1 Deadzone Compensation of an XY –Positioning Table Using Fuzzy Logic Adviser : Ying-Shieh Kung...

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Deadzone Compensation of an XY –Positioning Table Using Fuzzy Lo

gic

Adviser : Ying-Shieh Kung Student: Ping-Hung Huang

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

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Outline

Abstract Introduction Compensation of Deadzone Nonlinearity Adaptive Fuzzy-Logic Deadzone Compensation of

an XY - Positioning Table Simulation and Experimental Results Conclusion

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Abstract

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 bounded-parameter estimates.

The fuzzy-logic deadzone compensator is implemented on an XY –positioning table to show its efficacy.

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Introduction(1)

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

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Introduction(2)

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.

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Compensation of Deadzone Nonlinearity(1)

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 the system to small signals. It represents a “loss of information” when the signal falls into the deadband and can cause limit cycles, tracking errors, etc.

u. d ,d- u

d u d - 0,

d- u , d-- u

Dd(u) T(1)

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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+.

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The nonsymmetric deadzone may be written as

where the nonsymmetric saturation function is defined as

(u) u - (u) D T dd sat(2)

u. d, d

du d - u,

d - u , d

udsat

(3)

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To offset the deleterious effects of deadzone, one may place a precompensator as illustrated in following figure.

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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.

) then(positive), is ( If

du ) then(negative), is ( If

du (4)

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To make this intuitive notion mathematically precise for analysis, we define the membership functions

One may write the precompensator as

. ,

, ) (X

,

, ) (X

-

00

01

01

00

F u

(5)

(6)

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where is given by the rule base

The output of the fuzzy-logic system with this rule base is given by

F

d ω X

d ω X

F

F

ˆ then, ))((If

ˆthen,))((If

ωXωX

ωXd ωXd ωF

-

--

(7)

(8)

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1 )( )( - XX

)(ˆ Xd TF

where the fuzzy-logic basis-function vector is given by

)()( )(

- X

XX

(9)

(10)

(11)

14


The composite from w to T of the fuzzy-logic compensator plus the deadzone is

The fuzzy-logic compensator may be expressed as follows:

Given the fuzzy-logic compensator with rulebase, the throughput of the compensator plus deadzone is given by

)] (sat -[ ) ( )( FFF ddd DuDT

)(ˆ T Xdu F

TT )(~

- dXdT

(12)

(13)

(14)

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where the deadzone-width estimation error is given by

And the modeling mismatch term δ is bounded so that |δ| < δM for some scalar δM.

ddd ˆ~ (15)

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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

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

TTTxBxJ df (16)

(17)

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Given the desired trajectory xd, the tracking error is expressed by e = xd − x, and the filtered tracking error by

with Λ being a positive definite design parameter.

Differentiating and using (16)

eer

dr T f(q) -Br -T J

eer

(18)

(19)

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A robust compensation scheme for unknown terms in f(q) is provided by selecting the tracking controller

Deadzone compensation is provided using

With X(w) given by (11), which gives the overall feedforward throughout (14).The controller has a PD tracking loop with gains

v rKqf f - )(ˆ (20)

)(ˆ Xdu T

ΛeKe K rK fff

(21)

(22)

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Substituting (20) and (14) into (16) yields the closed-loop error dynamics

where the nonlinear functional estimation error is given by

It is assumed that the functional estimation error satisfies

(23)

(24)

(25)

] + + [ + ~

- )(~

+ ) + (- = vTfδdXdrBKrJ dTT

f

).(ˆ - )( ~

qfqff

)( ~

qff M

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Given the system (19), select the tracking control (20) plus deadzone-compensator (21), where X(w) is given by (11). Choose the robustifying signal

Then the tracking error r evolves with a practical bound

r

rqftv dM ))(()( (26)

(27)kBK

cr

f )(4

20

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Simulation and Experimental Results(1)

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

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The parameters of the XY table are estimated as

The gain of PD controllers are chosen as Kf = 1.5 and Λ = 3.0. The desired trajectory is selected by

m] [N 0.945 and ]m2 [kg 0.0143 BJ

m] [N 0.927 and ]m2 [kg 0.0135 B yJ y

6t5 ,2

51 ],2

)1(cos[2

10 ,2

)( tt

t

txd

65 ,2

51 ],2

)1(sin[2

10 ,2

)(

t

tt

t

tyd

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The deadzone is set at d1 = 0.03 and d2 = 0.024 for the X-axis, and d1 = 0.036 and d2 = 0.031 for the Y -axis. And set fM(q) = 0.018.

i) without compensation, ii) with compensation

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Experimental results for circle.


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The following shows modified circle and described it as

65 ,4.2

51 ],2

)1(5cos[

5

2]

2

)1(cos[2

10 ,4.2

)(

t

ttt

t

txd

65 ,0

51 ],2

)1(5sin[

5

2]

2

)1(sin[2

10 ,0

)(

t

ttt

t

tyd

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The simulation results of the modified circle.


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The experimental results of the modified circle.


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Conclusion

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.

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Thank you !

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