adaptive fault-tolerant control for time- varying failure ......adaptive fault-tolerant control for...

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: [email protected]

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



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


train systems


train systems


train systems

Figure 7. Position tracking error of the 7 th carriages in high-speed

train systems


train systems

Figure 9. Velocity tracking error of the 1th carriage in high-speed

train systems




train systems

Figure 11. Velocity tracking error of the 3 th carriage in high-speed

train systems


train systems


train systems


train systems


train systems


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.

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