IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 63, NO. 2, FEBRUARY 2016 201
Fault-Tolerant Control of a Class of Switched
Nonlinear Systems With Structural Uncertainties
Jun Fu, Tianyou Chai, Ying Jin, and Chun-Yi Su
Abstract—This brief considers fault-tolerant control of a class
of uncertain switched nonlinear systems against partial actuator
failures. This brief first presents a fault-tolerant control method by
using the common Lyapunov function technique and then applies
it to haptic display systems. Compared to the relevant literature
results, this proposed method is: 1) invariant to actuator fault
modes, in the sense that the proposed controllers are completely
independent from the fault modes, and 2) independent of switching
policies under certain conditions. Simulation studies on a practical
haptic display system with switched virtual environments are car-
ried out to demonstrate the effectiveness of the proposed method.
Index Terms—Fault-tolerant control, switched systems,
uncertainties.
I. INTRODUCTION
U
NCERTAINTIES are unavoidable in practical use of cir-
cuits and systems. If such uncertainties are ignored in the
course of control design, the resulting system usually exhibits
degraded performance or even instability [6]. Uncertainties can
be roughly classified into two groups: 1) parametric uncertain-
ties and 2) structural uncertainties [2], [3]. For the former, adap-
tive control can be designed, and for the latter, robust control
can be implemented [2], [4]. In the case of nonlinear subsys-
tems of switched systems, the uncertainties pertaining to the
subsystems make control design more interesting and challeng-
ing due to the hybrid nature of the switched systems [22]. How-
ever, most of the published results are concerned with switched
linear systems [8]–[10], with some recent publications on
switched nonlinear systems available [11], [12], [16]–[18],
[30]. However, in [16]–[18], the nonlinear part of the subsys-
tems of the switched systems is treated as nominal. From a
modeling point of view, this treatment has limitations, since it
ignores structural uncertainties [3], [13]. Since such structural
uncertainties can destabilize the whole system, considering
them is practically meaningful.
Manuscript received May 27, 2015; accepted August 9, 2015. Date of
publication August 14, 2015; date of current version January 28, 2016. This
work was supported in part by the National Natural Science Foundation of
China under Grants 61473063 and 61503065 and in part by the National
High-Tech Research and Development Program of China (863 Program) under
Grant 2015AA043802. This brief was recommended by Associate Editor
G. Millerioux.
J. Fu is with the State Key Laboratory of Synthetical Automation for Process
Industries, Northeastern University, Shenyang 110819, China, and also with the
Department of Mechanical Engineering, Massachusetts Institute of Technology
(MIT), Cambridge, MA 02139 USA (e-mail: fujuncontrol@gmail.com).
T. Chai and Y. Jin are with the State Key Laboratory of Synthetical Automa-
tion for Process Industries, Northeastern University, Shenyang 110819, China
(e-mail: tychai@mail.neu.edu.cn; jymontreal@gmail.com).
C.-Y. Su is with the Department of Mechanical Engineering, Concordia Uni-
versity, Montreal, QC H3H 1M8, Canada (e-mail: cysu@algol.concordia.ca).
Color versions of one or more of the figures in this brief are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCSII.2015.2468916
In addition, aging of actuator components is a continuous and
slow process. Therefore, considering resulting partial faults is
reasonable and useful in practical applications [14]–[21]. There
are a few results focusing on partial faults occurring in actuators
of standard nonlinear systems and switched systems [16], [17].
The authors in [16] and [17] developed fault-tolerant controllers
for switched systems with structural uncertainties. However, in
these references, the nonlinearities are connected to the consid-
ered system in a parallel manner, which makes the controller
design less difficult since it can be directly compensated for by
adding a proper term in the control signal. Wang [14] used the
multiple Lyapunov function method to study the fault-tolerant
control of a class of switched systems with the fault models
introduced in [18]–[20]. However, the references required that
the actuators be decomposed into two groups: one group robust
to faults and the other susceptible to faults. The classification
of the actuators to either group must be known before its
implementation.
From the motivation earlier and following our previous
work [12], this brief presents a fault-tolerant control method for
the switched cascade nonlinear systems, where the nonlinear
subsystem contains structural uncertainties and also whose ac-
tuators are allowed to partially fail. We first define the represen-
tation of partial actuator fault models and then find a common
Lyapunov function to design a controller to asymptotically
stabilize the switched systems.
The contributions of this brief are highlighted as follows.
First, the proposed method is invariant to actuator fault modes,
in the sense that the proposed controller is completely indepen-
dent from the fault modes; as a result,the proposed method works
well for all the possible actuator fault modes (except the case that
all actuators completely fail) without needing to modify the con-
troller. Second, the proposed method is independent of switch-
ing policies, provided that a common Lyapunov function is
found. Finally, the proposed method was verified using a simu-
lated haptic display system with switched virtual environments.
The rest of this brief is organized as follows: In Section II, the
problem statement is given. In Section III, the detailed design
is presented. In Section IV, the proposed method is applied
to a simulated haptic display system with switched virtual
environments. Section V concludes with brief remarks and an
outlook for future work.
II. P
ROBLEM STATEMENT
Consider the system
˙z = g
i
(z,x)+Δg
i
(z,x)
˙x =(A
i
+ΔA
i
)x +(B
i
+ΔB
i
)u
i
(1)
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