Transcript 下載/瀏覽

Deadzone Compensation of an
XY –Positioning Table Using
Fuzzy Logic
Jun Oh Jang; Industrial Electronics, IEEE Transactions on Volume 52, Issue 6, Dec.
2005 Page(s):1696 - 1701 Digital Object Identifier 10.1109/TIE.2005.858702
Adviser : Ying-Shieh Kung
Student: Ping-Hung Huang
1
Outline






2
Abstract
Introduction
Compensation of Deadzone Nonlinearity
Adaptive Fuzzy-Logic Deadzone Compensation of
an XY - Positioning Table
Simulation and Experimental Results
Conclusion
Abstract
3

Classification property of fuzzy-logic systems makes them a
natural candidate for the rejection of errors induced by the
deadzone

A tuning algorithm is given for the fuzzy-logic parameters,
so that the deadzone-compensation scheme becomes
adaptive, guaranteeing small tracking errors and boundedparameter estimates.

The fuzzy-logic deadzone compensator is implemented on
an XY –positioning table to show its efficacy.
Introduction(1)
4

Very Accurate control is required in mechanical devices
such as XY -positioning tables, overhead crane mechanisms,
robot manipulators, etc.

Precise positioning, in particular, control of very small
displacement, is an especially difficult problem for
micropositioning devices.

Actuator nonlinearities are typically defined in terms of
piecewise linear functions according to the region to which
the argument belongs
Introduction(2)
5

In this paper, present the deadzone-compensation method
in an XY -positioning table using fuzzy logic.

Derive a practical bound on the tracking error from the
analysis of the tracking-error dynamics and investigate the
performance of the fuzzy-logic deadzone compensator in an
XY -positioning table through the computer simulations.

The fuzzy-logic deadzone compensator is implemented on
an XY -positioning table to show its efficacy in canceling the
deleterious effects of system deadzones.
Compensation of Deadzone Nonlinearity(1)


6
This section provides a rigorous
framework for fuzzy-logic
applications in deadzone
compensation for a broad class of
XY -positioning tables.
Deadzone is a static nonlinearity
that describes the insensitivity of
u  du - d- ,
the system to small signals. It

T

Dd(u)

d-  u  d 
represents a “loss of information”
0,
u - ,
when the signal falls into the
d
d   u.

deadband and can cause limit cycles,
tracking errors, etc.
(1)
Compensation of Deadzone Nonlinearity(2)
7

One can see that there is no output as long as the input
signal is in the deadband defined by d− < u < d+.

When the signal falls into this band, the output signal is
zero, and one loses information about the input signal.

Most compensation schemes cover only the case of
symmetric deadzones, where d− = d+.
Compensation of Deadzone Nonlinearity(3)

The nonsymmetric deadzone may be written as
(2)
T  Dd (u)  u - sat d (u)

where the nonsymmetric saturation function is defined as
d  ,



u

sat d
 u,

 d,
8
u  dd-  u  d 
d   u.
(3)
Compensation of Deadzone Nonlinearity(4)

9
To offset the deleterious effects of deadzone, one may place
a precompensator as illustrated in following figure.
Compensation of Deadzone Nonlinearity(5)


The power of fuzzy-logic systems is that they allow one to
use intuition based on experience to design control systems,
then provide the mathematical machinery for rigorous
analysis and modification of the intuitive knowledge.
A deadzone precompensator using engineering experience
would be discontinuous and would depend on the region
within which w occurs.

If ( is positive), then( u    d  )

If ( is negative), then( u    d )
10
(4)
Compensation of Deadzone Nonlinearity(6)


11
To make this intuitive notion mathematically precise for
analysis, we define the membership functions
0,
X  ()  
1,
0
0 
1,
X-()  
0,
0
0  .
One may write the precompensator as
u    F
(5)
(6)
Compensation of Deadzone Nonlinearity(7)

where F is given by the rule base


( )) , then ωF  dˆ 
If (  X  ( )) , then ωF  dˆ
If (  X 


The output of the fuzzy-logic system with this rule base is
given by


d  X  ω d- X-ω
ωF 
X  ω  X-ω
12
(7)
(8)
Compensation of Deadzone Nonlinearity(8)
F  dˆ T X ( )
X  ()  X - ()  1
(9)
(10)
where the fuzzy-logic basis-function vector is given by
X ( )   X  ( )
 X - ( ) 
13
(11)
Compensation of Deadzone Nonlinearity(9)

The composite from w to T of the fuzzy-logic compensator
plus the deadzone is
T  Dd (u)  Dd (  F )    [F - satd (  F )]

(12)
The fuzzy-logic compensator may be expressed as follows:
u    F    dˆ T X ( )
(13)

Given the fuzzy-logic compensator with rulebase, the
throughput of the compensator plus deadzone is given by
~T
T   - d X ( )  d T
14
(14)
Compensation of Deadzone Nonlinearity(10)

where the deadzone-width estimation error is given by
~
d  d  dˆ

15
(15)
And the modeling mismatch term δ is bounded so that |δ| <
δM for some scalar δM.
Adaptive Fuzzy-Logic Deadzone
Compensation of an XY - Positioning Table(1)

The dynamics of the X-axis (similar to the Y -axis) system
with no vibratory modes can be written as
Jx  Bx  Tf  Td  T
16
(16)

where x(t) is the position, J is the inertia, B is the viscous
friction, Tf is the nonlinear friction, Td is the bounded
unknown disturbance, and T is the control input.

The unknown deadzone widths are bound so that
|d| < dM
(17)
Adaptive Fuzzy-Logic Deadzone
Compensation of an XY - Positioning Table(2)

Given the desired trajectory xd, the tracking error is
expressed by e = xd − x, and the filtered tracking error by
r  e  e
(18)
with Λ being a positive definite design parameter.

Differentiating r  e  e and using (16)
J r  -Br -T  f(q)  Td
17
(19)
Adaptive Fuzzy-Logic Deadzone
Compensation of an XY - Positioning Table(3)

A robust compensation scheme for unknown terms in f(q)
is provided by selecting the tracking controller
  fˆ ( q)  Kf r - v
(20)

Deadzone compensation is provided using
u    dˆ T X ( )

18
(21)
With X(w) given by (11), which gives the overall
feedforward throughout (14).The controller has a PD
tracking loop with gains
Kf r  Kf e  KfΛe
(22)
Adaptive Fuzzy-Logic Deadzone
Compensation of an XY - Positioning Table(4)

Substituting (20) and (14) into (16) yields the closed-loop
error dynamics
~
~
Jr = - ( Kf + B )r + d T X ( ) - d T δ + [ f + Td + v ]

where the nonlinear functional estimation error is given by
~
f  f ( q) - fˆ ( q).

(24)
It is assumed that the functional estimation error satisfies
~
f  fM ( q )
19
(23)
(25)
Adaptive Fuzzy-Logic Deadzone
Compensation of an XY - Positioning Table(5)

Given the system (19), select the tracking control (20) plus
deadzone-compensator (21), where X(w) is given by (11).
Choose the robustifying signal
v ( t )  ( f M ( q )   d )

(26)
Then the tracking error r evolves with a practical bound
c02
r
4( K f  B ) k
20
r
r
(27)
Simulation and Experimental Results(1)




21
The Y -axis of the XY table was placed on the X-axis.
The actuators of the XY table were two dc servo motors.
Ball screws were connected to the motors and allowed the
movement of the table.
The main control algorithm is implemented at a 100-Hz
Simulation and Experimental Results(2)

The parameters of the XY table are estimated as
J   0.0143 [kg  m2] and B   0.945 [N  m]
J y  0.0135 [kg  m2] and B y  0.927 [N  m]

22
The gain of PD controllers are chosen as Kf = 1.5 and Λ =
3.0. The desired trajectory is selected by
2,
 (t  1)

xd (t )  2 cos[
],
2
2,

0  t 1
1 t  5
5 t  6
2,
 (t  1)

yd (t )  2 sin[
],
2
2,

0  t 1
1 t  5
5 t  6
Simulation and Experimental Results(3)

The deadzone is set at d1 = 0.03 and d2 = 0.024 for the Xaxis, and d1 = 0.036 and d2 = 0.031 for the Y -axis. And set
fM(q) = 0.018.
i) without compensation, ii) with compensation
23
Simulation and Experimental Results(4)

Experimental results for circle.
i) without compensation, ii) with compensation
24
Simulation and Experimental Results(5)

The following shows modified circle and described it as
0  t 1
2.4,
 (t  1) 2
5 (t  1)

xd (t )  2 cos[
]  cos[
], 1  t  5
2
5
2
2.4,
5 t 6

0,
 (t  1) 2
5 (t  1)

yd (t )  2 sin[
]  sin[
],
2
5
2
0,

25
0  t 1
1 t  5
5t  6
Simulation and Experimental Results(6)

The simulation results of the modified circle.
i) without compensation, ii) with compensation
26
Simulation and Experimental Results(7)

The experimental results of the modified circle.
i) without compensation, ii) with compensation
27
Conclusion
28

The classification property of fuzzy-logic systems makes
them a natural candidate for offsetting this sort of actuator
nonlinearity that has a strong dependence on the region in
which the arguments occur.

It was shown how to tune the fuzzy-logic parameters so that
the unknown deadzone parameters are learned online,
resulting in an adaptive deadzone compensator.

Using nonlinear-stability techniques, the bound on the
tracking error is derived from the tracking error dynamics.
Thank you !
29