Adaptive Fault-Tolerant Control for Time- Varying ... - Semantic Scholar

JOURNAL OF NETWORKS, VOL. 8, NO. 12, DECEMBER 2013

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Adaptive Fault-Tolerant Control for TimeVarying 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 Highspeed 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.

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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:

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mi xi (t )  i ui (t )  f i c1 (t )  f i c (t )  f i p (t )  f i e (t )

By assuming a target speed, we derive the corresponding position, acceleration, and traction force: xi (t )  x0 (t )  xi (t )

 i ui (t )  k   xi (t )  xi 1 (t )   b( xi (t )  xi 1 (t ))  k   xi (t )  xi 1 (t )   b  xi (t )  xi 1 (t ) 

(1)

n

xi (t )  ai0 (t )  xi (t )  xi (t )

  c0  cv xi (t )  mi  ca xi2 (t ) (mi ) i 1

 9.98sin i mi  0.002mi dl / Rt

where xi is the position of ith carriage, mi is the mass of ith carriage, i ui represents the traction force provided by the powered carriages. i  1 indicates that the ith carriage is powered; otherwise the ith carriage is unpowered. c0 , cv , ca are obtained by experiments. The running resistance consists of the main resistance f i p and external resistance f i e . 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., 0.002mi dl / Rt ) and the ramp resistance (i.e., 9.98sin i mi ). d l is the length of the axle, and Rt 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. fi c , fi c 1 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: m1 x1 (t )  1u1 (t )  k   x1 (t )  x2 (t )  b  x1 (t )  x2 (t )    c0  cv x1 (t )  m1

(3)

n

ca x12 (t ) (mi )  9.98sin 1m1  0.002m1dl / Rt mi xi (t )  i ui (t )  k   xi (t )  xi 1 (t )   b( xi (t )   c0  cv xi (t )  mi  9.98sin i mi  0.002mi dl / Rt

(i  2,

(4)

, n  1)

mn xn (t )  n un (t )  k   xn (t )  xn 1 (t )   b  xn (t )  xn 1 (t )    c0  cv xn (t )  mn (5)  9.98sin  n mn  0.002mn dl / Rt

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ui (t )  ui0 (t )  ui (t ) The ideal motion equations of high-speed train systems are as follows: 4

u10   c0  cv v0  m1  ca v02 (mi ) i 1

ui0   c0  cv v0  mi (i  2, , n) Eqs. (3), (4) and (5) will be transformed into the following: n

m1x1 (t )  (cv m1  2ca v0 m1 )x1 (t )  k  x2 (t )

(6)

i 1

 k x1 (t )  bx2 (t )  bx1 (t )  1u1 (t ) 

mi xi (t )  cv mi xi (t )  2k  xi (t )  k  xi 1 (t )  k  xi 1 (t )  2bxi (t )

(7)

 bxi 1 (t )  bxi 1 (t )  i ui (t )

(i  2,

, n  1)



mn xn (t )  cv mn xn (t )  k xn (t )  k  xn 1 (t )

(8)  bxn (t )  bxn 1 (t )  n un (t ) The standard state space form of the high-speed train systems can be transformed into the following equation: x(t )  Ax(t )  Bu (t ) (9) y (t )  Cx(t ) x(t )   x1 (t ) ... xn (t ) x1 (t ) ... xn (t ) 

T

y(t )   x1 (t ) ... xn (t ) x1 (t ) ... xn (t )  The target system will be: xm (t )  Am xm (t )  Bm r (t )

T

ym (t )  Cm xm (t )

(10)

The control objective is lim e  0,(e  x  xm ) . t 

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. u (t ) Let be the fault input, where u (t )  [u1 , u2 ,

i 1

 xi 1 (t ))  k   xi (t )  xi 1 (t )   b  xi (t )  xi 1 (t ) 

xi (t )  v0 (t )  xi (t )

, um ]T , u j  u jl f jl (t )

(1)

where u jl is an unknown scalar, and f jl (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:

JOURNAL OF NETWORKS, VOL. 8, NO. 12, DECEMBER 2013

 x(t )  * * v(t )  K1*T    K 2 r (t )  k3 x ( t )  

Let   diag{ 1 ,  2 ,

(12)

, m }

1 (ui  ui ) 0 (ui  vi ) When a known fault occurs, the original system is transformed into the following:

i  

I  x   0   x(t )   0 0   v(t )     (u (t )  v(t ))     1 1     x(t )    A C  A B  x     

I  x   0  *T  x(t )   x(t )   0    K1      1  1    x(t )    A C  A B  x      x (t )  0 0 0    K 2* r (t )    k3*     (u (t )  v(t ))     0   1   A C

I    A1 B 

 0  *T    k1 j   j  j1 j p   j  

 0  *   k2 j rj (t )   j  j1 j p   j   0     u j (t ) j  j1 j p   j 

 x(t )     x(t ) 

 0  *   k3 j  j  j1 j p   j 

and then the control parameters satisfying the following equation will be taken as follows: I   0   0     k1*j T  1 1    A C  A B  j  j1 j p   j  (13) I  0    1  1   Am Cm  Am Bm   0  *  0  (14)   k2 j    j  j1 j p   j   m   0  *  0    k3 jl f jl (t )    u jl f jl (t )  0 (15)  j  j1 j p   j  j  j1 j p   j 



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:  x(t )  u (t )  K1T  (16)   K 2 r (t )  k3  x(t )  Define the parameter errors: k1 j (t )  K1 j  K1*j k2 j (t )  K2 j  K2* j k3 j  k3 jl  k3* jl The original system will be transformed into the follows:

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 xm (t )   0     1  xm (t )    A C  *T  K1 

I   xm (t )   0     I     A1 B   xm (t )    

 0 * 0  x(t )  *    K 2 r (t )     k3     u (t ) x ( t )     

  0  x(t )      I     K1T    K 2 r (t )  k3    x(t )     0   1   Am Cm

  x(t )   0    r (t )   A Bm   x(t )    m   0   0      k1Tj x(t )     k 2 j r (t ) j  j1 j p   j  j  j1 j p   j   0      k3 jl f jl (t ) j  j1 j p   j  I

1 m

We have the tracking error equation: I  0  T e e  0         k1 j x(t )     1 1  e    Am Cm  Am Bm   e  j  j1 j p  j   0   0      k2 j r (t )     k3 jl f jl (t ) j  j1 j p   j  j  j1 j p   j 

Consider the positive-definition equation: T  0  e e V ( x, K1 , K 2 , k3 )    P       k1Tj  j k1 j  e   e  j  j1 j p   j   0   0      k2 j  21j     k3 jl  31j j  j1 j p   j  j  j1 j p   j 

(17)

(18)

where P  R2n2n , P  PT  0 such that T

I  0 I   0 T   1  P  P  1   Q 1 1   Am Cm  Am Bm    Am Cm  Am Bm  for any constant Q  R2 n2 n , such that Q  QT  0 ,

 j  R 2 n2 n is constant such that  j  Tj  0 ,  2 j  0 ,

 3 j  0 are constant, j  1, , 2m . Then we have  0  e e e  e V ( x, K1 , k3 )    P      P    2    k1Tj  j k1 j e e e e         j  j1 j p   j  T





j  j1

T

 0   0  1 1   k2 j  2 j     k3 jl  3 j jp   j  j  j1 j p   j 

Choose the adaptive laws as follow: T

 x(t )  e   0  . K1,i  i    P   .  x(t )  e   i 

(19)

T

e 0  K 2   2 j r (t )   P    e   i 

(20)

T

e 0  k3   3 j f jl (t )   P    e   i  Then we have T  e e V ( x, K1 , k3 )     Q    0 e  e

(21)

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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: u j  5sin 0.2t (u jl  5, f jl (t )  sin 0.2t )

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

t fault  30s Parameters of CRH5 are adopted. The parameters of the high-speed train are shown in the table 1. TABLE I. Symbol n w mi (i=1,2,4,7,8) mi (i=3,5,6) c0 cv ca k b umax umin

Figure 1.

Figure 2.

Figure 3.

PARAMETERS OF HIGH-SPEED TRAIN SYSTEMS Implication No. of carriages No. of powered carriages No. of powered carriages Mass of powered carriages Mechanical resistance coefficient Mechanical resistance coefficient Aerodynamic drag coefficient Elasticity coefficient Damping coefficient Maximum traction input Minimum traction input

Value 8 5 8500 8095 5.2 0.038 0.00112 20×106 5×106 302 205

Unit null null kg kg N/kg N∙s/m∙kg N∙s2/m2∙kg N/m N∙s/m∙kg kN kN

Position tracking error of the 1th carriage in high-speed train systems

Position tracking error of the 2th carriage in high-speed train systems

Position tracking error of the 3 th carriage in high-speed train systems

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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

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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.

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

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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.

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