Fixed-time Consensus Algorithm for Second-order
Multi-agent Systems with Bounded Disturbances
Jian Liu, Qing Wang, and Yao Yu
School of Automation and Electrical Engineering,
University of Science and Technology Beijing
Beijing, 100083, P. R. China
Email: bkliujian@163.com; sunnyqing1020@163.com; yuyao@ustb.edu.cn
Abstract—This paper addresses the problem of fixed-time
consensus algorithm for the second-order multi-agent systems
with bounded disturbances. The proposed fixed-time protocol
can ensure a fixed -time consensus. First, the control law is
designed to make the true speed track the virtual speed in a fixed
time based on the fixed-time control method. Then, based on the
backstepping design method, the virtual speed is designed, which
makes the consensus be achieved in a fixed-time. Compared with
the finite-time consensus results, the fixed-time consensus results
can ensure the convergence settling time regardless of the initial
states of the agents. Finally, a example is presented to show the
effectiveness of the fixed-time consensus algorithm.
Keywords—fixed-time; consensus; second-order; multi-agent
systems; disturbances
I. INTRODUCTION
Cooperative control of the multi-agent systems is the main
research direction of current international context and leads to
lots of significant results such as formation control [1], [2],
flocking [3], [4], data fusion [5] and so on. As a basic prob-
lem of cooperative control, Consensus is a typical collective
behavior that requires the agents to converge to a common
value or a agreement by communicating with their neighbours
[6].
Noticeably, most of the existing works about the consensus
problem for multi-agent systems are asymptotic consensus
results which means that the consensus can only be achieved
as time approaches infinity [7]–[9]. In [7], the distributed
consensus algorithm was proposed for linear multi-agent sys-
tems with delays and noises in transmission channels. The
consensus problems with the fixed and switching topologies
were researched in [8], and two consensus protocols for the
networks with and without time-delays were introduced. For
the the external disturbances, input delays and nonlinear uncer-
tainties of the system model, the consensus control algorithm
for multi-agent systems has been researched in [9]. However,
for the consensus of multi-agent systems, convergence speed
is a very important performance property. This naturally leads
to the analysis and construction of convergence speed control
protocols for multi-agent systems.
Compared with the convergence speed of the asymptotic re-
sults, the finite-time consensus results are much better in prac-
tical aspplications. Motivated by these advantages of finite-
time control protocols, for instance, faster convergence speed,
higher accuracy and more robustness to nonlinear uncertainties
[10]. Finite-time consensus control of multi-agent systems
has attracted much attention in recent years [10]–[12]. The
finite-time consensus control problems for first-order multi-
agent systems were researched in [11]. By adding a power
integrator method, the authors proposed continuous finite-time
consensus protocols for leaderless and leader-follower second-
order multi-agent systems [12]. [11] and [12] were extended to
the high-order multi-agent systems in [13], where two finite-
time consensus protocols were proposed with state feedback
and output feedback for the normal form high-order systems.
However, in these researches, the finite-time consensus is
based on the initial conditions of the agents, in practical
applications, the initial states of the agents may be unavailable
in advance.
To deal with these constraints, some new works based on the
fixed-time stability [14] have been studied, which can ensure
the convergence settling time in spite of the initial states of the
agents. In [15], the continuous fixed-time consensus algorithm
was proposed for first-order multi-agent systems, and the
fixed-time pinning control problem also be taken into consider-
ation. The fixed-time leader-following problem was researched
for first-order multi-agent systems with nonlinear uncertainties
in [16]. In [17], the fixed-time consensus algorithms with
linear and nonlinear state measurements ware proposed for
first-order multi-agent systems. Due to the nonlinear property
of the fixed-time convergence controller, it is difficult to extend
the first-order results to the multi-agent systems with more
complex dynamics. The first attempt is studied in [18] based on
the terminal sliding mode control method, where the fixed-time
consensus tracking control algorithms for second-order multi-
agent systems was proposed. Note that, unknown dynamics
and disturbances were not considered in these linear multi-
agent systems.
Motivated by the existing works, in this brief, fixed-time
consensus algorithm for second-order multi-agent systems was
proposed. Compared with the previous works related to this
brief, the results shown in this paper have the following
features. First, the fixed-time consensus algorithm can avoid
the convergence time depends on the initial conditions of
the agents. Second, the multi-agent systems was modeled as
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