Short communicationDistributed event-triggered control of discrete-time heterogeneous multi-agent systems
Introduction
During the past years, distributed coordination control for multi-agent systems (MASs) has drawn increasing attention from various research communities due to its broad applications in many fields such as control engineering, cooperative control of unmanned air vehicles, rendezvous and so on. As one of the most typical collective behaviors of MASs, consensus has had a variety of applications in flocking, formation control, distributed sensor network, attitude synchronization in satellite swarms and so on [13], [30], [15], [29], [22], [1]. Generally speaking, consensus means that a group of agents converge to a consistent quantity of interest under some control protocols [7]. Recent years have witnessed remarkable contributions devoted to the consensus problem for the single-integrator dynamics and double-integrator kinematics MASs, such as [19], [12], [2], [31], [17], [18], [3], [25], [11], [5], [6], to name a few. For details, one can refer to the survey papers [13], [23], [14] which have provided a comprehensive overview of consensus protocols and references therein.
All the aforementioned references have focused on the homogeneous MASs, i.e., all the agents are assumed to have the same dynamic behaviors. However, in the practical systems, as stated in [32], the dynamics of the agents coupled with each other may be different due to various restrictions. Until now, only few available results [32], [10], [9], [21] have considered the heterogeneous case, where the agents have different dynamics. Ref. [9] investigated the output consensus of heterogeneous uncertain linear MASs and [21] studied the high-order consensus of heterogeneous MASs using frequency-domain analysis method. Furthermore, in [32], consensus problem in dynamic networks with different-order integrator agents was considered. Ref. [10] considered the consensus problem of the discrete-time heterogeneous MASs with communication delays.
On the other hand, decentralized consensus control for MASs is currently facilitated by recent technological advances on embedded devices, computing, communication resources and so on. Each agent can be equipped with a small embedded micro-processor, who will be responsible for collecting the information from its neighbors and updating the controller according to some pre-designed protocols [16]. However, the embedded processors are usually resource-limited. So one of the most important aspects in the implementation of decentralized consensus algorithms is the design of suitable communication and control actuation schemes in order to decrease energy consumption and meanwhile save the communication resource and decrease traffic through the multi-agent network.
In order to satisfy these requirements, a novel control strategy has been paid an increased attention in more recent years, where the sampled signals of the agents whether or not to be instantly transmitted to their neighbors or to the controller is determined by certain events that are triggered depending on some predesigned rulings. This approach is called event-triggered control strategy [4], [8], [16], [20] which provides a useful way of determining when the sampled signal is sent out. For example, in [20], the control actuation is updated whenever a certain error becomes large enough to overtake the norm of the state while the distributed tracking control design in [8] enforced each agent to update its control law whenever some measure of its state error is above a specified threshold.
As mentioned above, consensus for discrete-time heterogeneous MASs has been investigated in [10]. In this paper, we show our attention on the design of a reasonable discrete-time event-triggered communication scheme for such heterogeneous MASs composed of first-order and second-order agents in order to save the limited resource and reduce the communication burden while preserve the desired consensus performance. It is assumed that each agent broadcasts its state information or velocity information to its neighbors only when needed, which requires that each agent updates its control only when the values of some trigger functions are positive. The distributed event-triggered control strategy with several tuning parameters used in the current paper is designed in light of [27], which has advantages over other event-triggered communication schemes [27].
In this paper, the communication topology within the MASs is fixed and represented by a directed communication graph which is represented by the adjacency matrix A or the Laplacian L. Based on the Kronecker product and the well studied Lyapunov functional stability theory, a sufficient condition for the consensus is obtained in terms of LMI.
The main contributions of this paper can be summarized as follows:
(1) In most recent works concerning the study of design of event-triggered control strategy for MASs, continuous-time modes with single-integrator dynamics were employed [4], [16]. This paper presents the design of event-triggered consensus control for a class of discrete-time heterogeneous MASs containing both first-order agents and second-order agents, which are obviously more significant than those in [4], [16] from the point of view of practical applications. Since the complexity in the structure of the MASs investigated in this paper, the existing results in [4], [16] cannot be directly used.
(2) Different from [16], in this paper, the event triggers for both the states and velocities are considered separately, so the numbers of both the transmitted state signals and velocity signals through the multi-agent network are reduced by using the distributed event-triggered communication scheme proposed in this paper. Moreover, the event trigger conditions assumed in this paper have several parameters that can be tuned and are also state-dependent and velocity dependent (i.e., dynamic), which is more general than that static event-triggered communication scheme assumed in some existing works, such as [36], [37].
(3) Different from the existing works [28], [20] which have used the event-triggered communication scheme, a common assumption in them is that the feedback gain of the controller must be known in advance, in other words, if the feedback gain of the controller in them is not known as prior, the methods proposed in them are no longer valid. In this paper, the explicit expression of the feedback gain matrix K is derived and can be solved numerically by LMI toolbox in Matlab.
The remainder of this paper is organized as follows. Some background of graph theory and the consensus problem to be solved in this paper are given in Section 2. The discrete event-triggered control scheme is also proposed in Section 2. Section 3 presents the main result of this paper. A numerical example is given in Section 4 to validate the usefulness of the proposed method. Conclusion is given in Section 5.
Notations: Rn represents the n-dimensional Euclidean space, In is the identity matrix with dimension n. . The notation for means that the matrix P is positive definite. Notation denotes the diagonal matrix For matrix X and two real symmetric matrices A and B with appropriate dimensions, denotes a real symmetric matrix, where denotes the entries determined by symmetry.
Section snippets
Preliminaries
In this section, some basic concepts about the algebraic graph theory are introduced for the following sections.
The main result
The objective of this paper is to find the condition for the discrete-time heterogeneous MASs (1), (2) to reach consensus under the event-triggered communication scheme described above. For further analysis, some algebraic manipulation is needed.
According to Eqs. (20), (22), the event-triggered control algorithms (14), (19) can be rewritten as
Example
To illustrate the obtained theoretical results above, in this section we consider an applied practical heterogeneous MAS consists of four mobile robots moving in the two-dimensional Euclidean space with the purpose to reach both state and velocity consensus. The four mobile robots are indexed by 1, 2, 3, 4, respectively and categorized by the following two subsystems: the motion of the first two robots are governed by the following linear second-order dynamics:
Conclusion
To minimize the utilization of the communication resources and reduce the communication burden for the multi-agent network, in this paper, the consensus problem under event-triggered communication scheme for discrete-time heterogeneous multi-agent systems on a directed interconnection topology has been investigated. A sufficient condition in the form of linear matrix inequality has been derived by constructing a Lyapunov functional and using the Kronecker product technology. Finally, a simple
Acknowledgments
This work was supported by the Ph.D. Programs Foundation of Ministry of Education of China (2012AA062105) and National Natural Science Foundation of China (61074025, 60834002, 11226240 and 61104140) and the National High Technology Research and Development Program of China (863 Program) (No. 2012AA062105).
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