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10.1109/TAC.2015.2479535, IEEE Transactions on Automatic Control
1
Hierarchical Decomposition Based Consensus Tracking for Uncertain
Interconnected Systems via Distributed Adaptive Output Feedback Control
Wei Wang, Member, IEEE, Changyun Wen, Fellow, IEEE, Jiangshuai Huang, and Zhengguo Li, Senior Member, IEEE
Abstract—In this note, distributed adaptive controllers are developed
for output consensus tracking of multiple linear systems with unknown
parameters, uncertain subsystem interconnections and external disturbances.
The subsystems are allowed to have non-identical dynamics and the same yet
arbitrary order. It is assumed that only part of subsystems can have direct
access to the time-varying trajectory information and the subsystem states
are unmeasurable for local feedback control. In our design, the directed
graph representing the information transmission status among subsystems is
preprocessed by splitting it into a hierarchical structure. Then local adaptive
controllers of subsystems in different layers can be computed in a sequential
order and the difficulty on deriving mutually dependent local controls in
previous consensus works are successfully overcome. In each subsystem,
additional estimates are introduced to account for the unknown parameters
and states in its neighbors’ dynamics. Besides, only the information of local
outputs and inputs need be collected from the neighboring subsystems.
Certain robust terms are added in distributed adaptive laws to mitigate the
effects of uncertain subsystem interactions and disturbances. It is proved
that with our scheme, all closed-loop signals can be ensured bounded when
the strengths of uncertain subsystem interconnections are sufficiently weak.
The tracking errors for the entire group of subsystems will converge to a
compact set and the transient tracking error performance can be adjusted
by appropriately choosing design parameters.
Index Terms—Distributed adaptive control, consensus tracking, unknown
parameters, uncertain subsystem interconnections, output feedback.
I. INTRODUCTION
C
Onsensus [1], [2], [3], as a typical group behaviour, has received
significant attention in the past few decades. This is mainly due to
its wide potential applications in diversity of fields such as intelligent
robot networks, electrical power grids and transportation management.
According to different patterns of the final synchronization values,
consensus control issues can be classified as constant consensus and con-
sensus tracking with time-varying trajectories. As opposed to traditional
centralized tracking control, the main challenge in distributed consensus
tracking control lies in the constraint that only part of the subsystems
in the group can have direct access to the desired trajectory, which is
sometimes represented by the output of a reference system [4], [5], [6].
A number of effective distributed consensus tracking control algo-
rithms have been developed. One common feature of these algorithms
is that “active” subsystem couplings are designed purposely in local
controllers with the aid of information transmission among different
subsystems. In [4], [7], [8], [17], partial knowledge of the desired tra-
jectory is assumed available to all the subsystems. Distributed observers
are then constructed to account for the remaining uncertainties. In [11],
[12], although the references are totally unknown to a subset of the
subsystems, the effects of their bounds are counteracted by introducing
sign functions in local control inputs. Asymptotically consensus tracking
can also be guaranteed, but the invoked chattering phenomenon may
affect the practical implementation of the approaches. In [5], [9], the
control input u
i
of subsystem i is computed directly based on its
local state x
i
as well as the states x
j
and control signals u
j
collected
from its neighbours in the sense of information transmission graph,
i.e. u
i
= f
i
(x
i
, x
j
, u
j
), for j ∈ N
i
. Based on this, alternative
solutions to perfect consensus tracking are provided. However, u
i
and
W. Wang is with the School of Automation Science and Electrical Engineering,
Beihang University, Beijing 100191, China. (Corresponding Author. E-mail
address: w.wang@buaa.edu.cn)
C. Wen is with the School of Electrical and Electronic Engineering, Nanyang
Technological University, Nanyang Avenue, Singapore 639798, Singapore.
J. Huang is with the Department of Electrical and Computer Engineering,
National University of Singapore, Singapore 117576, Singapore.
Z. Li is with the Signal Processing Department, Institute for Infocomm
Research, Singapore 138632, Singapore.
This work was supported by the National Natural Science Foundation of China
under Grants 61203068 and 61290324.
u
j
are mutually dependent during the design process and such mutual
dependence will bring new problem if they are generated without a
prescribed priority. More concretely, to solve the resulting simultaneous
equations, information of the subsystems beyond the neighbouring areas
may be required to derive the final solution.
Besides, all the aforementioned results except for [17] and [9] are
developed based on precise system models. However, subsystems’
intrinsic uncertainties can hardly be avoided in modeling. It is well
known that an adaptive controller has the capability of self-tuning its
parameters based on system working conditions and thus it can adapt
to handle parametric uncertainties with robustness to non-parametric and
environmental uncertainties. However, in contrast to the fact with fruitful
achievements in adaptive control of single uncertain systems, the results
on distributed adaptive control are still very limited. A main reason
is perhaps due to the presence of the resulting parameter estimation
errors in the closed-loop system which make the design and analysis of
controllers much more difficult. This can be observed from [15], [17],
where synchronization problem for a group of first-order subsystems with
unmodeled dynamics is considered. In [15], a distributed neural adaptive
consensus control algorithm is proposed to achieve bounded consensus.
To treat the coupling terms in the derivative of a Lyapunov function,
which are related to both local consensus errors and also parameter
estimation errors in the neighbors’ dynamics, a technique in robust
control is incorporated. The results are extended to more general class
of systems with higher-order dynamics in [16]. In [17], similar coupling
terms are eliminated via extra transmission of the local consensus errors
among subsystems. Recently, a backstepping based distributed adaptive
control scheme is presented in [10] for multiple uncertain nonlinear
subsystems in parametric strict-feedback form. Perfect output consensus
tracking is guaranteed without additional transmission of either local
consensus errors [17] or parameter estimates [18].
In this note, distributed adaptive consensus tracking controllers will
be developed for a group of uncertain linear interconnected subsystems
with the same yet arbitrary order. The time-varying desired trajectory is
allowed to be completely unknown to part of the subsystems. Moreover,
the active information transmission conditions among different subsys-
tems are represented by a directed graph. The main contributions of this
note can be summarized as follows. (i) Unknown parameters, uncertain
subsystem interconnections and external disturbances are considered
simultaneously in each subsystem’s dynamic model. Actually, uncertain
subsystem interconnections are seldom considered in distributed con-
sensus control, though they can be frequently encountered in treating
practical systems such as electrical power systems and complex me-
chanical systems [23], [24]. Note that such type of subsystem couplings
are not restricted to exist among the neighboring subsystems as those
“actively” designed for consensus via information transmission. It will
be seen from later discussion that the entire closed-loop system could
be destabilized if they are not well handled. (ii) Different from most
of the existing results focusing on synchronization with state feedback
control, output consensus is explored in this note with only subsystem
outputs measurable for local feedback control. Distributed filters are
designed by utilizing only the output and input information exchanged
among different subsystems. Then the estimates of the unknown states
within the neighboring areas can be obtained in each subsystem. (iii)
According to the shortest possible path from each subsystem to the
desired trajectory regarded as a virtual leader node, the information
transmission graph is preprocessed by splitting it into a hierarchical
structure. Thus the controllers for the subsystems located in different
layers can be generated in a sequential order with only locally available
information. Thus the problem in computing the mutually dependent
control inputs in [5], [9] can be successfully bypassed. It is shown