Adaptive T he -Varying Sliding Mode Control for - Semantic Scholar
2004 8th lntemational Conference on Control, Automation, Robotics and Vision Kunming, China, 6-9th December 2004
Adaptive T h e -Varying Sliding Mode Control for Hydraulic Servo System Cheng Guan and Shanan Zhu College of Ekctrical Enginetiring,Zhejiang University Hangzhou, China, 3 IO027 Email: gch8zju.edu.cn
Abstract This paper studies the position control of an electro-hydraulic servo system. Because the dynamics of the system are highly nonlinear and have large extent of model. uncertainties including big changes in load and hydraulic parameters, which are unmatched, a time-varying sliding mode control approach combined with adaptive control is proposed based on Lyapunov
analysis . The time-varying sliding mode control avoids the reaching phase of conventional sliding mode control, thus the control method proposed can be robust all the time. Adaptive control is used to identify the system parameters to overcome the influence of the uncertain parameters and disturbances. Simulation results indicate that the control approach has nice global robusmess and improves position tracking accuracy considerably. Keywords: nonlinear hydraulic system, position tracking, time-varying sliding control, adaptive control
1 Introduction Sliding mode control is used widely because of its advantage that once the system state reaches sliding surface, the system dynamics remain insensitive to parameter uncertainties and disturbances. However, robust tracking is guaranteed only after the system state reaches the sliding surface but not during the reaching phase. If the parameters of the system vary too much or the disturbance is great enough, the advantage would not be guaranteed when a conventional time-invariant sliding mode control is used. Trying to overcome this problem, deerences [1][2][3]etc. proposed a time-varying sliding mode variable construct -control scheme for a class of nonlinear system, which makes the system initial state be on the sliding surface to eliminate the reaching phase. As a consequence, the control is robust with respect to external disturbances and parameter uncertainties from the very beginning of the system motion. However, the control methods need an assumption that the bounds of the system parameters and external disturbances are known. Since the accurate bounds of system parameters are difficult to get, the controller will usually be conservative and have discontinuous functions, thus the chattering will be made ,even though the system ,initial state be on the
0-7803-8653-11041$20.00 Q 2004 IEEE
sliding surface.
The advantage of adaptive control is that the control scheme does not need to know the bound of the parameters, so this paper presents a control method which combines the time-varying sliding mode control with adaptive control through introducing a parameter adaptation scheme based on Lyapunov analysis. In addition , the chattering phenomena is eliminated, since the adaptation law is continuous and the system state is on the sliding surface from the very beginning. We apply the control scheme in position tracking of hydraulic servo system. Simulation shows the global robustness and the nice tracking performance of the proposed control method.
2
System model dynamics
and
designed
For a hydraulic position tracking system which uses a four way spool servo valve to control a hydraulic motor, the whole system dynamics model is given by the following differential equations.
where Pi : pressure difference between the input and the output of motor. D, : motor displacement. Tg: external load twist. J,: inertia on the motor output shaft. 8:, output angle. B,: viscid damping coefficient of the motor. Qj: the load fluid flow. Be: effective bulk modulus. C,: coefficient of total intestine leakage. V,: total motor volume. C:, discharge coefficient. 0 : spool servo valve area gradient. x,: spool servo valve displacement P,: system pressure . U: input voltage to spool servo valve Choose system state: x I = 8, , x2 =e, , x j = Pl
1774
Then, the system state-space equation is: i,=x7
Y=x1 Therefore via state feedback controlkr (4), the nonlinear system (3) is converted to linear system (5). Now the goal has been switched to design ('Y" in the equ. (4) and eqn. (5).
= U is the input of system, y is the output of system. All states are measurable. At this stage, it is readily seen that the system is nonlinear due to static flow mappings of the servo valve, and has unmatched model B,,,, I@,, C,, uncertainties due to the variations of .If, V I , Cd and external load twist T,
3.2 Define time-varying sliding surface
U
Our goal is to design a controller to make the output angle track the desired angle. That is: lim(r-yd)=0
Define the tracking error: e = y - yd
Take into account
system ( 2 ) and ( 5 ) , w e have: a) q=Zl-yd
b)
-
-y
e j d e ~ i - ~ ) - (i-t)-,,~-l)=zi-y~-l)
Define time-varying sliding surface as: o=ke+Q(t)
,
i=2,3 (6a)
f*
where ydis the desired angle. Assume y d is known, and its first, second and third derivative are known and bound.
3
Design adaptive time-varying sliding mode controller
Not to lose generality, let k, =I , T>O is a constant, A , B and C are constants. Select parameter ki and kz so that k,p+k2p2+p" is Hunvitz poiynomid.
3.1 Linearization of the system 141 It is necessary to convert the system state-space equation to a normal form when sliding mode control is used.
Define :
{I. ' . .
x2 XI
]
Assume 1: the system's representative point belongs to the line of eqn. (6) at t=O,namely, fl (0) =O ,then:
- ~ I x Z+ a 2 ~ 3 - T ,
C=-k,e, (0)-k2e2(0)-q (0)
Therefore the system state-space equation (1) may be written in the normal form as:
i
i,=22
2 2 =z3
i3=a( X ) + P ( X ) U +d
where: C X ( X ) = ~ ! & + @ ~ ~P(+)=B-, , -J
d=alTl
2
O1=-al -alp1
(3)
,
(7) Assume 2: the right-hand side of eqn. (6) is continuous with its derivative, so that there is no instantaneous movement of the line or rapid change of its velocity at r = T , therefore, we have: (8) AT^ + B T + C = o 2AT+S=O (9) Take into account eqn. (71, ( 8 ) and ( 9 1 , we have:
, O2 =-a, a2-a2P2 ,
4 =-ad3 I
Since in hydraulic system choose the control law:
p(x)#O all the time, Then
(4) Substituting eqn. 4 1 into the system (3),we have:
3.3
~=% (0)-tk,e2 j ( ~ ) + e(OIJ ,
T
The design of time-varying sliding mode controller
When using the sliding mode control, input should force the states of system converge to sliding surface
1775
control for hydraulic position tracking control.
o ( x , t ) a to guarantee that fie fie stage of sliding motion is existent, and the tracking error converges to zero.
When the parameters are uncertain, ~3 can not be calculated from the system states accurately, thus the error e3 and controller v can not be calculated accurately. However, via Laplace transform, it will yield:
The derivate of eqn. ( 6 ) is:
(11)
Zj=Z2S=&S
If the measured x2 contain high frequency noise, 23 from equation (14) will include great disturbance, so that its accuracy will decrease. Here, we adopt a low-pass filter
Therefore :
to eliminate high frequency element, thus the following estimated .?3 is used to replace actual z 3 .
Substituting (5b) into the above eqn.:
hs
(12)
j 3=-Xz
s+h
We adopt the following control strategy:
where h is a positive constant. ( 13
If the parameters are not known exactly, their estimates can be used to replace the actual values in (4).The resulting control becomes
where c is a positive constant. Substituting the above eqn. into eqn. ( 12):
Dr=-caz
Adaptive T h e -Varying Sliding Mode Control for Hydraulic Servo System Cheng Guan and Shanan Zhu College of Ekctrical Enginetiring,Zhejiang University Hangzhou, China, 3 IO027 Email: gch8zju.edu.cn
Abstract This paper studies the position control of an electro-hydraulic servo system. Because the dynamics of the system are highly nonlinear and have large extent of model. uncertainties including big changes in load and hydraulic parameters, which are unmatched, a time-varying sliding mode control approach combined with adaptive control is proposed based on Lyapunov
analysis . The time-varying sliding mode control avoids the reaching phase of conventional sliding mode control, thus the control method proposed can be robust all the time. Adaptive control is used to identify the system parameters to overcome the influence of the uncertain parameters and disturbances. Simulation results indicate that the control approach has nice global robusmess and improves position tracking accuracy considerably. Keywords: nonlinear hydraulic system, position tracking, time-varying sliding control, adaptive control
1 Introduction Sliding mode control is used widely because of its advantage that once the system state reaches sliding surface, the system dynamics remain insensitive to parameter uncertainties and disturbances. However, robust tracking is guaranteed only after the system state reaches the sliding surface but not during the reaching phase. If the parameters of the system vary too much or the disturbance is great enough, the advantage would not be guaranteed when a conventional time-invariant sliding mode control is used. Trying to overcome this problem, deerences [1][2][3]etc. proposed a time-varying sliding mode variable construct -control scheme for a class of nonlinear system, which makes the system initial state be on the sliding surface to eliminate the reaching phase. As a consequence, the control is robust with respect to external disturbances and parameter uncertainties from the very beginning of the system motion. However, the control methods need an assumption that the bounds of the system parameters and external disturbances are known. Since the accurate bounds of system parameters are difficult to get, the controller will usually be conservative and have discontinuous functions, thus the chattering will be made ,even though the system ,initial state be on the
0-7803-8653-11041$20.00 Q 2004 IEEE
sliding surface.
The advantage of adaptive control is that the control scheme does not need to know the bound of the parameters, so this paper presents a control method which combines the time-varying sliding mode control with adaptive control through introducing a parameter adaptation scheme based on Lyapunov analysis. In addition , the chattering phenomena is eliminated, since the adaptation law is continuous and the system state is on the sliding surface from the very beginning. We apply the control scheme in position tracking of hydraulic servo system. Simulation shows the global robustness and the nice tracking performance of the proposed control method.
2
System model dynamics
and
designed
For a hydraulic position tracking system which uses a four way spool servo valve to control a hydraulic motor, the whole system dynamics model is given by the following differential equations.
where Pi : pressure difference between the input and the output of motor. D, : motor displacement. Tg: external load twist. J,: inertia on the motor output shaft. 8:, output angle. B,: viscid damping coefficient of the motor. Qj: the load fluid flow. Be: effective bulk modulus. C,: coefficient of total intestine leakage. V,: total motor volume. C:, discharge coefficient. 0 : spool servo valve area gradient. x,: spool servo valve displacement P,: system pressure . U: input voltage to spool servo valve Choose system state: x I = 8, , x2 =e, , x j = Pl
1774
Then, the system state-space equation is: i,=x7
Y=x1 Therefore via state feedback controlkr (4), the nonlinear system (3) is converted to linear system (5). Now the goal has been switched to design ('Y" in the equ. (4) and eqn. (5).
= U is the input of system, y is the output of system. All states are measurable. At this stage, it is readily seen that the system is nonlinear due to static flow mappings of the servo valve, and has unmatched model B,,,, I@,, C,, uncertainties due to the variations of .If, V I , Cd and external load twist T,
3.2 Define time-varying sliding surface
U
Our goal is to design a controller to make the output angle track the desired angle. That is: lim(r-yd)=0
Define the tracking error: e = y - yd
Take into account
system ( 2 ) and ( 5 ) , w e have: a) q=Zl-yd
b)
-
-y
e j d e ~ i - ~ ) - (i-t)-,,~-l)=zi-y~-l)
Define time-varying sliding surface as: o=ke+Q(t)
,
i=2,3 (6a)
f*
where ydis the desired angle. Assume y d is known, and its first, second and third derivative are known and bound.
3
Design adaptive time-varying sliding mode controller
Not to lose generality, let k, =I , T>O is a constant, A , B and C are constants. Select parameter ki and kz so that k,p+k2p2+p" is Hunvitz poiynomid.
3.1 Linearization of the system 141 It is necessary to convert the system state-space equation to a normal form when sliding mode control is used.
Define :
{I. ' . .
x2 XI
]
Assume 1: the system's representative point belongs to the line of eqn. (6) at t=O,namely, fl (0) =O ,then:
- ~ I x Z+ a 2 ~ 3 - T ,
C=-k,e, (0)-k2e2(0)-q (0)
Therefore the system state-space equation (1) may be written in the normal form as:
i
i,=22
2 2 =z3
i3=a( X ) + P ( X ) U +d
where: C X ( X ) = ~ ! & + @ ~ ~P(+)=B-, , -J
d=alTl
2
O1=-al -alp1
(3)
,
(7) Assume 2: the right-hand side of eqn. (6) is continuous with its derivative, so that there is no instantaneous movement of the line or rapid change of its velocity at r = T , therefore, we have: (8) AT^ + B T + C = o 2AT+S=O (9) Take into account eqn. (71, ( 8 ) and ( 9 1 , we have:
, O2 =-a, a2-a2P2 ,
4 =-ad3 I
Since in hydraulic system choose the control law:
p(x)#O all the time, Then
(4) Substituting eqn. 4 1 into the system (3),we have:
3.3
~=% (0)-tk,e2 j ( ~ ) + e(OIJ ,
T
The design of time-varying sliding mode controller
When using the sliding mode control, input should force the states of system converge to sliding surface
1775
control for hydraulic position tracking control.
o ( x , t ) a to guarantee that fie fie stage of sliding motion is existent, and the tracking error converges to zero.
When the parameters are uncertain, ~3 can not be calculated from the system states accurately, thus the error e3 and controller v can not be calculated accurately. However, via Laplace transform, it will yield:
The derivate of eqn. ( 6 ) is:
(11)
Zj=Z2S=&S
If the measured x2 contain high frequency noise, 23 from equation (14) will include great disturbance, so that its accuracy will decrease. Here, we adopt a low-pass filter
Therefore :
to eliminate high frequency element, thus the following estimated .?3 is used to replace actual z 3 .
Substituting (5b) into the above eqn.:
hs
(12)
j 3=-Xz
s+h
We adopt the following control strategy:
where h is a positive constant. ( 13
If the parameters are not known exactly, their estimates can be used to replace the actual values in (4).The resulting control becomes
where c is a positive constant. Substituting the above eqn. into eqn. ( 12):
Dr=-caz
