IET Control Theory & Applications
Research Article
Robust output synchronisation of
non-identical linear agents via
internal model principle
ISSN 1751-8644
Received on 6th November 2014
Revised on 16th January 2015
Accepted on 12th March 2015
doi: 10.1049/iet-cta.2014.1192
www.ietdl.org
Yuanqing Wu, Zhengguang Wu, Hongye Su
National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang Uni versity, Yuquan Campus,
Hangzhou Zhejiang, 310027, People’s Republic of China
E-mail: hysu@iipc.zju.edu.cn
Abstract: This study investigates the output synchronisation of a heterogeneous network of agents affected by parameter
perturbations. The authors use the output regulation theory to solve this problem. The main result is the construction
of the reference generators and the robust regulators, which can guarantee the outputs of the non-identical agents track
the output of the leader. The designed method includes two steps, which can be seen as a kind of separation principle.
Firstly, the outputs of a network of local identical reference generators are forced to asymptotically follow the output of
the leader. Secondly, under the control of an individual regulator, the output of each agent is forced to track the output of
the corresponding local reference generator, via robust output regulation theory.
1 Introduction
Recently, the synchronisation problem of multiagent systems has
attracted considerable attention in systems and control commu-
nity, because of its application to a wide range of problems, such
as sensor networks, rendezvous, formation control and flocking
control [1–14]. The agents exchange information through a com-
munication graph, which is a time-varying graph [15–18]ora
time-invariant graph [19–22]. The dynamics of individual agents
in the network can be identical [23–28] or non-identical [29–31].
The performances of the agents can be modified by a control law,
to perform collective behaviors, such as to reach synchronisation
about relevant variables of interest. In a heterogeneous network of
non-identical agents, output synchronisation among all agents can
be achieved via the theory of output regulation. As shown by the
internal model principle, if the outputs of the agents in the hetero-
geneous network track a trajectory, this trajectory needs to be the
output of autonomous system. Inspired by the authors [19, 20, 29],
we consider the leader-following consensus problem. Under the
dynamical control law, the outputs of the followers exponentially
track the output of a prescribed leader.
In the leader-following consensus problem, the leader’s motion
is independent of all the other agents and followed by them. We
solve this consensus problem in two stages, which can be treated
as a kind of separation principle. In the first stage, a homoge-
neous network of N identical reference generators which track
the dynamic of the leader is considered. The reference genera-
tors exchange information through a time-varying communication
graph. Under the distributed control law which depends on relative
information, the outputs of these local reference generators can
exponential synchronise to the output of the leader. The method
in this stage is to synchronise a set of networked identical linear
reference generators to reach a common steady state. In the second
stage, the kth of the reference generators designed in first stage is
to be treated as the exosystem for the kth agent. So the problem is
approached in the framework of the classical output regulation the-
ory. We design a decentralised dynamical control law for individual
non-identical agent in such a way that its output tracks the refer-
ence output of its corresponding generator. The individual regulator
includes a stabiliser and an internal model, if the stabiliser can
stabilise the augmented system, the output regulation problem is
solved. The method used in this stage has the internal model prop-
erty with respect to the local identical linear exosystem in steady
state. These two stages guarantee the outputs of heterogeneous
networked linear systems can achieve output consensus with the
output of the leader.
This design philosophy was originally proposed in [22, 29]. In
our approach, we extend the results of such works in two ways:
we require only exchange of information at the level of controlled
output (as opposite to exchange of full state information) and we
handle the cases in which the models of the followers are affected
by parameter perturbations. In this way, we take full advantage of
the potentials of the theory of robust output regulation.
The rest of this paper is organised as follows. In the following
section, we give the description of the communication graph, for-
mulate the investigated problem and specify the structure of the
control law. Section 3 presents the consensus of the diffusively-
coupled reference generators for the first stage. In Section 4,we
address the output consensus of the heterogeneous network for the
second stage, by means of internal model-based regulators. Some
simulation results will be given in Section 5 to present the effective
of the obtained method. Finally, Section 6 concludes this paper.
Notation: The notations used throughout this paper are fairly
standard. R
n
and R
m×n
denote the n-dimensional Euclidean space
and the set of all m × n real matrices, respectively. The notation
X > Y (X Y ), where X and Y are symmetric matrices, means
that X − Y is positive definite (positive semi-definite). I and 0
represent the identity matrix and a zero matrix, respectively. The
superscript ‘T’ represents the transpose, and diag{···} stands for a
block-diagonal matrix. ||·|| denotes the Euclidean norm of a vec-
tor and its induced norm of a matrix. λ
max
(Q) (λ
min
(Q)) denotes
the maximum (minimum) of the eigenvalue of a real symmetric
matrix Q. Matrices, if their dimensions are not explicitly stated, are
assumed to have compatible dimensions for algebraic operations.
The vector 1
N
denotes a column vector with all entries equal to 1.
A ⊗ B denotes the Kronecker product of the matrices A and B.
2 Preliminaries
2.1 Communication graphs
A digraph G(V, E, A(t)) contains a set of N vertices V =
{v
1
, v
2
, ..., v
N
}, an edge set E ⊆ V × V and an adjacency
matrix A(t) =[a
ij
(t)]
N ×N
. For simplicity, digraph G(V, E, A(t))
is denoted by G(t). If an edge (v
i
, v
j
) ∈ E, then the information
IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1755–1765
© The Institution of Engineering and Technology 2015 1755
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