Journal of Systems Engineering and Electronics
Vol. 23, No. 1, February 2012, pp.137–144
Improvement for consensus performance of multi-agent systems
based on delayed-state-derivative feedback
Zhihai Wu
∗
and Huajing Fang
Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
Abstract:
The delayed-state-derivative feedback (DSDF) is in-
troduced into the existing consensus protocol to simultaneously
improve the robustness to communication delay and accele-
rate the convergence speed of achieving the consensus. The
frequency-domain analysis, together with the algebra graph the-
ory, is employed to derive the sufficient and necessary condition
guaranteeing the average consensus. It is shown that introduc-
ing the DSDF with the proper intensity in the existing consensus
protocol can improve the robustness to communication delay. By
analyzing the effect of DSDF on the closed-loop poles, this pa-
per proves that for a supercritical-delay multi-agent system, this
strategy can also accelerate the convergence speed of achieving
the consensus with provided the proper intensity of the DSDF.
Simulations are provided to demonstrate the effectiveness of the
theoretical results.
Keyw ords: multi-agent system, consensus, robustness, conver-
gence speed, delayed-state-derivative feedback.
DOI: 10.1109/JSEE.2012.00017
1. Introduction
Recently, the distributed coordinated control of multi-
agent systems, due to its broad applications in many areas
including swarming, flocking, rendezvous, formation con-
trol, and distributed sensor networks, has attracted a great
deal of attention in many fields such as physics, biology,
robotics and control engineering. In the distributed coor-
dinated control of multi-agent systems, a critical problem
is to find control laws to enable all the agents to reach an
agreement on certain quantities of interest, which is u su-
ally called the consensus problem. In the past decade, con-
siderable efforts have been devoted to this problem. Vic-
sek et al. proposed a simple model for phase transition
of a group of self-driven particles and numerically demon-
strated complex dynamics of the model [1]. Jadbabaie et
al. provided a theoretical explanation for the consensus
Manuscript received March 29, 2010.
*Corresponding author.
This work was supported by the National Natural Science Foundation
of China (60574088; 60874053).
behavior of the Vicsek model using the graph theory [2].
In [3], Ren and Beard extended the results of [2] and
gave some more relaxed topology conditions. Olfati-Saber
et al. provided a theoretical framework for the analysis of
consensus algorithms and several methods of convergence
analysis for the algorithms [4]. More results can be seen in
the survey [5] and the references therein.
In practical applications, the disturbance of communica-
tion delay is unavoidable and might cause multi- agent sys-
tems to oscillate or diverge. Thus, the de lay effect on sta-
bility of multi-agent systems should be considered. Up to
now, the consen sus problems of multi-agent systems with
communication delay have been studied [6−10]. Olfati-
Saber and Murray investigated a network of single integra-
tor agents with equal communication delay, demonstrat-
ing that the maximum communication delay that can be
tolerated by the network of integrators applying a linear
consensus protocol is inversely proportional to the largest
eigenvalue of the network topology [6]. The existence of
the m aximum delay naturally leads to an important ques-
tion: how to improve the maximum delay to make multi-
agent systems have the better robustnessto communication
delay.
Based on the relationship between the maximum delay
and the largest eigenvalue of the network topology, a natu-
ral strategy is to change the existing network topology by
rewriting edges or to design a network topology by using
the sem idefinite programming approach to minimize the
largest eigenvalue under the condition that communication
cost is given. However, the strategy has the following sev-
eral disadvantages. First, rewriting edges is infeasible be-
cause keeping the stability of the communication topology
of multi-agent systems is a precondition. Second, design-
ing the network topology by using the semidefinite pro-
gramming approach is a difficult computational problem,
and in general NP-hard. Third, reducing the largest eigen-
value simultaneously decreases the second smallest eige n-
value, which is called a lgebraic connectivity and is a mea-