Post on 19-Apr-2020
Adaptive Fault-Tolerant Control for Time-
Varying Failure in High-Speed Train Computer
Systems
Tao Tao and Hongze Xu School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China
Email: 10111049@bjtu.edu.cn
Abstract—We investigate a novel adaptive fault-tolerant
control method for time-varying failure in high-speed train
computer systems and propose a fault model for such
systems. First, the dynamics of high-speed train systems are
analyzed and a multiple point-mass model is developed.
When actuator outputs deviate from the expected value, a
novel adaptive fault-tolerant control method based on
Lyapunov stable theory is automatically implemented to
compensate for the unknown fault effects and ensure system
stability and performance. The effectiveness of the proposed
approach is also confirmed through numerical simulations
by using a train model similar to China Railways High-
speed 5.
Index Terms—Fault-Tolerant Control; Adaptive Control;
High-Speed Train Systems
I. INTRODUCTION
To date, fault-tolerant control is widely applied in various industries such as chemical engineering [1-4],
nuclear engineering [5, 6], aerospace engineering [7, 8],
and automotive systems [9]. The application of advanced
fault-tolerant control theory is necessary for high-speed
train systems.
A train operation control system generally uses two
types of models, namely, single and multiple point-mass
models. The single point-mass model assumes that the couplers among carriages are stiff and all carriages can be
considered a united rigid body. Therefore, the single
point-mass model uses a classical control method to
design the cruise controller. Howlett [10-12] uses the
Pontryagin principle to find the nature of the optimal
strategy and applies this information to determine the
precise optimal strategy. Khmelnitsky [13] develops a
detailed program for traction and brake applications that minimizes the energy consumption during train
movement along a given route in a given time. Wang [14]
applies an iterative learning control theory into an
automatic train operation system to enable the train to
drive itself consistently with a given guidance trajectory
(including the velocity and coordinate trajectories). Gao
[15, 16] studies the one-level speed adjustment braking of
an automatic train operation system. Song [17] investigates the automatic control problem of high-speed
train systems under immeasurable aerodynamic drag.
Robust and adaptive control algorithms have been
developed to ensure the high precision tracking of train
position and velocity. Considering that the functions of
the couplers and the relative velocity among adjacent
vehicles are ignored, the controller based on the single
point-mass model will cause the unstable motion of the system when carriages are connected by a flexible
connector. Therefore, the multiple point-mass model
assumes that a train, which is composed of locomotives
and wagons (both referred to as carriages), is modeled as
carriages connected by couplers. A dynamic model that
explicitly reflects the interaction effects among vehicles
is established by using the multiple point-mass Newton
equations. Song [18, 19] investigates the position and velocity tracking control problem of high-speed trains
with multiple vehicles connected by couplers. Lin [20]
presents an analysis of the cruise control for high-speed
trains. A train system is designed to be safe and reliable
with high efficiency and fault tolerance. Output
regulation with measurement feedback has been proposed
to control heavy-haul trains. The objective of this
approach is to regulate the train velocity to a prescribed speed profile. Chou [21] proposes a closed-loop cruise
controller to minimize the running cost of heavy-haul
trains equipped with electronically controlled pneumatic
brake systems.
The rest of the paper is organized as follows. Section II
introduces the plant model, which characterizes basic
actuator failure, and the control objective. Section III
derives an adaptive fault-tolerant control design when some actuators deviate from the expected value. Section
IV presents the proposed adaptive fault-tolerant control
method. Section V introduces the simulation. Section VI
concludes.
II. DYNAMIC MODEL OF HIGH-SPEED TRAIN SYSTEMS
A carriage in high-speed train systems is subjected to
the traction/brake force, adjoining internal-forces of
carriges, the aerodynamic force, friction between train wheels and track, gravity force, and the curve resistance
during train operation. Such carriage can be modeled as
follows:
JOURNAL OF NETWORKS, VOL. 8, NO. 12, DECEMBER 2013 2823
© 2013 ACADEMY PUBLISHERdoi:10.4304/jnw.8.12.2823-2828
1
1
1
0
1
2
1
1
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
9.98sin 0.00
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
/2
c c p e
i i i i i i i i
i i i i
i i i i
i
i i
n
v a i i
i
i i
i
i
u k x x b x x
k x x b x x
c c x m c x
m x t u t f t f t f t f t
t t t t t
t
m
m
t t t
t t
m d R
l t
(1)
where ix is the position of ith carriage,
im is the mass of
ith carriage, i iu represents the traction force provided
by the powered carriages. 1i indicates that the ith
carriage is powered; otherwise the ith carriage is
unpowered. 0 , ,v ac c c are obtained by experiments.
The running resistance consists of the main resistance p
if and external resistance e
if . The main resistance
mainly refers to the friction between train wheels and the
track during operation, as well as the air resistance. The
external resistance generally includes the curve resistance
(i.e., t/0.002 i lm d R ) and the ramp resistance (i.e.,
9.98sin i im ). ld is the length of the axle, and
tR is the
turning radius of the carriage. Schetz [22] and
Raghunathan [23] indicate that air resistance is
concentrated at the front of the train and that the
mechanical resistance is distributed in all carriages.
1,c c
i if f represent the tension among carriages. This
force is mainly caused by the velocity difference between carriages. The connections between the trains mainly rely
on the degree of coupling flexibility, which can be
approximated as the elastic model. The deformation of
the connector is proportional to the force between
carriages. The tension can be described as follows:
( )f k (2)
The coefficient of elasticity k is always bounded,
i.e., k k k . By considering the worst case scenario,
we used a fixed factor k to build the model during the
calculation to enable the calculated results to handle all
cases with any k .
According to the above analysis, the motion equations
of n-body high-speed trains with distributed driving types
are as follows:
1 1 1 1 2
1 2 1 1
1
0
2
11 1 1
1
( ) ( ) ( ) ( )
( ) ( )
( ) 9.98sin 0.00
( )
( 2) /
v
n
a i
i
x t t t t
t t t
t
m u k x x
b x x c c x m
c x m m d Rm
l t
(3)
1
1 1
0
1
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
(
)
9.98sin
( ) ( )
( )
/ ( 2, , 1)0.002
i i i
i i i
i i i i
v i i
i i
i i
i
m x u k x x b x
x k x x b x x
t t t t t
t t t
c c x m
t t
t
m d R i n
m
l t
(4)
0
1
1
( ) ( )
9.98sin 0.00
( ) ( )
( ) ( )
2
) (
/
n n
n n
n n n n
v
n n
n n
n
m x u kt t x x
b x x c c x m
t t
t t t
m d Rm
l t
(5)
By assuming a target speed, we derive the
corresponding position, acceleration, and traction force:
0
0
0
0
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
i i
i i
i i i i
i i i
x t x t x t
x t v t x t
x t a t x t x t
u t u t u t
The ideal motion equations of high-speed train systems
are as follows:
42
0
0
0
0
1 0 0 1
1
0
( )
( 2, , )
v a i
i
i v i
u v
v
c c v m c m
u c c m i n
Eqs. (3), (4) and (5) will be transformed into the
following:
1 1 0 1 2
1 2 1 1 1
1 1
1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2n
a
i
vx t c v x t x t
x t b x t b x t u t
m m c m k
k
(6)
1 1
1 1
( ) ( ) 2 ( )
( ) ( ) 2 ( )
( ) ( )
( ) ( 2, , 1)
i v i i i
i i i
i
i i
i i
x t c x t x t
k x t k x t x t
b x t b x t
u t i
m m k
n
b
(7)
1
1
( ) ( ) ( ) ( )
( ) ( ) ( )
n v n n n n
n n n
n
n
x t c x t x t x t
x t
m m k k
b b x t u t
(8)
The standard state space form of the high-speed train
systems can be transformed into the following equation:
( ) ( ) ( )
( ) ( )
x t Ax t Bu t
y t Cx t
(9)
1 1
1 1
( ) ( ) ... ( ) ( ) ... ( )
( ) ( ) ... ( ) ( ) ... ( )
T
n n
T
n n
x t x t x t x t x t
y t x t x t x t x t
The target system will be:
( ) ( ) ( )
( ) ( )
m m m m
m m m
x t A x t B r t
y t C x t
(10)
The control objective is lim 0,( )mt
e e x x
.
During actuator faults, the high-speed train systems
will not meet the requirements and safety standards.
Therefore, we should adopt a fault-tolerant control to
ensure train security in high-speed train systems with
faults.
Let ( )u t be the fault input, where
1 2( ) [ , , , ]T
mu t u u u ,
( )j jl jlu u f t (1)
where jlu is an unknown scalar, and ( )jlf t is a known
signal.
III. CONTROL DESIGN
To develop adaptive control schemes for systems with
unknown actuator failures, we should determine the
appropriate controller structure and parameter for cases when system parameters and actuator failure parameters
are known. When the fault parameters are known, the
system will be stabilized by the following:
2824 JOURNAL OF NETWORKS, VOL. 8, NO. 12, DECEMBER 2013
© 2013 ACADEMY PUBLISHER
* * *
1 2 3
( )( ) ( )
( )
Tx t
v t K K r t kx t
(12)
Let
1 2{ , , , }
1 ( )
0 ( )
m
i i
i
i i
diag
u u
u v
When a known fault occurs, the original system is
transformed into the following:
1 1
( ) 0 0 0( ) ( ( ) ( ))
( )
x t I xv t u t v t
x t A C A B x
1
*
11 1
* *
2 3
*
11 1
2
( ) 0 0 ( )
( ) ( )
0 0 0( ) ( ( ) ( ))
00
0( )
( )
p
T
T
j
j j j j
j
j
x t I x x tK
x t A C A B x x t
K r t k u t v t
Ik
A C A B
x tk
x t
1 1
1
* *
3
0( )
0( )
p p
p
j j
j j j j j j j
j
j j j j
r t k
u t
and then the control parameters satisfying the following
equation will be taken as follows:
1
*
11 1
1 1
00
0
p
T
j
j j j j
m m m m
Ik
A C A B
I
A C A B
(13)
1
*
2
0 0
p
j
j j j j m
k
(14)
1 1
*
3
0 0( ) ( ) 0
p p
jl jl jl jl
j j j j j jj j
k f t u f t
(15)
and then fault system will be tracking the target system.
Now we develop an adaptive control scheme for the
system with unknown fault parameters. The direct
adaptive fault-tolerant control will be expressed as follows:
1 2 3
( )( ) ( )
( )
Tx t
u t K K r t kx t
(16)
Define the parameter errors: *
1 1 1( )j j jk t K K
*
2 2 2( )j j jk t K K
*
3 3 3j jl jlk k k
The original system will be transformed into the
follows:
1 1
* * *
1 2 3
1 2 3
1 1
( ) ( )0 0
( ) ( )
( ) 0 0( ) ( )
( )
0 ( )( )
( )
0 (
m m
m m
T
T
m m m m
x t x tII
x t x tA C A B
x tK K r t k u t
x t
x tI K K r t k
x t
I x
A C A B
1 1
1
1 2
3
0)( )
( )
0 0( ) ( )
0( )
p p
p
m
T
j j
j j j j j jj j
jl jl
j j j j
tr t
x t
k x t k r t
k f t
We have the tracking error equation:
1
1 1
11 1
2 3
00( )
0 0( ) ( )
p
p p
T
j
j j j jm m m m
j jl jl
j j j j j jj j
Ie ek x t
A C A Be e
k r t k f t
(17)
Consider the positive-definition equation:
1
1 1
1 2 3 1 1
1 1
2 2 3 3
0( , , , )
0 0
p
p p
T
T
j j j
j j j j
j j jl j
j j j j j jj j
e eV x K K k P k k
e e
k k
(18)
where 2 2n nP R , 0TP P such that
1 1 1 1
0 0T
T
m m m m m m m m
I IP P Q
A C A B A C A B
for any constant 2 2n nQ R , such that 0TQ Q ,
2 2n n
j R is constant such that 0T
j j , 2 0j ,
3 0j are constant, 1, ,2j m .
Then we have
1
1 1
1 3 1 1
1 1
2 2 3 3
0( , , ) 2
0 0
p
p p
T T
T
j j j
j j j j
j j jl j
j j j j j jj j
e e e eV x K k P P k k
e e e e
k k
Choose the adaptive laws as follow:
. 1,
0( )
( )
T
i i
i
x t eK P
x t e
. (19)
2 2
0( )
T
j
i
eK r t P
e
(20)
3 3
0( )
T
j jl
i
ek f t P
e
(21)
Then we have
1 3( , , ) 0
Te e
V x K k Qe e
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The adaptive controller with adaptive laws applies to
the system with actuator failures and guarantees that all
closed-loop signals are bounded. The tracking error goes
to zero as time goes to infinity.
IV. SIMULATION
In this section, we use the simulation results to verify
the effectiveness of our proposed adaptive control schemes. The failure model is adopted as follows:
5sin0.2 ( 5, ( ) sin0.2 )j jl jlu t u f t t
30faultt s
Parameters of CRH5 are adopted. The parameters of
the high-speed train are shown in the table 1.
TABLE I. PARAMETERS OF HIGH-SPEED TRAIN SYSTEMS
Symbol Implication Value Unit
n No. of carriages 8 null
w No. of powered carriages 5 null
mi (i=1,2,4,7,8) No. of powered carriages 8500 kg
mi (i=3,5,6) Mass of powered carriages 8095 kg
c0 Mechanical resistance coefficient 5.2 N/kg
cv Mechanical resistance coefficient 0.038 N∙s/m∙kg
ca Aerodynamic drag coefficient 0.00112 N∙s2/m2∙kg
k Elasticity coefficient 20×106 N/m
b Damping coefficient 5×106 N∙s/m∙kg
umax Maximum traction input 302 kN
umin Minimum traction input 205 kN
Figure 1. Position tracking error of the 1th carriage in high-speed train
systems
Figure 2. Position tracking error of the 2th carriage in high-speed train
systems
Figure 3. Position tracking error of the 3 th carriage in high-speed
train systems
Figure 4. Position tracking error of the 4 th carriage in high-speed
train systems
Figure 5. Position tracking error of the 5 th carriage in high-speed
train systems
Figure 6. Position tracking error of the 6 th carriage in high-speed
train systems
Figure 7. Position tracking error of the 7 th carriages in high-speed
train systems
Figure 8. Position tracking error of the 8 th carriage in high-speed
train systems
Figure 9. Velocity tracking error of the 1th carriage in high-speed
train systems
2826 JOURNAL OF NETWORKS, VOL. 8, NO. 12, DECEMBER 2013
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Figure 10. Velocity tracking error of the 2th carriage in high-speed
train systems
Figure 11. Velocity tracking error of the 3 th carriage in high-speed
train systems
Figure 12. Velocity tracking error of the 4 th carriage in high-speed
train systems
Figure 13. Velocity tracking error of the 5 th carriage in high-speed
train systems
Figure 14. Velocity tracking error of the 6 th carriage in high-speed
train systems
Figure 15. Velocity tracking error of the 7th carriage in high-speed
train systems
Figure 16. Velocity tracking error of the 8th carriage in high-speed
train systems
With two fading actuators (motor 1 and motor 7), the
fault-tolerant control algorithms (19), (20) and (21) are
tested and the results are presented, one can observe that
the proposed fault-tolerant control scheme performs well
even if some of the actuators lose their effectiveness
during the system operation. In a fault system with an
adaptive tracking controller, the actual position can track the target position well (Figure 1-8). The actual velocity
in the fault system can track the target velocity
satisfactorily and satisfy the system control objectives
(Figure 9-16).The design parameters of the controller can
be set quite arbi-trarily without the need for consistently
tuning by the designer for tracking stability, although
some trade-off is needed to accommodate the tracking
precision and the magnitude of the control effort. It is shown in the Fig.1-8, that the fault is occurred at 30s and
the position tracking error of the high-speed train systems
does not change much. And Correspondingly, the
velocity tracking error of the high-speed systems in Fig.
9-16 also converges to zero. It is shown that the
performance of the fault high-speed train systems is very
excellent. But we can see in the Fig.1-16 the position
tracking error and velocity tracking error do not converge
to zero exactly, because the coefficient of elasticity k
we used is the lower bound of the actual value k .
V. CONCLUSION
The position and velocity tracking control problems of high-speed train systems have been tested in the presence
of actuator time-varying failures. A novel adaptive state
feedback controller has been constructed to compensate
for unknown fault effects automatically. The proposed
algorithm guarantees the stability, asymptotic position,
and velocity tracking of high-speed train systems. The
simulation results demonstrate the efficiency of the
proposed algorithms and their applicability to the operation control of trains.
REFERENCES
[1] I. Nimmo, "Adequately address abnormal operations," Chemical engineering progress, vol. 91, pp. 36-45, 1995.
[2] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, "A review of process fault detection and
diagnosis: Part I: Quantitative model-based methods," Computers & Chemical Engineering, vol. 27, pp. 293-311, 2003.
[3] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, "A review of process fault detection and diagnosis: Part II: Quanlitative models and search strategies," Computers & Chemical Engineering, vol. 27, pp. 313-326, 2003.
JOURNAL OF NETWORKS, VOL. 8, NO. 12, DECEMBER 2013 2827
© 2013 ACADEMY PUBLISHER
[4] V. Venkatasubramanian, R. Rengaswamy, S. N. Kavuri, and K. Yin, "A review of process fault detection and
diagnosis: Part III: Process history based methods," Computers & Chemical Engineering, vol. 27, pp. 327-346, 2003.
[5] Q. Zhang, X. An, J. Gu, B. Zhao, D. Xu, and S. Xi, "Application of FBOLES—a prototype expert system for fault diagnosis in nuclear power plants," Reliability Engineering & System Safety, vol. 44, pp. 225-235, 1994.
[6] K. Keehoon and E. B. Bartlett, "Nuclear power plant fault diagnosis using neural networks with error estimation by series association," Nuclear Science, IEEE Transactions on, vol. 43, pp. 2373-2388, 1996.
[7] C. Favre, "Fly-by-wire for commercial aircraft: the Airbus experience," International Journal of Control, vol. 59, pp. 139-157, 1994.
[8] D. Briere and P. Traverse, "AIRBUS A320/A330/A340 electrical flight controls - A family of fault-tolerant systems," in Fault-Tolerant Computing, The Twenty-Third International Symposium on, 1993, pp. 616-623.
[9] R. Isermann, R. Schwarz and S. Stolzl, "Fault-tolerant drive-by-wire systems," Control Systems, IEEE, vol. 22, pp. 64-81, 2002.
[10] P. G. Howlett, P. J. Pudney and X. Vu, "Local energy minimization in optimal train control," Automatica, vol. 45,
pp. 2692-2698, 2009. [11] P. Howlett, "Optimal strategies for the control of a train,"
Automatica, vol. 32, pp. 519-532, 1996. [12] P. Howlett, "The Optimal Control of a Train," Annals of
Operations Research, vol. 98, pp. 65-87, 2000. [13] E. Khmelnitsky, "On an optimal control problem of train
operation," Automatic Control, IEEE Transactions on, vol. 45, pp. 1257-1266, 2000.
[14] [Y. Wang, Z. Hou and X. Li, "A novel automatic train operation algorithm based on iterative learning control theory," in Service Operations and Logistics, and
Informatics, 2008. IEEE International Conference on, 2008, pp. 1766-1770.
[15] G. Bing, D. Hairong and Z. Yanxin, "Speed adjustment braking of automatic train operation system based on fuzzy-PID switching control," in Fuzzy Systems and Knowledge Discovery, 2009. Sixth International Conference on, 2009, pp. 577-580.
[16] G. Bing, D. Hairong and N. Bin, "Speed Control Algorithm for Automatic Train Operation Systems," in Computational Intelligence and Software Engineering, 2009. International Conference on, 2009, pp. 1-4.
[17] Q. Song and Y. Song, "Robust and adaptive control of high speed train systems," in 2010 Chinese Control and
Decision Conference (CCDC), 2010, pp. 2469-2474. [18] Q. Song, Y. Song, T. Tang, and B. Ning, "Computationally
Inexpensive Tracking Control of High-Speed Trains With Traction/Braking Saturation," Intelligent Transportation Systems, IEEE Transactions on, vol. 12, pp. 1116-1125, 2011.
[19] Q. Song and Y. Song, "Data-Based Fault-Tolerant Control of High-Speed Trains With Traction/Braking Notch Nonlinearities and Actuator Failures," Neural Networks, IEEE Transactions on, vol. 22, pp. 2250-2261, 2011.
[20] C. Lin, C. Chen, S. Tsai, and T. S. Li, "Extended sliding-mode controller for high speed train," in System Science
and Engineering (ICSSE), 2010 International Conference on, 2010, pp. 475-480.
[21] M. Chou and X. Xia, "Optimal cruise control of heavy-haul trains equipped with electronically controlled pneumatic brake systems," Control Engineering Practice, vol. 15, pp. 511-519, 2007.
[22] J. A. Schetz, "Aerodynamics of High-speed Trains," Annual Review of Fluid Mechanics, vol. 33, pp. 371-414, 2001.
[23] R. S. Raghunathan, H. D. Kim and T. Setoguchi, "Aerodynamics of high-speed railway train," Progress in
Aerospace Sciences, vol. 38, pp. 469-514, 2002.
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