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The human operator as a servo system element, Part I and II


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This paper considers the role of human ekmeots in certainclosed loop control syvI. A qasatitative descriptoaofhuman dyrainka useful to control system

is eeisi to understanding and analysis of .uch systems. Accordingly. the human behavior description must be expressed in terms which are compatible with t1O0*l description. of COOUOL 5Yst componenti This compatibility is

achieved by the use of a quasi-linear mathematical model for thehuman operator. The model is composed of two componentsa describing function and remnant. The

diiribing fuaction, which for s linear system i. identical with the ccjiventional

trins-1er function, is established to charactei-ize that portion of the operator's output which h linearly correlated with his input. The input, upon which the describing function is baied, isselected on the basis 01m jirisri estimatesofthe nature of certain human

non-linear behaii.r. Human output power which cannot be characterized by the opera-tion of the describing funcopera-tion on the input is designated as the remnant.

After prtsenting the analytic basis foc measurements 01 human dynamics. steady-state dribing functions measured by the vinon, experimentersin the SeId are dis-cuused and the adaptive, optimalizing behavior of the human operator is demonstrated The rvmnaiits sirs also discussed and p'ausible sources foe their origin are postulated.

Kno ledge of the range of parameter adjustment of which the human operator is capable in his adaptation as well,as knowledge of his criteria loe adjustingthese parameters enables the designer to specify input Functions sind operatorcontrolled dynamics oempatible with both desirable human operator behavior and good system performance. By judiciously trading o system complexity against operator prefer-encra, while still making proper engineering use of the humanoperator'. adaptability,

a control system may be nitirnized for both performance and reliability. I$TlODUCToa

The control engineer achieves a logical connection between a system and its components by expressing component characteristics in termsof

abstract models, such as transfer functions or computer setups, and then combining these models to obtain a description of system behavior. The use of such techniques in the design of automatic control systems has niet with outstanding success. In this process, designers have pended to overlook one of the most versatile and, in many respects, most

Thisi ren.srch was supported in whole or in part by tht Usised States Air Forceunder WContracts AF 33(l6)-3080, AF 33(038)-10420, AF 33(616)-2804, AF 33(6t6)-3610, and r AF 33(616)-S522, monitored jointly b, the Flight Control L.aboratory ¡nd the Aero Medical

Laboratory, Wright Air Devtlopmcnt Center, Wright-Patterson Air Force lIase. Ohio. 'Systems Techno'ogy, Inc., Inglewood, Calif.

1The Franklin Institute Laboratories loi Research and De'elopment, PhiLadelphia, Pa. ' Pari Il will appear ¡n this Jovpj*I. loe June. 1939.




readily available and reliable components of allthe human controller. There arc fairly broad areas where a suitable manual control system represents either the optimum solution or merits detailed consideration as a good possible choice. Before control engineering concepts can be applied to such cases, however, one must have a catalog of human static and dynamic characteristics, preferably expressed in ternis readily understood by the control engineer. One of the principal purposes of this paper is to provide the reader with a synthesis, in control engineer-ing terminology, of the available data on the dynamic characteristics of human operators in certain continuous control tasks. This summary as to a Large extent based upon McRuer and Krendel (i). Much of the available static characteristic data can be found in (2 and 3).

Many agencies and individuals, both in the U.S.A. and abroad, have made substantial contributions to the state of knowledge of human re-sponse characteristics.

In the past these data have been compared

rarely if ever, since they were measured in different ways, different tasks were given the operator, diverse system inputs were utilized, and the data were presented in different forms. A review of the situation in the light of servo analysis indicates that consistencies exist for these data, from which useful generalizations can be hypothesized about the human operator dynamics.

The human response measurements of concern here are those in-volved in dosed loop control tasks which yield operator transfer charac-teristics for routine, essentially continuous functions such as stabiliza-tion and tracking. Even for these simple tasks we cannot expect to obtain a unique transfer characteristic since the operator's responses will depend upon at least the following factors:

The dynamic characteristics of the controlled elements, that is, the dynamics between the manipulated variable and the display. (The operator, to be successful, must adapt his transfer characteristics to a form required for adequate stability and performance of the entire cloeed ioop system.)

The type of input or forcing function driving the system.

The individual reaction delays, thresholds, etc,, of the human during the particular operation. (There are variations between indi-viduals and within individualS which are functions of both physiological differences and task experience.. Any set of transfer characteristics ulti-mately proposed would have to allow for a variation in parameters of this type.)

The motivation, attention, previous training, and general psycho-logical and physiopsycho-logical condition of the human at the time of the operation.

These considerations specify a human mechanism which adapts itself

$ The bo'dface numbers in parentheica re{er to the rdercncr appended to Part I of this



in sorne way as a function of the system inputs and controlled elements, and which may have wide individual variations in the values of the pa-rameters adopted. Individual parameter variations cannot be specified at present since there are too few data dealing with adequately large populations. There do exist adequate data to demonstrate the adapt-ability of human transfer characteristics to both forcing functions and controlled elements and such data provide a rationale for classifying past experiments.

The selection of the form of the mathematical model to describe the operator's transfer characteristic is based upon several practical con-siderations: the ease with which it lends itself to conventional servo analysis methods; the convenience with which experimental data may be obtained to define such a model; and the extent to which the

rneasur-ing experiments are capable of generalization. These factors limit human operator models to the following:

A linearized model consisting of an equivalent quasi-linear transfer function (which is correlated linearly with the system input or forcing function) and a remnant or residual output (which represents all output content which cannot be ascribed to a linear operation on the input).

Simulation, using analog computer elements.

A series of nonlinear or piece-wise linear equations which can be solved by either analog or digital means.

To date only the first two have been used successfully as the basis ci experimental techniques. The observation that human operator

con-trol responses are discontinuous has led to the conjecture that a non-linear relay servo model would provide a more natural characterization for human control behavior. Although certain preliminary experi-mental results indicated the existence of possible dual or multi-mode relay type servo behavior on the part of the human, other data shed doubt on this behavior (4, 5). The literature demonstrates that the model lacks convenient applicability to the analysis of continuous closed loop systems with continuous inputs (ó, 7).

To provide an adequate representation of the operator in a closed loop controller application, the operator mathematical models should provide answers to two questions:

How does (will) the operator perform in a given situation, that is, what specific quasi-linear transfer function and remnant, or analog computer set-up, are required to describe his conduct?

How well can the operator perfonn and how do we make him behave that way, that is, what is the "desirable" operator characteristic and under what circumstances s it achieved?

The answer to the first of these questions is essential for manual control system analysis, while that of the second provides the starting point for


T. McRuri AND E. S. KRENDEL [J. F. L

nthesis. The first question is answered by the formulation of an laptive-optimatizing niodel developed to be consistent with all of the railable data. This model can be used, with caution, for stability íd limited performance predictions of manual control systems. The cond question is answered to the extent of defining a desirable system rin and the human operator's response when cperating with this form. his "desirable" operator characteristic refers to a "best" characteristic thin the adaptive properties of a so-called "average" human operator, id not to an ideal super-human-machine combination. What is de-'able from the system designer's lights may not be preferred from the ¶itroller's viewpoint.

In developing the answers to the above questions this paper is ganized in four main parts. The first Ls primarily background infor-ation on the types of manual control systems of interest to this paper, íd the general methods for the description of quasi-linear operation. he second presents a summary of presently available human dynamic sponse data, gathered from experiments directly involving the meas-ement of human describing functions from tracking systems which splay the error alone. Past experimental efforts are reviewed, and e approximate mathematical models evolved

to "fit" the data are


The third and fourth

parts generalize from the experi--ntal data.

The hypothetical models discussed are in two categories:a general pothetical model attempting a mathematical description of the human -rator for various system considerations (that is, various controlled ment coñfigurations and types of system inputs); and a "desirable" man operator form determined by a minimum rms error criterion of -tern performance. Since the form of the highly adaptive human -rator is determined by the control problem, the controlledelements the system can be subjectedto design change to elicit desirable

human -rator characteristics. By having both a general

hypothetical oper-.r model and a preferred operator model, the control system analyst

evaluate the performances of particular system configurations with t of a more or less theoretical opt imuiri, and can gain an idea of the of modification required to the rest of the system if system error ormance alone were the sole design objective. Since other factors h as reliability, maintainability and cost are often as important as amic performance, the designer must effect a compromise in order achieve an over-all optimum.


DESCRiPTION ator Environmental Factors Due to the Control


The fact of human adaptation makes description of the operator rmously complex when viewed in the large, and necessitates the


May, J959.]


setup of simplified, constant situations within whichone may have some hope of obtaining a simple behavioral model of engineering value.

For the control situation to be analyzed withconfidence the assigned task or goals of the humancontroller, and the constraints or rules govern-ing his interactions with the system, must remain reasonably constant over the duration of interest. In the present instance

we are concerned with manual processes for minimizing

visually perceived errors by exer-cising essentially continuous control so as to match

visually presented input and output signals. This type of process is often called tracking. One can classify tracking problems for conceptualconvenience. Our criteria for classification will be based

on the type of information which the stimulus to be tracked and the controller's response present to the controller as a basis for futuretracking decisions.

In application, these criteria result in three limiting types of classifications: precognitive,

pursuit and compensatory.

These classifications arise as interactions among training, the geometry of the display, and the nature of the input. As such, they can readily exist in mixed forms.

Precognitive. This condition exists when the operator has

complete information about the input's future and a stimulus can trigger off a repertory of practiced, properly sequenced responses.

In a sense, the situation itself may be the stimulus. Thus throwing

a baseball at a target is precognitive behavior, as is steering a car

out of a skid, or

navigational flying under VFR conditions. The operator doesn't need to maintain a frequent check on the individual responses in a aequence. Instead, end product responses are monitored; such as, "Did I hit the target?", "Am I still skidding?" or "Have I passed over a given fix?" In that continuous

close control is not maintained on the perceived error, precognitive behavior is not tracking. However, we have

included this classification becauseoccasionally tracking

approaches these conditions. One might characterize precognitive behavior as discrete control with long sampling intervals such that

open ioop control is in effect during these intervals.

Pursuit. In pursuit behavior past experience provides

the tracker with information with which to predict the future input, but he must operate in a closed loop fashion with visual feedbackabout hisresponses. In this for.n of tracking the display effects of the operator's corrective respoflses can be distinguished from his input.

These responses are displayed after they have been modified by the system dynamics. A gunner aiming for a moving target using an open sight is an example of pursuit tracking in which the system

dynamics are those of the me-chanicalsystem composed of gun, ann, and body.

Compensatory. Compensatory tracking

is the same as pursuitexcept that the visually

displayed effects of the controller's responses are not distinguishable from the system's input.


'-6 D. T. MCRL-ER ANT) E. S. KRENDEL [J. F. t.

e target using a telescopic sight with a small field of view, we would ve a compensatory situation. Since in compensatory tracking the

ual display is the system forcing function minus the inodilied control ponse, the operator can deterniine the effects of control motion alone

ly under zero input conditions.

The precognitive classification, representing a skilled operator re-anse or calibrated, open bop system, is not readily amenable to servo alysis techniques. The pursuit system is a particular type of open de, closed cycle servo system, and requires a more complicated servo alysis (8). The compensatory classification, representing continuous

,se-control tasks, can be attacked by simple servo methods. The

rticular type of manual control system to be considered here falls into

e compensatory Category.

The general nature of a compensatory display is represented in

g. 1. The operator is presented with an input consisting only of an



C a. C E 'Q) '(t) i(i)

-FIG. 1. Pure ccnlpcnaI«.r dispì.v.

icator showing the difference, or error, e(i), between a system forcing ction, or command signal, i(i), arid the system output r(I). The 'rator's task is to rninimiz.t the error signal presented by trying to

p the circle superimposed on the stationary dot.

Besides the display, other fundamental factors in the control

situa-n are the type of elemesitua-nt beisitua-ng cositua-ntrolled by the operator (such as

aircr;iut, automobile, etc.), atid the actual means of exerting control

ch as a control stick or wheel with their associated restraints, for

mplc, springs, dampers, etc.). All of these characteristics will be

iped)nto the general classification of the "controlled element." In

my cases the controlled element can be described suitably by transfer


functions, though in others nonhinearities may be present and require description.

The functional block diagrariis of the simple control situation

con-sidered, witi the human operator as an element of a closed loop system. is shown ¡n Fig. 2.


r(g) O VT P II?


FIG. 2. Function&l block diagram of simple

comnsatory manual control system.

Quasi-Linear Operaior Descripwn

If the characteristics of the human operator for a given over-all task are assumed to be capable of quasi-linear description, the operator mathematical model will consist of a describing function plus an addi-tional quantity inserted as an input into the system by theoperator. Then the functiona! block diagram of Fig. 2 can be made into an equiva-lent block diagram showing more detailed operations of the system. This is done in Fig. 3, where the symbols describe the various quantities

Qi.s..4ety I o. N, I <' I I


i(j) OS I + E I

I e(i) os c ° r(t) o. R


PuNCfloa i y,,(,,,,) I ILIMINT





FIG. 3. Block diagram showing human operat.of in continuous control task.

as time functions and as Fourier translonns. To simplify the structure, the dynamic characteristicsof the display are lumped with the controlled element, and the actual forcingfunction is modified, ifnecessary, to an equivalent one. The linear transfer characteristics of the human opet

ating On his presented input art' described by the weighting functior

or the Fourier transform, Y.Uù,). Since all of the



ptu aro. .,taAo.





D T. McRur.i AND E. S. KRENDEL (J. F. I.

)utput is not described by the action of a linear transfer characteristic, rn additional term, ti,(i), is added at his linear output to complete the lescription of the total operator output c(t). This additional term is :alled the remnant. The location of n,(t) at the operator's output is

trbitrary, and does not necessarily imply that such a quantity is

hysically inserted at that point.

The block diagram of Fig. 3 illustrates clearly the servo system :haracteristics of manual control systems in general tracking tasks. lhis allows us to apply the whole body of servo theory in our attack jpon human behavior in such manual control systems. As servo sys-ems with single inputs, the compensatory system is of extremely simple, iingle feedback form. As a closed loop system, the operator's output ¡nd system error are given in ternis of Fourier transforms and transfer unctions by,



Y,(jw)I(j) +N.(jw)

i + Y,(jw)Y.(jc)



i (ti Os o ?smi FUnCtaon I (j..,) os tho Vevrar trAform of i (t)

I(jca) -


FIG. 4. Linear system represenwx,n.

'ypei of Mat hetnahcal Models Useful in Describi,i Continuous Man-Machin System Be)uizwr

A review of some of the mathematical models which are useful to the ontrols engineer as means of describing physical devices leads directly o the concept of a quasi-linear describing function plus a reliluant as ic tools for describing human dynamic characteristics. This concept lows the retention of the linear methods so successfully eniployed in rvo analytical techniques.

The reninant so dvelolxl is

seen from i to be an element in the system error and output equations. In

any manual control systems, it can be the dominant operator output uantity in the generation of tracking errors.

With physical systems containing only elements which behave in a

iray deseribable by linear constant-coeflicient differential equations, the

(1) - k OUTPUT CHARACTERIZED 5y: 1. r (t) as o hrn. tvnctson Z. R(1& o m. Fourrer ?ren$Por,n of r(t)



appropriate mathematical model is the element's weighting or transfer function. If the system were represented by the block diagram of Fig. 4, where the weighting function, h(T), is the time response of thr system when an impulse function is applied at zero time, then the relationship between the response and the input would be given by the so-called superposition or convolution integral,



f h(r)1(t - r)dr.


If the Fourier transform is used, the transformation of Eq. 2becomes


= H(ja,)I(jw), (3)

where R(jw), I(jw), and H(jw) are the Fourier transforms of r(t), 1(t), and k(r), respectively. Laplace transforms can, of course, be used in a

similar way.

The introduction of the weighting function, or its transform, the transfer function, allows us to describe the performance of a ¡1,ear system element completely. If the convolution integral is not valid, and the behavior of the system is a function of the particular inputs and initial conditions the analyst must define system operation by input-response pairs. 'ortunately many nonlinear systems o( interest have specific input-specific response pairs which appear to be very similar to input-response pairs for linear systems. Thus the performance ofsome nonlinear elements, for particular inputs, can be approximated by a linear element called the describing function plus an additional quantity called the remnant.4

A familiar example of this concept is the sI,susoldal i,iist dscrlbing_ Lisna1on which is derived from consideration of the harmonic response of the nonlinear element to a sinusoidal input at various frequencies and amplitudes. Consider a sine wave applied to the input of a nonlinear element having a single input and output. The output will very likely be a nonsinusoidal periodic wave with the same period as the input

wave. If the output waveform is analyzed interms of its Fourier

com-ponents, the fundamental component will bear a relationship to the input sine wave which can be described ¡n amplitude ratio and phase angle terms.

Thescnbingunçtio_wilI be the ratio of the

funda-mental to the input in the same way as used in a linear system. The temnant will be composed otalt the higher harmonics.

Since the describing function concept is based upon a particular input type, describing functions can be defined for a particular non-linear element simply by considering different types of inputs. The

r-While the notion of moldinga nonh.earity into a. equivalent hopar element is quitt old

and has been used by many writers, the 6rt instance o( major exploitation of the technique



D T.


:nusoidal input describing function already mentioned

is the most

idely known example. \Vher the type of transient input is known rid simply described in analytic form, a transient describing function iay be used; for example, the step input dcscribiiig function. In ises where system input forms may not be simpk steps or sil.ewaves,

more useful describing function may be one based upon statistical

Iputs (io, ii).

Such a describing function, particularly for inputs

iaracterized by Gaussian amplitude distributions, is useful in nonlinear introl problems where the inputs expected in the application of the intro! system are statistical, and can be considered as a basis for a irly general type of equivalent linearization. The Gaussian input de-ribing function will be shown here to be of particular value in

describ-human response.

As a general comment on describing functions, it should be noted at, since the inputs are different, the only thing the various describing ctions have in common is the nonlinear element they intend to ap-ximate. As the elements become less nonlinear, the various describ-functions for a given element tend to approachone another. When nonlinearities are entirely removed, all of the describing functions .me identical to the element's transfer function.

Alternate approaches to the nonlinear problem exist. The most ctical alternate for human response measurements is time domain - lysis by the use of an operator analog comprised of analog computer ments. Using the very wide range of nonlinear and linear analog ments available, a computer setup can be made by cut-and-try

pro-ures and adjusted until the "analog operator" responses to particular uts are similar to those of the actual operator. If an aH-compre-ding analog model were ever achieved, the input type would not be


To achieve a practical and reasonably simple computer p in the case of the human operator, however, a separate analog i uired for different inputs. The analog model technique is of great e in instances where a point by point prediction of operator response esired, for studies including nonlinear control effects, and as a means roviding insight into some of the types of nonlinear behavior which ht occur in the operator. When nonlinear or time-varying transfer racteristics become exceedingly important it is probably the only

tical approach. The analog technique is fairly straightforward, so urther discussion of the general method need be includedhere. nUlles Involved in Human Response Measurements

uman operator characteristics are strong functions of the type of em input, so the describing function concept is necessary ilwe wish

tain and apply linear iucthods of analysisto manual control systems. ost all experimental research to specify operator dynamics in manual rol tasks has been based on this approach. A key factor in the



39 J

determination of the operator's response characteristics is the general predictability of the system input. Specifically, the operator's mode of' response with either a unity controlled clement or a thoroughly learned control element is different for single sine waves (which arc essentially completely predictable once recognized), step functions (which have known fInal values immediately after initiation of the stimulus), and "random appearing functions." The random appearing inputs are not restricted to functions which are known only by their statistics, but include those which can be made up by as few as three nonharnionically related sine waves. In most of the experimental work, the random appearing inputs have been either such a sum of sine waves, or time functions which had essentially Gaussian amplitude distributions. Therefore the types of describing functions of general interest in human response measurement are those for step, sinusoidal, and Gaussian in-puts. All of these describing Linctions fall into a category which can be called quasi-linear because they tend to be linear under a fixed set of conditions, such as given inputs, yet nonlinear when changes in these conditions are considered. Since the statistical forms are usually the

closest to reality in practical control

systems, we shall confine our

attention to transfer characteristics

derived for random appearing

inputs. When random inputs are used, the most common techniques of describing the characteristics of the time stationary inputs and out-puts are by means of correlation functions or power spectral densities. The describing functions are obtained from cross-correlation or cross-spectral density functions, and theremnant and system forcing functions are normally expressed as spectral densities. While most of the under-lying theory on the measurement and interpretation of these quan-tities is beyond our present scope (see, for example, 12 for details), it is necessary to develop some relationship among the fundamental quantities which are used to measure and define the operator's response characteristics.

The various quantities in the compensatory system of Fig. 3 which are important in the determination of operator characteristics àre given as a nomenclature summary, in the accompanying tabulation.

The operator's output and the system error for the compensatory system were previously given in Eq. 1. The cross-spectral density functions 4 and 4 are defined as follows (where the asterisk indicates a conjugate),

= liin[IC]

. I

hrn - [1(jw)C(jw)]



392 D. T. McRu ANt) E. S. KRENDEL [J. F. L Ezpceed as Exprrwd in a Time Ezpi-tsed sa a Spectral Quantity Function Fourier Transform

System forcing function i(i)

J(jw) or I

Error signal of the system; e(i)

E(j) or E

input to the operator

Operator output c(i)

C(jw) or C

System output response r(t)

R(jce) or R

Remnant at the operator's n4(t)

N,(jw) or N4


Operator's weighting and y,(r)

Y,(jw) or Y, describing functions

Controlled element weighting y,(ir)

Y.(jw) or Y,

and transfer functions

Closed loop weighting andde- h(r)

H(j4i) or H

cribing function between forcing function and

opera-tor's output

Cross spectral density of

forc-+14 ing function and operator


Cross spectral density of

sys-416 tern error and operator


Remnant expressedas a 'i.'(t)


or N4'


closed loop quantity 'So that the various time

(unctOnl, which may be storbastic, may be consideredto have

Fourier transforms, it should be understood that th

time (unction used in the tran*form is identical to the actual time function in the intervals - T ( ¡ ( T, and is zero elsewhere. The Fourier transform of a typical signal,fi(t), will then bedefined as

F(jc.i) ff(s)e-1_4o4; where f(s)

fi(S); - T <i < T

f(s) - O elsewhere.

"The power spectral density

of a typical signal f,(1), is given by


The cross spectrum oftwo sigrials,f(t) and gt(l) will be

lrn-F(w)G(y..). «r more detaili. on these quantities see, (or example, 13.


Using Eqs. J and 4, the cross-spectral density ,. is seen to be



+ Y.1Ç1



.... T

T-. T


The quantity, um - [l'I] is recognized as the power spectral densityof T-..


I Jim I

1"Him '[IS!]

4. 12


+ v.v,I




the forcing function, ,.(w). The quantity Hm

[NJ'] is similarly

recognized as the cross spectral density between the forcing function and the remnant,


This cross spectrum is zero, since by definition the remnant has no linear coherence with the forcing function. ThenEq. 6



[I +. Y,YI]


The open ioop describing function Y, can be obtained by dividing Eq. 7 by Eq. 8, and is given by


Equation 9 is fundamental in the measurement of the transfer charac-teristic of stable system elements. It is applicable to either random or periodic time furctions, which must be stationary. If the system is linear and the forcing function a sine wave, the determination of Y, re-duces to a conventional frequency response measurement.

The spectral density of N. is often an important property" of an equivalent linear system. To find this quantity we can forni the oper-ator output power spectral density, noting, as before, that the forcing function remnant cross spectra are zero,

4>,. Hrn

[CCJ = hrn



1' [

:] J



May, 1959.1


, +


Similarly, it can be shown that the croes spectral density between the error and the input is








t is apparent from Eq. 12 that for a linear system (remnant, 4'.,., equal o zero) the value of p would be unity. In general, the value of p is a :ood measure of the importance of the remnant. As will be seen

ater, the linear correlation is not necessarilya good measure of linearity, .lthough a value near unity does imply that the system is "almost"near.

All of the quantities presented above are applicable to most general ystenis where servo analysis methods have a reasonable chance of

uccess. They are essential, of course, in attempting to describe human

peration because of the enormous adaptability of man as a control

le ment.

394 D. T. McRuEk AND E. S. KRENDrL ti I.

In most cases it is easiest to consider the total output power in terms of that portion linearly coherent with the forcing function (the first term bovc) and a remainder or "closed loop"remnant. Since Y,/(1 + Y,}Ç)

LS recognized as a closed loop describing function, JI, and defining

I 1/(l + Y,Y) I! 4'.,.,, as the closed loop remnant power spectrum 4'.,,., we can write Eq. 10 as

= IIIl2. + 4',.,..


The possible sources and points of injection of the actual quantities :ornposing the closed loop remnant will be discussed later. For the resent, the term 4',.. is selected to represent, without regard to origin.

that portion of the operator's output power density which is

not inearly coherent with the forcing function.

Besides the describing function and the remnant, it is desirable to have some alternate means of relating the amount of operator output power "explained" by the action of the quasi-linear describing function to the total output power. We can define such a quantity by forming the ratio of the linearly correlated output power and total output power spectra. This fraction is the square of the linear correlation, p, and is riven by






Fhe linear correlation can also be expressed as








A. Tustin first noted that operators in manual control systems with random appearing forcing functions exhibit a type of behavior directly analogous to that of the equalizing eletrients inserted intoservo systems to improve the over-all performance of the system. By its nature, this equalization appears to be a more or less continuous operation per-formed on data observed by the external senses. Since the operator must adapt his equalizing behavior to cope with controlled elements having widely diverse dynamic characteristics, it is reasonable to classify describing function experiments on the basis of the controlled elements which "mold" the operator's characteristics. The discussion of oper-ator describing functions in compensoper-atory tasks has therefore been

divided into two major parts: the first will

treat past experimental describing function data on compensatory systems with unity gain controlled element dynamics, that is, Y. 1, and the second will con-sider describing fuflctions for operators in systems with more complex controlled element dynamic characteristics. To complete the

charac-terization of the operator, data

on the remnant are also presented, though pertinent data are sparse.

Operoor Describing F,r4chons in Cornper$atory Tasks with No ConlroUed Ek,nenl Dynamics

The experimental data considered for Y.

= i were taken directly

from the efforts of three investigators and their colleagi.es, specifically J. I. Elkind (14, 15), L. Russell (io) and E. S. Krendel (17). The pri-mary experimental variables in these researches were the type of con-troller and the forcing function characteristics.

The details of the

forcing functions used are summarized later inTables II and III. These experimental data have been replotted in Bode diagram form (when the original data were not in this form) and describing functions for the operator have been obtained from

curve fits to these plots.

The describing functions as derived from these curves are the basis for making analytical estimates from which one can determine the influence of the forcing function upon the human operator's dynamic charac-teristics. To the extent that thecurve fitted analytical model

approxi-mates the operator, we have insight into the

laws which actually govern the over-all behavior of the operator.

The most desirable types of linear models are transfer functions made up of ratios of rational polynomials, with a pure time delay term also allowed. Such transfer function forms are favored because they are simple, well understood, and adequate for approximating any of a large n umber of frequency response characteristics. Unfortunately, SÍflCC the experimental data covered a restricted frequency range, with


i---96 D. T. McRuz AND E. S. KRENDFL (J. F. I. )Oor ft lution, it is pussibk to fit functions to the limited number of lata points to an arbitrarily defined degree ofaccuracy. A painstaking Lnd accurate fitting procedure is not warranted because of both the neasureiuent and data reduction errors and the variation between sub-ects whose data were averaged. The appropriate model

to use in

itting a set of data is, therefore, the simplest transfer function form vhich is reasonably consistent with all of

the data trends and is

ap-)ropriate for both amplitude and phase characteristics. The need for x)nsistency between amplitude and phase forces compromises in the stimates of parameters.

As general criteria for the types of transfer function forms acceptable ve list the following which are consistent with all known human operator tata in compensatory tasks of variouskinds.

The operator's describing function must approach zero at infinite 'requency, that is, the transfer characteristics are fundamentally those

a low pass filter.

The describing function must be finite at zero frequency because f physical limitations on the force-displacement capabilities of the perator.

A pure time delay, e', must be included to account for a reaction rne delay.

When the experimental closed loop system is stable, the approxi-ate fitted describing function must define a stable closed loop system. As an example of this fitting procedure, we shall consider the case f Elkind's data for forcing functions with rectangular spectra. (The

al describing functions for this case are summarized in Table III.)' he simplest transfer function form that satisfied the foregoing

require-ents for these data is

Ke "

(T.jci, + 1) (14)

here K is the d-c. gain, r1 a pure time delay, and T. is a time constant presenting a low frequency lag. Since the pure time delay

e-" does

t affect the amplitude characteristics, the amplitude ratio curves are



t,,l -


proximate values for K and T, can be determined from the raw data


$ The original data upon whkh all of our discusahon is based can be found in the original rces in various forms. They are also presented in (1) in Bode diagram form.

Because of

e!K)rmous amount of data involved (Tables II and Ill summarizeover forty separate

ex-mental averages derived (rom over 300 ind&vidua! runs), it is patently ¡rnpoasible to show small variety here.


amplitude ratio plots with the chief criterion the visual appearance of a good fit.

The phase associated with thc term (Tejw ± 1)-' can then be

sub-tracted from the raw data phase to isolate the phase

lag due to e".

On a linear plot of phase versus frequency, this residual phase should be approximately linear with a slope of

- r1.

When the residual phase is not approximately linear, it is necessary to select other values

of T.

(and hence K), and repeat the

process until satisfactory results are attained.

While the low frequency approximation

of Eq. 14 fits much of

Elkind's data quite well, an analytical closed loop transfer function developed from this equation, with numbers derived from the experi-mental data points, would indicate an unstable condition at frequencies well beyond the measurement bandwidth. This was the case for all of Elkind's lower cutoff forcing functions. Since there was no evidence of a closed loop instability in the experimental data, the operator pre-sumably introduces more attenuation at high frequencies than is indi-cated by Eq. 15. Although the experimental data do not define the high frequency portion of an approximate transfer function, it is ap-propriate to "stabilize" the closed loop system by adding a lag to Eq. 14 and modifying tie value of r to r, that is, changing Eq. 14 to,



+ l)(Tijw + 1)


1f l/T is much greater than any of the frequencies measured, then the low frequency behavior of the transfer function of Eq. 16 can be made the same as that of Eq. 14 if T1 + r is approximately equal to r1. This addition of a simple lag, Tjjw

+ 1, is not a unique way to place

a stability bound upon the possible higher frequency behaviors; any combination of lags and leads leading to a stable system would be suit-able if it satisfied the relationships,


where the T. are lags, the Ta are leads and w.. is the.forcing function cutoff frequency (or an approximation thereto) in rad/second. The single lag represented by T1 is, however, the simplest form which can be added to Eq. 14 to assure stability and consistency with the measured low frequency response. Even with this procedure it is unfortunately not possible tu find unique values of r and T1 from the low frequency ex-perimental data.

The results of processing the various experimental data for systems with unity controlled element dynamics in the fashion described above, are shown in Tables 11 and III.' These tables delineate the

experi-The symbol $ Las been used in aU s4immary tabks fojw.

i i

»w,. )>w,.

T1 (17)



Í) '1


k A\I i k'tslH [J F T

ental cudit ions. the fït'qUeflLy ranges of nleasurcnit'nt, the average near correlations, and the appro\imate describing function fits to the


ï'abk I

is also included to provide a summary of operator

re-IAlti : l.--Sitntmary of Opt'rahir t)es.ibin Fundtons in Compessatory Tasks.

.Sitipfr Trakcr, Y. I, Srnaple Fundws.

G.ntral Contro' Ti.k

imple Following with pencil,

sticI, or wheel (forctng

function a)

imple following with pencil or wheel (forcing func-tion b)

a. Step functions

'lkw F,t' Ilunn Oprr*to lnvr,tìgtoa

Trnfr Function ftndRrmaki

Closed ioop tr4nsfer functioit


73 2Tss N

K '6.8- 7.8 sec'. Cheatham

r 0.25 c 1-0ß42 sec T, 1/K Mayrse

Ellson & Hill

"NoMs ywchresnoio" response. closed ioop



with constants as above Sywchronous response,

closed ioop K (o.)

Mmple following with pencil, Same as for Goodyear (19, 35) stick, or wheel (Forcing single steps Mayor

function r) Though rcan approach 0.20 sec as small range effect; steps get closer together and Seark & Taylor

appear more random Elison & Wheeler


Types oI forcing functions:



b. Sine waves and square


Goodyear (35)


Searte & Taylor

Goodyear (19,35)

Mayne, Filson & Gray, Cheatharn

c. Sequence of steps

;ponse to simple forcing functions. These are not considered in detail here, but have &me general interest and are also of some value in ob-taining our hypothetical human behavior models.

Tables Il and III shows a wide variety of forcing function conditions

represented by their power spectral densities, 4. The broadest

roverage is due to Elkind (Table III), who had remarkable success establishing relationships between the fitted describing function



White noise through

third order

bi-al (jUez' giving available



nom SIc. ot I, Z, and 4 rad/sec,

e. Superposition of four sinusoids, rad ¡sec




Operator Describsa

Fwsclio.j lis Compessoiory Tash:,

SsmpSe Trahe,, Y, - I, Raisdom Appearint

Forciag Fiisclioiss.

Gener*1 Conteol luk


lt,' human Op.r.tor

Trunder Fuuictlon

Frquenry ane d Humai Operaioe Me.aur,m.ni,

Average I.ivi',.r Ciwrtlatton Invt*tigatoi'i and Remark.

Simple tracker with apnng



to 4.0 rad/sec.

0.7 to 0.8

Franklin(17, 1), these data

restrained aircraft control stick in aircraft cockpit mockup (forcing func- tion d)

(Ti: + 1) (T: + 1)

Corner Freq. 1/Ti

1/TN I/Ti,



taken for 'lateral'



con-trol was exercised simul- taneously

1 .04 1.5 0.5 0.25 lOO 2 .11 4.55 2.0 0.20 40 4 0.2 11.0 3.0 0.25 1.5

Simple tracker handwheel


Same frequencies as shown


Ru sac Il

type control with no re- straints (forcing Íun- tion e)

+ I) (TN: + 1) "0.3 TN effect included in e in forcing function Forcing Function 2/Ti lIT,, K "4 -t'-a 'I. Low speed .103 Medium speed .37 High speed .62 13.3 5.0 3.7 52.5 11.0 20

Simple tracker aircraft type handwh,el control, with pring restraint, operated in pitch (push-pull)

di-rection (forcing


0.5 to 3 rad/sec,

L-10; 0.91 to 0.6


Hall, lateral axis airplane dynamic.



controlled...erage de- creased with írequeiic

(T,: + 1) r '0.2



Controlled Element Gain lIT,

I/Ti, K 0.88 tu 0.45 5 .027 2.7 10 '/ -.4 10 .027 2.7 7

Type. of forcing functions





(Loe) .66 2.17 (HIgh) /og .35 6.0.1 ÌtMtdI.a) .741 ISO (Low) 1,61 4.77 (Med.) 3,34 11.91 (l'tIgli) f. White noise through thiri roder

iinmial filter, cur iier it




eters and other parameters deflning the forcing function characteristics. These are also given in Table Ill. Some work has been donc (i) in an endeavor to find underlying criteria which may lead to these relation-ships. While these attempts were not completely successful, one can show that the conventional servo synthesis method of parameter ad-justment to minimize the rius error (13) leads to analogous results.

ii. llI.-Siim,nary of Operator Descrìbag Fu,,ctions iii Compeiisatory Tasks Showing Ejects nf Forcisg Fs*ttoe,. Stmpk Tracker, F. i, Random Appearing Forcing Fuaejion. These Dote

Taken front Eikind, in a Contro! Task Consisting of Staif4e Fo!krwu.g with a Pip Trapper.

Typ, . PorUng Pnwtho.

inction made up of 40-120 sinus-ods, giving a.ny

desirabk rectan-gula: spectra 1" ruu amplitude

(we g below)

)proi mate white noiw through Ist.

2nd, and 3rd order banoanial íi1te sa thown below. Highest fre-quency in forcing functions was 2.18 cpa. 1" rina implitude (we k bow) where:


"bern Fit' Human Operator Trsser Functoa

K'-'. (T,s + 1) (Tres + 1)


K-22/1..'; K9.42T, A chez! fitted data:





AcftioJ fitted data: Forcing


Frrqueucy Range at Humas Operator MeaauremeiiU

Highest frequency equal to forcing function bandwidth, lowest fre-quencies given below in rad/sec. [J. F. 1. Average LAmai Corrrfauoe. p. R. 16-.995 R.24-.99 R 40-.995 R.64-.98 R.96-.92 Rl .6-75 R14-.58 -I Forc-ing Func- -R. 16-.lS R. 24-. 15 R.40-.25 tion K(db) r 1/Ti 1/TN R.64-.25 R. 96-. 23 R.16 34.5 .110 .22 1.883 R1.6-.3 R-24 31.3 .104 .314 6.22 R2.4-.3 R.40 22.5 .133 .785 12.3 R-64 13.0 .150 1.73 30.3 R.96 6.5 .139 3.65 .. RI.6 -0.6 .122 3.77 R2.4 -3.0 .116 1,185


,rn as or Range Filter p. K 'i. 9.42T, I' ,f Filter

let order (rd/sec).25 to 18 2ridoi-dcrlit order 090.96 2ndorder .25to 12 3rd order 0.97


3rd order .25 to 6

j df

and Filter K(db) 1/Te f

FI; ist order 10.5 1.13 .13Q

F2; 2ndorder 25.0 0.314 .126


0i May ¡959 1


f..-16,.24,.40,.64. .96, 1.6, 2.4 cpa é 1, 1.5, 2.5. 4. 6, 10, IS rad/sec.


TAKLE III.-Coaue4. Foec-¡ng Func-K(db) tion e-(To be aonh*sed)

M -

- -u Typr ol Fc1ug

Funtioi, 'ri Fit Hunw, Opr..toeTranct Fiinct on Prrn'.nc k.,ng 4 IIUN'.an

Oerator Aetsr LlnciiCorrdaiion.p..

24 to 144 siiivaoidal

componenta giv- tui + 1) ( TNS + I)

ing forcing func-tion spectra shown (No. 01

corn poneii ta show u with

pu1-J where Y(s) was translated to ' the origin (or B3, 88, B9, 1110

where: Rl 112 RS 84 85 .6to6rad/sec. .6to3and6.3to9 J to9 .6to8.7 .6to9 BI 82 83 84 B5 .97 .91 .92 .90 .92 tkular spectra in parentheses.) I"

ruts (see i below) Kè9.42T,; 86Bi

8.8 .6to9 .6to3 3to6 86 87 88 .90 .99 .97

Acft..rilfi2led dosa: B9BIO 6to9 89 .91

9to12 810 .70 l/T, 1/TN t Bl 9.0 4.78 .133 82 1.5 5.03 .107 8.3 -1.0 12.6 .278 84 -03 12.6 .150 B5 11.1 1.88 .128 8.6 17.7 1.00 .149 87 23.2 .88 17.8 .100 88 9.0 3.14 6.28 .219 89 0.4 12.6 .390 810 -11.8 2.82 1.14



D. T. McRcEE AND E. S. KRENDEL, "Dynamic Response of Human Operators," WADC TR 56-524, August, 1957.

"Handbook of Human Engineering Data," Tufts College Institute for Applied

Experi-mental Psychology, Human Engineering Report SDC 119-I (Nay Exot, P-643), second edition, December, 1951.

R. G. E. EPI"LE, "The Human Pilot," Northrop Aircraft, Inc., BuAer Report AE-61-4111, August, 1954.

J. W. Salloras AND L. J. FoGEL, "The Human as a Pua! Mode Relay Servomechanism."

Man-Machine Servo Conference, Franklin Institute, Feb. 14-15, 1955.

(S) H. L. Pt.ATZER AND E. S. KRENDEL, "Nonlinear Techniques for Studying Human Operator

Performance," Man-Machine Servo Conference, Franklin Institute, Feb. 14- 15, 1955.

H. L. PLATZER, "The Phase-Plane as a Tool for the Study of Human Behavior in Tracking Problems," WADC TR 55-444, November, 1955.

H. L. PLATZER, "A Nonlinear Approach to Human Tracking," Franklin Institute Report I-2490--1, Contract Nay 1571(00), December, 1955.

J. R. MOORE, "Combination Open-Cycle, Closed-Cycle System," Proc. IRE, Vol. 39,

pp. 1421-1432 (1951).

N. KPYL0FF AND N. BOGOUUIIOFF, "Introduction to Nonlinear Mechanics," translated by S. Lepschetx, Princeton University Press, 1943.

R. C. Bo0T0N, Je., "The Analysis of Nnlìncar Control Systems sith Random Inputs,"

Proc. Symp. Nonlingar Circuit Analysis, Brooklyn Polytechnic Institute, April, 1953. R. C. B.00toN, Je., "Nonlinear Control Systems with Random Inputs." Trans. IRE

PGCT, Vol. CT-I. pp. 9-17 (1Q54).

H. PRESS AND J. W. Tuaay, "Power Spectral Methods of Analysis and Their Application to Problems in Airplane Dynamics." Part IV-C, Vol. IV, AGARE) Flight Test Manual. pp. 1-41, Juhe 1956 (reprinted as Bell System Monograph No. 2606).

H. M. JaMas, N. B. NIcHoLS AND R. S. PHILLIPS, "Theory of Servomechanisms," New York, McGraw-Hill Book Co., Inc., 1947.

J. I. ELEIND, "Tracking Response Characteristics of the Human Operator," Human

Factors Operation Research Laboratories Report HFORL Memo No. 40, PRP.2, September, 1953.

J. 1. ELK!t'w, "Characteristics of Simple Manual Control Systems," Technical Report

No. 111, MIT Lincoln Laboratory, April 6, 1956.

LINDSAY RL5SEI.L, "Characteristics of the Human as a Linear Servo-Element," M. S. thesis, MIT, May 18, 1951.

E. S. KIENDEL airo G. H. B.se'sts, "Interim Report on Human Frequency Response Studies," WADC TR 54-370, June, 1954.

A. TustIs, "The Nature of the Operator's Response in Manual Control and Its

Implica-tions for Controller Design," J. fasi. Ei«. E*grs., Vol.94, Part Il A, pp. 190-202 (1947).

"Final Report Human Dynamic Study." Goodyear Aircraft Corp., Report No.

GER-4750, April 8, 1952.

"Investigation of Vestibular and Body Reactions to the Dynamic Response of the Human Operator," Final Report, BuAer Report AE-61--6 and Goodyear Aircraft Corp., Report GER 5452, November 25, 1953.

"Investigation of Control 'FetI' Effects on the Dynamics of a Piloted Aircraft System,"

Goodyear Aircraft Corp. Report GER 6726. April 25, 1055.

N. D. DI.AMANTIDES AND A. J. CAcIorI'o, "Human Response Dynamics; Geda Computer

Application," Goodyear Aircraft Corp. Report GEk 8033, January 8, 1957.

I. A. M. HALL, "Effect of Controlled Element on the Ifurtian Pilot," %VADC TR 57- 509, October, 1957.

t EDITOR'S Nota: The majority of the above reports are available to qualified users of .ASTIA. Arlington Hall Station, Arlington 12, Va.



(24) H. P. BtRMINGHAM ANt)F. V. TALOt, 'A Ilunian Engineering Approach to the [)esign of Man-Operated Continuous Coiitrol Systems," NRL Report 43333, April 7, 1954.

4.. (25) H. P. BIaMINGIIAM *.s,n F. V. TAYLOS, "A I)esign Philosophy for Man-Machine Control Systems," Proc. ¡RE, Vol. 42, pp. 174*-1758 (1954).

H. P. BISMINGstAM, A. KAFIN AND F. V. TAYLOR, "A Demonstration of the Effects of Quickening in Multiple.Coordinate Control Tasks," NR!. Report 4380,June 23, 1954.

J. G. HOLLAND AND J. B. I1ENSON,"Transfer of Training Between Quickened and

Un-quickened Tracking Systems,' NRL Report 4103, February 3, 1956.

W. D. GARVEY AND L. L. MITNICR, "An Analysis of Tracking Rehavior in Ternis of

Lead-Lag Errors." NRL Report 4707, February 16, 1956.

L. V. SF.asLE AND F. V. TAYLOR, "Studies of Tracking Behavior: I. Rate and Time Characteristics of Simple Corrective Movements," J. ErMI. Psych., Vol. 38, pp. 613-63! (1948).

D. G. ELLSON AND H. HILL, "The Interaction of Response to Step Function Stimuli:

I. Opposed Steps of Constant Amplitude," AMC Memo Report No. MCREXD-694-2P, November 19, 1948.

R. MAYNE,"Some Engineering Aspoets of the Mechanism of Body Control," Ejec. Esg.,

Vol. 70, pp. 207-212 (1931).

D. C. CREATHAM, "A Study of the Characteristics of Human Pilot Control Response to

Sünulated Aircraft Lateral Motions,' NACA Report 1197,1954.

D. G. Eu.aoN AND L. WHEELER, "The Rangt Effect," Air Force Technical Report 5813, May, 1949.

C. W. Si..acE, "Some Characteristics of the 'Range Effect'," J. Expel. Psych., Vol. 46,

pp. 76-.O (1953).

"A Proposa! to Study the Dynamic Characteristics of Man as Related to the Man-Aircraft System," Goodyear Man-Aircraft Corp. Report GER-3006-A, May 2, 1950.

D. G. Eu.soN AND F. GRAY, "Frequency Responsesof Human Operators Following a

Sine Wave Input," Memo Report MCREXD-694--2N, !JSAF AMC. 1948.







Operator Describing Functions in Compensatory Tasks with Controlled Element Dynamics

The next step in a logical development after exhausting the Y i case would be to delineate the effects upon the operator describing functions due to increasingly complex controlled element dynamics, together with a full coverage of forcing function conditions. Unfor-tunately a complete experimental basis for such a presentation does not yetexist.

The primary experimental work with controlled elements has been done by Tustin (is),' Russell (ió), Goodyear Aircraft (19-22), Hall (23), and The Franklin Institute (i, 17).

Tustin's study was a pioneering effort, but was quite short and

limited in scope. While three different controlled elements were used, they had very similar dynamics in the frequency range of interest. He introduced the remnant concept, presented some information on it,

but left the total remnant picture relatively incomplete.

Russell's study, which is a logical extension of some of Tust in's notions, considered a progression of increasingly complex controlled element characteristics, with the primary emphasis of the experimental work being placedupon the operator's describing function. Remnant characteristics, in a spec-tral form, were not thoroughly explored but their effects were revealed to some extent in some cases by total remnant power and error power data. The Franklin Institute F-80 Simulator studies are completely documented in ternis of both describing function and remnant charac-teristics. In these tests the controlled element dynamics were those of

an F-80 aircraft and target in a constant range tracking taskby far

the most complex situation studied in detail to date. Finally, Hall's data fill in many oí the gaps in controlled element characteristic types,

This research was supported in whole or in part by the United States Air Force under Contracts AF 33(616)-3080, AF 33(038) 10420, AF J3(616)-2804, AF 33(616)-3610, and AF 3.3(616)-5822, monitored jointly by the Flight Control Laboratory and the Aero Medical Laboratory. Wright Air Development Center. Wright-Patterson Air Force Base, Ohio.

Systems Technology, Inc., Ingkwood, Calif.

'The Franklin Institute Laboratories for Research and Development, Philadelphia, Pa.

' Part I appeared in thisJOUINAL (or May, 1959.

'The boldface numbers in parentheses refer to the re(erenes apja'utded to Part I o( this



CneLr-oiied Eltinent. Human Operator Deicrab.rrg

1.0) FuUo, 1,(,)

and includu a survey for conventional short period airframe charac-teristics. (These latter are not included in the present paper.) His data, being the latest gathered chronologically, include several system parameters not measured previously as well as pilot opinion ratings for

the various controlled elements.

The results of fitting these data using criteria similar to those dc-scribed previously together with other pertinent information, are shown in Tables IV and V. The controlled element transfer functions are

TALe IV.Summary of Operator Describing Frsna:ons in Conpensaiory Tasks with Various Sim pie Cowirolird EIeme,rts, Random Appearuig Forcing Furict,ow.

¡ni-,,tigator. Forctng Function. ControL Task. sad Pamarka Russell: Superposition of four sinusoids, w...66, 1.68, 2.87,

and 4.27 rad/sec. Amplitude

enve)ope ' ist order binomial

filter with corner at LS rad/nec; Handwheel control with no


Hafl; White noise through 3rd

order binomial hiter with corner

at i rad/nec. Aireraf t

hand-wheel control with spring

re-straint operated (push-pull) for simulated aircraft longitudinal control, lateral axis Navion

air-c-raf t d, ria mica were controlled simultaneously. Range of rneas ure-ments 5 rad/nec to 3 rad/.ec

Average p decreases from .9 to .4

with increasing frequency.

Russell; Superposition of four

sinusoids, u,.66. 1.68, 2.87,

end 4.27 rad/sec. Amplitude

envelope Ist order binomial

uiltev with coi $l(r at .85 radIare;

IJandwhc-el control nith no

restraint s

4Tbe measured points (rom which the describing function esperiniental dat.i iii Tables IV and V were obtained are in (1).


June, i9S).l huMAs Ori.it.ToI<s 1c Stivo SYSTI:MS 513 TABLE LVCo,,lissed.

ControAki Element. Iltiman OperatorI)txiibtng lase,elgator. Forcing Function.

Functioti. Y,(s) Control Taak. ann Rrmarki

s s s Is s (s +1) z G±


s + 10 lOs + lo 20s + I

Hall; Wbite no.e through 3rd

order tjnoqiiial Idier with corner at I rad/sec. Aircraft

hand-wheel control with spnitg

re-straint operated (push-pull) for simulated aircraft longitudinal control, lateral axis Navion air-craft dynamics were cotitrolled

simultaneously. Range of

meas-urements .5 rad/sec to 3 rad/sec.

Average p decreases from .85 to .43 with increaalng frequency.

Russell Superposition of four

sinusoids, ,277, .741. 1.21,

and 1.90 rid/sec. Amplitude

envelope ' ist order binomial

filter with corner at .36 rad/sec; Handwheel control with no



Avei-at, - .9

çorrect over a wide frequency band, but the human operator describing functions are not, since either several particular frequencies or a narrow band of frequencies were used in their determination. To present the measurements in a more precise Context, Tables IV and V also show the controHed elcnìentpproximate characteristics in the frequency region covered by the human response data. The Goodyear results shown in Table V were actually obtained as an analog computer setup. These hiv' lw's'ii converted into Giissin ¡nntit dsirihing fiinrtin form tnkrs comparable with the data from othersources.

Tables I through V give a completesummary of the describing func-tion results in mathematical ternis. To impart a better appreciationof the effects of controlled clement charactcristic upon the operator, we


-.3--t-- --- ...,.n rSIT j. ,. Is.Kr.ur.I.

i Alti V - Sunivary of Operuor flec rsb:sg Figsagoiisiii Corn pt'zsaLorv 7a.sks wil' &'arsoas Cern/dez Conlro/l,dFJe,,:'nts,Random .lppcarii.gForcsisg frsaw*i.

Contofled Flernent. L. I (;9' + i



200e u. + i) i \ 2(.25)s +11




<+ I)[()i+2(45

+ (T.s + I) [VTJ

Itr.an Ojrr.tor Deicr*biii.i Investigator. Foictng l"unctlon.

Functson, ),(j) Control TaaS. and Remarki

se + i)


( + i)

Russell; Superposition of four

sinusoids, w. .66. 1.68, 2.87, and 4.27 rad/sec. Amp1tude

envelope I st order binomial

filter with corner at .85 rad/sec; Handwheel control with no

restraints. AveragePi.7S

Tustin Superposition of three

sinusoids, w, .113, .34, .68

rad/sec for slow input. w,-.17,

.49, .98 rad/sec for fast input. SimuLated tank turret tracking with spade grip handwheel;

Ye(s) I/s over the frequency

range of measurements.

Aver-age p -.95. The indicated 'anafec function does not agree exactly with that given by Tusun.


Tustin; same as foregoing but

no p estimate was made.



Goodyear (19);Superpoaitiono(four wave shapes such that the

( s)

forcing function was roughly

- equivalent to white


through s 3rd order binomial ñltei with corner at 2 rad/sec. Longitudinal control of simu-lated aircraft, stick withspring

mtrainta. Describing (unction

is an approximation to an analog computer set up.

Goodyear (20); White noise

through Ist order binomial filter

with corner at .33 rad/sec and

then through another Ist order binomial filter withcorner

fre-quency at . 4 rad/sec. Longi-tudinal control of simulated

aircraft, moving base pitching simulator, stick with inertial spring and damping restraints.

Describing function form isan

approximation froni an anak


Controlird FIrmn. "(4 L.


+II TAatY V- Contiaurd

human Oi.'rat0 Dmacr,bng Function. Y,(s)


w,.1 rad/.ec


...-2 rad/sec


_-4 rad/sec + i) '.6;


...-1rad/sec '.6;


i...-4 rad/sec

an equatixing form fit; w..-4 rad/sec a'as not ackìeved.

Ieat.gator. FQICIOC Fuj.ction.

Control Taak. and I&,rnj.rkm

Franklin Institute (1); White

noise through 3rd ordcr

Lii-noniat ftlter giving 3 available

corner frequencies, w,. I, 2 and

4 rad/sec; Lateral control of simulatedaircraft in tail chase,

stkk with spring restraints.

Longitudinal axis was under simultaneous control. Ranre

of measurements .6 to 3.6

rad/sec. Averagep .5.

Frznklin lnstitut.e (I); Same

forcing function and range of measurements as above, differ-ent stick spring loading. Longi-tudinal control of simulated

airaaft in tail chase. Lateral


Average p".6, A sgn (unction waa an equallygood fit tothat indicated for w..l and 2

rad/sec, and Y, .6e" was "best" fit (or ..-4rad/sec.

can make sorne general

comments and observations on some of the data for the three simplest

controlled element configurations.

Effect of Gain Changes in Cotarcijed

Element; Y, K,,. For a

given forcing function the effect of gain changes in the display or the control, either during a run or from run to run, is almost entirely corn-pertsated for by the operator, who tends to set his gain insuch a way as to hold the over-all loop gain at some given value.

Slight changes in over-all ioop gain can be effected by changes in the display gain, but nowhere near the amount that would octur had the operator

held his gain constant. An increase in loop gain

can be the result of instruc-tions to "put more into it" or "try harder."

Effects of Insertion of a Simple Thg; Y. = 1/Ts + 1).

\Vhen simple lags are placed into the control loop the operator's

transfer characteristics change to a marked degree.

In all cases the major

June, 1959j

I-1U l N S,ivo SYSTN



effects were the operator's development of a lead term in a fashion tending to compensate for the introduced lag, a reduction in gain, and probably a change in his first lag time constant.

The following should also be noted:

\\rhefl the lag time constant was smaller than about 0.05 second, the inserted lag had no effect on the describing function. In addition, the operator could scarcely notice the effect of the inserted lag on his stimulus.

As the controlled element time constant was increased above 'OS the operator could detect a distinct sluggishness in thesystem. This became more and more prominent until values of controlled element lag time constant, T, of 2 seconds or larger were used, when the controlled element appeared to the operator

to take on the

characteristics of a pure integrator.

When lags are initially inserted into the loop the operator tends to overshoot somewhat, occasionally to the point of system instability, until he acquires some practice. After practice for a minute or so, he adapts his characteristics to suitable values for stability.

The reduction in loop gain when the lag is inserted is larger than required to maintain stability.

3. Effect of Insertion of Pure Integration; Y.

= K,/s.

With a con trolled element consisting of a simple pure integrator the operator lowers his gain and introduces a lead which generally tendsto compen-sate for the e. This change allows the over-all system to be stable, but is also in the direction to increase the mean square tracking error over that with no controlLed element dynamics. The closed loop low frequency response is improved because of the higher over-all system amplitude ratio at low frequencies due to the s' term. With a change in K,, the operator modifies his gain to keep the crossover frequency

(or open loop gain) substantially constant.

Operator Remnant Dala

By definition, the remnant is that portion of the operator's output which is not linearly coherent with the forcing function, that is, which cannot be "explained" as the result of a linear operation on the system input. Since most of the available data were taken for random appear-, ing forcing functions, it is convenient to express the reniflant in terms of a power spectral density, 4g.,. In a given situation, components of

... could result from the following sources:

1. Operator responses to inputs other than the supposed system forcing (unction. These responses could exist in two categories,


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