Distributed robust consensus control in directed networks of agents with time-delay☆
Introduction
Recently, distributed coordinated control of multiple agents has attracted a great deal of attention in many fields such as biology, physics, robotics and control engineering. A critical problem in distributed coordinated control of multiple agents is to find control laws that enable all agents to reach an agreement on certain quantities of interest. This problem is usually called the consensus problem. It has been paid attention for a long time by computer scientists, particularly in the field of automata theory and distributed computation [1], [2], [3].
In the past decade, numerous interesting results have been obtained for consensus problems [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Vicsek et al. proposed a simple model for phase transition of a group of self-driven particles and numerically demonstrated complex dynamics of the model [4]. Jadbabaie et al. provided a theoretical explanation for the consensus behavior of the Vicsek model using graph theory [5]. Moreau used a set-valued Lyapunov approach to study consensus problems with unidirectional time-dependent communication links [6]. In [7], a systematical framework of consensus problem in networks of agents was investigated. Three consensus problems were discussed, directed networks with fixed topology, directed networks with switching topology, undirected networks with communication time-delay and fixed topology. Kingston et al. developed a distributed algorithm for average consensus in discrete-time networks based on a formal matrix limit definition of average consensus [11]. The authors of [14], [15] considered a group of mobile second-order agents moving in the plane and introduced a set of control laws that enable the group to achieve a common velocity while avoiding collisions. In [16], Moshtagh et al. studied the problem of flocking and velocity alignment in a group of kinematic nonholonomic agents in 2 and 3 dimensions. In [17], Dimarogonas et al. introduced a decentralized feedback control strategy which drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation. Also, in [18], [19], strategies for connectivity preserving were proposed where connectivity is not just an assumption but an invariant property.
In reality, some variables of the agents in a multi-agent network system may not be able to be measured precisely due to various disturbances, such as time-delay, model uncertainty, external disturbances and asynchronism of clocks, which might cause the network system to diverge or oscillate. Some related problems have been studied by many researchers [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Y. Hong et al. considered the tracking control for multi-agent consensus with an active leader and gave a local controller together with a neighbor-based state-estimation rule [20]. P. Ögren et al. presented a stable control strategy for enabling the agents to climb the gradient of a noisy environment [21]. L. Xiao et al. considered a stochastic model for distributed average consensus with zero mean noise [23]. P. Bliman et al. extended the results of [7] to the case of nonuniform time-delay [24]. M. Cao et al. investigated a discrete-time model with delayed information and explained how convergence to a common heading is achieved [27]. In addition, from the Laplacian eigenvalue standpoint, Y. Kim et al. focused on maximizing the second smallest eigenvalue of a state-dependent graph Laplacian to improve the robustness of the dynamic systems [28].
With this background, we investigate consensus problems of the multi-agent system on directed graphs with external disturbances and model uncertainty in absence and presence of time-delay. Here, each agent can neither get precise information from its neighbors nor perform accurately in accordance with the control input. Under these conditions, we present rules which allow us to find distributed consensus linear protocols. In doing the analysis, we first perform a model transformation and turn the original system into a reduced-order system; then based on this system, we derive conditions to guarantee all agents reach consensus while satisfying the desired performance on fixed and switching topologies.
The paper is organized as follows. In Section 2, we introduce some basic results in graph theory and robust control theory. The problem under investigation is described in Section 3. The main results are presented in Section 4. Following that, Section 5 gives numerical simulations, and finally, some conclusions are drawn in Section 6.
Section snippets
Graph theory
Let be a directed graph of order , where is the set of nodes, is the set of edges, and is a weighted adjacency matrix. The node indexes belong to a finite index set . An edge of is denoted by , where the first element of the is said to be the tail of the edge and the other to be the head. The adjacency elements associated with the edges are positive, i.e., . Moreover, it is assumed that for all .
The problem description
Suppose that the network system under consideration consists of agents. Each agent is regarded as a node in a directed graph, . Each edge corresponds to an available information link from agent to agent . Moreover, each agent updates its current state based upon the information received from its neighbors.
Let be the state of the th agent. Suppose the th agent () has the dynamics as follows: with initial condition , ,
Main results
In this section, we will give rules for designing the feedback matrix in directed networks of agents with and without time-delay.
Before presenting the main results, the following lemmas are introduced. Lemma 5 WriteThe following statements hold. (1) The eigenvalues of are with multiplicity and 0 with multiplicity 1 . The vectors and are the left and the right eigenvectors of associated with the zero eigenvalue, respectively. (2) There exists an
Simulations
Numerical simulations will be given to illustrate the theoretical results obtained in the previous section. Fig. 1 shows four different graphs each with 4 nodes. All graphs in this figure have spanning trees. In Fig. 2, a finite state machine is shown with four states which denote the states of a network with switching topology, and it starts at , and switches every 0.01 s to the next state. Suppose that all initial conditions are zero, and the uncertainty of each edge satisfies
Conclusions
This paper studies consensus problems for directed networks of agents with external disturbances and model uncertainty on fixed and switching topologies. Both networks with and without time-delay are taken into consideration. Several conditions are presented to ensure all agents to reach consensus while satisfying performance. Simulation results are provided to demonstrate the effectiveness of our theoretical results. It is worth noting that the communication topology considered is required
Acknowledgements
The authors gratefully acknowledge suggestions and comments by the anonymous reviewers.
References (39)
- et al.
Decentralized control of vehicle formations
System Control Lett.
(2005) - et al.
Tracking control for multi-agent consensus with an active leader and variable topology
Automatica
(2006) - et al.
Fast linear iterations for distributed averaging
System Control Lett.
(2004) - et al.
Distributed average consensus with least-mean-square deviation
J. Parallel Distr. Comput.
(2007) - et al.
Average-consensus in networks of multi-agents with both switching topology and coupling time-delay
Physica A
(2008) - et al.
Robust output-feedback controller design via local BMI optimization
Automatica
(2004) - et al.
Distributed asynchronous deterministic and stochastic gradient optimization algorithms
IEEE Trans. Automat. Control
(1986) - et al.
Parallel and Distributed Computation
(1989) Distributed Algorithms
(1997)- et al.
Novel type of phase transition in a system of self-driven particles
Phys. Rev. Lett.
(1995)
Coordination of groups of mobile autonomous agents using nearest neighbor rules
IEEE Trans. Automat. Control
Stability of multi-agent systems with time-dependent communication links
IEEE Trans. Automat. Control
Consensus problems in networks of agents with switching topology and time-delays
IEEE Trans. Automat. Control
Consensus and cooperation in networked multi-agent systems
Proc. IEEE
Information flow and cooperative control of vehicle formations
IEEE Trans. Automat. Control
Consensus seeking in multi-agent systems under dynamically changing interaction topologies
IEEE Trans. Automat. Control
Flocking for multi-agent dynamic systems: Algorithms and theory
IEEE Trans. Automat. Control
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This work was supported by the NSFC (60374001, 60334030, 60774003), the COSTIND (A2120061303) and the National 973 Program (2005CB321902).