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Backstepping sliding mode tracking control of a vane-type air motor X–Y table motion system
ISA Transactions 50 (2011) 278–286
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Backstepping sliding mode tracking control of a vane-type air motor X –Y table motion system Chia-Hua Lu a,∗ , Yean-Ren Hwang a,b , Yu-Ta Shen a a
Department of Mechanical Engineering, National Central University, Chung-Li 320, Taiwan
b
The Institute of Opto-Mechatronics Engineering, National Central University, Chung-Li 320, Taiwan
article
info
Article history: Received 24 February 2010 Received in revised form 30 November 2010 Accepted 23 December 2010 Available online 26 January 2011 Keywords: Air motor Pneumatic PID controller Backstepping sliding mode controller
abstract Air motors are increasingly being used in pneumatic related industries because of their advantages of low operating cost and low maintenance. The DSP controller and the backstepping sliding mode control method were utilized in this study to control an X –Y pneumatic table for tracking trajectory. Due to the effects of the compressibility of air, friction between the motor and ball screw table and the dead-zone effect caused by the proportional valve, the system will yield different responses even with the same inlet pressure and will chatter at low speed. Hence under certain conditions, this method of backstepping sliding mode control can be applied to achieve better results than with the PID controller, such as for tracking circle error and tracking error of the two axes. According to the results, a steady-state error of 0.5 µm can be achieved. The proposed method of backstepping sliding mode control can accomplish accurate tracking circle trajectory performance, offering an improvement in the tracking error of more than 50% over that of the PID controller. © 2010 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction Pneumatic actuators are increasingly being used in air motors and cylinders in pneumatic related industries. Unlike conventional electric and hydraulic actuators, the compressibility of the air allows pneumatic actuators to offer highly nonlinear performance. Their major advantages are their high payload-to-weight and payload-to-volume ratios, and high speed and force capabilities. Their reasonable cost is another reason that makes them appealing for use in industry. Unlike the easily controllable conventional electric DC motors used with ball screw drivers which have been widely applied in the past, pneumatic drives require sophisticated controllers, due to the compressibility of air, adverse friction characteristics and low damping. However, they also offer several advantages: the compressibility of air provides a ‘‘cushioning’’ effect which is important in applications like blow molding or glass forming. This also results in highly nonlinear pressure dynamic characteristics. As a result, conventional PID feedback controllers [1,2] are not so effective even for position control. There have been a number of investigations and analyses of the dynamics of vane-type air motors [2–5] carried out to date, most for different types of control, like PID control [2], MRAC [3], and sliding mode control [5]. In
∗
Corresponding author. Tel.: +886 3 4227151x34342; fax: +886 3 4254501. E-mail address: [email protected] (C.-H. Lu).
addition, a servo pneumatic system (cylinder) has been proposed for position control [6,7]. In backstepping sliding mode control, the nonlinear system is divided into many subsystems. Sliding mode control is designed for each subsystem with the Lyapunov function to guarantee the convergence of the position tracking error for all possible initial conditions. The integral action is mixed with backstepping control. The addition of an integrator strengthens the system’s robustness against modeling uncertainties and external disturbances, thus improving steady-state control accuracy [8]. An adaptive backstepping sliding mode controller is proposed in [9] to control the position of a linear induction motor (LIM) drive. The device is able to compensate for uncertainties including the force of friction. A tracking position control system based on the backstepping design has been proposed for an electropneumatic device, consisting of a cylinder and two valves [10]. Synchronous control for mutual positioning of two vertical-type pneumatic servo systems is proposed in [11]. A fuzzy controller is adopted in each pneumatic servo system to let the system track the reference input. Air motors have uncertain behaviors. In most practical motion systems, the flow of air from the inlet of the storage to the air motor will decrease after each experimental result. The four-vane-type air motor used in the experiments is shown in Fig. 1. Each vane will cause friction, especially at low speed. The air pressure entering the air motor will decrease in a non-regular fashion due to the various volumes of the two vanes. To counteract these uncertain behaviors and eliminate the dead zone caused by friction, modelreferencing adaptive control (MRAC) with fuzzy control [3] is
0019-0578/$ – see front matter © 2010 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2010.12.008
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Fig. 2. Schematic diagram of the air motor system. Fig. 1. Representation of the principles of vane-type air motor operation.
proposed. Fuzzy rules combined with sliding mode control are used in the aforementioned study [5] to improve the steady response and build robustness with an air motor ball screw table. With a pneumatic cylinder [10], backstepping control provides robustness to cope with the compressibility of air. In order to overcome uncertain behaviors the proposed control scheme for an air motor ball screw table system utilizes backstepping sliding mode control for a second-order uncertain system. This controller is designed by adopting sliding mode control with two Lyapunov functions to guarantee that the error will converge to zero as t → ∞. The system is asymptotically stable even if external force disturbances and frictional forces exist. The experimental results demonstrate that the overshoot phenomenon will not occur with this system and can maintain steady-state position error in position tracking compared with the PID controller. 2. Air motor servo system Fig. 1 shows a rough representation of a vane-type air motor. The motor has a rotational drive shaft with four slots, each of which is fitted with a freely sliding rectangular vane. When the drive shaft starts to rotate, the vanes tend to slide outward due to centrifugal force, but are limited by the shape of the rotor housing. Depending on the flow direction, the motor will rotate in either a clockwise or counterclockwise direction. A schematic diagram of the air motor system is shown in Fig. 2. The system consists of an air motor (GAST 1AM), an air tank, an electronic proportional directional control valve (FESTO MPVE), X –Y table (ISEL, anti-backlash ball nut design), a filter/regulator with lubricant (SHAKO FRL-600), an optical linear scale (A1-0600) the accuracy of which is 0.5 µm, and a digital signal processor (DSP, TI C240). The TMS320C240 device is a member of a family of DSP controllers based on the TMS320C2xx generation of 16-bit fixedpoint digital signal processors. This family is optimized for digital motor/motion control applications. The DSP controllers combine the enhanced TMS320 architectural design of the C2xLP core CPU for low-cost, high-performance processing capabilities and several advanced peripherals optimized for motor/motion control applications. These peripherals include the event manager module, which provides general-purpose timers and compare registers to
Fig. 3. Photograph of the experimental air motor system.
generate up to 12 PWM output, and a dual 10-bit analog-to-digital converter (ADC), which can perform two simultaneous conversions within 6.1 µs. The control loop time used in the experiments is 5 ms and the software ‘‘core composer v.2 (ccs c2000)’’ is adopted to write the control program. A DSP emulator XDS510 utilizing a standard JTAG plug acts as the Scan-Based Emulator. Since this is a plug-and-play emulator, integrated debugger programs can be executed from both on or off chip memory without additional delay under any clock speed. The airflow path starts from the air tank, moving first through the filter, then the control valve, finally entering the air motor. The airflow entering the motor is determined by the position of the valve, which is controlled by an externally applied voltage. The experimental air motor ball screw table system is shown in Fig. 3. 3. Controller design A method of fuzzy sliding mode position control incorporating a ball screw driven by a vane-type air motor has been proposed in [5] in which fuzzy logic control is applied to avoid the disadvantages of disturbances and chattering. However this study did not show the sinusoidal tracking error. The experiment results only showed that the fuzzy rules could stabilize the system. There is no state of the art method focusing on X –Y table control. In this current study,
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Fig. 4. X –Y table performance under the same inlet pressure: (a) curves of X axis; (b) curves of Y axis.
Fig. 5. Experimental results after adjusting parameters: (a) k1 ; (b) c1 ; and (c) h.
two controllers are designed and compared: a PID controller and a backstepping controller. The Ziegler–Nichols tuning method is adopted to tune the parameters of the PID controller, by setting the Integral (I) and Derivative (D) gains to zero. The Proportional (P) gain is increased until it reaches the ultimate gain Ku , at which point the output of the control loop oscillates with constant amplitude. Ku and the oscillation period Tu are used to set the KP , KI , and KD gains depending on the Ziegler–Nichols tuning method (KP = 0.6Ku , KI = 2Kp /Tu , KD = 0.125KP Tu ). Finally, the variable parameters Ku and Tu are adjusted to approximately 38.33 and 0.07, respectively. Fig. 4 shows the open loop step responses for backstepping control of a vane-type air motor installed on X –Y table with an inlet pressure of 3 bar. It is obvious that two tables can move nearly 200 mm within 10 s. Because the weight of the X axis and Y axis are different, there will be a slight difference in the output response for each experimental result for each axis but the variation is small. The compressibility of air, overall mechanical friction (which includes ball screw table friction), nonlinear behavior of the servovalves, and pressure loss are the main problems associated with the highly nonlinear characteristics of pneumatic actuation, namely air motors. The overall system will demonstrate nonlinear phenomena. Given the results, the vane-type air motor connected
with a X –Y table is considered to be a second-order system so that the dynamics of the system can be described by the following equation for considering friction: As can be seen in Fig. 1, the air flow enters the vane-type air motor from chamber A and exhausts from chamber B. r is the inner rotor radius; R is the radius of the motor body; d is the difference between R and r (d = R − r ); θ is the angle of motor rotation; xr is working radius of the vane measured from the center of the rotor which is given by xr = d cos θ +
R2 − d2 sin2 θ ,
(1)
(assuming that d sin θ << R ). Considering friction and the different payloads, and using Newton’s second law of angular motion we get 2
2
M − Mc S (θ˙ ) − Mf θ˙ = J θ¨ ,
2
(2)
where θ˙ is the angle velocity; θ¨ represents the angular acceleration; Mc is the stiction coefficient; Mf is the friction coefficient; and S (θ˙ ) is described as
1, ˙ S (θ ) =
sign(θ˙ ),
θ˙ = 0 θ˙ ̸= 0.
(3)
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Fig. 6. Experimental results for PID control: (a) position results within 10 s; (b) output voltages for two axes; (c) maximum overshoot between 1 and 3.5 s.
Fig. 7. Experimental results for backstepping sliding mode control: (a) position results within 10 s ; (b) output voltages for two axes; (c) maximum overshoot between 2 and 3.5 s.
The driving torque is determined by the difference in pressure acting on the vane between the drive and exhaust chambers, and is given by M = (Pa − Pb )(xr − r )Lxr = (Pa − Pb )(x2r − rxr )L.
(4)
Assuming that x is the displacement of the ball screw table and Ps is the screw pitch: the displacement x = ps θ ; the velocity x˙ = ps θ˙ . The dynamics of the system can be described by the following equation considering friction:
x¨ = −
Mf Jm
x˙ +
Ps Jm
L(x2r − rxr )(1P ) −
Ps Jm
Mc S
x˙ Ps
−
Ps Jm
Tl ,
(5)
where 1P is the (Pa − Pb ); Jm is the total system inertia (motor and ball screw); Tl is the load torque. Eq. (5) can be derived as follows: x¨ = −
Mf Jm
x˙ +
kp ps Jm
L(x2r − rxr )(u) + Des , p
where Des = − J s Mc S m
x˙ ps
−
ps T Jm l
, U = Pa − Pb .
(6)
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Fig. 8. Experimental results for random step trajectory tracking position (a) tracking position results; (b) output voltages.
Fig. 9. Experimental results for PID control (frequency is 0.05 Hz): (a) tracking circle trajectory with a radius of 15 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error.
X1 defines the displacement; X2 is the moving velocity. Eq. (6) can now be rearranged as [9]
X˙ 1 = X2 X˙ = (A + 1A)X2 + (B + 1B)U + Des (t ) 2 Y = X1 .
(7)
z˙1 = Y˙ − Y˙d = X2 − Y˙d . (8)
(10)
(9)
(11)
Define the stable item
α1 = c1 z1 ,
where F is the uncertainty F = 1AX2 + 1BUDes (t )
z1 = Y − Yd . Then
Eq. (7) can be derived as follows: X˙ 2 = AX2 + BU + F ,
and |F | ≤ F¯ , 1A is the variable uncertainty of the system. Suppose the variation of the variable uncertainty and external disturbance are not so quick that F˙ = 0. The tracking error between the system output Yd and the position command signal Y can be defined as
where c1 is a positive constant.
(12)
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Fig. 10. Experimental results for backstepping sliding mode control (frequency is 0.05 Hz): (a) tracking circle trajectory with a radius of 15 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error.
Define the Lyapunov function as V1 =
1 2
z12 .
The control input can be designed as follows: (13)
Defining z2 = z˙1 + α1 = X2 − Y˙d + α1 , V˙ 1 = z1 X2 − Y˙d = z1 (z2 − α1 ) = z1 z2 − c1 z12
(14)
z˙2 = X˙ 2 − Y¨d + α˙ 1 = AX2 + BU + F − Y¨d + α˙ 1 .
(15)
Define the Lyapunov function as 1
s , 2 where s is the sliding surface. Define the sliding function as V2 = V1 +
2
s = k1 z1 + z2 ,
U = B−1 [−k1 (z2 − c1 z1 ) − A(z2 + Y˙d − α1 )
− F¯ sgn(s) + Y¨d − α˙ 1 − h(s + β sgn(s))], where h and β are positive constants. Substituting Eq. (19) into Eq. (18) we get V˙ 2 = z1 z2 − c1 z12 − hs2 − hβ |s| + Fs − F¯ |s|
≤ −c1 z12 + z1 z2 − hs2 − hβ |s| + |s| (|F | − F¯ ) ≤ −c1 z12 + z1 z2 − hs2 − hβ |s| .
(16)
(20)
Define Q as a positive definite symmetric matrix
c1 + hk21
Q =
(17)
where k1 > 0; then
hk1 −
hk1 −
1
1
2 ,
h
2
(21)
then
V˙ 2 = V˙ 1 + ss˙ = z1 z2 − c1 z12 + ss˙ = z1 z2 − c1 z12 + s (k1 z1 + z˙2 ) z T Qz = z1
= z1 z2 − c1 z12 + s[k1 (z2 − c1 z1 ) + AX2 + BU + F − Y¨d + α˙ 1 ] = z1 z2 − c1 z12 + s[k1 (z2 − c1 z1 ) + A z2 + Y˙d − α1 + BU + F − Y¨d + α˙ 1 ].
(19)
c1 + hk21
z2
hk1 −
1
hk1 −
1
2 z
1
z2
T
h 2 = c1 z12 + hk21 z12 + 2hk1 z1 z2 − z1 z2 + hz22 (18)
= c1 z12 − z1 z2 + hs2 ,
(22)
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Fig. 11. Experimental results for backstepping sliding mode control (frequency is 0.05 Hz): (a) tracking circle trajectory with a radius of 30 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error.
where z = z1
z2 and Eq. (20) can now be written as
V˙ 2 ≤ −z Qz − hβ |s| . T
Then
|Q | = h(c1 +
hk21
) − hk1 −
1 2
2
1
= h(c1 + k1 ) − . 4
(23)
Now choose the values of h, c1 , k1 to let |Q | > 0. Then V˙ 2 ≤ 0.
(24)
Therefore, z1 and z2 will converge to zero. The stability of the proposed backstepping sliding mode control system can be guaranteed. From Eq. (19), the initial experimental conditions are designed as follows: A and B for two axes (X and Y ) are −7.865, 42, and −7.865, 36, respectively; F¯ is approximately 0.6; k1 , c1 and h are assumed to be 8, 8 and 15, respectively. The MATLAB system identification toolbox is used to measure the values of A and B. System identification based on MATLAB [12] is highly efficient. First of all, a second-order system is built with the System Simulation Toolbox of SIMULINK, and then the parameters of the system are identified using the System Identification
Toolbox. The input data (pressure) and output data (distance of table movement) were obtained from the experimental results. These data are input into the System Identification Toolbox. Parameters A and B in Eq. (7) will be identified. According to the control input design in Eq. (19), k1 affects the transient response, c1 has an influence when the system approaches the sliding surface, and h affects the steady-state response. If these three parameters are chosen to be too large or too small, the system will have large overshoot and steady-state error or a have slow rising time and setting time. The experimental results obtained after adjusting the k1 , c1 and h parameters are shown in Fig. 5. Fig. 5(a) shows the experimental results for the backstepping sliding mode controller with k1 . Since this parameter affects the transient response, different k1 values will have an effect on the rise time. Parameter c1 affects the approach of the sliding surface. The results are shown in Fig. 5(b). For a gradual increase in c1 , a longer time is needed to approach the sliding surface. As shown in Fig. 5(c), the dissimilarity of parameter h affects the various chattering phenomena of the pneumatic system. The steady response leads to a serious chattering phenomenon. According to the experimental results, when the values of k1 , c1 and h are 8, 8 and 15, respectively, this will produce a better response for the transient and steady-state errors.
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Fig. 12. Experimental results for backstepping sliding mode control (frequency is 0.1 Hz): (a) tracking circle trajectory with a radius of 10 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error. Table 1 Comparison of different controllers for position control.
Table 2 Comparison of different controllers for tracking circle trajectory.
Position control at 50 mm
Tracking circle trajectory with radius 15 mm
PID controller
Backstepping sliding mode controller
Steady-state position (Pss)
Steady-state position (Pss)
X = 49.9 mm
X = 50.0005 mm
Y = 50.02 mm
Rising time (Tr) X = 2.395 s
Y = 49.9995 mm
Rising time (Tr) Y = 2.375 s
X = 2.38 s
Y = 2.365 s
Maximum overshoot (Mo)
Maximum overshoot (Mo)
X = 1.431%
X = 0%
Y = 1.1%
Y = 0%
Steady-state error (Ess)
Steady-state error (Ess)
X = 0.1 mm
X = 0.0005 mm
Y = 0.02 mm
Y = 0.0005 mm
4. Experimental results This section is divided into two parts, one discussing the position tracking results and the other performance analysis and discussion. 4.1. Position tracking results In this experimental study, the main focus is on position control at 50 mm and the tracking circular trajectory with a radius R =
PID controller
Backstepping sliding mode controller
Frequency is 0.05 Hz
Frequency is 0.05 Hz
Tracking error X axis Y axis Circle trajectory
Tracking error
±0.8 mm ±0.6 mm ±0.6 mm
X axis Y axis Circle trajectory
+0.3 mm to −0.1 mm ±0.1 mm ±0.25 mm
15 mm. The operating pressure is regulated to be a constant value of 3 bar. The pressure is the same for each experimental run, so there is little variation in system nonlinearities. The performance of the proposed PID control is compared to the backstepping sliding mode control. The experimental results with PID control and backstepping sliding mode control at position 50 mm are shown in Figs. 6 and 7, respectively. As can be seen in Fig. 6(a) the steady-state response for the X axis is 49.9 mm and steady-state response for the Y axis is 50.02 mm, hence the experimental steady-state error for these two axes are 0.1 mm and 0.02 mm, respectively. On the other hand, the experimental results show that there is serious overshoot phenomenon with PID control. Fig. 6(b) shows the output voltage for the two axes. It can be seen in Fig. 6(c) that the maximum overshoot of the X axis is 1.431% and for the Y axis it is 1.1%.
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Table 3 Experimental results for different tracking circle trajectories. Tracking circle trajectory with backstepping sliding mode controller Frequency is 0.05 Hz (radius 30 mm)
Frequency is 0.1 Hz (Radius 10 mm)
Tracking error
Tracking error
X axis Y axis Circle trajectory
±0.2 mm to −0.1 mm +0.2 mm to −0.1 mm ±0.2 mm
Fig. 7(a) shows the steady-state responses of the X axis and Y axis. Compared with the experimental results for PID control (shown in Fig. 6(a)), the steady-state error can be maintained within 0.5 µm for both the X and Y axes, and the maximum overshoot can be obviously reduced. In other words, the experimental results show no overshoot phenomenon. Fig. 8 shows the experimental results for the random step trajectory tracking position. For X axis tracking, for each step target position, the steady-state errors are 5.999 mm, 15.9995 mm, 0.0005 mm, and 5.9995 mm. For Y axis tracking, for each step target position, the steady-state errors are 10.9995 mm, 25.999 mm, 5.999 mm, and 20.999 mm. The results show that random step trajectory tracking positioning can be accomplished for both axes with the backstepping sliding mode controller, to maintain the steady-state error within 1 µm. We now discuss tracking circle trajectory control for the pneumatic servo system. In the experiment, the X axis is designed to track the wave such that the equation is x(t ) = x0 + R cos(2π ft ) and the Y axis is designed for tracking y(t ) = y0 + R sin(2π ft ). The initial experimental conditions are set as follows: x0 and y0 are adjusted to zero; the radius is R = 15 mm; the frequency is f = 0.05 Hz. The goal of the experiments is to compare performances of two proposed controllers, a PID controller and the backstepping sliding mode controller. Fig. 9(a) shows the tracking circle trajectory with a frequency of 0.05 Hz. The trajectory consists of two axes: the tracking sinusoidal wave (cosine) of the X axis and the sinusoidal wave (sine) of the Y axis, as shown in Fig. 9(b) and Fig. 9(c), respectively. There is serious chattering phenomenon under the PID controller. According to the experimental results, the trajectory error for the X axis is approximately ±0.8 mm and that for the Y axis is approximately ±0.6 mm, as shown in Fig. 9(e). Fig. 9(f) shows the tracking error of the circle trajectory to be approximately ±0.6 mm. The compressibility of air and the friction make the overall system for the X –Y table nonlinear. Backstepping sliding mode control is proposed in order to overcome the chattering phenomenon caused by this disturbance and decrease the tracking error. The experimental results (tracking circle trajectory with a radius of 15 mm), the tracking sinusoidal wave (cosine) of the X axis and the sinusoidal wave (sine) of Y axis, for the backstepping sliding mode control are shown in Fig. 10(a)–(c), respectively. Compared with the results shown in Fig. 9(a)–(c) for PID control, it can be seen that the proposed controller not only restrains the chattering phenomenon but also maintains the tracking error at under 0.3 mm. According to the results shown in Fig. 10(e)–(f), the trajectory error for the X axis is approximately +0.3 to −0.1 mm; the trajectory error for the Y axis is approximately ±0.1 mm; the tracking error for the circle trajectory is approximately ±0.25 mm. Figs. 9 and 10 show a comparison between the two different
X axis Y axis Circle trajectory
+0.1 mm to −0.2 mm +0.3 mm to −0.1 mm ±0.25 mm
controller performances in order to prove that the system can track more circle trajectories that are with different radii and frequencies. Fig. 11 shows the tracking results for a circle with a radius of 30 mm; the frequency is 0.05 Hz. Fig. 12 shows the tracking results for a circle with a radius of 10 mm; the frequency is 0.1 Hz. 4.2. Analysis of performance and discussion A comparison of different controllers for position control is shown in Table 1. Position control by the backstepping sliding mode controller can maintain excellent transient response performance (rising time), eliminate the overshoot and keep the steady-state error under 0.5 µm. The results for different tracking circle trajectory control are shown in Tables 2 and 3. Although PID controller is popularly used to control most motion systems, some pneumatic nonlinearity due to compressibility cannot be eliminated. According to the experimental results in Table 2 the proposed backstepping sliding mode controller offers an improvement in the tracking error of more than 50% compared to the PID controller. This means that this proposed controller is able to overcome pneumatic nonlinearity better. References [1] Zhang Y, Nishi A. Low-pressure air motor for wall-climbing robot actuation. Mechatronics 2003;13(4):377–92. [2] Shen YD, Hwang YR. Dynamic modeling and controller design for air motor. In: SICE–ICASE, 2006. International joint conference. 2006. p. 461–6. [3] Hwang YR, Shen YD, Jen KK. Fuzzy MRAC controller design for vane-type air motor systems. Journal of Mechanical Science and Technology 2008;22(3): 497–505. [4] Wang J, Pu J, Moore PR. Modeling study and servo-control of air motor systems. International Journal of Control 1998;71(3):459–76. [5] Song J, Ishida Y. A robust sliding mode control for pneumatic servo systems. International Journal of Engineering Science 1997;35(8):711–23. [6] Taghizadeh M, Ghaffari A, Najafi F. Improving dynamic performancees of PWM-driven servo-pneumatic systems via a novel pneumatic circuit. ISA Transactions 2009;48(4):512–8. [7] Shin MC, Lu CS. Fuzzy sliding mode position control of a ball screw driven by pneumatic servomotor. Mechatronics 1995;5(4):421–31. [8] Tan YL, Chang J, Tan HL, Hu J. Integral backstepping control and experimental implementation for motion system. In: Proceedings of the 2000 IEEE internation conference on control application. p. 25–7. [9] Lin FJ, Shen PH, Hsu SP. Adaptive backstepping sliding mode control for linear induction motor drive. IEE Proceedings—Electric Power Applications 2002; 193(4):184–94. [10] Smaoui M, Brun X, Thomasset D. A study on tracking position control of an electropneumatic system using backstepping design. Control Engineering Practice 2006;14(8):923–33. [11] Shibata S, Yamamoto T, Jindai M. A synchronous mutual position control for vertical pneumatic servo system. JSME International Journal. Series C 2006; 49(1):197–204. [12] Li M, Chen C, Liu W. Identification based on MATLAB. In: Proceedings of the 2009 international workshop on information security and application. IWISA 2009. 2009. p. 523–5.
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Backstepping sliding mode tracking control of a vane-type air motor X –Y table motion system Chia-Hua Lu a,∗ , Yean-Ren Hwang a,b , Yu-Ta Shen a a
Department of Mechanical Engineering, National Central University, Chung-Li 320, Taiwan
b
The Institute of Opto-Mechatronics Engineering, National Central University, Chung-Li 320, Taiwan
article
info
Article history: Received 24 February 2010 Received in revised form 30 November 2010 Accepted 23 December 2010 Available online 26 January 2011 Keywords: Air motor Pneumatic PID controller Backstepping sliding mode controller
abstract Air motors are increasingly being used in pneumatic related industries because of their advantages of low operating cost and low maintenance. The DSP controller and the backstepping sliding mode control method were utilized in this study to control an X –Y pneumatic table for tracking trajectory. Due to the effects of the compressibility of air, friction between the motor and ball screw table and the dead-zone effect caused by the proportional valve, the system will yield different responses even with the same inlet pressure and will chatter at low speed. Hence under certain conditions, this method of backstepping sliding mode control can be applied to achieve better results than with the PID controller, such as for tracking circle error and tracking error of the two axes. According to the results, a steady-state error of 0.5 µm can be achieved. The proposed method of backstepping sliding mode control can accomplish accurate tracking circle trajectory performance, offering an improvement in the tracking error of more than 50% over that of the PID controller. © 2010 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction Pneumatic actuators are increasingly being used in air motors and cylinders in pneumatic related industries. Unlike conventional electric and hydraulic actuators, the compressibility of the air allows pneumatic actuators to offer highly nonlinear performance. Their major advantages are their high payload-to-weight and payload-to-volume ratios, and high speed and force capabilities. Their reasonable cost is another reason that makes them appealing for use in industry. Unlike the easily controllable conventional electric DC motors used with ball screw drivers which have been widely applied in the past, pneumatic drives require sophisticated controllers, due to the compressibility of air, adverse friction characteristics and low damping. However, they also offer several advantages: the compressibility of air provides a ‘‘cushioning’’ effect which is important in applications like blow molding or glass forming. This also results in highly nonlinear pressure dynamic characteristics. As a result, conventional PID feedback controllers [1,2] are not so effective even for position control. There have been a number of investigations and analyses of the dynamics of vane-type air motors [2–5] carried out to date, most for different types of control, like PID control [2], MRAC [3], and sliding mode control [5]. In
∗
Corresponding author. Tel.: +886 3 4227151x34342; fax: +886 3 4254501. E-mail address: [email protected] (C.-H. Lu).
addition, a servo pneumatic system (cylinder) has been proposed for position control [6,7]. In backstepping sliding mode control, the nonlinear system is divided into many subsystems. Sliding mode control is designed for each subsystem with the Lyapunov function to guarantee the convergence of the position tracking error for all possible initial conditions. The integral action is mixed with backstepping control. The addition of an integrator strengthens the system’s robustness against modeling uncertainties and external disturbances, thus improving steady-state control accuracy [8]. An adaptive backstepping sliding mode controller is proposed in [9] to control the position of a linear induction motor (LIM) drive. The device is able to compensate for uncertainties including the force of friction. A tracking position control system based on the backstepping design has been proposed for an electropneumatic device, consisting of a cylinder and two valves [10]. Synchronous control for mutual positioning of two vertical-type pneumatic servo systems is proposed in [11]. A fuzzy controller is adopted in each pneumatic servo system to let the system track the reference input. Air motors have uncertain behaviors. In most practical motion systems, the flow of air from the inlet of the storage to the air motor will decrease after each experimental result. The four-vane-type air motor used in the experiments is shown in Fig. 1. Each vane will cause friction, especially at low speed. The air pressure entering the air motor will decrease in a non-regular fashion due to the various volumes of the two vanes. To counteract these uncertain behaviors and eliminate the dead zone caused by friction, modelreferencing adaptive control (MRAC) with fuzzy control [3] is
0019-0578/$ – see front matter © 2010 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2010.12.008
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Fig. 2. Schematic diagram of the air motor system. Fig. 1. Representation of the principles of vane-type air motor operation.
proposed. Fuzzy rules combined with sliding mode control are used in the aforementioned study [5] to improve the steady response and build robustness with an air motor ball screw table. With a pneumatic cylinder [10], backstepping control provides robustness to cope with the compressibility of air. In order to overcome uncertain behaviors the proposed control scheme for an air motor ball screw table system utilizes backstepping sliding mode control for a second-order uncertain system. This controller is designed by adopting sliding mode control with two Lyapunov functions to guarantee that the error will converge to zero as t → ∞. The system is asymptotically stable even if external force disturbances and frictional forces exist. The experimental results demonstrate that the overshoot phenomenon will not occur with this system and can maintain steady-state position error in position tracking compared with the PID controller. 2. Air motor servo system Fig. 1 shows a rough representation of a vane-type air motor. The motor has a rotational drive shaft with four slots, each of which is fitted with a freely sliding rectangular vane. When the drive shaft starts to rotate, the vanes tend to slide outward due to centrifugal force, but are limited by the shape of the rotor housing. Depending on the flow direction, the motor will rotate in either a clockwise or counterclockwise direction. A schematic diagram of the air motor system is shown in Fig. 2. The system consists of an air motor (GAST 1AM), an air tank, an electronic proportional directional control valve (FESTO MPVE), X –Y table (ISEL, anti-backlash ball nut design), a filter/regulator with lubricant (SHAKO FRL-600), an optical linear scale (A1-0600) the accuracy of which is 0.5 µm, and a digital signal processor (DSP, TI C240). The TMS320C240 device is a member of a family of DSP controllers based on the TMS320C2xx generation of 16-bit fixedpoint digital signal processors. This family is optimized for digital motor/motion control applications. The DSP controllers combine the enhanced TMS320 architectural design of the C2xLP core CPU for low-cost, high-performance processing capabilities and several advanced peripherals optimized for motor/motion control applications. These peripherals include the event manager module, which provides general-purpose timers and compare registers to
Fig. 3. Photograph of the experimental air motor system.
generate up to 12 PWM output, and a dual 10-bit analog-to-digital converter (ADC), which can perform two simultaneous conversions within 6.1 µs. The control loop time used in the experiments is 5 ms and the software ‘‘core composer v.2 (ccs c2000)’’ is adopted to write the control program. A DSP emulator XDS510 utilizing a standard JTAG plug acts as the Scan-Based Emulator. Since this is a plug-and-play emulator, integrated debugger programs can be executed from both on or off chip memory without additional delay under any clock speed. The airflow path starts from the air tank, moving first through the filter, then the control valve, finally entering the air motor. The airflow entering the motor is determined by the position of the valve, which is controlled by an externally applied voltage. The experimental air motor ball screw table system is shown in Fig. 3. 3. Controller design A method of fuzzy sliding mode position control incorporating a ball screw driven by a vane-type air motor has been proposed in [5] in which fuzzy logic control is applied to avoid the disadvantages of disturbances and chattering. However this study did not show the sinusoidal tracking error. The experiment results only showed that the fuzzy rules could stabilize the system. There is no state of the art method focusing on X –Y table control. In this current study,
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Fig. 4. X –Y table performance under the same inlet pressure: (a) curves of X axis; (b) curves of Y axis.
Fig. 5. Experimental results after adjusting parameters: (a) k1 ; (b) c1 ; and (c) h.
two controllers are designed and compared: a PID controller and a backstepping controller. The Ziegler–Nichols tuning method is adopted to tune the parameters of the PID controller, by setting the Integral (I) and Derivative (D) gains to zero. The Proportional (P) gain is increased until it reaches the ultimate gain Ku , at which point the output of the control loop oscillates with constant amplitude. Ku and the oscillation period Tu are used to set the KP , KI , and KD gains depending on the Ziegler–Nichols tuning method (KP = 0.6Ku , KI = 2Kp /Tu , KD = 0.125KP Tu ). Finally, the variable parameters Ku and Tu are adjusted to approximately 38.33 and 0.07, respectively. Fig. 4 shows the open loop step responses for backstepping control of a vane-type air motor installed on X –Y table with an inlet pressure of 3 bar. It is obvious that two tables can move nearly 200 mm within 10 s. Because the weight of the X axis and Y axis are different, there will be a slight difference in the output response for each experimental result for each axis but the variation is small. The compressibility of air, overall mechanical friction (which includes ball screw table friction), nonlinear behavior of the servovalves, and pressure loss are the main problems associated with the highly nonlinear characteristics of pneumatic actuation, namely air motors. The overall system will demonstrate nonlinear phenomena. Given the results, the vane-type air motor connected
with a X –Y table is considered to be a second-order system so that the dynamics of the system can be described by the following equation for considering friction: As can be seen in Fig. 1, the air flow enters the vane-type air motor from chamber A and exhausts from chamber B. r is the inner rotor radius; R is the radius of the motor body; d is the difference between R and r (d = R − r ); θ is the angle of motor rotation; xr is working radius of the vane measured from the center of the rotor which is given by xr = d cos θ +
R2 − d2 sin2 θ ,
(1)
(assuming that d sin θ << R ). Considering friction and the different payloads, and using Newton’s second law of angular motion we get 2
2
M − Mc S (θ˙ ) − Mf θ˙ = J θ¨ ,
2
(2)
where θ˙ is the angle velocity; θ¨ represents the angular acceleration; Mc is the stiction coefficient; Mf is the friction coefficient; and S (θ˙ ) is described as
1, ˙ S (θ ) =
sign(θ˙ ),
θ˙ = 0 θ˙ ̸= 0.
(3)
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Fig. 6. Experimental results for PID control: (a) position results within 10 s; (b) output voltages for two axes; (c) maximum overshoot between 1 and 3.5 s.
Fig. 7. Experimental results for backstepping sliding mode control: (a) position results within 10 s ; (b) output voltages for two axes; (c) maximum overshoot between 2 and 3.5 s.
The driving torque is determined by the difference in pressure acting on the vane between the drive and exhaust chambers, and is given by M = (Pa − Pb )(xr − r )Lxr = (Pa − Pb )(x2r − rxr )L.
(4)
Assuming that x is the displacement of the ball screw table and Ps is the screw pitch: the displacement x = ps θ ; the velocity x˙ = ps θ˙ . The dynamics of the system can be described by the following equation considering friction:
x¨ = −
Mf Jm
x˙ +
Ps Jm
L(x2r − rxr )(1P ) −
Ps Jm
Mc S
x˙ Ps
−
Ps Jm
Tl ,
(5)
where 1P is the (Pa − Pb ); Jm is the total system inertia (motor and ball screw); Tl is the load torque. Eq. (5) can be derived as follows: x¨ = −
Mf Jm
x˙ +
kp ps Jm
L(x2r − rxr )(u) + Des , p
where Des = − J s Mc S m
x˙ ps
−
ps T Jm l
, U = Pa − Pb .
(6)
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Fig. 8. Experimental results for random step trajectory tracking position (a) tracking position results; (b) output voltages.
Fig. 9. Experimental results for PID control (frequency is 0.05 Hz): (a) tracking circle trajectory with a radius of 15 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error.
X1 defines the displacement; X2 is the moving velocity. Eq. (6) can now be rearranged as [9]
X˙ 1 = X2 X˙ = (A + 1A)X2 + (B + 1B)U + Des (t ) 2 Y = X1 .
(7)
z˙1 = Y˙ − Y˙d = X2 − Y˙d . (8)
(10)
(9)
(11)
Define the stable item
α1 = c1 z1 ,
where F is the uncertainty F = 1AX2 + 1BUDes (t )
z1 = Y − Yd . Then
Eq. (7) can be derived as follows: X˙ 2 = AX2 + BU + F ,
and |F | ≤ F¯ , 1A is the variable uncertainty of the system. Suppose the variation of the variable uncertainty and external disturbance are not so quick that F˙ = 0. The tracking error between the system output Yd and the position command signal Y can be defined as
where c1 is a positive constant.
(12)
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Fig. 10. Experimental results for backstepping sliding mode control (frequency is 0.05 Hz): (a) tracking circle trajectory with a radius of 15 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error.
Define the Lyapunov function as V1 =
1 2
z12 .
The control input can be designed as follows: (13)
Defining z2 = z˙1 + α1 = X2 − Y˙d + α1 , V˙ 1 = z1 X2 − Y˙d = z1 (z2 − α1 ) = z1 z2 − c1 z12
(14)
z˙2 = X˙ 2 − Y¨d + α˙ 1 = AX2 + BU + F − Y¨d + α˙ 1 .
(15)
Define the Lyapunov function as 1
s , 2 where s is the sliding surface. Define the sliding function as V2 = V1 +
2
s = k1 z1 + z2 ,
U = B−1 [−k1 (z2 − c1 z1 ) − A(z2 + Y˙d − α1 )
− F¯ sgn(s) + Y¨d − α˙ 1 − h(s + β sgn(s))], where h and β are positive constants. Substituting Eq. (19) into Eq. (18) we get V˙ 2 = z1 z2 − c1 z12 − hs2 − hβ |s| + Fs − F¯ |s|
≤ −c1 z12 + z1 z2 − hs2 − hβ |s| + |s| (|F | − F¯ ) ≤ −c1 z12 + z1 z2 − hs2 − hβ |s| .
(16)
(20)
Define Q as a positive definite symmetric matrix
c1 + hk21
Q =
(17)
where k1 > 0; then
hk1 −
hk1 −
1
1
2 ,
h
2
(21)
then
V˙ 2 = V˙ 1 + ss˙ = z1 z2 − c1 z12 + ss˙ = z1 z2 − c1 z12 + s (k1 z1 + z˙2 ) z T Qz = z1
= z1 z2 − c1 z12 + s[k1 (z2 − c1 z1 ) + AX2 + BU + F − Y¨d + α˙ 1 ] = z1 z2 − c1 z12 + s[k1 (z2 − c1 z1 ) + A z2 + Y˙d − α1 + BU + F − Y¨d + α˙ 1 ].
(19)
c1 + hk21
z2
hk1 −
1
hk1 −
1
2 z
1
z2
T
h 2 = c1 z12 + hk21 z12 + 2hk1 z1 z2 − z1 z2 + hz22 (18)
= c1 z12 − z1 z2 + hs2 ,
(22)
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Fig. 11. Experimental results for backstepping sliding mode control (frequency is 0.05 Hz): (a) tracking circle trajectory with a radius of 30 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error.
where z = z1
z2 and Eq. (20) can now be written as
V˙ 2 ≤ −z Qz − hβ |s| . T
Then
|Q | = h(c1 +
hk21
) − hk1 −
1 2
2
1
= h(c1 + k1 ) − . 4
(23)
Now choose the values of h, c1 , k1 to let |Q | > 0. Then V˙ 2 ≤ 0.
(24)
Therefore, z1 and z2 will converge to zero. The stability of the proposed backstepping sliding mode control system can be guaranteed. From Eq. (19), the initial experimental conditions are designed as follows: A and B for two axes (X and Y ) are −7.865, 42, and −7.865, 36, respectively; F¯ is approximately 0.6; k1 , c1 and h are assumed to be 8, 8 and 15, respectively. The MATLAB system identification toolbox is used to measure the values of A and B. System identification based on MATLAB [12] is highly efficient. First of all, a second-order system is built with the System Simulation Toolbox of SIMULINK, and then the parameters of the system are identified using the System Identification
Toolbox. The input data (pressure) and output data (distance of table movement) were obtained from the experimental results. These data are input into the System Identification Toolbox. Parameters A and B in Eq. (7) will be identified. According to the control input design in Eq. (19), k1 affects the transient response, c1 has an influence when the system approaches the sliding surface, and h affects the steady-state response. If these three parameters are chosen to be too large or too small, the system will have large overshoot and steady-state error or a have slow rising time and setting time. The experimental results obtained after adjusting the k1 , c1 and h parameters are shown in Fig. 5. Fig. 5(a) shows the experimental results for the backstepping sliding mode controller with k1 . Since this parameter affects the transient response, different k1 values will have an effect on the rise time. Parameter c1 affects the approach of the sliding surface. The results are shown in Fig. 5(b). For a gradual increase in c1 , a longer time is needed to approach the sliding surface. As shown in Fig. 5(c), the dissimilarity of parameter h affects the various chattering phenomena of the pneumatic system. The steady response leads to a serious chattering phenomenon. According to the experimental results, when the values of k1 , c1 and h are 8, 8 and 15, respectively, this will produce a better response for the transient and steady-state errors.
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Fig. 12. Experimental results for backstepping sliding mode control (frequency is 0.1 Hz): (a) tracking circle trajectory with a radius of 10 mm ; (b) tracking trajectory of X axis; (c) tracking trajectory of Y axis; (d) output voltages for two axes; (e) tracking errors for two axes; (f) tracking circuit error. Table 1 Comparison of different controllers for position control.
Table 2 Comparison of different controllers for tracking circle trajectory.
Position control at 50 mm
Tracking circle trajectory with radius 15 mm
PID controller
Backstepping sliding mode controller
Steady-state position (Pss)
Steady-state position (Pss)
X = 49.9 mm
X = 50.0005 mm
Y = 50.02 mm
Rising time (Tr) X = 2.395 s
Y = 49.9995 mm
Rising time (Tr) Y = 2.375 s
X = 2.38 s
Y = 2.365 s
Maximum overshoot (Mo)
Maximum overshoot (Mo)
X = 1.431%
X = 0%
Y = 1.1%
Y = 0%
Steady-state error (Ess)
Steady-state error (Ess)
X = 0.1 mm
X = 0.0005 mm
Y = 0.02 mm
Y = 0.0005 mm
4. Experimental results This section is divided into two parts, one discussing the position tracking results and the other performance analysis and discussion. 4.1. Position tracking results In this experimental study, the main focus is on position control at 50 mm and the tracking circular trajectory with a radius R =
PID controller
Backstepping sliding mode controller
Frequency is 0.05 Hz
Frequency is 0.05 Hz
Tracking error X axis Y axis Circle trajectory
Tracking error
±0.8 mm ±0.6 mm ±0.6 mm
X axis Y axis Circle trajectory
+0.3 mm to −0.1 mm ±0.1 mm ±0.25 mm
15 mm. The operating pressure is regulated to be a constant value of 3 bar. The pressure is the same for each experimental run, so there is little variation in system nonlinearities. The performance of the proposed PID control is compared to the backstepping sliding mode control. The experimental results with PID control and backstepping sliding mode control at position 50 mm are shown in Figs. 6 and 7, respectively. As can be seen in Fig. 6(a) the steady-state response for the X axis is 49.9 mm and steady-state response for the Y axis is 50.02 mm, hence the experimental steady-state error for these two axes are 0.1 mm and 0.02 mm, respectively. On the other hand, the experimental results show that there is serious overshoot phenomenon with PID control. Fig. 6(b) shows the output voltage for the two axes. It can be seen in Fig. 6(c) that the maximum overshoot of the X axis is 1.431% and for the Y axis it is 1.1%.
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Table 3 Experimental results for different tracking circle trajectories. Tracking circle trajectory with backstepping sliding mode controller Frequency is 0.05 Hz (radius 30 mm)
Frequency is 0.1 Hz (Radius 10 mm)
Tracking error
Tracking error
X axis Y axis Circle trajectory
±0.2 mm to −0.1 mm +0.2 mm to −0.1 mm ±0.2 mm
Fig. 7(a) shows the steady-state responses of the X axis and Y axis. Compared with the experimental results for PID control (shown in Fig. 6(a)), the steady-state error can be maintained within 0.5 µm for both the X and Y axes, and the maximum overshoot can be obviously reduced. In other words, the experimental results show no overshoot phenomenon. Fig. 8 shows the experimental results for the random step trajectory tracking position. For X axis tracking, for each step target position, the steady-state errors are 5.999 mm, 15.9995 mm, 0.0005 mm, and 5.9995 mm. For Y axis tracking, for each step target position, the steady-state errors are 10.9995 mm, 25.999 mm, 5.999 mm, and 20.999 mm. The results show that random step trajectory tracking positioning can be accomplished for both axes with the backstepping sliding mode controller, to maintain the steady-state error within 1 µm. We now discuss tracking circle trajectory control for the pneumatic servo system. In the experiment, the X axis is designed to track the wave such that the equation is x(t ) = x0 + R cos(2π ft ) and the Y axis is designed for tracking y(t ) = y0 + R sin(2π ft ). The initial experimental conditions are set as follows: x0 and y0 are adjusted to zero; the radius is R = 15 mm; the frequency is f = 0.05 Hz. The goal of the experiments is to compare performances of two proposed controllers, a PID controller and the backstepping sliding mode controller. Fig. 9(a) shows the tracking circle trajectory with a frequency of 0.05 Hz. The trajectory consists of two axes: the tracking sinusoidal wave (cosine) of the X axis and the sinusoidal wave (sine) of the Y axis, as shown in Fig. 9(b) and Fig. 9(c), respectively. There is serious chattering phenomenon under the PID controller. According to the experimental results, the trajectory error for the X axis is approximately ±0.8 mm and that for the Y axis is approximately ±0.6 mm, as shown in Fig. 9(e). Fig. 9(f) shows the tracking error of the circle trajectory to be approximately ±0.6 mm. The compressibility of air and the friction make the overall system for the X –Y table nonlinear. Backstepping sliding mode control is proposed in order to overcome the chattering phenomenon caused by this disturbance and decrease the tracking error. The experimental results (tracking circle trajectory with a radius of 15 mm), the tracking sinusoidal wave (cosine) of the X axis and the sinusoidal wave (sine) of Y axis, for the backstepping sliding mode control are shown in Fig. 10(a)–(c), respectively. Compared with the results shown in Fig. 9(a)–(c) for PID control, it can be seen that the proposed controller not only restrains the chattering phenomenon but also maintains the tracking error at under 0.3 mm. According to the results shown in Fig. 10(e)–(f), the trajectory error for the X axis is approximately +0.3 to −0.1 mm; the trajectory error for the Y axis is approximately ±0.1 mm; the tracking error for the circle trajectory is approximately ±0.25 mm. Figs. 9 and 10 show a comparison between the two different
X axis Y axis Circle trajectory
+0.1 mm to −0.2 mm +0.3 mm to −0.1 mm ±0.25 mm
controller performances in order to prove that the system can track more circle trajectories that are with different radii and frequencies. Fig. 11 shows the tracking results for a circle with a radius of 30 mm; the frequency is 0.05 Hz. Fig. 12 shows the tracking results for a circle with a radius of 10 mm; the frequency is 0.1 Hz. 4.2. Analysis of performance and discussion A comparison of different controllers for position control is shown in Table 1. Position control by the backstepping sliding mode controller can maintain excellent transient response performance (rising time), eliminate the overshoot and keep the steady-state error under 0.5 µm. The results for different tracking circle trajectory control are shown in Tables 2 and 3. Although PID controller is popularly used to control most motion systems, some pneumatic nonlinearity due to compressibility cannot be eliminated. According to the experimental results in Table 2 the proposed backstepping sliding mode controller offers an improvement in the tracking error of more than 50% compared to the PID controller. This means that this proposed controller is able to overcome pneumatic nonlinearity better. References [1] Zhang Y, Nishi A. Low-pressure air motor for wall-climbing robot actuation. Mechatronics 2003;13(4):377–92. [2] Shen YD, Hwang YR. Dynamic modeling and controller design for air motor. In: SICE–ICASE, 2006. International joint conference. 2006. p. 461–6. [3] Hwang YR, Shen YD, Jen KK. Fuzzy MRAC controller design for vane-type air motor systems. Journal of Mechanical Science and Technology 2008;22(3): 497–505. [4] Wang J, Pu J, Moore PR. Modeling study and servo-control of air motor systems. International Journal of Control 1998;71(3):459–76. [5] Song J, Ishida Y. A robust sliding mode control for pneumatic servo systems. International Journal of Engineering Science 1997;35(8):711–23. [6] Taghizadeh M, Ghaffari A, Najafi F. Improving dynamic performancees of PWM-driven servo-pneumatic systems via a novel pneumatic circuit. ISA Transactions 2009;48(4):512–8. [7] Shin MC, Lu CS. Fuzzy sliding mode position control of a ball screw driven by pneumatic servomotor. Mechatronics 1995;5(4):421–31. [8] Tan YL, Chang J, Tan HL, Hu J. Integral backstepping control and experimental implementation for motion system. In: Proceedings of the 2000 IEEE internation conference on control application. p. 25–7. [9] Lin FJ, Shen PH, Hsu SP. Adaptive backstepping sliding mode control for linear induction motor drive. IEE Proceedings—Electric Power Applications 2002; 193(4):184–94. [10] Smaoui M, Brun X, Thomasset D. A study on tracking position control of an electropneumatic system using backstepping design. Control Engineering Practice 2006;14(8):923–33. [11] Shibata S, Yamamoto T, Jindai M. A synchronous mutual position control for vertical pneumatic servo system. JSME International Journal. Series C 2006; 49(1):197–204. [12] Li M, Chen C, Liu W. Identification based on MATLAB. In: Proceedings of the 2009 international workshop on information security and application. IWISA 2009. 2009. p. 523–5.