Bài báo trình bày phương pháp thiết kế bộ điều khiển bám thích nghi bền vững cho hệ truyền động qua bánh răng trên cơ sở sử dụng nguyên tắc điều khiển trượt và thích nghi giả định rõ. Chất lượng bám ổn định tiệm cận luôn được đảm bảo và không phụ thuộc vào sự xuất hiện của các thành phần bất định trong hệ thống gồm khe hở, ma sát và độ không cứng vững của các bánh răng. Kết quả mô phỏng đã chứng minh khả năng bám thích nghi bền vững rất tốt của hệ kín. “nội dung được trích dẫn từ 123doc.org - cộng đồng mua bán chia sẻ tài liệu hàng đầu Việt Nam”
Trang 1Robust 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 Ha1), Nguyen Thi Chinh2), Nguyen Doan Phuoc3) 1)
, 2) Thai Nguyen University of Technology; 3) Hanoi University of Science and Technology
e-mail: 1)hahien1977@gmail.com, 2)nguyenthichinh-tdh@tnut.edu.vn, 3)phuocnd-ac@mail.hut.edu.vn
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 principle The asymptotic tracking behavior of the system in the presence of all uncertainties caused by backlash, friction or cogwheel elasticity is proved The simulation results are provided to illustrate the satisfactory performance of the closed loop system
Keywords: Adaptive control; sliding mode; nonlinear system; backlash; cogwheel elasticity; friction
Tóm tắt
Bài báo trình bày phương pháp thiết kế bộ điều khiển bám thích nghi bền vững cho hệ truyền động qua bánh răng trên cơ sở sử dụng nguyên tắc điều khiển trượt và thích nghi giả định rõ Chất lượng bám ổn định tiệm cận luôn được đảm bảo và không phụ thuộc vào sự xuất hiện của các thành phần bất định trong hệ thống gồm khe hở, ma sát và độ không cứng vững của các bánh răng Kết quả mô phỏng đã chứng minh khả năng bám thích nghi bền vững rất tốt của hệ kín
Từ khóa: Điều khiển thích nghi; điều khiển trượt; hệ phi tuyến; khe hở; bánh răng; ma sát
1 Introduction
The uncertainties that usually limit the
performance of a gearing transmission control
system in many practical applications are mainly
caused by immeasurable friction, unpredictable
elasticity of shafts and imprecise description of
backlash between cogwheels[1], [3], [7], [9]
Those inevitable uncertainties can reduce the
lifetime of the whole system or even disturb the
system behavior Therefore, damping the torsional
vibration due to the shaft or cogwheel elasticity
and suppressing the effect of friction or backlash
are the most important control problems of
mechanical systems in general and of gearing
transmission systems in particular
Conventionally, self-tuning PI controllers are
often used to approach these problems [8]
However, only using such PI controllers cannot
damp torsional vibrations effectively [1]
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
Trang 2The sliding mode control is one of the robust
control theories to suppress the effect of bounded
noises or disturbances in systems In addition, the
certainty equivalence is also the most successfully
used principle in adaptive controller designs for
uncertain nonlinear systems in the presence of
unknown constants in the systems' model In this
connection, the paper combines both the sliding
mode technique and the certainty equivalence
principle for designing an adaptive robust tracking
controller for gearing transmission systems, in
which the unpredictable elasticity of cogwheels
and the imprecise description of backlash between
cogwheels are considered as unknown constant
parameters, whereas immeasurable shaft friction
and the load capacity are regarded as bounded
time dependent noises and disturbances in the
system
This paper is organized as follows In section 2 the
mathematical model of gearing transmission
systems is included The section 3 describes
design procedure of robust adaptive controller for
system In section 4 the experimental results and
simulations are inculdes Finally are included in
section 5 some conclutions and commentaries
about future research
2 Model of Gearing Transmission Systems
2.1 Euler-Lagrange model
Consider a gearing transmission system with a
controller as depicted in Figure 1 The driving
motor provides a control torque M d which is
transmitted to the load M c through two wheel
gears 1 and 2 and two elastic shafts Let M f1 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])
c f
where r1 and r2 are the outer radii of
corresponding wheels 1 and 2, i12 i211
is the transmission rate of the two wheels and J J1, 2, J d
are the inertia moments of wheel 1, wheel 2 and
the driving motor respectively and J1J dJ1
denotes the sum of inertia moments of wheel 1
and the driving motor
system While J J1, 2, i12, i21, r1 and r2 in Euler-Lagrange model (1) can be considered as known parameters, the other parameters such as shaft elasticityc,friction moments M f1, M f2,load moment M c,backlash are all uncertainties or disturbances of the system
2.2 States space model
In the following, all unknown constant parameters
of the model will be denoted by k, whereas disturbances by d k By using
cos , cos
where
1, 2
b b known constants,
1, 2
unknown constants,
T p
,
T q
,
( )k
x kthderivative of x, ,
p q finite positive integers,
1 ( 1 , ) , 2 ( 2 , )
d t d t unknown disturbances, the Euler-Lagrange model (1) becomes
d
From the second equation of (2), it is easy to see that
with
d i d J i b
and
(4)
where d4 d3 , d5d4 From (2), (3) and (4), it follows that
1
f
Load
c
c
d
M 1
2
2
f
M
M
Controller
1
2
1
2
Trang 3
(4)
d
b i J d d b d d
Next, let states vector x, truncated states vector
x , input control signal u, vector of unknown
constants f , unknown constant g and unknown
disturbance d( , )x t be defined as follows
x
x
x
x
x x x
1 3
1
f
b i
J
1 5 1 3 1 4 1
1 3
1
( , )
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
( , )
T
with d( , )x t being bounded by a number 0,
that is
( , )
d x t for all x ,t (6)
3 Robust and Adaptive Tracking Controller
3.1 Sliding mode controller
Let w t( ) be the reference signal, so the reference
trajectory for system (5) will be
w w w w , , , T
w
and the vector of reference error is
e e e e , , , T
e
where
e w x w
To control the states vector x( )t of (5) to
asymptotically track the reference trajectory w( )t
based on sliding mode control, first the following
sliding surface is used
s e a e a e a e e a e (7)
where all elements a a1, 2, a3 of vector
1 , 2 , 3 , 1T
a
are chosen such that the following polynomial
( )
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
( ) 0
s e
and ( ) for 0
s e t Now consider the following candidate control Lyapunov function (CLF)
2
1 ( ) 2
with its derivative being given by
3
1
( , ) .
i
V ss s a e a e a e w x
s a e w x d x t u
Therefore, if the following controller is used
3
1
sgn( )
k
x , (10) then
3
1
( , ) sgn ( ) ( , ) sgn ( ) 0,
i
s
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 and
g
To overcome this limitation, the certainty equivalence principle will be employed
First, the unknown constants f and g in (10) are replaced by time functions ( )
f t
and ( )
g t , respectively, yielding
3
1
sgn( )
k
where is any chosen constant
Trang 4With this replacement, the derivative of the sliding
surface (7) is now given by
(4)
(4)
3
1
3
1
3
1
3
( ) 1
( , )
( , )
( , )
k
k
k
k
T
k
k k
k
s a e a e a e e
a e a e a e w x
a e w
a e
(4) sgn( )
( , ) sgn( ) ( , ) sgn( )
T f
T
T
x
where f f f
and g g g
It can be noted further that
f f
and gg
because of constancy of f and g
Second, by using an adaptive CLF candidate
T
T
V V
s
F
F
where FR3 3 is any symmetric positive definite
matrix and is an arbitrary positive constant
By using (12) and (13), one subsequently obtains:
1
1
1
1
1
( , ) sgn( ) 1
( , ) sgn( ) 1
( , ) sgn( )
T
T
T
T
T
T
V ss
F
F
F
F
1
(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
F
(15)
the derivative V
becomes negative definite
( , ) sgn( ) ( , ) 0
s
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 y x 12 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
Estimate of according to (6) Choose three constants a a1, 2, a3 so that the polynomial (8) is Hurwitz
Construct the sliding surface s e( ) according to (7) Choose any symmetric positive defined matrix
3 3
F R and a positive constant Construct the adaptive adjustor according to (15) Construct the sliding mode controller according to (11)
u
,
f g
Plant
(2)
Controller
(11)
Adjustor
(15)
Trang 54 Numerical Example
Consider a gearing transmission system as
described in Figure 1 where d( , )x t is a white
noise with d 0.1
and the reference signal is
given by w t( )sin(0.1 )t
Let design parameters be chosen as follows:
3
50
F I , where I3 is the unity matrix in R3 3
0.1
1 125, 2 75, 3 15
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
unknown parameters f , g and the bounded
disturbance d( , )x t
-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 1.4
Figure 6 Adjusted parameter g
-25 -20 -15 -10 -5 0 5 10 15 20 25
time dependent uncertainties
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
Figure 8 Adjusted parameters f[1]
compared with
[1]( )
-1.5
-1
-0.5
0
0.5
1
1.5
f
( )
w t
2( )t
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
2
e w
Trang 60 10 20 30 40 50 60 70
-2
-1.5
-1
-0.5
0
0.5
1
Figure 9 Adjusted parameters f[2]
compared with
[2]( )
It should be noted that the adjusted parameters f
and g
do not tend to the actual values of
unknown parameters f and g In fact, in this
example, the plant (5) was simulated with
1 2 3T
f
and g 1
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( ),t g( )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 0.4 sin(0.5 )
[2] 1 0.2sin(0.5 )
[3] 1 0.2 sin(0.5 )
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Figure 10 Adjusted parameters f[3]
compared with
[3]( )
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 a a a1, 2, 3 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 In the case, that the constant has to be choosen less than , the controller (11) can be revised as
3
1
( , ) sgn( )
k
g
u
x x
where d( , )x t is an estimate of d( , )x t such that
,
sup ( , ) ( , )
t
d t d t
x
The function d( , )x t can be obtained easily by using, for example, a neural network
References
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torsional vibration in high dynamic drivers 8
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2012
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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
as head of department in 2008 Her research interests include modeling of mechanical systems and controller design for Euler-Lagrange systems
Nguyen Thi Chinh received her
B.S degree from Hanoi University
of Science and Technology in
2003 and M.S degree from Thai Nguyen University of Technology
in 2007 Since 2003 she is working as Uni lecture in Industrial Automation Department of TNUT Viet Nam Her research interests are Fuzzy Control and Neural Networking
Nguyen Doan Phưoc received his Dipl.-Ing and Dr.-Ing degree from Institut für Steuerungs- und Regelungstheorie, TU Dresden, Germany in 1982 and 1994 From
1994 to 1996 he has worked with Fraunhofer Institut Dresden on Modelling and Simulation Since 1997 he has been with Automatic Control Department at HUST Viet Nam, where he is nominated as associate professor in 2003 His research interests are adaptive and robust control, optimization and optimal control