1EEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL iNSTRUMENTATION, VOL. IECI-19, NO. 4, NOVEMBER 1972

acteristic allows us, therefore, to keep the system re-sponse linear for large dynamic intervals of the quantityunder control, when the sensor response is linear.

4) The nonlinearity effects of the sensors whose re-sponse is exponential can be compensated by takingadvantage of the circuit nonlinearity effects through thematching with an appropriate interface. Such compensa-tion extends over large intervals of the quantity undermeasurement. Especially for a thermistor, the system-response linearity over intervals of the order of 100°C isparticularly interesting.

ACKNOWLEDGMENTThe authors wish to thank Prof. A. Alberigi Quaranta

for his constant interest and support in this work, Dr. E.Pupillo, M. Dondi, and R. Volta for valuable help inperforming the electronic circuit and taking the mea-

surements. They also wish to thank Prof. F. Baracchifor some helpful suggestions.

REFERENCES[1] R. A. Klososky, "Computer architecture for process control,"

IEEE Trans. Ind. Electron. Contr. Instrum., vol. IECI-17. pp.277-281, June 1970.

[2] D. M. Considine, Ed., Process Instruments and Controls Hand-book. New York: McGraw-Hill, 1957.

[3] M. Prudenziati, A. Taroni, and G. Zanarini, "Semiconductorsensors: I-Thermoresistive devices," IEEE Trans. Ind. Elec-tron. Contr. Instrum., vol. IECI-17, pp. 407-414, Nov. 1970.

[4] A. Taroni, M. Prudenziati, and G. Zanarini, "Semiconductorsensors: II -Piezoresistive devices," IEEE Trans. Ind. Electron.Contr. Instrum., vol. IECI-17, pp. 415-421, Nov. 1970.

[5] K. Arthur, Transducer Measurements, Tektronix, Inc., Beaver-ton, Ore., 1970.

[6] H. S. Black, Modulation Theory. Princeton, N. J.: Van Nostrand,1953.

[7] J. C. Hancock, An Introduction to the Principles of Communica-tion Theory. New York: McGraw-Hill, 1961.

[8] C. E. Shannon, "A mathematical theory of communications,"Bell Syst. Tech. J., vol. 27, pp. 3.79-656, Oct. 1948.

Optimum Design of a Position Detection System witha Sinusoidal Perturlation Signal

J. H. AYLOR, EDWARD A. PARRISH, JR., ANI) GERALD COOK

Abstract-In a paper by McVey and Chen [I] the effect of theamplitude of a sinusoidal perturbtion signal on the accuracy of aposition detection system was presented. However, the amplitudesconsidered were chosen arbitrarily, and no optimum value wasspecified. This paper presents a method for determining an optimumamplitude for a sinusoidal or, for that matter, any other perturba-tion signal. In addition, the method may be used to eliminate certainfunctional forms of a perturbation signal from consideration for agiven receptor geometry.

Manuscript received August 24, 1971; revised July 16, 1972. Theresearch on which this paper is based was supported by the ResearchInstitute of the U.S. Army Engineer Topographic Laboratories,Department of the Army, Fort Belvoir, Va., under Contract DAAK02-70-C-0280.

The authors are with the Department of Electrical Engineering,School of Engineering and Applied Science, University of Virginia,Charlottesville, Va.

I NT1RODUCTIONN A PAPER by McVey and Clheni [1], a method ofimproving detection accuracy of a discrete-outputreceptor composed of photosensitive elements was

presented. The output of each element is assumed to be1 if illuminated by an amount equal to or greater thansome threshold, and 0 otherwise. This being the case,the transfer characteristic of an infinitely long, one-di-mensional receptor is nonlinear.The system proposed in [1 ] to smooth the nonlinear

characteristic is shown in Fig. 1. The elements are as-sumed square (MXM) and are separated by an amountg. Smoothing is obtained by moving the target imageback and forth across the receptor with a perturbation

114

AYLOR et al.: DESIGN OF POSITION DETECTION SYSTEM

POSITION REFERENCE LINE

p0N

z

01-(n00-

0F-

z

0

10

Fig. 1. Infinitely long, one-dimensional receptor and system.

pINi PINNI ~~~~~~~~N5

pIN (ACTUAL POSITION)

Fig. 2. Illustration of method for determiningthe performance index.

signal of 0 average value. The frequency of this signal isassumed to be much greater than the velocity of thetarget image across the receptor. The receptor outputsare weighted, summed, and fed to a low-pass filter andan average value of position is obtained.

In this initial work by McVey and Clhen, the per-turbation signal was assumed to be sinusoidal fornathematical convenience. For this form of perturba-tion signal, results were obtained for various values ofperturbation amplitude, threshold setting, element spac-ing, and target-image size. Also, in a recent paper byParrish and Aylor [2 ], it was shown that the smoothingeffect could be enhanced by using a different set ofweights than those used in [1].The values for the perturbation amplitude were chosen

arbitrarily in [1]. To provide more insight into the prac-ticality of sinusoidal perturbation, the present paperpresents a method for determining an optimum valuefor the perturbation amplitude, relative to yielding theleast position error [3]. In this way, a judgement can bemade as to the value of sinusoidal perturbation in con-junction with different receptor geometries.

OPTIMIZATION TECHNIQUEThe input to the receptor is

7Mz0

()

0w

o 3M

z

iM-4M -3M -2M -M M 2M 3M 4M

ACTUAL POSITIOr*-IM

--3M

--5M

--7M

Fig. 3. Zero-deadband transfer characteristic withoutperturbation for alternate weighting scheme.

N ini Poi2J = E __

i==l iPi, Po.(2)

e(t) = Pi, + B sinG

where Pin is the position of the target-image center,assumed constant over a perturbation cycle, B is theperturbation amplitude, and 6=wt. The problem istherefore one of finding a perturbation amplitude Bsuch that, when used in the detection system, a linear(ideally) transfer characteristic of indicated position T0

versus actual position Pi. is formed. Thus a criterionfunction (or index of performance) which expresses

mathematically the desired effect of B must be deter-mined.One such performance index (PI), J, can be defined as

a function of the error between the actual characteristicand the ideal characteristic. That is

NPoi and Pin;

PinN

PO

total number of quantum steps of Pi.;the ith sample points along the Po axisand Pin axis, respectively;maximum value of Pin

2 N

=N-E (po).N j77

It is clear that when the actual transfer characteristicis linear, the PI goes to zero; therefore, the optimizationproblem is to minimize (2). The normalization processin (2) is done to insure that transfer characteristics withlow gain will not be rewarded by the PI. The term -POis included so that the ideal transfer characteristic is the

(1) where

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, NOVEMBER 1972

Ob- MYMg =o

a = 0.5M

M = 2

.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

PERTURBATION AMPLITUDE

Fig. 4. Performance index for zero-deadband case.

0

°b =

g = 0

a = 0.5M

M = 2B = 3.8 =I.9M

z

0-

0

a.

/

6'.

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

ACTUAL POSITION

Fig. 5. Zero-deadband transfer characteristic for B = 3.8.

closest linear approximation to the nonlinear (actual)characteristic. This prevents the slope of the transfercharacteristic from becoming a parameter of the prob-lem. Fig. 2 is an illustration of this method.

Since the optimization involves only one variable, a

plot of the PI versus perturbation amplitude can be ob-tained. The value of B corresponding to the minimumvalue of J will yield the smoothest transfer characteris-tic for the receptor geometry used. The process is thenrepeated for each different receptor geometry.

°b = MXM

g = 0

a = 0.5M= 2

B = 1.0 = 0.5M

z

0

0

a.OoJn f-Wa

0-

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

ACTUAL POSITION

Fig. 6. Zero-deadband transfer characteristic for B = 1.0.

RESULTSThe problem was programmed for computer solution

and plots of the PI versus B and P7O versus Pin, were ob-tained. The weighting scheme proposed in [2 ] and shownin Fig. 3 was used in the simulation of the system.The plot of the PI versus perturbation amplitude for

the case

object size Ob = M X M;deadband g = 0;

00(0)j

f'r)00o _

o Nx X)xw0

z 4.w o

0

Z 0

0.0

\

ro.o-

0.5

0.5

116

AYLOR et al.: DESIGN OF POSITION DETECTION SYSTEM

z0

a-cn

a.

C])

0zI

Ob = MXMg = 0o = 0.5M

M = 2

B = 2.8 = 1.4M

ACTUAL POSITION

Fig. 7. Zero-deadband transfer characteristic for B = 2.8.

U)Cti

N

xwa o7

w -

Z a)

CE0

0*

L Cq

w

111Q-

0

=b- MXM

g = O.5M

a = 0.5M

M = 2

t.O 0.5 1.0 i.5 2.0 2.5 3:0 3.5 4.0PERTURBATION AMPLITUDE

Fig. 8. Performance index for Case 1.

threshold a = 0.5 M;M = 2;

is shown in Fig. 4. It is obvious that amplitude is an im-portant factor. It should be noted that the minimumvalue of the PI occurs at more than one value of B in therange of B considered.

Choosing one of the values of B(B =3.8) correspond-ing to the minimum value of the PI, a transfer charac-teristic was obtained and is shown in Fig. 5. Transfercharacteristics for two suboptimum values of B were ob-

4.5 5.0 5.5 6.0

tained and are shown in Figs. 6 and 7. As expected, itcan be seen that the transfer characteristics for the non-optimum values of B are more nonlinear than that ofFig. 5. This verifies the usefulness of (2) as a quantita-tive measure of least position error.The behavior of the PI for two nonzero-deadband

cases was also investigated. The cases were the following.

Case 1Ob = M X Mg = 0.5M

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, NOVEMBER 1972

Ob = MXM

9i = 0.5M

a = 0.5MM = 28 = 1.2 = 0.6M

'0.0 1.0

Fig. 9. Tranisfer characteristic for Case 1 with B = 1.2.

°b = MxM

g = 0.2M

a = 0.5M

M = 2

0.5 1.0 1.5 2.0PERTURBATION AMPLITUDE

Fig. 10. Performance index for Case 2.

a =-0.5MM=2.0.

Case 2

Ob-=MXM

g =0.2M

a = 0.5M

M=2.0.

5.0 5.5 60

Fig. 8 shows the behavior of J for Case 1. This case

exhibits a global minimum, and the transfer characteris-tic for the corresponding value of B(B = 1.2) is shown inFig. 9. The PI for Case 2 is shown in Fig. 10. The opti-mum transfer characteristic for this case is shown inFig. 11. It should be noted that in both cases the op-timum values of B did not yield single-valued transfercharacteristics, although they do provide the fewestnonlinearities for the respective receptor geometries.

118

z0

0a.a-w

C-0Z

t0

OI0

c'J

w0z- O

z

C0

LL

W(D(L O*

o- ;

c'J

0

0.0

AYLOR et al.: DESIGN OF POSITION DETECTION SYSTEM

z0

in-04

0

N

Ob = MXMg = 0.2Ma = 0.5MM = 2

B = 2.3 - 1.65M

ACTUAL POSITION

Fig. 11. Transfer characteristic for Case 2 with B = 2.3.

Therefore, this approach is not only useful for finding anoptimum value of B for a given receptor geometry but,since transfer characteristics can be obtained, also pro-

vides information on how effective sinusoidal perturba-tion is for that receptor.

CONCLUSIONIn the work by McVey and Chen [I], the effect of

different perturbation amplitudes on the transfer char-acteristic was noted, but no systematic method for de-termining a best possible value was defined. A methodhias been presented here for determining an optimumamplitude for a sinusoidal perturbation signal or, forthat matter, any perturbation signal.

Furthermore, the optimization procedure will rule outcertain perturbation waveforms for a given receptorgeometry. This is manifested in the study of the systemwith deadband. Regardless of the amplitude of the sinu-soidal signal, the resulting transfer characteristic doesnot become single-valued, much less approach linearity.Clearly, one necessary condition for accurate positiondetection is that the system transfer characteristic besingle-valued.

REFERENCES

[11 E. S. McVey anid P. F. Chen, 'Improvement of position andvelocity detecting accuracy by signal perturbation,' IEEE Trans.Ind. Electron. Contr. Instrum. (Special Issue on Transducers),vol. IECI-16, pp. 94-98, July 1969.

[2] E. A. Parrish, Jr., and J. H. Aylor, "Comment on 'Improvementof position and velocity detecting accuracy by signal perturba-tion,' " IEEE Trans. Ind. Electron. Contr. Instrum. (Short Notes),vol. IECI-18, pp. 20-22, Feb. 1971.[3] J. H. Aylor, "The investigation of signal perturbation theory for

position and velocity detection," M.S. thesis, Univ. of Virginia,Charlottesville, June 1971.

119