Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 25

Mã bài: 07
Robust and Adaptive Tracking Control of Two-Wheel-Gearing
Transmission Systems
Điều khiển thích nghi bền vững hệ truyền động qua
một cặp bánh răng
Le Thi Thu Ha
1)
, Nguyen Thi Chinh
2)
, Nguyen Doan Phuoc
3)

1)
,
2)
Thai Nguyen University of Technology;
3)
Hanoi University of Science and Technology
e-mail:
1)
,
2)
,
3)Abstract
This paper proposes a new design procedure of an adaptive and robust tracking controller for gearing
mechanical transmission systems by using the sliding mode control technique and the certainty equivalence

Furthermore, desired results in suppression of the
effect of the shaft elasticity or backlash between
cogwheels at once at damping torsional vibrations
cannot be achieved without additional states
feedback [7]. Therefore, many attempts of using
additional states feedback controller to improve
the performance of mechanical systems with shaft
elasticity or backlash have been carried out during
the last few years (see, for examples, [3] and
[10]).
Such proposed controllers, however, can only be
used either for systems with shaft elasticity or
with backlash separately [12]. Moreover, a good
tracking performance of systems, in which all
uncertainties like immeasurable friction,
unpredictable elasticity of shafts and backlash are
simultaneous present, cannot be achieved with
such nonadaptive states feedback controller.
To overcome this problem, the adaptive robust
control based on the sliding mode technique (see,
for example, [11]) and the certainty equivalence
principle (see, for example, [6]) is applied to
improve the overall tracking performance of the
closed loop system.
26 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc

VCM2012
The sliding mode control is one of the robust
control theories to suppress the effect of bounded
noises or disturbances in systems. In addition, the

which is
transmitted to the load
c
M
through two wheel
gears 1 and 2 and two elastic shafts. Let
1
f
M
and
2
f
M
denote the friction moment on each shaft.
Both shafts have the same elasticity factor denoted
by
c
. Let
1

and
2

be the rotational angles of
corresponding shaft and

the backlash between
cogwheels. The Euler-Lagrange model of this
gearing transmission system is given as follows
(see, for example, [2]).

12 21
i i

 is the
transmission rate of the two wheels and
1 2
, ,
d
J J J

are the inertia moments of wheel 1, wheel 2 and
the driving motor respectively and
1 1
d
J J J
 

denotes the sum of inertia moments of wheel 1
and the driving motor.
Figure 1. Configuration of a gearing transmission
system
While
1 2 12 21 1
, , , ,

, whereas
disturbances by
k
d
. By using

2 2 1 2 2
1 1 2 2
1 1 1 1 1 2 2 2 2 2
cos , cos
( , ) , ( , )
f c f
cr cr
M b d t M M b d t
   
   

 
     
 

where

1 2
,
b b
 known constants,

1 2
,


( )
k
x

th
k
derivative of
x
,

,
p q
 finite positive integers,

1 1 2 2
( , ) , ( , )
d t d t
 
 unknown disturbances,
the Euler-Lagrange model (1) becomes

1 1 1 1 12 2 1 1 1
1 1
2 2 2 2 12 1 2 2 2
( )
( )
d
J i M b d
J i b d

(3)
with

3 12 2 2 3 2 2 4 12 2 2
, ,
d i d J i b
    
  

and

1 3 2 4 2 12 2 4
(4)
1 3 4 2 12 2 5
2
i d
i d
     
     
   
   
   
  
(4)
where
4 3 5 4
,
d d d d
 
 

Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 27

Mã bài: 07




 
 
(4)
1 3 1 4 1 3 2 1 4 1 3 1 12 2
2
1 4 1 12 2 1 5 1 3 1 4 1

d
M J J b b J i
b i J d d b d d
        
   
      
     
 


Next, let states vector
x
, truncated states vector
x
, input control signal
u

   
   
   
   



x ,
2 2
3 2
4 2
x
x
x



   
   
 
   
   
   



x ,
d
u M



 
1 5 1 3 1 4 1
1 3
1
( , )
d t J d d b d d
J


    x
The Euler-Lagrange model (2) of the gearing
transmission system, can now be rewritten in the
form of uncertain states model (5)

1
4
if 1 3
( , )
k k
T
f g
x x k
x d t u


  




3.1 Sliding mode controller
Let
( )
w t
be the reference signal, so the reference
trajectory for system (5) will be

 
, , ,
T
w w w w

  
w
and the vector of reference error is

 
, , ,
T
e e e e

  
e
where

1 2
e w x w

   
.

T
a a aa
are chosen such that the following polynomial

2 3
1 2 3
( )p a a a
   
   
(8)
will be Hurwitz. Note that, by using sliding
surface (7), in order to ensure the asymptotic
tracking performance


0
e
and
 
e

the nesecessary and sufficient condition is

( ) 0
s e

.
Thus, the initial tracking control aim can now be
replaced with



     
 
    
 
 


    
i T
i f g
i
V ss s a e a e a e w x
s a e w d t u

x x

Therefore, if the following controller is used

3
1 ( ) (4)
1
sgn( )
k T
g k f
k
u a e w s
 





 

 
    
 
 
  
 
  



x
x
x

which sufficiently ensures the boundedness of
( )
s e
as well as the asymptotic decay to zero of
( )
s e
.
3.2 Adaptive Parameters Adjustment
In practice, the controller (10), however, cannot be
used because of the unknown parameters
f


sgn( )
k T
g k f
k
u a e w s
 


 
   
 
 
 

 

x (11)
where
 

is any chosen constant .
28 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc

VCM2012
With this replacement, the derivative of the sliding
surface (7) is now given by

 
 
(4)

k
k
k
s a e a e a e e
a e a e a e w x
a e w d t u
a e w
d t u u
a e w d t u
a e

  
 




   
    
 
    
 
  
 
    
 

      



 

  


     
   

 

 

x
x x
x x (12)
where
f f f
 

  
and
g g g
  
 

.
It can be noted further that

f f


T
f f g
V V
s
 





     
  
F
F
   
   
 

where
3 3


F
R
is any symmetric positive definite
matrix and

is an arbitrary positive constant.
f g
T
f f g g
T
f f
V ss
s u d t s
s u d t s
sd t s s s
 

 
 

 
 






  
 
   
 
 
 
   
 

 
 
 
 


(14)
Now, by using the following adaptive adjustments
for the time functions
( )

f
t

and
( )


g
t
of
controller (11)

( )
( )
f
g
s e
s e u
 



x x

which is sufficient for ensuring that ( )
s e
 
and
( ) 0
s e

.
3.3 Controller Design Procedure
Figure 2. shows the main configuration of the
closed loop system, in which the designed
controller, including sliding mode controller (11)
and adaptive parameters laws (15), always drives
the output
1 2
y x

 
to asymptotically converge
to any four times differentiable desired trajectory
( )
w t
.
To obtain this closed loop system’s tracking
performance, in summary, the following steps
should be executed.

,
f g

 


Plant
(
2
)

Controller

(
11
)

Adjustor
(15)
Tuyển tập công trình Hội nghị Cơ điện tử toàn quốc lần thứ 6 29

Mã bài: 07
4. Numerical Example
Consider a gearing transmission system as
described in Figure 1. where
( , )
d t
x
is a white
noise with

a a a
  
, sliding surface
constants

0.5


is infinite norm of disturbance

1


is parameter for controller (11)
The tracking error and the system output are
shown in Figure 3. and Figure 4. ; three elements
of the vector
f


and
g


from the adaptive
adjustors, are also given in Figure 5. and Figure
6. , respectively.
From the simulation results, it can be seen that the
system output asymptotically converges to the
desired trajectory even in the presence of the

Figure 4. Desired trajectory and system output

0 10 20 30 40 50 60 70
-80
-60
-40
-20
0
20
40
60
80
100

Figure 5. Adjusted parameters
f



0 10 20 30 40 50 60 70
0
0.2
0.4
0.6
0.8
1
1.2

0 10 20 30 40 50 60 70 80 90 100
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4

Figure 8. Adjusted parameters
[1]
f


compared with
[1]
( )
f
t


0 10 20 30 40 50 60 70
-1.5
-1
-0.5
0



2
e w

 
30 Le Thi Thu Ha, Nguyen Thi Chinh, Nguyen Doan Phuoc

VCM2012
0 10 20 30 40 50 60 70
-2
-1.5
-1
-0.5
0
0.5
1

Figure 9. Adjusted parameters
[2]
f


compared with
[2]
( )
f
t



.
However, this does not affect the tracking
performance of the system. In addition, although
the asymptotic tracking convergence of system is
theoretical proved under assumtion that
uncertainties
,
f g


are constants, this
performance is still keeping even in the case of
time dependent uncertain vector
( ), ( )
f g
t t

as the
experimental simulation results has shown in
Figure 7. Figure 10. , whichs are carried out for
system (5) with the time dependent functions of
three uncertainties:

[1] 1 0.4sin(0.5 )
f
t

  

[2] 1 0.2sin(0.5 )

t


5. Conclusion
The adaptive and robust controller, which is
designed by using the procedure proposed in this
paper, obviously satisfies the tracking requirement
of the system. This satisfaction has been proved
theoretically and numerically. In fact, the
controller can effectively attenuate the disturbance
and suppess the effect of parameter uncertainties.
Note that although the tracking error is guaranteed
to be zero at its steady state, its value during the
transient period cannot be constrained in a
predetermined range. This limitation can be
avoided by using a barrier CLF instead of (9) and
choosing
1 2 3
, ,
a a a
of the sliding surface (7)
appropriately.
Furthermore, as a consequence of using sliding
mode control, there still exists the chattering in the
system. In oder to damp this undesired behavior,
the constant

should be chosen as small as
possible but not less than


( , )
d t

x
is an estimate of
( , )
d t
x
such that

,
sup ( , ) ( , )
t
d t d t
 
  

x
x x
The function
( , )
d t

x
can be obtained easily by
using, for example, a neural network.
References
[1] Eutebach, T. and Pacas, J.M.: Damping of
torsional vibration in high dynamic drivers. 8.
European Conference on Power Electronics

Estimated Torsional Torque. IEEE trans. on
Industial Electronics, Vol.43, No.1, pp. 56-64,
1996.
[9] Szabat,K. and Orlowska,K.T.: Vibration
suppenssion in two mass drive system using PI
speed controller and additional feedbacks -
comparative study. IEEE trans. on Industial
Electronics, Vol.54, No.2, pp. 1193-1206,
2007.
[10] Szabat,K. and Orlowska,K.T.: Performance
Improvement of the Indusrial Drivers with
mechanical Elasticity using nonlinear
adaptive Kalman Filter. IEEE trans. on
Industial Electronics, Vol.55, No.3, pp. 1075-
1084, 2008.
[11] Utkin, V.: Sliding Modes in Optimization and
Control. Springer Verlag New York, 1992.
[12] Walha, L.; Fakhfakh, T. and Haddar, M.:
Nonlinear dynamic of two stage gear system
with mesh stiffness fluctuation, bearing
flexibility and backlash. Mechanism and
Machine 44, pp.1058-1069, 2009.
Le Thi Thu Ha received B.S. and
M.S. degrees from Thai Nguyen
University of Technology in 1999
and 2003 respectively, all in
automation technology. Since
2000 she has been with Electrical
Engineering Department at TNUT
Viet Nam, where she is nominated