there are reports available on various group consensus control protocols; see, e.g., Refs. [20]–[22].

Despite significant developments in the group consensus theory of multi-agent systems with fixed and switching topologies, most studies are unconcerned with the influence of disturbances on every agent. However, the agents of a realistic multi-agent system are usually located in an environment subject to external disturbances,^[23] thereby affecting the consensus behavior to a certain extent. To reduce the effects of these disturbances on the system’s output to a prescribed level, Qin et al.^[24] adopted the control concept^[25] for single-integrator multi-agent systems with external disturbances. It was proven in Ref. [24] that the group consensus can be guaranteed provided the intra-cluster coupling strengths are large enough. In Ref. [26], Wang and Jia considered a class of nonlinear multi-agent systems with both parameter uncertainties and external disturbances, and developed sufficient conditions for ensuring the leader-following group consensus by means of robust control theory. Large-scale multi-agent systems consisting of heterogeneous interconnected subsystems were discussed by Cui et al. in Ref. [27], where an event-triggered pinning control strategy was employed to achieve the group consensus and a pre-scheduled performance simultaneously under the Markovian switching communication topologies.

It is noted that the control protocols in both Refs. [26] and [27] were grounded in the assumption that all agent states are available, while in practice full measurement of the agent state variables is often inconvenient and can even be infeasible.^[28,29] Besides, the group consensus results available in the literature are limited to the deterministic setting, whereas, as pointed out in Ref. [1], in a number of engineering areas including maneuvering target tracking and unmanned aerial vehicle (UAV) cooperative control, there is a need to model the dynamics of agents by stochastic differential equations. Motivated by the above analysis, this paper is concerned with the issue of couple-group consensus for a class of discrete-time multi-agent systems subject to both external disturbances and additive stochastic noises. Both fixed communication topology and Markovian switching communication topologies are discussed. The purpose of the current study is to determine output-feedback control protocols so that the multi-agent system reaches couple-group consensus asymptotically in mean square and possesses a prescribed performance bound. It is shown that the closed-loop systems are able to be converted into reduced-order systems via linear transformations. With the aid of the Lyapunov stability theory and the properties of mathematical expectation, new criteria for the mean-square asymptotic stability and performance of the reduced-order systems are proposed. On the basis of the criteria obtained and by means of linear matrix inequality (LMI) techniques, constructive approaches for the design of the output-feedback control protocols are developed for the fixed communication topology and the Markovian switching communication topologies, respectively. Finally, two numerical examples are employed to verify the applicability of the present theoretical results.

Notations We let be the expectation operator and ‖·‖ be the Euclidean norm. Define G = (V, E) as a directed graph with n + m nodes in which V = v₁,v₂,…, v_n+m and E stand for the node set and the edge set, respectively. The adjacency matrix associated with G is represented by A_G = [a_ij] ∈ R^(n+m)×(n+m), where a_ij > 0 for i ≠ j and a_ij = 0 for i = j. The Laplacian matrix associated with A_G is defined as , where and l_ij = −a_ij for i ≠ j. Denote by the symmetric matrix with Φ_ii = (n − 1)/n and Φ_ij = −1/n for all i ≠ j, by the symmetric matrix with and for all i ≠ j, by He(X) the sum of a matrix X and its transpose X^T (i.e., He(X) = X + X^T), by e_p the p × 1 column vector of all ones, and by 0_{p × q} the p × q-dimensional zero matrix. Furthermore, we use symbol * to represent the block that can be clearly inferred by symmetry.

Consider a discrete-time stochastic multi-agent system consisting of n + m agents, each with dynamic model

We can now give the definition of mean-square couple-group consensus:

In this section, we consider the issue of couple-group consensus for multi-agent system Eq. (1) under the fixed communication topology and the Markovian switching communication topologies, respectively.

3.1. Fixed topology case

When the communication topology is fixed (i.e, G(k) ≡ G), the control protocol to be designed is given by

where is the control gain to be solved, μ_1i and μ_2i are the same as those given in Eq. (4). The control protocol considered herein is in the form of static output feedback.^[34] For dynamic output-feedback control protocol, one may refer to Refs. [35]–[37].

As Ref. [20], the two subgraphs G₁ and G₂ are assumed to satisfy a condition as follows.

In this section, two numerical example are given to show the validity of the present control protocols.

The issue of couple-group consensus for a class of discrete-time stochastic multi-agent systems via output-feedback control has been addressed in this paper. Both fixed communication topology and Markovian switching communication topologies have been discussed. It has been shown that via employing linear transformations, the closed-loop systems are able to be converted into reduced-order systems and, at the same time, the couple-group consensus issue can be changed into a stochastic control problem. New conditions for the mean-square asymptotic stability and performance of the reduced-order system have been proposed. On the basis of these conditions, constructive approaches for the design of the output-feedback control protocols are developed for the fixed communication topology and the Markovian switching communication topologies, respectively. Finally, two numerical examples have been employed to illustrate the applicability of the present control protocols. Future work in the field of group consensus for multi-agent systems will focus on finite-time control^[38–40] with consideration of sensor saturations and sensor failures.^[41–43]