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Citation for published version
Mao, Zehui and Yan, Xinggang and Jiang, Bin and Chen, Mou (2019) Adaptive Fault-Tolerant
Sliding-Mode Control for High-Speed Trains with Actuator Faults and Uncertainties. IEEE Transactions
on Intelligent Transportation Systems .
DOI
Link to record in KAR
https://kar.kent.ac.uk/74378/
Document Version
Author's Accepted Manuscript
1
Adaptive Fault-Tolerant Sliding-Mode Control for
High-Speed Trains with Actuator Faults and
Uncertainties
Zehui Mao, Xing-Gang Yan, Bin Jiang, Senior Member, IEEE, Mou Chen, Member, IEEE
Abstract—In this paper, a novel adaptive fault-tolerant sliding-
mode control scheme is proposed for high-speed trains, where the
longitudinal dynamical model is focused, and the disturbances
and actuator faults are considered. Considering the disturbances
in traction force generated by the traction system, a dynamic
model with actuator uncertainties modelled as input distribution
matrix uncertainty is established. Then, a new sliding-mode
controller with design conditions is proposed for the healthy train
system, which can drive the tracking error dynamical system
to a pre-designed sliding surface in finite time and maintain
the sliding motion on it thereafter. In order to deal with the
actuator uncertainties and unknown faults simultaneously, the
adaptive technique is combined with the fault-tolerant sliding-
mode control design together to guarantee that the asymptotical
convergence of the tracking errors is achieved. Furthermore, the
proposed adaptive fault-tolerant sliding-mode control scheme is
extended to the cases of the actuator uncertainties with unknown
bounds and the unparameterized actuator faults. Finally, case
studies on a real train dynamic model are presented to explain
the developed fault-tolerant control scheme. Simulation results
show the effectiveness and feasibility of the proposed method.
Index Terms—Actuator faults, fault-tolerant sliding-mode con-
trol, adaptive control, actuator uncertainty, high-speed train.
I. INTRODUCTION
Due to the increasing requirements of the reliability and
safety of the modern control systems, fault detection and
fault-tolerant control design have attracted more and more
researchers and engineers (see [1]- [5]). High-speed trains with
their high loading capacities, fast and on schedule, have been
one of the most important transportation means. Similar to
the other large-scale and complex control systems, faults also
exist in high-speed trains, which motivates the studies of the
fault detection and fault-tolerant control design for high-speed
trains (see [6]- [9]).
Uncertainty, including modelling uncertainty and distur-
bance, widely exists in real physical systems, and thus it is es-
sential to consider various uncertainties in control design, fault
detection and fault-tolerant control design. For high-speed
trains, there exist some internal and external uncertainties,
This work was supported in part by the National Natural Science Foundation
of China under Grant 61490703, Grant 61573180, Qing Lan Project and the
Fundamental Research Funds for the Central Universities No. NE2019101.
(Corresponding author: Bin Jiang.)
Z. Mao, B. Jiang and M. Chen are with College of Automation Engineering,
Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China,
Email: zehuimao@nuaa.edu.cn (Z. Mao), binjiang@nuaa.edu.cn (B. Jiang),
chenmou@nuaa.edu.cn (M. Chen).
X. Yan is with School of Engineering and Digital Arts, University of Kent,
Canterbury, Kent CT2 7NT, United Kingdom, Email: x.yan@kent.ac.uk.
such as modelling uncertainties from the electric equipments
and mechanical installations, and disturbances from the track
irregularities, tunnels and slopes. It should be noted that the
external disturbances can be modelled as an additional signal
for the system model, while the internal uncertainties should
be modelled as state or input/actuator uncertainties in the
system differential dynamical equation.
It is well known that the input saturation, deadzone and
hysteresis are popular problems for actuator uncertainties, see
[10]- [11], for which the input signals are limited and bounded.
It should be noted that the internal uncertainties cannot be
considered as the external uncertainties in system modelling,
since the boundedness of the system states should be ensured
by the controller design, which are always used in the designed
controller and cannot be assumed to be bounded, a priori.
Actually, the complex coupling between the input distribution
uncertainties and the control signal makes the control design
full of challenges. Among the existing results for the controller
or fault-tolerant controller design of high-speed trains, the
external disturbances, which are modelled as an additional
signal for the system model, are widely investigated [12]-
[15]. However, the internal uncertainties, which are modelled
as state or input/actuator distribution matrix uncertainties in
the system differential dynamical equation, are rarely taken
into considerations. Thus, the fault-tolerant control for high-
speed train with actuator uncertainties is of both theoretical
challenge and practical importance.
For the faulty system, the fault-tolerant control is an es-
sential and effective technique to guarantee system stability
and/or some performances (such as asymptotic tracking), in
the presence of faults. Due to the unknown fault, adaptive
techniques are always used to deal with this case to achieve
the desired tracking performance (see [16]- [20]). As the
position/speed tracking is the main task for trains to guarantee
the on-time schedule, the adaptive technique is pertinent to
high-speed trains with unknown faults. Moreover, the results
about the adaptive fault-tolerant sliding-mode control are rare,
although there are some works for the aircrafts [21], [22].
This paper is focused on the fault-tolerant control problem
for the longitudinal dynamical model of high-speed trains
with traction system actuator faults and uncertainties. Both the
traction system actuator uncertainties and external disturbances
are considered, which are modelled as the input distribution
matrix uncertainties and additional disturbances in the high-
speed train. For the healthy and different faulty cases, the adap-
tive fault-tolerant sliding-mode control schemes are proposed
2
with the controller structure, design conditions, and adaptive
laws being derived. The main contributions of this paper are
summarized as follows:
(i) Considering the traction system actuator uncertainties
and external disturbances, a model with input distribu-
tion matrix uncertainty and additional disturbances is
introduced to describe the dynamic properties of the
high-speed trains.
(ii) A set of conditions and the controller structure are
developed for the healthy case such that the designed
novel sliding-mode controller can drive the tracking
error dynamical system to a pre-designed sliding surface
in finite time and maintains the sliding motion on it
thereafter, even in the presence of input distribution
matrix uncertainty.
(iii) For different cases (the bound of the actuator uncer-
tainty is unknown; the actuator fault is unparameterized),
the fault-tolerant sliding-mode controllers with adaptive
laws are developed for the longitudinal dynamical model
of high-speed trains, respectively.
The rest of this paper is organized as follows: In Section II,
the longitudinal dynamical model of high-speed trains with
actuator uncertainties is presented, and the actuator fault-
tolerant control problem is formulated. In Section III, a sliding-
mode controller with the design condition is developed for
the healthy system with actuator uncertainties and external
disturbances, to achieve the displacement and speed tracking.
In Section IV, a new fault-tolerant sliding-mode controller with
adaptive laws is proposed for the faulty system with the known
bound of fault. In Section V and VI, the proposed fault-tolerant
sliding-mode controller is extended to the cases of the actuator
uncertainties with unknown bound and unparameterized fault,
respectively. In Section VII, simulations for four cases (health
and faulty cases) are presented, and the effectiveness of the
fault-tolerant control scheme is verified. Finally, Section VIII
concludes the paper.
II. PROBLEM FORMULATION
For high-speed trains, the general dynamical model of
longitudinal motion can be described as [6], [23], [24]
M(t)¨x(t) = F
t
(t) − M(t)(a + bv(t) + cv
2
(t)) + d(t), (1)
where x(t) is the displacement of the train, M (t) is the
mass of the train, F
t
(t) is the traction force generated by the
traction system, the parameters a, b and c are resistive force
coefficients of the Davis equation, d(t) models the external
disturbances from weather conditions or rail conditions (ramp,
tunnel, curvature, etc.).
Remark 1: It should be noted that the slope and curvature
rails can induce additional resistances. In order to achieve
a high speed for a high-speed train, the railway should be
smooth, and the slope angle and the degree of curvature
should be as small as possible. According to [29], under the
speed 300km/h, the minimum curve radius is 4500m, and
the maximum slop is 12‰. In connection with this, the train
moves in a one-dimensional space, with slope and curvature
resistances considered as disturbances, which are modeled as
(1). In China, a plenty of bridges are built to make the railway
straight. On the other hand, the suspension system model is
always used to describe the lateral and roll dynamics, which
can be decoupled from the longitudinal dynamic model (1).
Thus, the considering that train moves in a one-dimensional
space and modelled as a rigid body, is reasonable. 2
Actuator uncertainty. According to [6], the mass of a train
can be considered as varying with respect to the stations and
keeps constants between two consecutive stations. Therefore,
it is reasonable to express the mass of train in the dynamics (1)
as M (t) =
¯
M +∆
M
(t), where
¯
M is a constant determined by
the loadings of train, ∆
M
(t) is also a constant during the two
stations and only changed at the stopping stations. According
to the maximum loading of a train, ∆
M
(t) is bounded and its
bound can be estimated in reality.
The traction system generates the traction force, which is
considered as the actuators in high-speed trains, and consists of
traction motors, inverters, PWMs (pulse width modulations),
rectifiers, and related mechanical drives, etc. The uncertainties
widely exist in these equipments. In this paper, considering
the actuator uncertainties, a dynamics model is introduced to
express the taction force F
t
(t) as follows:
F
t
(t) =(1 + ∆
f
(t))F (t) + ∆F (t), (2)
where ∆
f
(t) and ∆F (t) are time-varying functions to rep-
resent the uncertainties in the traction system, F (t) is the
force that the motors provide. The traction force model (2)
contains both additive and multiplicative uncertainties, which
are used to express the most of the actuator uncertainties.
Moreover, these two terms ∆
f
(t) and ∆F (t) are bounded
with their bounds obtained from the maximum traction force
and mechanical installation.
Remark 2: For high-speed trains, both the input saturation
and deadzone exist in the actuators. Because the breaking
system is working when the traction system starts, the input
deadzone can be avoided, as traction force is applied to the
train when the motors in the traction system work normally.
Moreover, the allowed maximin speed decides the maximin
traction forces and the redundances of the traction system.
Then, the high-speed train cannot be operated under the
input saturation. Thus, the presented traction force model (2)
can mainly display the uncertainties in the high-speed train
actuator. 2
Dynamic model of high-speed trains. Let x
1
= x,
x
2
= ˙x, m = 1/
¯
M, ∆m(t) = (1 + ∆
f
(t))/M(t) − 1/
¯
M
and
¯
d(t) = (d(t) + ∆F (t))/M(t). Due to the known bounds
of ∆
f
(t), ∆F (t) and M (t), the bounds of ∆m(t) and
¯
d(t)
can be calculated easily. The longitudinal motion dynamics
(1) with (2) can be expressed as
˙x
1
(t) = x
2
(t), (3)
˙x
2
(t) = (m + ∆m(t))F (t) − a − bx
2
(t) − cx
2
2
(t) +
¯
d(t),(4)
where m, a, b and c are known system parameters, ∆m(t)
and
¯
d(t) satisfy the following conditions:
0 ≤ ∆m(t) ≤ m
b
< m, |
¯
d(t)| ≤ d
b
, (5)
3
with m
b
and d
b
> 0 being known constants.
Actuator faults. The general faults for traction system are
motor faults, IGBT faults in rectifier and inverter, mechanical
faults, and so on. In modelling, most of these faults can
be equivalent to the effectiveness loss of the motor, and the
traction force can be considered as the sum of the motor forces.
The parametric fault model for one motor can be expressed as
(see, e.g. [9] and [20])
F
i
(t) =
¯
F
i
(t) = f
i0
+
l
i
X
ρ=1
f
iρ
s
iρ
(t), t ≥ t
i
, (6)
for some i ∈ {1, 2, . . . , n}, where n is the number of motors.
Here, t
i
is the fault occurring time instant, i is the fault index,
f
i0
and f
iρ
are constants, which are all unknown. The basis
signals s
iρ
(t) are known, and l
i
are the number of the basis
signals of the ith actuator fault.
This fault model (6) covers several practical fault conditions
of the high-speed train actuators, which is shown as follows:
1) Totaly fault. The motor stopping fault is a total fault.
Then, Eq. (6) can be written as F
i
(t) =
¯
F
i
(t) = f
i0
= 0, with
f
iρ
= 0, for ρ = 1, . . . , l
i
.
2) Constant fault. The mechanical drives locked fault can
lead the constant torque, which is a constant actuator fault.
Then, Eq. (6) can be written as F
i
(t) =
¯
F
i
(t) = f
i0
=
non-zero constant, with f
iρ
= 0, for ρ = i, . . . , l
i
.
3) Periodic fault. The IGBT (Insulated Gate Bipolar Tran-
sistor) fault (from PWM) can lead the periodic fault with
approximately known frequency, which could be a sine func-
tion. Then, Eq. (6) can be written as F
i
(t) =
¯
F
i
(t) =
f
i1
sin(wt) for some known w, with f
i0
= 0, f
i1
=
non-zero unknown constant and f
iρ
= 0, for ρ = 2, . . . , l
i
.
In some cases, a completely parameterized fault model
may be an ideal model for some time-varying actuator faults,
as the knowledge of the basis functions f
iρ
(t) may not be
available for some applications. In such cases, approximations
of the basis functions f
iρ
(t) can be employed to achieve
approximate compensation of actuator faults. Some commonly
used approximation methods, such as Taylor series and neural
networks, are employed to approximate the unknown actuator
faults. The approximation for the actuator fault, usually will
result in a bounded approximation error, the whose magnitude
can be very small by proper choices of the basis functions
used in approximation.
Consider that there are n motors. From (6), the input of
system (3)-(4) can be rewritten as
F (t) = σ
ν
ν(t) + ϑ
T
ζ(t), (7)
ϑ = [ϑ
T
1
, ϑ
T
2
, . . . , ϑ
T
n
]
T
,
ϑ
i
= [f
i0
, f
i1
, . . . , f
il
i
]
T
∈ R
l
i
+1
, i = 1, . . . , n, (8)
ζ(t) = [1, s
11
(t), . . . , s
1l
1
(t), . . . , 1,
s
i1
(t), . . . , s
il
i
(t), . . . , 1, s
n1
(t), . . . , s
nl
n
(t)]
T
, (9)
where ν(t) is the control input, σ
ν
is the number of the
remaining healthy actuators, ϑ and ζ(t) are the actuator fault
pattern parameters describing the types of faults. The vector
ϑ could change with the fault evolution, but is fixed in a time
interval.
For actuator fault-tolerant control design of high-speed
trains, the assumption for faults is given as: (A1) there is no
more than ¯n (¯n < n) actuators fail, and the fault parameter ϑ
is bounded and satisfies ||ϑ||
2
≤ ϑ
0
, where ϑ
0
> 0 is a known
constant. It implies that for n − ¯n ≤ σ
ν
≤ n, the remaining
healthy actuators can still achieve the desired control objective.
Objective. The objective of this paper is to develop an
adaptive fault-tolerant sliding-mode control scheme for the
high-speed trains described by (3) and (4), to guarantee the
stability and asymptotic tracking properties, in the present of
the actuator uncertainty ∆m(t) and actuator faults modeled in
(7)-(9).
III. SLIDING-MODE CONTROLLER DESIGN FOR HEALTHY
CASE
In this section, a controller is to be designed to make
the close-loop system (3)-(4) stable and achieve the tracking
performance. For high-speed trains, the Curve-To-Go is always
achieved through speed tracking. Let the desired speed trajec-
tory be x
d
(t), and the desired displacement trajectory y
d
(t).
Then, ˙y
d
(t) = x
d
(t).
Sliding-surface design. Denote the tracking errors e
1
(t) =
x
1
(t) − y
d
(t) and e
2
(t) = x
2
(t) − x
d
(t). From (3)-(4), the
tracking error dynamic equation can be written as:
˙e
1
(t) = e
2
(t), (10)
˙e
2
(t) = (m + ∆m(t))F (t) − a − bx
2
(t) − cx
2
2
(t)
+
¯
d(t) − ˙x
d
(t). (11)
For error dynamical system (10)-(11), design a sliding
function:
δ(e
1
, e
2
) = ke
1
(t) + e
2
(t), (12)
where k > 0 is a design parameter. The sliding surface δ(t) =
0 can be described by
e
2
(t) = −ke
1
(t). (13)
From the structure of system (10)-(11), it is straight forward
to see that system (10) dominates the sliding motion of the
system (10)-(11) with respect to the sliding surface (13). From
(10) and (13), the corresponding sliding mode dynamics can
be described by
˙e
1
(t) = −ke
1
(t), (14)
which implies
e
1
(t) = e
−kt
e
1
(0), e
1
(0) = x(0) − y
d
(0). (15)
Due to k > 0, it is clear to see that lim
t→∞
e
1
(t) = 0.
The analysis above shows that the sliding motion of the error
dynamical system (10)-(11) associated with the sliding surface
(13) is asymptotically stable. Therefore, after sliding motion
occurs, it has lim
t→∞
(x
1
(t) − y
d
(t)) = 0, which implies
that x
1
(t) tracks the desired signal y
d
(t) asymptotically. The
objective now is to design a sliding-mode controller such that
the error system (10)-(11) can be driven to the sliding surface
(13) in finite time and maintains the sliding motion thereafter.
4
Sliding-mode controller design. For train dynamic system
(3)-(4), consider the controller
F (t) = −
1
m
F
0
(t) −
1
m
d
b
−
r(t)
m
sgn
k(x
1
(t) − y
d
(t))
+x
2
(t) − x
d
(t)
, (16)
where
F
0
(t) = k(x
2
(t) − x
d
(t)) − a − bx
2
(t)
−cx
2
2
(t) − ˙x
d
(t), (17)
r(t) is a nonnegative time varying gain to be designed later,
and d
b
satisfies (5).
Then, the following result is ready to be presented.
Theorem 1: The sliding-mode control in (16) drives the
error dynamical system (10)-(11) to the sliding surface (13)
in finite time and maintains a sliding motion on it thereafter
if m
b
< m and the control gain r(t) in (16) satisfies
r(t) ≥
m
m − m
b
η +
m
b
m
(|F
0
(t)| + d
b
)
, (18)
for η > 0.
Proof: From (12) and (10)-(11), the dynamic equation of
sliding surface can be given by
˙
δ(t) = k ˙e
1
(t) + ˙e
2
(t)
= k(x
2
(t) − x
d
(t)) + (m + ∆m(t))F (t)
−a − bx
2
(t) − cx
2
2
(t) +
¯
d(t) − ˙x
d
(t). (19)
Substituting (16) into equation (19) yields
˙
δ(t) = ∆m(t)
−
1
m
F
0
(t) −
1
m
d
b
−
r(t)
m
sgn(δ(t))
−d
b
+
¯
d(t) − r(t)sgn(δ(t)), (20)
where δ(t) is the sliding function defined in (12).
From (20) and δ(t)sgn(δ(t)) = |δ(t)|, it follows that
δ(t)
˙
δ(t)
= δ(t)∆m
−
1
m
F
0
(t) −
1
m
d
b
−
r(t)
m
sgn(δ(t))
−r(t)|δ(t)| − δ(t)(d
b
−
¯
d(t)). (21)
From (5), (21) and r(t) > 0,
δ(t)
˙
δ(t)
≤ |δ(t)|m
b
1
m
|F
0
(t)| +
1
m
d
b
+
r(t)
m
− r(t)|δ(t)|
= −
−
m
b
m
(|F
0
(t)| + d
b
+ r(t)) + r(t)
|δ(t)|. (22)
From (18), it has
m − m
b
m
r(t) ≥ η +
m
b
m
(|F
0
(t)| + d
b
). (23)
The inequality (23) can be rewritten as
r(t) −
m
b
m
r(t) −
m
b
m
(|F
0
(t)| + d
b
) ≥ η, (24)
which implies that
r(t) −
m
b
m
(r(t) + |F
0
(t)| + d
b
) ≥ η. (25)
Substituting (25) into (21), yields
δ(t)
˙
δ(t) ≤ −η|δ(t)|. (26)
Therefore, the reachability condition holds and hence the
result follows. ∇
The proposed sliding-mode controller (16) with F
0
defined
in (17), can drive the error dynamics (10)-(11) to the sliding
surface (13) in finite time. Since the sliding motion has
been asymptotically stable as analysed earlier, it follows that
lim
t→∞
e
1
(t) = 0 and lim
t→∞
e
2
(t) = 0. Thus, the proposed
controller (16) can guarantee the tracking errors of health train
system (3)-(4) converge to zero asymptotically.
Remark 3: In Theorem 1, the right hand side of the
inequality (18) is a function dependent on the system state
x
2
(t) and desired signal x
d
(t). It is not reasonable to assume
x
2
(t) is bounded, a priori. Thus, the r(t) is designed to be
a positive function dependent on the system state x
2
(t) and
desired trajectory x
d
(t) and ˙x
d
(t). For different faulty cases
discussed in the following sections, the controller parameter
r(t) is also a function dependent on the system states, desired
trajectory, basic function of fault, etc. 2
Remark 4: The sliding-mode control has been used ex-
tensively to deal with fault-tolerant control (see, e.g. [21]-
[22], [25]- [27]). However, the uncertainty existing in the
input distribution matrix are rarely considered in the existing
work, and specifically, the associate result for high-speed train
has not been available. It should be emphasized that such a
class of uncertainties is interacted with control signal and thus
the traditional design method cannot be applied. This paper
provides the contribution for high-speed train in this regard
for the first time. 2
IV. FAULT-TOLERANT SLIDING-MODE CONTROLLER
DESIGN
In this section, a fault-tolerant controller will be designed
for the train dynamic model (3)-(4) with the actuator fault
described by (7). For the actuator fault model (7), the fault
parameter ϑ could be changed, and be a constant during a
certain time instant. According to Assumption (A1) that the
remaining healthy actuators can achieve the control perfor-
mance, the fault parameter ϑ can be assumed to be bounded.
Faulty system. From (3)-(4) and (7), the dynamics of the
faulty system can be rewritten as:
˙x
1
(t) = x
2
(t), (27)
˙x
2
(t) = (m + ∆m(t))(σ
ν
ν(t) + ϑ
T
ζ(t)) − a − bx
2
(t)
−cx
2
2
(t) +
¯
d(t), −cx
2
2
(t) +
¯
d(t), (28)
where ν(t) is system input, σ
ν
is the number of the remaining
health actuators and satisfies n − ¯n ≤ σ
ν
≤ n, ϑ and ζ(t) are
defined in (8) and (9), and ||ϑ||
2
≤ ϑ
0
with ϑ
0
being a known
constant.
With the tracking errors e
1
(t) = x
1
(t) − y
d
(t) and e
2
(t) =
x
2
(t) − x
d
(t), The error dynamic equation can be written as:
˙e
1
(t) = e
2
(t), (29)
˙e
2
(t) = (m + ∆m(t))(σ
ν
ν(t) + ϑ
T
ζ(t)) − a − b x
2
(t)
−cx
2
2
(t) +
¯
d(t) − ˙x
d
(t). (30)
