Output consensus for heterogeneous multi-agent systems with linear dynamics
Introduction
The consensus problem as one of the most fundamental research topics in the field of coordination control of multi-agent systems has attracted considerable attention over the past few years due to its extensive applications in cooperative control of mobile autonomous robots, the design of distributed sensor networks, spacecraft formation flying, and other areas.
Much of the attention has been devoted to achieving state consensus in homogeneous networks, i.e., networks where the agent models are identical and with the same state dimensions. The consensus problems of homogeneous multi-agent systems, such as consensus of systems with second-order dynamics [1], [2], [3], [4] and high-order dynamics [5], [6], distributed containment control [7], [8], [9], consensus of agents with time-delay [10], [11], and consensus with switching topology [12], [13], [14], just to mention a few, have been intensively studied recently.
In practical applications, it is often impossible to require all agents are identical. Therefore, the difference between the agents which can be called the heterogeneity cannot be neglected in the design of consensus protocol. In this case, the individual agents are not identical and in particular the state dimensions may be different, then state consensus among all agents cannot be achieved. Therefore, the objective is to realize output consensus. This problem is challenging, and there are few available results on output consensus of heterogeneous system can be found, see [15], [16], [17], [18], [19], [20], [21], [22] for instance. In [15], leader-following consensus of heterogeneous agents with nonlinear intrinsic dynamics was addressed with a fuzzy disturbance observer. In [16], [17], it has been shown that the agents have to possess an internal model of the consensus trajectory. Also based on the internal model principle, a constructive method for designing local dynamic controllers with relative information was proposed in [18]. In [19], [20], by using of the notion of system inclusion and system intersection, the agents can be synchronized by an appropriate networked controller. By means of state transformation, the new agents in [21] were almost identical except for different exponentially decaying signals and then a decentralized controller was designed. A similar state transformation approach was taken in [22], in which the output synchronization problem for a heterogeneous network of non-introspective agents was considered.
In this paper, we address output consensus problem of heterogeneous multi-agent systems. The cases of leaderless and leader-following are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by full-developed technique in homogenous multi-agent systems. And it can be seen that there is a close link with the parameters in homogeneous system and the eigenvalue of topology matrix. Furthermore, dynamic regulators based on the state observers also are presented which is suitable for the case that the system states are not accessible.
Notations. Throughout this paper, for real symmetric matrices X and Y, the notation X ≥ Y (respectively, X > Y) means that the matrix is positive semi-definite (respectively, positive definite). I denotes an identity matrix of appropriate dimension. be the vector with all entries being 1. The notation ‘*’ is used as an ellipsis for terms that are induced by symmetry. Λ(X) denotes the set of the eigenvalues of X and the set of the eigenvalues with positive real part of X. The Kronecker product of matrices X and Y is denoted as X⊗Y. ℜ(x) represent the real parts of a complex number x.
Section snippets
Preliminaries
Let be a weighted directed graph with the set of nodes the set of directed edges and a weighted adjacency matrix . A directed edge eij in network is denoted by the ordered pair of nodes (i, j), meaning that node j can receive information from node i. The elements of the adjacency matrix are defined as aij > 0 if and only if there is a directed edge (j, i) in ; otherwise, .
A directed path is a sequence of nodes 1, 2, … , r such that
Output consensus without a leader
From [16], if the heterogeneous agents (1) achieve output consensus, then the all individual systems are able to track the same virtual exosystem defined by the dynamics matrix S and the output matrix D, which can be regarded as an internal model of the virtual exosystem. To reach output consensus of the heterogeneous agents, the following protocol is proposed: where F ∈ Rk × r and Q ∈ Rr × k are needed to be designed. S ∈ Rr × r
Output consensus with a leader
In this section, we consider output consensus problem of heterogeneous agents with a leader. The dynamics of the leader is given by If agent i can access the leader, a virtual edge (i, 0) is said to exist with weighting gain ; otherwise, . Denote the weighting matrix as We provide the following output consensus protocol: where M ∈ Rr × r and μ ∈ R are needed to be designed. S ∈ Rr ×
Numerical examples
In this section, we provide two examples to demonstrate the effectiveness of the proposed methods.
Example 1 Consider the heterogeneous multi-agent system (1) with the topology shown in Fig. 1 with the parameters given by:
Suppose that the matrix pair (S, Q) read
Then Πi, Γi, for the linear regulator Eq. (3) can be obtained as:
Conclusions
In this paper, output consensus problem of heterogeneous multi-agent systems is studied. The cases of leaderless and leader-following are considered. For each case, a dynamic consensus protocol is proposed, in which the parameters are dependent on the solution of a regulation equation and a homogeneous system. The homogeneous system can be analyzed by full-developed technique in homogenous multi-agent systems. Furthermore, dynamic regulators based on the state observers also are presented which
Acknowledgments
This work was supported by the National Natural Science Foundation of China under grant nos. 61403199, 61403178, 61503189, 61374153, the Natural Science Foundation of Jiangsu Province under grant nos. BK20140770 and BK20150926.
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