A Nonlinear Backstepping Control Design for Ball and Beam System

Abdulhakim A. Ezzabi, Ka C Cheok and Fatma A. Alazabi

Electrical and Computer Engineering Department, Oakland University Rochester, Michigan, USA 48309-4401

Email: { aaezzabi, cheok, faalazab } @oakland.edu

Abstract In this paper, a technique for designing a Ball and Beam System controller based on a nonlinear backstepping design is presented. A control law is developed, and asymptotic stability based on a Lyapunov stability criterion is satisfied. The goal is to design a controller for the ball position with the least overall energy consumption and minimum overshoot. In order to illustrate the efficiency of the proposed control strategy, the simulations are demonstrated and compared with [1]. The results show that the nonlinear backstepping design gives a smoother performance and needs less input magnitude compared to the LQR design. Moreover, the robustness of the proposed design with respect to parameter variations and different reference inputs is examined, and the simulations validated the control scheme.

Key words: Ball and Beam System, Nonlinear BackStepping (BS) control, Lyapunov stability, Trajectory tracking.

I. INTRODUCTION The Ball and Beam System is one of the most widely used systems [1],[4]. The nonlinearity and instability of the system opened the window for studying the control systems [1]. Changing of the ball position without limit for fixed input of beam angle makes the system an open loop unstable system. In order to stabilize the system, many control techniques such as gain scheduling PID [3], fuzzy control [4], and LQR design [1] have been applied to the system.

The paper is organized as follows. In Section 2, the Ball and Beam System is described and modeled, Section 3 is devoted to illustrate the backstepping controller design in detail, in Section 4, the Linear Quadratic Regulator (LQR) controllers is designed, and finally, the simulation results between both controllers and the conclusion are demonstrated in Sections 5 and 6, respectively.

II. BALL AND BEAM STATE SPACE MODELING This section provides a brief description of the modeling of

the Ball and Beam System. The system consists of five main parts: ball, beam, two arms, gear, and DC servo motor. The

beam is tilted by a DC servo motor together with a ball rolling back and forth on the track of the beam as shown in Figure (1). One side of the long beam is supported by a pinned arm, and other side is connected by a level arm which is coupled to a gear driven by a DC servo motor with a reducing gearbox [1]. The position of the ball can be obtained by measuring the output voltage of the potentiometers.

Fig.1 Ball and Beam system

The dynamic model of the Ball and Beam System can be written as [1]:

( ) s in ( ) cos ( )m x t m g t m g t = (1) Figure (1) shows the body diagram of the Ball and Beam System. From the body diagram, the analysis of the force balance is obtained by Newtons law where , , , ,x m g and represent the ball position on the beam, beam angle from the horizontal position, ball mass, gravity acceleration, and the friction constant, respectively. By assuming that the beam angle is too small ( )( ) 10t < and there is no friction applied to the system, ( ) ( )mx t mg t= is determined. The relationship between the beam angle and the gear angle can be approximated as:

( ) ( )rt tL

(2)

where r is gear radius, and L is is beam length. By manipulating the dynamic equation (1) and substituting the assumptions and equation (2) into (1), the transfer function can be represented in the state-space form and output

1318978-1-4799-0066-4/13/$31.00 2013 IEEE

equation as stated in (3) and (4) where defined variables are as follows [1]:

1 2 3 4[ ] [ ( ) ( ) ( ) ( ) ]T Tx x x x x t x t t t = = x

and [ ]T denotes transpose 1 1

2 2

3 3

4 4

0 1 0 0 000 0 0

( )00 0 0 110 0 0 0

x xg rx x

u tLx xx x

= +

(3)

[ ]( ) 1 0 0 0 ( )y t t= x (4) where 4x D R is a state vector, y is the output of the ball position , and u R is the input voltage of the motor for the Ball and Beam System. In addition, we have chosen to omit the dynamic behavior of the motor [2]:

motorm m m

ddiL R i k Vdt dt

+ + = (5)

motor ik i = (6) where , , ,m m m iL R k k are motor constants, motor is a motor angle, V and i are motors voltage and current. We have assumed the motor dynamics (5) to be much faster than the upcoming backstepping control system for (3).

III. BACKSTEPPING DESING Nonlinear backstepping control is a design approach for

the Ball and Beam System model. A systematic construction of both feedback control laws and associated Lyapunov functions has been involved in the backstepping control. backstepping control consists of designing a series of control laws recursively by using some of the state variables in a system as virtual controls [5],[ 6], [7]. It is seen from the model equation (3) that ball position can be controlled by measuring the output voltage of the potentiometer. In order to design the backstepping control system, the new state variables are defined as: 1 1 dz x x= (7) 2 2 1 1( )z x z= (8) 3 3 2 1 2( , )z x z z= (9) 4 4 3 1 2 3( , , )z x z z z= (10) where dx is the desired position input and, 1 2, , and 3 are stabilizing functions for the new state variables. Step1. From equation (7), the state space equation for 1z is

1 2 1 1( ) dz z z x= + (11) 1 1( )z should be selected through the Control Lyapunov

Function (CLF) to guarantee the stability of the control system as

21 1 1

1( )2

V z z= (12)

Then, 1 1 1 1 1 1 1 1 2( ) ( ( ) )dV z z z z z x z z= = (13)

From equation (13), if 1 1 1 1( ) dz c z x = + , equation (11) can be exponential stable as t where 1c is a design parameter. Step2. From equation (8), the state space equation for 2z is:

2 3 2 1 2 1 1( , ) ( )z z z z z = + (14) where 1 1 1 1 1 2 1 1( ) ( )d dz c z x c z c z x = + = + Since equation (14) includes the information of equation (11), the CLF is selected as:

22 1 2 1 1 2

1( , ) ( )2

V z z V z z= + (15)

2 1 2 1 1 2 22

1 1 2 3 2 1 2 1 2 1 1

( , ) ( )

( ( , ) ( ))

V z z V z z z

c z z z z z z z z

= +

= + + +

(16)

If the stabilized function 2 in equation (16) is defined as

2 1 2 2 2 1 1 1( , ) ( )z z c z z z = + , where 2 0c > is a design parameter, then 2 can be rearranged as:

22 1 2 1 2 2 1 1( , ) ( ) (1 ) dz z c c z c z x = + + (17) Therefore, 2 22 1 2 2 3 1 1 2 2( , )V z z z z c z c z= (18) Step3. From equation (9), the state space equation for 3z is

3 4 3 1 2 3 2 1 2( , , ) ( , )z z z z z z z = + (19) where 2 1 2 2 2 1 1( , ) ,z z c z z = + 21 1 1 1 2 dc z c z x = + Since equation (19) includes the information of equations (11) and (14), the CLF is selected as

23 1 2 3 2 1 2 3

1( , , ) ( , )2

V z z z V z z z= + (20)

Then,3 1 2 3 2 1 2 3 3

2 21 1 2 2 3 4 3 2 3 1 2 3 2 1 2

( , , ) ( , )

( ( , , ) ( , ))

V z z z V z z z z

c z c z z z z z z z z z z

= +

= + + +

(21)

If the stabilized function 3 in equation (21) is defined as

3 1 2 3 3 3 2 1 2 2( , , ) ( , )z z z c z z z z = + , where 3 0c > is a design parameter, then 3 can be rearranged as: 3 2 2( , , ) ( 2 ) (2 ) ( )3 1 2 3 1 2 1 2 3 31 1 2 1 2z z z c c c z c c z c c c z xd = + + + + (22)

Therefore, 2 2 23 1 2 3 1 1 2 2 3 3 3 4( , , )V z z z c z c z c z z z= + (23) Step4. From equation (10), the state space equation for 4z is:

4 4 3 1 2 3

3 1 2 3

( , , )

( , , )

z x z z z

u z z z

=

=

(24)

where 3 2 2

3 1 2 3 1 1 2 1 1 2 2 1 2 3 3( , , ) ( 2 ) (2 ) ( ) dz z z c c c z c c z c c c z x = + + + + By substituting 3 1 2 3( , , )z z z into equation (24), we have

3 2 24 4 1 1 2 1 1 2 2 1 2 3 3( 2 ) (2 ) ( ) dz x c c c z c c z c c c z x= + + + + + + (25)

Since equation (24) includes the information of 1 2, ,z z and 3z, the CLF is selected as

24 1 2 3 4 3 1 2 3 4

1( , , , ) ( , , )2

V z z z z V z z z z= + (26)

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4 1 2 3 4 3 1 2 3 4 42 2 2

1 1 2 2 3 3 4 3 3 1 2 3

( , , , ) ( , , )

( ( , , ))

V z z z z V z z z z z

c z c z c z z z u z z z

= +

= + + +

(27)

To satisfy the system stability in equation (27), the backstepping control law can be selected as 4 4 3 3 1 2 3( , , )u c z z z z z= + (28)

IV. LQR CONTROL DESIGNS In [1], the LQR design for the Ball and Beam model is demonstrated in discrete time. For the reference purpose, LQR law ( )u that minimizes the performance measure can be defined as[8]-[ 9]

( )0

( ) ( ) ( ) ( )T TJ x t Q x t u t R u t d t

= + (29) where Q is a symmetric positive semidefinite weighting matrix and R is a symmetric positive definite weighting matrix

( ) ( )r refu kx t k y t= + (30)

where k is a control gain matrix, and rk is a control tracking gain matrix. The optimal control of (30) that minimizes J can be expressed as

11 1( ) ( ) ( ) ( )Top cl refu t R B Px t H A B y t

= + (31) where clA is the closed loop system matrix of (3), ( )x t and

( )refy t are the position and the position reference of the ball respectively, H is a picking matrix which equals the measurement of ball position, and P is the symmetric positive definite solution of the Algebric Riccati Equation (ARE)

1 0T TA P P A P B R B P Q+ + = (32)

Figure (2) shows the block diagram for Ball and Beam System with LQR design.

Fig.2. LQR control structure

V. SIMULATION AND RESULTS A. Systems Parameters The parameters of the Ball and Beam System are shown in Table 1[1]. Symbol Parameter Value Unit Radius of the gear 0.045 m Length of the Beam 0.441 m g Gravity acceleration 9.8 m/s2

B. Simulation Results In this section, the simulation results of the proposed

controller, which is performed on the model of the Ball and Beam System, are presented. Moreover, a rectangular wave is required in order for the ball position to follow the reference value between 0.15m and 0.3m [1]. To examine the effectiveness of the proposed trajectory tracking control methodology, the simulations for the Ball and Beam System were performed in MATLAB. The simulation results of LQR design and nonlinear backstepping design of tracking control performance for the ball position are shown in Figure (3). The LQR parameters are chosen as ( )( ) 1500,1000,15,1diag Q = and

1R = , and the backstepping controller parameters are chosen as 1 2 3 40.8, 3, 4, 20c c c c= = = = . The control input simulations for both designs are also considered to decide which control design required less effort. The results proved that nonlinear backstepping design requires less effort than LQR as shown in Figure (4). Figure (5) shows the performance of the gear angle for both LQR and backstepping designs.

C. Robustness of the Proposed Controller In order to validate the proposed control scheme for the Ball and Beam system, a sinusoidal reference input is considered as following: sin(0.25 ) sin(0.05 )dx t t = + (33) and it was assumed that the beam length parameter of the system has a perturbation of +25% of its original value. Figure (6) shows the ball position tracking for sinusoidal input without perturbation. Figure (7) displays the and ability of the proposed design to track the sinusoidal reference in the presence of the parameter perturbation.

Fig.3. Tracking control performance by LQR and BS designs.

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Fig. 4.Control input for LQR and (BS) design.

Fig. 5 Gear angle performances for LQR and BS design.

Fig.6. Tracking of BS for the sinusoidal reference input without perturbation.

Fig.7. Tracking of BS for the sinusoidal reference input with perturbation.

VII. CONCLUSION This paper demonstrates a nonlinear backstepping design for controlling the ball position of the ball and beam dynamic system. The design procedures are discussed in detail, and the comparison with the LQR design is provided. The results show that the nonlinear backstepping design delivers a better performance in terms of transient and steady state responses, and needs less control effort than the LQR design. In addition, the robustness of the proposed design with respect to parameter variations and different reference inputs is examined to validate the control scheme. Our future work will consider the adaptive backstepping design with genetic algorithms to handle more complex models.

REFRENCES [1] Z.-H. Pang, G. Zheng and C-X. Luo, Augmented State Estimation and

LQR Control for a Ball and Beam System, IEEE Conference on Industrial Electronics and Applications (ICIEA), pp. 1328 - 1332, 2011.

[2] M. Keshmiri, A. Jahromi, A. Mohebbi, M. Amoozgar, W-F Xie, Modeling and Control of Ball and Beam System Using Model Based and Non-Model Based Control Approaches , International Journal on Smart Sensing and Intelligent systems, vol. 5, pp.14-35, 2012.

[3] B. Krishna, S. Gangopadhyay, J. George, Design and Simulation of Gain Scheduling PID Controller for Ball and Beam System, International Conference on Systems, Signal Processing and Electronics Engineering, 2012.

[4] M. Amjad, M.I. Kashif, S.S. B. Abdullah, Z. Shareef, Fuzzy Logic Control of Ball and Beam System, International Conference on Eduction Technology and Computer, v3, pp.489-493, 2010.

[5] M. Krstic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley and Sons, 1995

[6] R. Wai, Kun-lum L. Chuang, Design of Backstepping Particle-Swarmoptimisation Control for Maglev Transportation System,IEEE on IET Control Theory & Applications vol.4, pp.625 645, 2010.

[7] F.Mazenc,A.Astolfi, R. Lozano, Lyapunov Function for the Ball and Beam:Robustness Property,IEEE Conference on Decision and Control, vol.2, pp.1208 1213.1999.

[8] S. Zak, Systems and Control, Oxford University Press, USA, 2002. [9] C. Kumar, S. Lal, N. Patra, K. Halder, M. Reza, Optimal Controller

Design for Inverted Pendulum System based on LQR method, IEEE international conference on ICACCCT, pp.259 263, 2012.

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