an improved technique for transfer function

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0 IEEE TRANSACTIONS ON AUTOMATIC COWTROL, AUGUST 1970111. EXPERINEKTAL VERIFICATION REFERENCES

A digital computer hLbS een prepared w-hich, at the [I ] s. m. Golomb. Ed. . Digi ta l COr(lmuniCQti0nS Sp acz App l ica t ion s.Englewood Clifh, X. J.: Prent,lee-Hall, 1964, p. 76.[21 J. L.W. Churchill and K. D. Boardman Experience in the use ofrandom probing signals to determine fkquency response of full1 ) generates the maximum length sequence from a given size of signalTesting, 3 1 ~ ~965.ize industrial plant.. presented at the IEE Colloquium on Randomshift register; 131 D. A. Bell, The recovery of information from noisy measurements,2 ) calculates the values of A, i an d C$e for the sequence of 1) H I K. D. Smith and J. C. Hamilton, The logical design ofa digitalT r a n s . SOC.nstr. T e c h . , vol. 18, pp.,K2-102, June 1966.

using (3) ; pseudorandom noise generator. IEEE T r a n s . u c l e a rc i e n c e ,3) processes measwed response dat a rom he system being 151 J. S. Bendat.,and A. G . Pieml. Measurement and analysis Orol. NS-13. pp. 351-381, February 1966.skudied for the npu t sequence of 1) to provide eitheror (61 J. W. Co,oley and J. W . Tukcy. A n algorithm for the machinerandom data, Ne w Pork: Wiley, 1965.both computatlon of complex Founereries, M a t h . of C o mp ut a t zo n ,a) t.he cross correlation function, Le., th e impulse response vol. 19, p. 297, April 1965.

for carefully selected noise/system characteristics,b) frequency response parameters A,o and C$m using (6).we are concerned m+th I), 2 ), and 3b). The noise source is

e user, does one or more of th e following:

IO-stage shift register giving, therefore, a sequence length u ofTabIe I shows the values of A ,; and ~ $ ~ ialculated for ths

2 ) of the digital program. Each uni t of frequencyTableI is %/uA.t, where At is th e clock pulse time of the sequence,d he values of A,i havebeen normalized with respect t.o the

e a t wl.Th e system chosen for esamination has the ransfer funct.ion

T =0.1 second. The clock speed used is 200 Hz or A t = 5 ms,d hence 01 =frequency increment = 1.228 radis.The results or the syst.em of (8) are summarized by Fig. 1,is shown along wit.h the values

f&). The valueshave no t been corrected for t,he errorby normalization at. wl,where the true amplit.ude ratio

1.015 and not 1.0, nor for the variat ion in AT;,as this is verym e r he frequency range of interest (Table I ) .

IV. CONCLUSIOSShas been shown how readily the frequency response of a systembe obtained from i ts response to a pseudorandom binary sequenceconsideration of t.he spectral properties of sucha timulus.the method proposed avoids the calculation of th e cross

function, its subsequent transformat.ion, and correctiona r th e nonwhite noise characteristics of the input., considerable

saving is achieved, particularly where only a relatively smallof frequency po int s may be required. Also, errors due to

Fourier transformation of a. truncat.ed cross correlat.ion funct.ione avoided. A wide range of systems m ay be examined by suitable

tions of the sequence length and/or clock speed, both of-will va ry he Iocation of thespect ral lines of the npu t

Noise and harmonic rejection will be good and in the usual wayenhanced by analysis of a number of input cycles. In digitizing

e continuous data , however, care must be exercised in samplingavoid aliasing high frequencies with low frequencies [S . This

e of error does not occur ait .h analog proms ing .While a special purpose equipment can be nvisaged which would

he signal generation and the analysis, it. is felt that theful applicat.ion of t.he technique lies in he associat.ion with

digitd computer either on- or off-line, perhaps incorporating theurie r trans form (FFT) algorithm of Cooley and Tukey [SI.

J. D. LAMB

An Improved Technique for Transfer FunctionSynthesis from Frequency Response Data

Abstract-An improved curve fitting technique, based on modi-fication of work by Levy andSanathananand Koerner,z incor-porating known physical constraints is described. A comparisonwith experimental r esults on high-order systems shows tha t ther eis a much better guarantee of obtaining s table transfer functionsand of predicting responses in other domains.

I . IXTRODUCTIONDuring the analysis and synthesis of a complex control syst.em,

it s often necessary to employ frequencydomain curve fitkingtechniques i n order to est imate the coefficients of a transfer func-tion.Often heform of th e unknom transfer funct.ion maybedetermined by using imple xperimentalechniques and hisinformation may then be wed to advantage to const.rain t.he fittedt.ransfer funct.ion.

11. ANALYSISThe transfer functionwhich is to be fit.ted t o a set of frequency-doma.in data is

N ( w ) A+A I (jw) +A ? ( ~ ) ~- .-+A,( J U ) ~G(jw) =_ _ -D( w ) Bo +- Bl( jU)+ I i z ( j w ) 2 +.. +B,( j w Pand a solution to the curve fitting problem is given by Levy1 ina weighted least-squares sense. The weight.ing funct ion D ( w ) maybe removed by the iterat ive method of Sana thanan and Koerner.zIf in addit.ion to frequency response data t.he steady-state responseof the system tosuch inputs as the unit. st.ep, ramp, parabola, etc.,is determined, the n use mag be made of th e relationships betn7eenthe transfer functioncoefficients and the steady -state errors.As an example, consider a system which exhibits zero error to almit step input and a finite steady-state error y to a unit rampinput. Equation (1) may then be modified as follows:

If th e procedure given in Levy1 s followed, th e equations to besolved become


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CORRESPONDENCE

where

and

c u2+ - 2s2x 4

0-x60- ;T58 6

-- s

wherewi; X-th experimental requencyn to ta l number of experiment.al frequenciesR I , real part of G ( jwk)

[cl =

... T3... -s4... - T s". s6... T l

u 4... 0... - u s... 0

...

... Us

481

111. EXAMPLETheparticular technique previously described has been pro-

grammed for a digital computer and experimental results obt.ainedfrom a complex electrohydraulic servomechanism have been usedto obtain a t.ransfer function. The results are illustrated in Fig. 1,where a comparison is made with the curve fit. obtained where noprior knowledge of the velocity error characterist.ic is assumed.The improvement in the comt.rained curve fit is clearly discernible.In order tomake use of the ransfer unct ionobtained,onemight. eqec t to be able to predict. t.he system transient responseand the two impulse responses given by the transfer functions re-lat,ing to Fig. 1, are shown in Fig. 2, where they are compared n4t.hthe measured system cross correlation function (which is a goodapproximation t o he impulse response). The impulse responsegiven by the constrained curve fit compares very favorably withthe measured data, whereas that obt.ained from the unconstrainedcwve fit is unstable.

The tendency for the transfe r functionpredicted from freauencv

In the case of a system exhibiting zero step and ramp error b ut a constraintsarebuilt nto hecwve fit technique. Investigationfinite parabolic error, a similar solution is clearly available. of the results given by Sanathanan and Koernerl show tha t bot h


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T W S F E R FUNCTi0:i C O E F F I C i E h T SCO:UST&iiPED UNCONSTiUIXED

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Comparison between constrained and unconstrained curve fits.

0 EXPERIMENTAL POINTSFROM A CROSS CORRELATOR.

i ;

Fig. 2. Impulse responses resulting from two t,ransfer functions compared with cross correlation results.


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CORRESPONDEWCE 483denominator polynomials obtainedby t.heir technique, andbyt.hat.of Levy give rise to right half-plane poles. In fact., of severalhundred curve fits carried outby heauthor recently,about. 30percent resulted in unstable transfer functions when t.he techniqueof Levy' and Sanat.hanan and Koerner* was employed. Whent.he t.ransfer functions fit,ted were constrained t.0 haveparticularvelocity error characterist,ics, t,he percent.age of unstable systemsfell to 1 percent.

IV. CONCLUSIONSThe resultsquoted here clearly show t.he improvement tha t

maybe gained by adopting the technique described. This im-provement is not only one of improved accuracy of fit to theexperi-ment.al dat,a but also that of a reduced tendency for the predictedtransfer function t o cont.ain right half-plane poles. The improvementin the prediction of the system response in other domains is a.lsoevident.

P. 8. .4YWEDynamic Analysis Group

University of WalesInst . of Sci. and Tech.

Cardiff, Wales

A Generalized Root Locus Following TechniqueAbstract-A generalized root locus following technique is pre-

sented. Fundamentally, the concept involved is hat of approxi-mating a locus by a straight line in order to predict another pointon the locus. The method is applicable to a large class of root lociproblems and yields expressions for calculating the real and im-aginary parts of th e roots in erms of the parameter for whichthe locus is desired.

ISTRODUCTIONOft,en in the study of a system it, becomes necessary to find the

loci of the rootsof t he characteristic equation as a functicn of someparameter. This parameter could be open-loop gain in regular rootloci or sampling period in modfied root. loci. (A m odi j ed root locusis a locus that is a function of sampling period [SI.) It could evenbe t.he gain of a minor loop of some control system, or it could bereal time if the system has t.ime-varying elements.

In the pas t t,he plot. of t.he root loci as a function of open-loopgain has been considered in detail [I]. A regular root locus (a rootlocus tha t is a funct,ion of open-loop gain) can be achieved wit.h Bspirule; however, as he comp1exit.y of t.he syst.em increases thi smethod becomes impractical because of t ,he amount. of labor involved.This has led several people to consider digit.al computer methods

The simplest of these met.hods uses the computer t,o find th eroots of the characteristic equation as the parameter is varied [4],For a simple system this met.hodsuffices. For more complicatedsyst,ems several problems such as inconsistency in rootprogramconvergence or excess computer omputation t,ime may result.In circumventing hese difficulties a better approach might, be alocus following technique. The advantages of amethod of this

Pl-C71.

type are1) reduced computation time,2) greatly increased probability of convergence by he oot

program, and3) single locus study without, having t.o obta in heothers in.

the study of many systems this is a desirable charact.erist,ic).The suggestion of an algorithm which will accomplish this t.ask wasrecent.ly published [2]. However, its applicat.ion is limited to theregular root. locus. I t is t.he purpose of this correspondence to developa procedure which is applicable to a regular root locus as well as toseveral other types.

AiPPROXID.IATIONOF A LOCUS BY A STRAIGHT LINEThe characterist,ic equation of a typical feedback control system

is of the form1 f G(s ,T) =0 (1)

where G(s ,T) s the open-loop transfer f u nd on and T is the param-eter for which t,he locus is being made. The necessary and sufficientconditions for a point. SO tQ be a point. on the locus are

i G(so,T) i =1 (2)and

LG(so,T) = RTwhere 71 = 1,3,5,. - - for negative feedback. T he open-loop transferfundon can be nTitt.en as

G(x,y,T) = Gn(z,y,T)+G~(x ,y ,T ) (4)in which GR(x,y,T) and GI(x,y,T) are t,he real and imaginary partsof G(x,y,T) ,respectively, and x and y are the real and imaginarypar ts of t,he complex variable s. The angle relation of (4) s

whereas the magnihde-squared relat,ion of (4) s

Equations (5 ) and (6) represent t.wo hypersurfaces which arefunctions of the three variables x, , and T . The manifold of thesehypersurfaces is composed of several lines, namely, the t.hree-dimensional lines of the root loci. It is desired to find a line whichis tangent to the line cf intersection of these t a o surfaces. This isdone by first finding ihe gradients of the tw o surfaces. These are

and

where i,j , and k are unit vectors.Taking the cross product of the two gradients results in

an improved technique for transfer function

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