(McRuer) Human Pilot Dynamics in Compensatory Systems

8/12/2019 (McRuer) Human Pilot Dynamics in Compensatory Systems

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SECURITYMARKINGThe classified or limited status of this report appliesto each page, unless otherwise marked,Separale page printouts MUST be marked accordingly.

THIS DOCUMENT CONTAINS INFORMATION AFFECTING THE NATIONAL DEFENSE OFTHE UNITED STATES WITHIN THE NBANING OF THE ESPIONAGE LAWS, TITLE 18,U.S.C., SECTIONS 793 AND 794. THE TRANSMISSION OR THE REVELATION OFITS CONTENTS IN ANY MANNER TO AN UNAUTHORIZED PERSON IS PROHIBITED BYLAW.

NOTICE: When government or other drawings, specifications or otherdata are used for any purpose other than in connection with a defi-nitely related government procurement operation, the U. S Governmentthereby incurs no responsibility, nor any obligation whatsoever; andthe fact that the Government may have formulated, furnished, or in anyway supplied the said drawings, specifications, or other data is notto be regarded by implication or otherwise as In any manner licensingthe holder or any other person or corporation, or conveyi-g any rightsor permission to manufacture, use or sell any patented invention thatmay in any way be related thereto.

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AFFDL-TR-65-15

HUMAN PILOT DYNAMICS IN COMPENSATORY SYSTEMSTHEORY, MODELS, AND EXPERIMENTS WITH CONTROLLED ELEMENT

AND FORCING FUNCTION VARIATIONS

DUANE McRUER ,DUNSTAN GRAHAML ' SYSTEMS TECHNOLOGY, INC,

EZRA KRENDEL A, - . WILLIAM REISENER, 1R. 4THE FRANKLIN INSTITUTE

TECHNICAL REPORT Nr, AFFDL-TR-65-15

JULY 1965

AIR FORCE FLIGHT DYNAMICS LABORATORYRESEARCH AND TECHNOLOGY DIVISION k:. AIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OHIO

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'IAFFDL-TR-65-15

I,HUMAN PILOT DYNAMICS IN COMPENSATORY SYSTEMS

THEORY, MODELS, AND EXPERIMENTS WITH CONTROLLED ELEMENTAND FORCING FUNCTION VARIATIONS

DUANE MoRUERDUNSTAN GRAHAM

SYSTEMS TECHNOLOGY, INC.

EZRA KRENDELWILLIAM REISENER, ]it.

THE FRANKLIN INSTITUTE

"A,


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IThe description of human pilot dynamic characteristics inmathematical terms compatible with flight control engineeringpractice is an essential prerequisite to the analytical treat-ment of manual vehicular control systems. The enormouslyadaptive nature of the human pilot makes such a descriptionexceedingly difficult to obtain, although a quasi-linear modelwith parameters which vary with the system task variables hasbeen successfully applied to many flight situations. Theprimary purposes of the experimental series reported are thevalidation of an existing quasi-linear pilot model, and the

extension of this model in accuracy and detail.To this end the influences of controlled-element dynamicsand system forcing function characteristics on the pilot'sAdynamic characteristics are investigated using a five-stageprocess: (I) Pre-experiment analyses are conducted using theexisting model to predict the outcome of experiments which areespecially contrived to exercise the model to its limits;(2) controlled-element dynamics which are both crucial task

variables (per the pre-experiment analyses) and idealizationsof aircraft, booster, and space vehicle dynamics are delineated;(3) describing function and remnant measurements are taken inan extensive experimental program involving eight differentcontrolled element forms and three forcing functions;(.) analytical abstractions of the data are made by Purve-fitting procedures; (5) variations in the pilot's chd cter-istics due to controlled element and/or forcing functionchanges are described in terms of the parameters of the curvefits. The outco're of this process is a substantially refinedand extended adaptive and optimalizing model of human pilotdynamic characteristics. Models corresponding to three levelsof precision and complexity are developed, the several aspectsof pilots' adaptation to controlled element and forcing func-tion changes are detailed, the selective variability nature ofhuman pilot dynamics is presented, key remnant sources are dis-covered, and many other aspects of human pilot dynamics aretreated with a combined experimental-analytical approach.

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FOIMWThe investigations reported here are an element in a U. S. Air Force

research nrokrm to eamonre And n1nit +,hf +.hnAn voht,.hie1t nv.m'io.handling qualities are, to a large extent, dependent on the action ofthe pilot as a control element in the pilot-vehicle closed-loop systeml.In this concept, control analynis techniques and pilot dynamic responsemodels are used in the study and optimization of man-vehicle systems.Such procedures promise to g.eatly enhance the processes involved in thedesign of manned vehicles.

This report documents an analytical and experimental investigationof human pilot dynamics accomplished under Contract AF 33(616)-7501,Project No. 8219, Task No. 821905, sponsored by the Flight Control Divi-sion of the Air Force Flight Dynamics Laboratory. The researcn was per-formed by Systems Technology, Inc., at both its Hawthorne, California,and Princeton, New Jersey, offices, and, under subcontract, by Th eFranklin Institute Laboratories for Research and Development, Philadelphia,Pennsylvania. The project principal investigators were D. T. MoRuer andD. Graham, of STI, and E. S. Krendel, of FIL. The Flight Control Divisionproject engineer through most of the program was R. J. Wasicko, succeededin the last phases by R. H. Smith and P. N. Pietrzak.

As in most team efforts, many have contributed to the results reportedhere. The major contributors, who all participated in the analytical,planning, experimental execution, data interpretation, and reportingphases, are listed as authors. An indispensable portion of the program,the design and development of analysis apparatus, was accomplished meticu-lously by R. A. Pcters and K. A. Ferrick of STI. Important contributions,'were also made by R. J. Wasicko in experimental planning, and by R. E.Magdaleno of STI in the interpretation of the results. Special mentionand thanks are due: R. P. Harper, Jr., of Cornell Aeronautical Laboratory;G. E. Cooper, of Ames Research Center, NASA; and Lt. Comdrs. M. Johnson andT. Kastner, Capt. B. Baker, and Lts. G. Augustine, F. Hoerner, and J. Tibbs,of the Naval Air Test Center, Patuxent River, Maryland, for their interestand assistance as subjects. The authors would also like to thank theirco-workers Diana Fackenthal, S. H. Greene, and M. M. Solow, of FIL, fortheir contributions to running the experiments, reducing the data andassisting in its analysis; and H. B. Grudberg, A. V. Phatak, and D. B.McElwain, of STI, for their assistance in pre-experiment predictions,and/or postexperiment data analysis and interpretation. Acknowledgment isdue to BolL Beranek and Newman, Inc., of Cambridge, Massachusetts, for theamplitude distribution data processing and some goodness of fit analyses.Finally, the report has been substantially improved by the incorporationof many suggestions due to the extremely careful review by R. 0. Anderson,P. E. Pietrzak, and R. J. Woodcock of FDCC.

This technical report has been roviewed and In. appr.-

W. A. SLOANJColonel, USA LChioer, Ylighit Control DivisionVF Flight Dynamais Laboratory


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am.

PageI. IV DUCTI N . . . . . . . . . . . . . . 1

A. Project Background and Purpose ...... 1B. Outline of the Report .. ........... ..... 4C. A Guide for the Reader .... .... ........... 6

II. PILOT MODEL SUM4ARY AND EXPERII4MAL OBJECTIVES . . . 7A. The Analytical-Verbal Model ..... ... ......... 7

1. General Nature of the Model .... ... ........ 72. Review of Existing Quasi-Linear System DescribingFunction Plus Remnant Model for Single-Loop Tasks. 103. Rationale for the Existing Model ..... ...... 23

B. Objects of the Experiments .... .......... 291. Model Validation ...... ............ 292. Model Extension .......... ............ 30

III. PRE-EXPERIMENT ANALYSES ...... ............ 35A. Equalization Adjustment .......... ........... 35

1. Y a Kc.............. .............. 362. Yc = I /jw........... .............. 413. Y0 - Kc/(jo)2 . . . . . . . . . . . . . . . . 42

B. Equalization Adjustment for Conditionally StableSystems .......... ................ 471. Yc , Kc/(jw - 1/T) .......... .......... 482. Yc - Kc/jW(jo I/T) ......... .......... 51

C. Performance Measures and Minimization Adjustments . . 55IV. MEASUR0E4NT CONDITIONS, FORCING FUNCTIONS, AND TECHNIQUES

FOR EQUIPMENT USAGE ......... ............ .59A. Physical Layout and Equipment...... ......... 65B. Description of the Forcing Function ............ 67

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PageC. Measurement Situation and Data PresentationxConventions ........... ............... 76D. Measurement Techniques ... ......... .... 79

V. EXPERIMENTAL DESCRIBING FUNCTION DATA .. ..... . 87A. Experimental Plan ............. 87B. Pure Gain Controlled Element and Input Amplitude 92C. Variability of Describing Function Data ..... ..... 91

I. Intrapilot Variability--Run-to-Run andDifferential Kc Effects ... ......... 96

2. Intersubject Variability ... ......... 100D. Grand-Average Describing Functions and StatisticalAnalysis ......... ................ 102

1. Describing Function Averages ... ........ 1022. Statistical Comparisons of the Data ...... 110

VI. EXPERIMENTAL REMNANT DATA .... .......... .. 119A. General .......... ................ 119B. Relative Remnant Data, pa .... .......... 120C. Remnant Power Spectral Densities and Correlation

Coefficients ......... .............. 122I. Effect of Kc Variation... ........... .... 1282. Effect of Forcing Function Bandwidth ..... .123. Effect of Controlled Element Variation on Remnant. 1.51

D. Amplitude Distributions of Signals in the Loop . . . 131E. Output Power Spectral Densities . ........ 136

VII. INTERPRETATION AND ANALYTICAL APPROXIMATIONS FORDESCRIBING FUNCTION AND PERFORMANCE DATA ... ...... 4i1A. Introduction ............. .............. I'l1B. The Crossover Model ...... ............ 14'.C. The Extended Crossover Model .... ........ I'.

I. Extended .CossoverModels for Yc = Kc/Jwl,Ke/(Jo- 2), and Kc/(ja)) 2 . . . . . . . . .. . . .1,2

2. Extended Crossover Models for Yc m Kc/JWOU(J - 1/T) 162vi


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D. rrecision model . 164E. wc Regression, Performance Measures, and Te

Trajectories . . . . . . . . . . . . . 171. a, Regression . . . . . . . . . . . . 1732. Performance Measures .......... 1763. Te Trajectories and Phase Margin Adjustment 179

VIII. GENERAL SUMMARY ANM CONCLUSIONS ......... 183A. The Data in General .......... ............ 183B. Status of the Existing (Circa 1960) Analytical-VerbalDescribing Function M4odel........ .......... 185C. Extensions to the Anulytieal-Verbal Desc ribingFunction Model ........... ... ........... 1 86D. Status of Remnant Data ...... ........... 188

REFERENCES ............. .... .................. 190

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Page1. Single-Loop Manual Control System ...... . 112. Equivalent Block Diagram of the Human Operatorin a Conmensatory Tracking Task ......... 133. Weighting Function for Bode's Amplitude Ratio Slope:Phase Relationship ............. 264. Bode Plots of Estimated and Crossover Model Open-LoopDescribing Functions for Yc w Kc System .. ...... 385. Crossover Frequency Estimation for Yc o Kc ..... 4o6. Detail Analysis of Y. = Ka/s 2 Man-Machine System

for 1/TLu < 1 .............. 437. Crossover Frequency Estimation for Yc - yc/(jw) 2 . . . 468. Detail Analysis of Yc - ic/(s - 1/T) Man-Nachine System . 499. Detail Analysis of Yc 0 Kc/s(o - 1/T) Man-Machine System. 52

10. Elements of ej Calculations. .... .......... 5511. Mean-Squared Error Based on Crossover Model ... ..... 5812. General Measurements and Task Variables .. ...... 6013. Watt Hour Meter Analyzer Control Rack and FunctionGenerator ............. .... .............. 6514. Watt Hour Meter Box .............. ........... 6515. Tape Recording Equipment ......... ......... 6616. Cross Spectral Analyzer ........ .......... 6617. Controlled Element Simulator ...... ........ 6818. Manipulator.............. .............. 6819. Accuracy and Compatibility of Analyzers .. ...... 6920. Measured Input Power Spectra Magnitudes ...... 7121. Measured Cumulative Probability Distribution of

Forcing Function Amplitude .... .......... 72viii


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22. AverageX2 Values for Augmented 1.5, 2.5, and 4 o . . 7423. Comparison of Elkind's Data for Rectangular (R.4o) and

Augmentec (3-6) Forcing Functions ....... ....... 7524. Stick Manipulator ......... .............. 7825. Internal Consistency Check, Yc Amplitude and Phase. 8526. Internal Consistency Check, Yp Amplitude And Phase. . 8627. Experimental Plan......... .............. ... 88-8928. Typical Proficiency Curve ...... ........... 9129. Comparison of Yc - Kc Data with Elkind's Data .... 9330. Comparison of Y, foy Different Forcing Function Ampli-

tudes;Yc= K-c/(J) U',~1.= 5.. ..... ......... 9531. Run-to-Bun Variability; Yq w 5//Jw, ci w 2.5 ..... .. 9832. Run-to-Hun Variability; Yc - 5/(jw - 3), mi - 2.5 . . . 9833. 1Rn-to-Run Variability; Y. _ 5/(jo))2 , aJi - 2.5 . . . . 9834. Run-to-Run Variability; Yc - 5/JW(%jc- 1.5), w - 1.5,1/4"................................... 9835. Effect of Kc Variation; Yo a Kq/jw, aIj - 2.5 .... .. 9936. Effect of Kc Variation; Yc a Kc/(Jw) 2 , a 2.5 . . . 9937. Pilot-to-Pilot Variabilty; Yc w 5/j)a, iO- 2.5 . ... 10138. Pilot-to-Pilot Variability; Yc - Kg/(Jm - 2), u)1 - 2.5 . 10139. Pilot-to-Pilot Variability; Yo - 5/(jU) 2 , i 2.5. . . 10140. Pilot-to-Pilot Variability; Yc - Kc/jc(jc- 1.5)1I5, /4k .............. .. 10141. Open-Loop Describing Fanctions for Yc K- /jw . . . 10.42. Open-Loop Describing Functions for Yo m Kc/(Jw - 2) 0543. Open-Loop Describing Functions for Yo r Kc/(jw) 2 10 644. Open-Loop Describing Functions for Yc = Kc/ja)(Jm - I/T),11 I/"ift ......................... 10745. Crossover Frequency Variations ... ......... 108

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46. Phase Margin Variations ......... ......... .. 10947. Typical Cumulative Distributions of Amplitude Ratioand Phase . . . . . . . . . . . . . . . . 112 |48. Averaged Open-Loop Describing Functions forYc a Kc/(j with a is Parameter ...... . 11549. Averaged Open-Loop Describing Functions for

Yc = Kc/(Jm - 2) with c1 as Parameter ........ 11650. Averaged Open-Loop Describing Functions for

Yc Kc/( Jw)~ with uw as Parameter....... ..... .. .... 11751. Averaged Open-Loop Describing Functions forYe - Kc/Jcu(Jci - I/T), ai - 1.5, 1/4", with 1/T asParameter .................. ............. 11852. Relative Remnant Versus K. for Yc w Ke/,jeu.... .... . 12353. Relative Remnant Versus Kc for Yc - Kc/(Jo))2. . . . . . 12354. Relative Remnant Versus Forcing Function Bandwidth; I

Yec uK0 /j....... ..... .. .... .. .... .... .. .... 12455. Relative Remnant Versus Forcing Function Bandwidth;"c K/(Jo 2) ........ .............. ..... 456. relative Remnant Versus Forcing Function Bandwidth;Yc a Kc/(Jcu) 2 . . . . . . . . . . . . . . . . . . . 12457. Relative Remnant Versus 1/T for Yc - Kc/Jo(jc- I/T),a) 1., 1/4" ......... ............... 12 558. Effect of Kc Variation on Remnant; Yc - Kc/JW,w- a 2.5, Pilot 8..... ............ .... 12959. Effect of Kc Variation on Remnant; Yc a Kc/(JaO) 2 ,wI w 2.5, Pilot 8 ......... ........... 12960. Effect of wi Variation on Remnant; Yc - 6/1j, Pilot 7. . 13061. Effect of c*1 Variation on Remnant; Yc - 5/(Jco) 2 , Pilot 2. 13062. Effect of wi Variation on Remnant; Y. - 5/(Jai- 2),Pilot ................ ................. 13063. Effect of Y. Variation on Remnant; Y. - Ko/j-

wj a 2.5, Pilot 8............... .......... 13264. Effect of Y. and Pilot Varistionb on Remnant; cw1 2.5 1352

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22. AverageX2 Values for Au6mented 1.5, 2.5, and 4.0 . . . 7423. Comparison of Elkind's Data for Rectangiular (R.40) andA1m, nt (_-6) Fer.ing Fn"-t~on-...................75 I24. Stick Mknipulator ....... .............. 7825. Internal Consistency Check, Yo Anplitude and Phase. . . 8526. Internal Consistency Check, Yp Amplitude and Phase. . . 8627. Experimental Plan ..................... 88-8928. Typical Proficiency Curve .... .......... 9129. Comprison of Yc w Yc Data with Elkind's ae.ta . . .. 9330. Comparison of Yp fol Different Forcing Function Ampli-tudes; Yc Ko/Uwl ) ,) wi - I.5 . .............. 95,31. Run-to-Run Variability; Yc w 5/jw, oA = 2.5 ..... .. 9832. Rim-to-Run Variability; Yc - 5/(Jw - 2), pj n 2.5 . . . 98 733. Run-to-Run Variability; Yc - 5/(Jeo)2 , Ua, U 2.5 . ... 9834. Run-to-Run Variability; Yc u 5/Jc(jo - 1.5), - 1.51,1/4''......... ..... .. .... .. .... .. .... .......98 *35. Effect of Kc Variation; Yc a Kc/.jc, aj - 2.5 ..... .. 9956. Effect of Kc Variation; Yc " Kc/(jc) 2 , wi - 2.5 ... 9937. Pilot-to-Pilot Variability; Y. a 5/Jc') w - 2.5. . . . 10 138. Pilot-to-Pilot Variability; Yc - Kc/(jco - 2), A1 - 2.5 10 139. Pilot-to-Pilot Variability; Ye 5 /(Jw) 2 , aj 2.5. . . 10140. Pilot-to-Pilot Variability; Yc Kc/jm(jco - 1 .5),

- 1.5, 1/4 ......................... 10141. Open-Loop Describing Fuctions for Y-c /I .. .. 1012. Open-Loop Describing Functions for Yc - Kc/(Je- 2) . . 10 544. Open-Loop Describing Functions for Yc U K/(j)2 . . . 10644. Open-Loop Describing Functions for Yc - KI/jw(jw - 6/T),S-1.5, /4" ........................ . . o10745" Crossover Frequency Variations .. ......... 108

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65. Effect of Yq Variation on Remnant; Yc * Ke/Jw(Jw - I/T),(L4 = 1.5. 11/4" P4Im. F; ....... ............ . e66. Amplitude Distributions ............ 1567. Output Power Spectrum; Y. a K 1, ac .4, 1", Pilot 2 . 13868. Output Power Spectrum; Y * Kc -5, at 4, 1 , Pilot 6 13869. Output Power Spectrum; Y. - Kc " 10, aR1 4,1 , ilot 3. 13970. Output Power Spectrum; Ye -=% - 1, R14, I", Pilot 1 . 13971. Output Power Spectrum; Yc = 5/(J(0) 2 , o y w 1 .5, 1/2",Pilot 2 .......... .... ........... .. .. 14o72. Output Power Spectrum; Y. _ 5/(Jm) 2 , ai _ 1.5, 1/2",

Pilot 7 .............. ................. 14073. Outptst Power Spectrum; Yc - 10/(Jw) 2 , wi = 1 .5, 1/2",Pilot 8 .............. ................ 14074. Variations of Crossover Frequency with Forcing Function

Bandwidth .......... ................ 14875. Dependence of Incremental Time Delay on Forcing FunctionBandwidth ............ .............. . 14 876. Extended Crossover Models for Yc a Kc/Jc. ........ .. 15477. Extended Crossover Models for Yc - Kc/(J5 - 2) 15678. Extended Crossover Models for 1c w Kc/(j)) 2 . . . . . 1587?- Variations of Effective Time Delays with ForcingFunction Bandwidth ......... ............. 15980. Dependence of Incremental Time Delay on ForcingFunction Bandwidth ........ ............. 16181. Variations of Low Frequency Incremental Phase lagwith Forcing Function Bandwidth ... ......... 16 182. Extended Crossover Models for Yc - Kc/jw(Jw - 1/T)

cji =-1 .5.. ............. ................. 16 383. Variation of Effective Tim, Delay and Low Fre uency

Incremental Phase Lag with 1/T for Yc - Kc/jWIM /T) . 16 584. Residual Amplitude Ratio and Phase Giving High FrequencyNeuromuscular System Characteristics for Yc a Kc/(Jw 2) 17 0

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85. Comparison of Precision Model with Data fore - 0 r- . . . . . . . . . . . .. .

86. Maind's Data Super iose& on Mean-Squared ErroriCrossover Frequency Plot ........... 17787. Cooverison of Actual Mean-Squared Error with One-ThirdVaw Estimates, Yo a C .V. .u.t . . . . . . ... 17788. Comparison of Actual Mean-Squared Error with EstimatedValues Based on Various Assumptions, Yc w Kc/jw) . . . 17789. CoB~arison of Actual Mean-Squared Error with EstimatedValues Based on Various Assumptions, Ye - Kc/(ja))2 . . . 17890. Pilot Optimization Adjustments of Time Delay for

Ye =KJjw andYe Kc/(Jw- 2) ... ......... 18091. Sketch of Closed-Loop Error/Forcing Funotion TransferFunction for System Based on Simple Crossover Model . . 182

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1. Human Operator Nonlinearities and Nonstationarities. . 8II. Summary of Experiments in Human Response for Stationary

Situations .................... . 14III. Equalizer Forms for Different Controlled Elements . . 20

IV. Open- and Closed-Loop Transfer Functions of a "Good"Servomechanism .............. 24V. Operator Describing Function and Crossover Model. . . 28

VI. Statistical Assessment of Selective VariabilityProperty ......... ............... 113VII. Conditions Selected for Detailed Analysis ..... .. 126

VIII. Phase Margins and Crossover Frequencies ...... 149IX. Su ry (,, to and a . . . . . . . . . . . . . . . 150

X. Sumary of Describing Function Constants for

XI. Summry of Describing Function Constants forSYc Kc/(J)-2) ....... ...... 1

XII. Sunmery of Describing Function Constants forYc -Y,,/(JW2. .. . . . .. .. . 157XIII. Swmary of Describing Function Constants for

Yc - KY/,jo)(Jcu - lI/T),pi - 1..5..... ........ 164XIV. Normalized Mean-Squared-Error Data Derived from

Elkind's Ye * 1 Experiments. ... ......... 173

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GIUOLMThreshold value of the indifference threshold nonlinearity IAi Expected number of observations in a category for a Gaussiandistribution

c(t) Operator output time function, limb positionC(Jc) Fourier transform of operator outpute(t) Error time functionei(t) Component of error due to forcing functionE(jw) Fourier transform of errorF(t) Limb-applied forcePs Stick force/degree stickG Total open-loop describing function, Y.YoOie Closed-loop describing function, error/forcing functionaim Closed-loop describinu function, system output/forcing functionH Closed-loop describing function, pilot output/forcing functioni(t) Forcing function time functionI(U) Fourier transform of forcing functionJw Imginary part of the conmlex variable, c jk ExponentK Open-loop gainKc Controlled element gainKf Control sensitivity- inches signal def'lection (on displly)/stickforceKp Human pilot gainKe Control sensitivity-- inches (display)/stick motionSGain of indifference threshold describing functiona Integer

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Im(t) System output time functionN IntegerM(jci) Fourier transform of system outputn Integer

Mean-squared remnant, to % dnc(t) Operator remnant time functionN Number of subjectsNc(jw) Fourier transform of operator remnant01 Number of observed measurements in a categoryo( ) Order of ( )p Significance levelPC Output amplitude distribution first probability density function)y Error emplitude distribition first probability density function)

R.,(r) Autocorrelation of a general time signal x(t)s Corplex variable, s m a + jco; Laplace transform variablet TimeT Time constantTI General lag time constant of human pilot describing functionTK, TK Lead and lag time constants in precision model of human pilot

describing functiona General lag and lead time constantsTleadiJ

TL General lead time constant of human pilot describing functionTN First-order lag time constant approximation of the neuromuscularsystem

TNI First-order lag time constant of the neuromuscular systemTR Run lengthu Btandard scorev Ratio of standard deviations

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IX(t) A gen-er-al time aignalx2 M.an-squared value of x(t)j jY Transfer functionYc (jw) Controlled element (machine and display) transfer functionYP Pilot describing function

a ow frequency phase approximation parameterI ITT for Ye - K0 ; /TL% for Yc " Kl/(Jc) 2

8 Dirac delta functionDamping ratio

CL Closed-loop damping ratioDamping ratio of second-order component of the neuromuscularsystem

P orrolation coefficient, Iicl i , liel li'e,Ph Relative remnant at -pilot's output, f - 2/2a Standard deviationc0 Standard deviation of pilot output, c(t)a's Standard deviation of error signal, e(t)Cri rmer alue of the forcing function

SStandard deviation of '~P'db or Yp atOT rms value of input to the indifference threshold

Standard deviation of the general time signal x(t)Pure time delay

&T Incremental time delay'Ve Effective time delay

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o E e te d

To Effective time delay for zero forcing function bandwidth, IO ~Effeotive time delay for zero forcing function bandwidth whenweis oons tant, To . n//M.,SPhase angle49 w Incremental low frequency phase anglePhase marginPeak amplitude of a sinusoidal component of c(t) at frequency a~

cpi(u) Peak amplitude of a sinusoidal component of i(t) at frequency anpx( ) Peak amplitude of a sinusoidal component of x(t) at frequency o

,Ph Closed-loop remnant power spectral density, specifically a1cotinuous density, at pilotts output

9nnc Open-loop remnant power spectral density, specifically acontinuous density, at pilot's output%n Open-loop remnant power spectral density, specifically a

continuo density, at system errori ~xContinuous component of power spectral density of x(t)

cc(a)) Pilot's output power spectral densitytee(w) Error power spectral densitytee Cross power spectral density between e and c. ic Cross power spectral density between i and cOic Cross power spectral density betwedn i and eOii(w) Forcing function power spectral densityOmn(w) System output power spectral densityInn Closed-loop remnant spectral density, at pilot's outputnnc pen-loop remnant spectral density, at pilot's output,n 1 + YPYO I60

onne Open-loop emnant spectral density, at system error,1 + p nn/l p

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Oxx Power spectral density of the general time signal x(t) composedof E random component annd M einueoide

O(W %. 40 + A M~ q~;

X Chi-squared distribution variablex Mean value of X*m Phase angleU)Angular frequency, rad/sec( a Undamped natural frequency for a critically damped mode10c System crossover frequency, i.e., frequency at which lYpYcIL Effective bandwidth of the spectral and cross-spectral analyzer

(0.141 rad/sec for processing of lower forcing function frequen-cies; 0.628 rad/sec for high frequencies)6 c Incremental crossover frequency

Average crossover frequencywoo System crossover frequency for zero forcing function bandwidth

A closed-loop inverse time constant01i Forcing function bandwidth if 2ie Effeotive bandwidth of forcing function, Jo"-----)--

M Undamped natural frequencyFrequency of forcing functiun sinusoidal componentUndamped natural frequency of second-order part of the neuro-muscular system

WO Sampler frequency0-)A Crossover frequency for neutral stability

Approximately equal toAngle of

db Decibels; 10 log 10 ( ) if a power quantity, e.g., spectrum;20 loglo if an amplitude quantity, e.g., Ypxviii


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I I'gnitudeC) mean value

F- * Coniplex oonju~pte7 Inches

Inverse Laplace trmnsform

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A., P1OFA MIWO AIM IVMiIThe effective use of manned flight vehicles has always required a

satisfactory match of vehicle characteristics (which include vehicledynazmcs, control manipulator, display, etc.) with the human pilot'scharacteristics as a flight controller. An agreeable meting is notinherent in the design process, and the provision of proper vehiclehandling qualities has often posed serious problems which the vehiclesystem designer must solve.

Classically, handling qualities concepts were based on engineeringknowledge of vehicle characteristics, leavened by pilot opinion ratings.The opinion ratings were subjective expressions of the over-all sulta-bility of the manual control system consisting of the pilot and thevehicle. A convenient way to express the handling qualities was ascatalogs of vehicle dynamic parameters given as functions of pilot rat-Inge. In spite of their reliance on the subtleties of subjective pilotratings, such catalogs of handling qualities characteristics have beenevolved in the past, and will continue to be expanded in the future.But, in a fundamental sense, these catalogs are only reports of specifictest, results--they fail to adequately explain the mutual interactionsbetween the pilot and the vehicle, and they are difficult, if not impos-sible, to extrapolate to new situations and novel vehicle characteristics.

To achieve understanding and the capability to extrapolate requiresa mathematical theory which can be used to explain old findings and topredict new ones. For handling qualities a theory of this kind has beenin the process of construction, refinement, and successful applicationfor some years. It is based on tho methods of control engineering, andtreats the pilot-vehicle system as a closed-loop (in general, a multi-loop) entity. The sine gua non of the theory is a model of pilot dynamiccharacteristics in a form suitable for application using relatively i


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conventional control engineering techniques. Further, the applicationsto which the theory can, with confidence, be employed is limited funda-mentally by the level of pilot-model knwledge.

An. dequate description of a pilot's dynamic response characteristicsis not easily obtained because of the pilot's inherent adaptability andvaplcity for learning. Nevertheless, suitable experinental techniqueshave been devised and employed in the past to provide a limited amountof data. The quality of some of these data, however, has been open toquestion because of real or imagined deficiencies and uncertainties inthe experimental task variables and in the necessary analysis equipmentsuch as spectral and cross-spectral analyzers. In many cases theanalyzer calibrations were tnsutficient to establish fully the accuracyof the data reduction methods (Ref. 51). Furthermore, most experimentershad not made certain measurements which, it turned out, were of crucialimportance in the context of pilot-vehicle control system analysis. Inspite of such imperfections, the collation of all the existing data,expanded by ultraconservative extrapolations based only on the limitedhigh-reliability data, yielded a data base from which a serviceable, butincomplete, mathematical model was evolved (Ref.. 28, 34-36). Th eextensive use of this pilot dynamics model in handling qualities andpiloLt-vehicle system analysis (Refa. 1-4, 8, 11, 12, 17, 27, 33-36, 43 ,149, 55, 56, and 58) heightened the desire for, and increased the potentialimportance of, a more complete understanding of the mathematically describ-'able aspects of human dynamics in vehicle control systems. The nature ofdesired improvements in the model was fundamentally one of degree ratherthan kind, i.e., increased scope and precision. Such an expanded viewcould not be evolved from existing data, which had been used in the con-struction of the model; instead, new data were requi.red. Fortunately, by1960 enough effort had been devoted to model building and application togive a more definitive notion of Low the data were to be used. This,coupled with a very much better appreciation for the data reduction prob-lem, made possible the planning and initial execution of a research programto meet the real needs.

Work on the project was initiated I July 1960, and almost two yearswere e=dent in the design, construction, and calibration of the data

2I


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I

reduction machines. The results of this effort have been reported

elsewhere (Refs. 9, 10, 16, 44- 46). Tests with human operators wereinitiated in May 1962. It in primarily to the task of recording and1. interpreting thedata from some of these experiments that this reportis addressed.

The prirary results desired from the program from a vehicle controlstandpoint are, of course, better pilot models to use in handling quali-ties and manual control system analyses. Although not treated here, theapplications of these models are expected to be far-reaching in thefuture. One reason for this anticipation are the extensive uses of themuch less precise and more vague circa 1960 model. For example, in thereferences cited above thzs model has been used to :

1. Estimate human pilot and over-all pilot-vehiclesystem dynamic response, stability, and averageperformance.2. Determine barely controllable vehicle dynamics and

controllability boundaries.3. Delineate those features of the vehicle dynamicswhich are most likely to affect the vehicle

handling qualities.4. Indicate the type of additional system equalization

(to be achieved via the display, the manipulator,or by vehicle modifications) desirable to achievebetter pilot control-as well as the effects onthe pilot characteristics of such modifications.5. Find the maximum forcing function bandwidth com-oftible with reasonable control action on the partof the pilot.

The new models developed here are intended to be used for the same sortsof things, but with far greater confidence and considerably better pre-cision. The refined models are expected to be very useful in other waysalso. For example, the characteristics of the human pilot's "actuator"and "sensor" dynamics, which were previously lumped into a mid-frequencyapproximation to lower and higher frequency effects, are distinctlyreflected in the new data and models. This new knowledge should havesignificant impact both on the content and nature of the informationdisplayed to the pilot and on the design of the manipulative deviceswith which the pilot imparts his desires to the vehicle.


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A: noted above, Macor DurPOCC Of thG Gxoian~ ~cgzuLdiscussed here was to provide data for the extension and validation of theexisting pilo- model. Therefore a necessary preliminary .s1Bo consider tihestatus of the mthemtioal model of the human operator at the time theexperiments were planned. This is accomplished in the next chapter, whichcomprises a statement of the best that was known about human operatorbehavior in compensatory tracking in 1960 together with a discussion ofthose areas in which specific knowledge oi' operator behavior and itsmeasurement was either totally lacking or substantially incomplete.Previous knowledge of human operator behavior in compensatory tracking issummarized in the first section- of Chapter II entitled "The Analytical-Verbal Model." while the discussion of what it appeared necessary to findout follows in a treatment entitled "Objects of the Experimbnts."

A jajor part of the plan for the model validation aspects was to makeanalytical predictions of human operator performance in compensatorytracking and then to observe whether or not these predictions could beoonfirmed with experimental results. To this end the analytical-verbalmodel, reviewed in Chapter 1, is bravely put to use in Chapter III.Although it was recognized that additional data were clearly required tosubstantiate some of the conjectures on which the model was founded,predictions critical to model validity were made which could later becompared with experimental results. These pre-experiment analyses alsoprovide excellent concrete examples of the techniques involved inapplying the human pilot models.

The really essential portion of this program is experimntal, for thepotential of all existing data as sources for model builder- 4nd elaborationhad been exhausted. Past experimental efforts have seW rthyexamples of precision. In the new experimental seri(- .. . empr-tant desired feature was the provision of supplement-: ;iontechniques and method. which would maximize flexibilit, Asdze thechances of experlmental error. The apparatus was desig make theseobjectives feasible, but changing feasibility to actua,.. requires excep-tional experimental and data reduction procedures whir -Anonst mount to

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trade secrets. Some of these are described in Chapter IV, together witha description of the experimental and analytical apparatus.

The experimental objectives of Chapter II plus the pro-experimentanalyses of Chapter 71Z lead directly to a desired experimental programplan. An outline of this program in resented. in the first section ofChapter V.

Chapters V and VI present the experimental data--describing functionsin Chapter V and remnants in Chapter VI. The datA are aggregated in vari-ous ways to illustrate the gross and detailed effects of changes in theforcing function and controlled element task variables. Other aggregationsare used to illustrate the effects of intra- and intersubject variations.The describing function chapter also includes statistical treatments ofcertain major conclusions. A special feature of the remnant discussion isa presentation of data whinh indicate the likely major remnant source, aswell as data tending to indicate what the remnant is not.

Chapter VII is devoted to detailed analyses of the data and to thedevelopment of interpretations and rationalizations. In extending theexisting analytical-verbal model it is concluded that, by and large,the hypotheses and extrapolations made from the limited data previouslyavailable were generally correct, and reasonably explained. Further,the updated models developed here, which subsume the old model and con-form to the new data as well, answer many other questions concerning+' human behavior. These models are of three levels of precision. Th efirst is relatively crude, and is intended for use in the region ofcrossover only. The second is sufficiently precise to be suitable formost handling qualities analyses, which primarily emphasize vehicledynamics, including those involving statically unstable vehicles.Finally, the third is a precision model which is capable of representingthe high and very low frequency actuator and sensor dynamic characteris-tics of the human pilot. Thus, the three models provide a range of com-plexity and utility which is analogous to the several degrees of modelcomplexity used in ordinary automatic pilot design.

Finally, Chapter VIII summarizes the general conclusions and findingsof the study.

' + + ++m : -............


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0. A W = ZMM M.Becaus: this i a very long report, with mny involutions, it is

appropriate to give some words of advice for readers with variousinterests. For the casual reader, this introduction and the generalsumary of Chapter VIII give the gist of the effort. If Just a bi tmore is desired, add Chapter II,

The applications-oriented engineer should staft with Chapter II,Section A, for a review of the existing model, followed by Chapter V,Section A for the experimental plan outline and Section D for the grand-average describing function data, and then on to Chapter VIII for thegeneral summary of results. He will then probably wish to absorb moredetails on the new models, as developed in Chapter VII, and may desireto examine the pre-experimental analyses of Chapter III as concreteexamples of applications.

Those who are interested primarily in data can proceed directly toChapters V and VI, although the experimentalist will also wish to coverChapter IV .

The model builders can turn directly to Chapter VII, and, if theyhave models to test against the realities of data, to Chapters V and VI.

Pinally, for the reader who is interested in the entire effort th eway is directly through the report as laid out, although many of thedetails can be sloughed over lightly on a first reading. These diligentsouls have both the authors' syMathy and blessings l


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-rmzz

A. D AM ZOULPN,,AL HM"W"I*Oes.tufe at the HOWaThe primary objective of most of the past experimental and analyticalprograms to develop mathematical descriptions for pilot response charac-

teristics has been to achieve reasonable descriptions of the pilot as acomponent in engineering systems. Major efforts in model building hnvethus been placed on the evolution of models which can predict pilotdynamic response characteristics of engineering significance, but whichare otherwise of minimam analytical coplexity* Such models are oonoep-tual descriptions of the human. They are intended to exhibit analogouscause-and-effect behavior rather than to be analogs in any structumlsense. The models are valid to the extent that their behavioral proper-ties resemble the performance of the actual human operator. They gain inscope and descriptive power if certain of their features can also beidentified structurally, although they cannot be rejected because of anyfailure to satisfy this desirable quality.As a control component the human exhibits a bewildering variety ofnonlinear and time-varying b bavior. Table I lists some of these. Anappropriate type of engineers mathematical description for nonlinearcontrol elements of this na1,u.e is some kind of quasi-linear system.This is an equivalent syste' in which the relationships between some,but not necessarily all, pe t.nent measures of system input and outputsignals have "linear-like" features for fixed input conditions in spiteof the presence of nonlinear elements.

The quasi-linear system concept originally evolved from theobservation that a great many nonlinear systems have responses tospecific inputs which appear similar to responses of equivalent linear

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TABLE I.1 K~~~uNAPmwoa~ NowLflmarmgI AM~ NONaTATIONOafMIE1. Adaptation an& Iearning

The adaptive humn responds to changes in external environmentby modiW ng iharaoteristics so as to Unrove performnoe over thatwhich would be achieved if the characteristics used In the originalenvl1ronment were maintained in the new one.

The l human, after operating experience in a givenexternal i-nVirormient, modifies characteristics to achieve betterperformance than previously exhibited in the given environment.2. Set Changes

"Set" characterizes the temporary operating points or baselineconditions of the human subsystems involved in the control tasks.Ohanges in these internal conditions facilitate a certain more-or.leos specific type of activity or response in the adaptation andlearning processes. Set changes include:

Internal system topography changes, i.e., use of different feed-back and feedforward pathsVariation in steady-state muscle tensionVariation in force rangesVariation in indifference threshold

D. Series NonlinearitiesSensory thresholdsMakximum force and displacement limits

i . Fluctuations in Attention, Motivation, Etc.; General Drifting ofCharacteristiosI


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Isystems to the same specific inputs.* For such ombinatnn.r Om speacificinputs and nonlinear systems, the response of the nonlinear system can bedivided into two parts- one cozponent which corresponds to the response ofthe equivalent linear element driven by the particular input, and an addi.tional quantity which represents the difference between the output of theactual nonlinear system and the equivalent linear element. This second compo-nent is called the "remnant" because it is left over from the portion of thesystem response representable by a linear element. Quasi-linear equivalents tothe nonlinear system, for the specific input of interest, are characterizedmathematically by a describing function (which is the equivalent linearelement) and the remnant. An essential feature is that the quasi-linearsystem has a response to the input in question identical to that of theoriginal nonlinear system$ so the quasi-linear system is an exact cause-effect representation of the nonlinear system for the specific kinds ofinputs and responses considered. When the inputs are changed, the quasi-linear'model also changes. If the device were such that its quasi-linearsystem remained the mame for all kinds of forcing functions, and if, further,the remnant were zero) then the system would have a constant-coefficientlinear nature.

The most oomon quasi-linear system element in engineering use is thesinusoidal-input describing function, which is of such great value instability studies of nonlinear servomechanisms. Here the action of thedescribing function on a sinusoidal input results in an output which isthe fundamental of the output of the aoAal nonlinear system. The remnant,which must be added to the output fundamental to achieve equivalence withthe nonlinear system, is made up of P11 the higher harmonics resulting fromthe passage of the sinusoid through the nonlinearity. Describing functionscan also be defined for transient inputs, such as step functions, and forrandom.-inputs. In principle the systems can be time-varying as well asnonlinear. Pandom-input quasi-linear systems representing the human oper-ator for certain conditions are the type pertinent to the operator datatreated here.

"Wn;y- exts on nonlinear control theory treat aspects of quasi-linearsystems. Chapters 3- 6 of Ref. 21 are especially pertinent as backgroundin the context of this reports9


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1.Rdvof biseiig 'Qmei-L1ftia1' @stm~ Duseri1bingIMU~nto PUW~ Rumt NuOW for S ngIe-LM~ kib IIn term of idealimations, the simplest manual control system is thatshown in the block diagram of Fig. 1. This is also the most commonly

occurring system structure in practical manual control systems- eitheras a total system by itself or as a component part of a more complexmultiloop system. To control engineers, this is a single-loop feedbacksystem (except for feedbacks internal to the operator)j to engineeringpsychologists, it s a compensatory system. The important single-loopfeatures are the solitary stimulus (the error) and the random-appearingnature of the forcing function (the system input). If either of theseis changed, the whole complexion of the task also changes. For instance,if the pilot is shown the system's input and output directly (a pursuitdisplay) instead of their differences, he is often able to take advantageof the additional information and thereby to improve over-all systemperformance. Also, if the forcing function is not random-appearing, butperhaps periodic over relatively short time intervals, the pilot can Voften detect and anticipate the repetitive or deterministic nature ofthe input and adjust his response accordingly. Both of these higherorder types of behavior amount to the presence in the system of furthersignal paths and a more complex than single-loop structure.

For the system shown in Fig. 1 there are three task variables thathave a major effect on the pilot's dynamics- the forcing function char-acteristics, the controlled-element dynamics, and the manipulator. Many 4other factors are implicitly involved. These include operator-centeredvariables such as training, fatigue, and motivation, and external environ-mental characteristics such as ambient illumination and temperature.Ideally, all of these implicit (or "procedural") variables should betaken into account, and someday perhaps they will.* But for the present

*Chapter VII of Ref. 34 presents a preliminary discussion of theeffects of a few implicit variables on pilot response measures. Refer-ence 48 provides an excellent detailed example of pilot describing func-tion changes accompanying changes in a typical environmental variible-in this case, the pilot's effective "g" field.

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_ _l

: t

_ __t It l

II A


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the central aim is to explore operator dynamics in specialized situationswherein the vast majority of these procedural variables are held constant.This Is simple enough to do for the external environmental factors, but for -the operator-centered factors the best that can be hoped for is the estiab-olivsent of reasonably stable levels of tationaritn. Statnonargty in theover-all exierfmentallstuatbl i s ofhationaiityn tateonriing funthtions to signals possessing stationary characteristics, and by using highlytrained and motivated subjects drawn from a narrowly limited population forwhich high-grade skill in manual control is an essential feature.

Tustin (Ref. 59) first noted, in a formal way, that operators in manualcontrol systems, responding to random-appearing visual forcing functions,exhibit a type of behavior which in analogous to the behavior of equalizingelements inserted into a servo system to iprove the over-all dynamic per-formance. Since then, a number of measurements of human response to visualinputs have been .made in situations such as the one illustrated by the con-trol system block diagram of Fig. I. For the actual measurement situationsthe human being is represented by his quasi-linear model, that is, as adescribing function plus a remnant. The dynamics of the display and othersystem elements are lumped into a "controlled element," and the system forc-ing function is modified (if necessary) into an equivalent forcing function.The equivalent block diagram then takes the form shown in Fig. 2. The con-trol loop signals are represented as time functions and their Fourier trans-form, e.g., e(t) and E(jw), and also as power spectra or power spectraldensities, e.g., 00e(w). The linear constant-coefficient controllea elementis totally defined by its transfer function, Yo(jw), whereas the nonlineartime-varying humn requires the describing function, Yp, and the remnantpower spectrum, %hnc, o provide an adequate dynamic description in th esense that the power spectral densities of the signals in the actual andthe quasi-linear equivalent system are the same.

The number of conditions studied by the principal investigators ofhuman operator describing functions, through the year 1960, are sumna-rized in Table 1I. The table is organized with respect to task variables.The most influential of these turned out to be the forcing function and

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f.I -jI ,-I ii II- -

I..... ... I ~illi

I.4,-.1I. ..........


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04I4-B~

to

10

-y CUj cmr-lt

Cd0u- C U co

I1


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. .

the dynamics of the controlled element. By comparing the results fromall theee e.-*erimentt, the influence of the manipulator "rAn ho m to bcunimportant for the ranges of frequency measured and manipulators tested.

The dependence of the human operator describing function on forcingfunction and controlled element dynamics has tended, in the actualexperimental situations, to be obscured by the limited frequency range(bandwidth) and the run-to-run variability of the measurements. Refer-ence 34, however, shows hows by considering averaged data, it is possibleto folmulate a fairly simple analytical describing function form whichcan be adjusted to describe the main features of human behavior. Althoughthe Hall data (Refs. 24 and 25) were not available at the time Ref. 34was written, Ref. 36 represents an effort to bring these data into conjo-nance with the results considered earlier, and the circa 1960 statementof the model below takes account of those results. Thus, when the

/describing function data for all the experiments represented in Table II'are considered as a whole, they serve as the data base for evolution ofa 'servo model describing human operation and adaptation for compensatorytracking with a visually presented, random-appearing forcing function.This model is the key element in a dynamic description of the human oper-ator's capabilities in such tasks (the other element being the remnant).It characterizes the predominant majority of all the experimental results.

The model comprisee two elements:a. A generalized describing function formb. A series of "adjustment rules" which specify how

to "set" the parameters in tne generalizeddescribing function so that it becomes anapproximate model of human behavior for theparticular situation of interest

The most extensive and generalized describing function form for one-and two-dimensional compensatory control tasks developed in Ref. A4 is.,

j~e (TzjJ~T 2uJ] 4 (T] 1 cn

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where -P gainT reaction time delay (transport lag)(T)iW- 1

. (Tiw) - equalization characteristic

I ndifferenoe threshold describingLrTJ function (I - N(73 aT/cFT when&T/


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Ideveloped in Ref 34 has only fir;t-oder neurowsmular lag term, It + -I where T% A TNl + (2N/o,). IIith the e imlfioition discussed above, the general low frequencyi1describing function of Eq 1 becomes approximately

Y9 Tjc+1 (2)(TIjcu +I)(TNjm +I)

V While Eq 2 can be shown to have a very considerable range of validityfor a variety of operators; forcing functions, controlled elements, andmanipulators, only the form of the describing function has this over-allvalidity. Mobt of the paramaeters in the describing function are adjust-able as needed to make the system output follow the forcing function-i.e., the parameters as adjusted reflect the pilot's efforts to make theover-all system (including himself) stable and the error small.

The pure time delay represented by the -JOT term is due to sensorexcitation (the retina in the visual case), nerve conductions co-uta-tional lags, and other data processing activities in the central nervoussystem. It is closely related to, but not identical with) certain kindsof classical reaction times. T is currently takcn to be a conatantbecause it appears to be essentially invariant with forcing function andcontrolled element dynamics for either single or dual random-appearinginput tasks. However both inter- and intra-subject T variations occur.Observed V'5 run as low as about 0.1 see Fnd Pts high as 0.2 see.

The zneuromusoular lag, T%, is partially adjustable for the task.The nature of the adjustment is somewhat obscure due to the lack of highfrequency data, although the general trend is a monotonic decrease in TNIwith increasing forcing function bandwidth (see Table 13, Ref. 34). Theobserved variation of T with forcina function bandwidth ranges fromless than 0.1 sec to somewhat greater than 0.6 sec. Because the detailsof the TN variation with forcing function bandwidth are not known, thisinortant variation has often been ignored in applications and typicalvalues of TN near 0.1 sea have been used.

The equalizing characteristics, (TL,Jw 1)/(Tljw + 1 ) coupled withthe gains Kp, are the major elements in that adaptive capability of the

17

,,. .,,;


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human which allows him to control many differing dvnAmin dAvyie . T1hpo,function is the modification of the stimulus signal into a suitableneuromeasulr somuand which is p'operly p.aled .fno phased.for. sAtableover-all n~n-machine system operation. For given input and controlledelement characteristics, the form of the e4ualizer is adapted to coupon-sate for the controlled element dynamics and the pilot's reaction timedelay.

The describing function adjustment rules are not simply stated sincethey depend intimately on interactions of the elements in the man-machinesystem. In general, the adjustments can artificially be divided into twocategories - adaptation and optimalization. Broadly speaking. adaptationis the selection by the operator of a specific form (lag-lead, lead-lag,pure lead, pure lag, or pure gain) for the equalization characteristicsjand optimalization is the adjustment of the parameters of the selecbedform to satisty some internally generated criteria. The result of theadaptation process is fairly well understood, since the form selected isone compatible with good low frequency, closed-loop response and theabsolute stability of the system. The internal optimalizing criteriaare not known, although they appear to be generally compatible with theminimization of the rms error (Refs. 31 and 34).

The known adjustment rules for the human operator's describingfunction, in decreasing order of their certainty, can be summarized asfollows (Refs. 34, 36, and 37):

1. *bablIA.I: The human adapts the form of his equalizingcharacteristics to achieve stable control.2. ftm lsdeatiu- LaW Yquseaye : The human adapts the form of

his equalizing characteristics to achieve good low frequencyclosed-loop system response to the forcing function. A low frequencylag, TI, is generated when both of the following conditions appl4:a. The lag would improve the aystem low frequency

characteristics.b. The controlled element characteristics are such thatthe introduction of the low frequency lag will notresult in destabilizing effects at higher frequencies which cannot beovercome by a single first-order lead, TL, of somewhat indefinite butmodest size.

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3. Yom 5eMaotim- Dead: After good low frequency characteristicsare assured, within the above conditions, lead is generated whenL the controlled element characteristics together with the reaction timedelay are such that a lead term would be essential to retain or imrovehigh frequency system stability.ho. smbet Adjustue: After adaptation of the equalizing form,the describing function parameters are adjusted so that:

a. Closed-loop low frequency performance in operating onthe forcing function is optimum in some sense analogousto that of minimum mean-squared tracking error.b. System phase margin, %4, lies somewhere between 00 and500 when the forcing function bandwidth, ki, is much less

than the system crossover frequency, o-); and from 500 to 1100 when cm isnear ca,. This strong effect of forcing function bandwidth on the phasemargin is associated with the variation of TN with the same task variable**C%. IZnvartaaoe prooetles

a. a%--K Zndendenee: After initial adjustment) changesin controlled element gain, Kc, are offset by changes inpilot gain, Xp; i.e., system crossover frequency, %c, is invariant with K0 .b. %-% devpeedenae: System crossover frequency does notdepend on forcing function bandwidth for q < ukt. (0oh isthat value of w. adopted for qA


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It turns out that the operator Uc;cribirng 1'Urt wilwdw w Lor agiven task is very similar to the one that a control engineer would select

if he were gLVen an element to control together withi a controller""blaokbox;' having within it elements making up the describing function givenby Eq 2, and knobs on the outside for the adjustment of T1 , TL, and Ky.Thus, the adaptation of equalization form covered by Adjustment Rules 2and 3 my be foreign to the reader who is not thoroughly grounded incontrol theory since it is analogous to operations which havo an artisticflavor even in conventional contr'ol system synthesis. Examples often help, soTable III is presented to illustrate the equalizer form taken for severalsinple controlled elements.

TABLE IIIEQUALIZER FOR3 FOR DIFFERENT CONTROOLLED ELDEMNT8

WHEN TSE THE OPERATOR' SCONTROLLI ELMAT EQUALIZER FORMTRANSFER AUO'TION, Yo, IS ADAPTD ISKc Pure gain, IpKI Lag-lead, T, >> TL

(j~) 2 Lead-Ing, TL> TIKe Lead-lag (if a% 4< 2/,r)

(jc)2+%(co +lag-lead (i >2/1)

This, then) is the analytical-verbal describing function model ofthe human operator which existed prior to the current study, Th eanalytical portion of the model is the expression of Eq 2, and theverbal portion comprises the adjustment rules given in the numberedstatements above*

While the describing function is the critical factor in determiningoperator-system stability, the uncorrelated - in a linear sense -portionof the operator's output$ the remnant, can be important for estimates of

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system prefosnrmnn^ ot~w1h k mam tn- re error, Thie eitto ftriaeabecause the remnant, represented as a power spectral density, 11=(w), Iis one of two additive terms in the expressiou for the spectral densityof the operator's output.

Considering that no disturbances are present, the mean-squared error

I CO2 Y0 0 )2-~ ~ a '-OJjit(CiO)d ft Ji +Y ~O ne(to)LD

where A (I)1 =J he error/input describing function I IWYpYc( a)

and Oji(o) - the power spectral density of the inputnno(N) the power spectral density of the remnant expressed

0 an an "equivalent" open-loop input applied at thepilot's output

The first term in Eq 4 derives from the describing function portion ofthe pilot model operating on the forcing function. At the operator'soutput this component can be represented by a power speotre3 densityIHtoii(w), where H = p/(l + YpYc) is he closed-loop describing f'unc-tion relating pilot's output to the system forcing function. The powerspectral density of the remnant can also be expressed in closed-loop formam Onn . I,/(, + ypyo) 2lOnn0 . Then the total power spectral density ofthe operator's output is

000((o) " 12li(*) + Dn(e)

The ratio of the linearly correlated pilot-output power to the totalpilot-output power is the square of the "correlation coefficient,1" P:

IHIii nn (6)The meaning of p in a specific instance is dependent on the datanalysis apparatus and on the nature of the aystem forcing function.

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For our experiments p is found using an analyzer which mechanizesspectral and cross-spectral measurements using multiplications andvery low pass filters (Chapter IV). If the forcing hunction is a sumof sinusoids, Oil will be a mum of delta functions (i.e., a seriesof line spectra whioh exist only at the frequencies of the individualforcing function sinusoidal coponents). Then, in general, the output,0o00 will be a sum of delta functions at the same frequencies as thosein oil, plus delta functions at other discrete frequencies (if non-linearities or constant rate sampling are present), i1us a continuouspower spectral density component representing random fluctuatione inthe output. At the frequencies for which they exist the delta functioncomponents will generally overpower the random component, and the pmeasured at forcing function frequencies will generally be 1.0 unless lowfrequency time variations in H result in additional power within themeasurement filter bandwidth. In fact, p will be 1 .0 even in the pres-ence of many kinds of system nonlinearities. At other frequencies pwill be undefined since ilj s zero.

For forcing functions which are samples of random processeso thepower spectral densities in Eq 6 arm all continuous. The meaning of pfor this case is quite different; Jts value relative to 1 .0 becomesprimarily a measure of the relative importance of thu remnant. Near-unity p values indicate a linear constant-coefficient sersttm, whereslesser values imply nonlinearity and/or nonstatiorarity and/or "noise"injection (Ref. 34). Thus p for the random-input case is not as dis-criminating a measure as when the input is made up of simple sinusoils.

All past human operator data for which p values exist used forringfunctions which were samples of random processes (Refs. 24, 29, 34,and 50), or very many, closely spaced in frequency, sinumoids (Ref. I1)which were not separable in the analysis technique used. Therefore ipast p data fall into the random-input category. Much of the existingdata show correlation coefficients near 1 .0, although considerably lowervalues were not uncomaon. In general, the larger the correlation coeffi-cient, the smaller was the renmant and the mean-squared trac'ing errors.The smallest observed remnants occurred in connection with controlledelements which had the least energy storage, i.e., Y. '- Ko. Here

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the foz of the operator's equalizer characteristio s either a lowfroquenay USg or a pure gain,, Finally, the least remnant Is assnociatedwith forcing functions having the least bandwidth rr high frequencyoomponents.

Unfortunately, mechanisms for the description of the remnant are notnearly as well "understood" in an equivalent mathematical model form asthose for the describing function. Observed remnants in one-dimensionalcontrol tasks have, however, been "explained" in three possibly equallylikely ways (Ref. 3)a

a. Random "noise," with a mean-squared value proportionalto mean-cquartd linear output, superimposed on theoperator's linear output

b. Nonsteady opera-tor behavior. that 4s, variation of theopeiator's describing function during the course of ameasurement rc nc. N~on~intar anticipation or relay-like opersationp super-imposed on the operator's linear output

Remnant data are far more sparse than lescribing function information;and the data available are not especially reliable. Consequently) thesunmary statements above are about all that should be said in connectionwith the oizoa 1960 model.3. eafte for the tshingl Mo*el

It may be well to remark that the analytical-verbal model is in parta hypothosis whiuh derives from whatd after all, can only be describedas limited data. The rationale of the model, however, does not restexclusively on observation. The adiustment rules are partly an exprem-sLen of practical synthesis procedurss for inanimate servomechanisms(Rees, 7P 22, a~id 26).

Desirable properties oi a "good" feedback control system are to:a. Provide opeclf.ed conmand-response relationsah jb. Suppress unwanted inputs and disturbancesc. Reduce efectb of variations and uncertainties inelements of the control loop

23.S---i-


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It can be shown (Ref. 7, for example) that these three functions are Iaoocqlishod in a single-loop manual or automtic feedback control Isyntax by making the oonplex gain of the open-loop system, verylarge over the range of frequencies in which the cowand input and loaddisturbances have substantial components, and very small outside thiirane. .For the unity feedback system of Fig. 2, the desired closed-loop

transfer function is the transfer function of a low pass filter. Theentire range of positive real frequencies may then be divided into threeregions of principal interest in terms of the magnitude of the open-loopsystem transfer function. These are displayed in Table IV .

TABLE IVOPEN- AND CLOSED-LOOP TRANSFER FUNOCIONSOF A "GOOD" SERVOMECHANISM

MQUENC'Y OPEN-L4OO CLOSED-LOOPTRANSFER FUNCTION TRANS1IR FUNCTION

1l I + YPY1

I ~ "Y-u 1 -yya 1


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?Inear crossover frequency determines the dynamics of the dominant modes

Of yrauam wvusyvw. Tha uuzial.uz ur z~ui--znra otscillabur'Y j~i yi

oproand *9 Yc -- 9 when gn Y.pYc(0) -+ (7)a:dHance, neutrally st~able or unstable dominant modes are moot often avoided

: by adjusting the system so that

SIYp~c --n when IYPY c l " I

These are the often quoted conditions of positive gain margin and phaoemargin (e.g., Ref. 22). They are an expression for saimle minimum phaseor minimum phase plus transport lag systems of the Nyquist stabilitycriterion. Since stability is quite litorally the mine ua non of afeedback control system, these requirements for gain and phase margin,ay severely restrict the choice of the crossover frequency.

The phase angle at a frequency a) of a transfer function, YpYc, whichcontains a transport la&, T, but which i.s otherwise minimum phase, interms of amplitude ratio slopes is (Refs. 7 and 2)

S[c)- dIlp(c)I] ()1r oot--.--..-d(

where ~ ~~h loed YPYOI/l(t/oc)],rare expYeswedInbdead.Allustrated by Fi. 5, he l 0oth/2)Il(/e)1 term in the inte (aapplies a large weighting to slope changes in the immediate vicinity of a..and greatly attenuates the effects of the integriand of slope changes else-where. Consequently, the phase at %c is affected prirsrily by v , by thelocal db amplitude ratio slope (the second term in the expression), and by

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+TN230 -3.0 --

ij~cIJ

CJ

01 0.2 0A 0.6 1.0 2.0 4.0 6.0 10.0Figure 3. Weighting Function for Bode's Amplitude Ratio Slope,Phase Relabionship

local changes in this slope (the integral term). If the db amplitude ratioslope is esaentially constant over a wide region about w, the expressionreduces approximately to th~e second term alone plus the transport lag'scontribution. In this event the phase associated with a constant amplituderatio slope of -20n db/decade will be simply -- rut-nrac/2 rad.

For low pass open-loop transfer functions the amplitude rat .o slope atgain crossover is ne ative, so a positive phase margin can usually existonly when d YpYcl1 , /dl (w/b)] in the imuediate vicinity of crossover isless (numericslly) than -40 db/decade, the local changes in slope are moder-"ate, and the Ta) contrLibulion is ninor. The available crobsover reg.inns formost transf-v functions are, therefore, confined to areas where the localamplitude ratio slope fulfills these conditions, and the choi.ce of crossoverfrequency is delindt;d accordingly.

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These considerations are embodied in the P Rule of Thumb ofauz u ayiL a L a rLvuLuh. 220. 1b/a.uwl"slope for the amplitude ratio, and then make it the crossover region byputting the 0 ab line through it" [i.e., make the gain such that Iypyo(jm) 1where YpYc(jO ) i/J(COV%)]. This crude prescription for stability and goodresponse is generally adequate for minimum phase systems. It can beextended directly to the transport lag case by adding a prescription fora positive phase margin. Typically, the phase margin, qq, for a well-adjusted regulator is approximately 30 -400; a somewhat higher value iscustomarily used for servomechanisms, which must follow commands as wellas suppress distu-bances. The human has his own ideas (Adjustment Rule 4b)l

Finally, then, for a "good" feedback control system the operator'sdescribing function, Yp(Jci), must be adjusted so that the crossoverfrequency, wo, exceeds the highest important frequency in the input, wjand so that IpYc(jcu) conforms et low frequencies, high frequencies, andcrossover frequency to the requirements noted in Table IV.

Many of the above remarks about the rationale of equalization adoptedby the pilot can be made more concrete and understandable by the defini-tion of an approximate "crossover model" for manual control systems.This has been done in Ref. 38, where it is pointed out that considera-tion of the requirements of "good" feedback system performance leadsdirectly to the conclusion that the pilot adjusts his describing functionso that the open-loop function, YpYa, in the vicinity of the gain cross-over frequency, wo, is closely approximated by

ii aMbe-j'e4 . Y Y e "; ( I0

This crossover model is not a replacement for the analytical-verbal model,but is instead a convenient approximation suitable for many engineeringpurposes. While it is a better description of amplitude ratio character-istics than of phase characteristics, it often describes the most signif-icant features of operator behavior adequately. This is because theactual shape of the open-loop function away from the gain crossover

27


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frequency is usually almost irrelevant to the closed-loop performance.It is, therefore, often unnecessary to, retain terms in the describingfunction whose influance is uot felt in the immediate vicinity of gaincrossovers

Table V shows the application of the adjustment rules to th eprediction of a pilot's describing function and to the formulation ofthe crossover model for several simple limiting-case controlled elementsu.

TABLE'VOPERATOR DESCRIBING FUNCTION AND CROSSOVER MODEL

WHEN THEWHEN T THE OPERATOR' S THE CROSSOVERCONTROLLED ELEMENT DESCRIBING FUNCTION IS MODELTRA18FER FUNCTION IS (w about ac)(Ye) (Yp) IS YpYcKKp- Kpe-Ji"e ,1Kce-ce

Kpe-Je 1 ,K, e-Jane(TIJ+1) T


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|I3. mN i in 1

The primary purposes of the experimental series are the y |of the existing analytical-verbal model and the uesen of this modelin accuracy and detail.

In the sense in which It is used here, "validation" is intended tomean the confirmation of the extrapolations based on limited data. Thedelineation of crucial experiments designed to test the validity of themodel involve the following preliminary predictions:

a. Describing function form adjustment. For a variety ofcontrolled elements the adjustment rules should be usedto predict the operator's describing function form. Thesecontrolled elements should be such as to require a completerange of operator equalization form adaptationj and they shouldbe "new" in that they have not previously been examined experi-mentally in detail.

b. Mean-squared error minimization and variation. Estimatesof mean-squared error due to system forcing f~mction, andits variation with forcing function characteristics, should bederived. Special control situations should be selected whichmost concretely illuminate the experimental consequences ofdescribing function adjustment for minimum mean-squared trackingerror, and the variation of this quantity with forcing functionmodifications.

With these preliminary analyses and predictions in band, an experimentalseries can readily be delineated to provide data which either confirm ordeny the predictions of the model.

As already indicated in Table II1, the simple controlled elementsYe * Lc, Kr/jo, and Ky/(Jw) 2 evoke a complete range of equalizer formadjustment, so these become prime candidates for critical validation

experiments. Presuming that Elkind's experiments (Ref. I1)with avariety of forcing functions for the simple controlled element Yccu Kccould be repeated, in part, in the cur'rent serie, then his data as awhole could be considered in direct context wi~n our own. The crucialcontrolled element form for model validation would then be Ya u KC',o

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and KC/(Jc) 2 . Limited tests have been accomplished for the first ofthese (Refs. 24 and 47), but no data are available for the second.These controlled element forms are further attractive in that theyrepresent, in the vicinity of crossover, limiting oases of a greatvariety of other possib le con trolled elements of far greater complexity.

As another aspect of validation, it is necessary to distinguishbetween those features of the model in any particular situation whichmust be evidenced, and which therefore would tend to be invariant withdifferent operators, and those features which might be, to an extent,a matter of indifference. It will become apparent later that individualoperators may display what can only be described as a particular trackingstyle. The style of one operator may be somewhat different from thestyle of another operator, and this is reflected in changes in thedescribing function model in those regions where the form of the modelis not critical to good tracking performance. On the other hand, wherethe describing function form is critical to good tracking performance,most particularly in the vicinity of the crossover frequency, it is tobe expected that the operator will not exercise a choice and that thetracking performance will be tightly constrained. The three controlledelements mentioned above are not particularly constraining except in theregion of crossover. However, certain unstable controlled elements tendto require a more uniform behavior.2. Model hbensiosn

The primary limitations to the existing model are due to limitationsin the experimental data on which the model is based. In essence, th emodel Is about as sophisticated as it can be without further empiricalknowledge. So, the question becomes "Where are the existing datadeficient?"

a. The vast majority of the data are based on forcingfunction bandwidths which were very low in frequencyrelative to ut. Hence, most of the important crossover.features of the model are based on extiapolations and thedoctrine of compatibility between date sourcee.

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bi Th_,.,e leow frequency fcrcin-function bandvidth data dvnot disclose any differences between manipulators.

ar. o achieve rndom-appearing forcing functions, noisei generators or their equivalent were used. This choicei of forcing function, while satisfying the randomnappearing

criterion, introduced other problems:(1) Sapling variability (i.e., effects of finite runlength) introduced several uncertainties.(2) Data processing equipment was conplex and oftennotoriously unreliable.(3) The remnant, of course, came out as a continuouspower spectral density. Consequently it was of

such form that it was not possible to discriminate among theseveral possible remnant sources (i.e., remnant can be due tosexpling, nonlinearities, nonstationary behavior, actual randomnoise injection, etc., but the distinctive features correspond-ing to each of these tended to be lost in a general smoothed-over slush). Also, precious little remnant data were available.d. With the notable exception of Elkind's results, the effectsof variation in forcing function bandwidth were not system-atically explored. Also, the consequences of chunges in forcingfunction amplitude on open-loop describing function data were notknown in detail.

Many other data deficiencies exist, in both scope and kind# but thefour major items listed above were the most influential in the programplanning phases of the experiments to be reported here. A major correc-tion of these data deficiencies was desired, at least in the areas mostsignificant for man-imchine system analysis. Thus, beyond the criticalexperiments for the validation of the analytical-verbal model, it wasexpected that experimental phases of the program would yield additionaldata which would permit:

a. The refinement of statements concerning the adjustmentrules and the form of the describing function in thevicinity of the crossover frequency. Such data might include:(I) Phase margin data (and gain margin data worethis parameter is pertinent)(2) Closed-loop dominant mode characteristics, suchas closed-loop damping ratio, CC


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(3) Tighter Limits on vp'ues of T and/or - +TI(4) Better understanding of and limits on the aregression conditioni ) '.'Better ,lmits on the meius value of leadtins constant# TLi, and .lag time constantp TI

b. An improved understanding of the general effects offorcing function amplitude and bandwidth.c. An Improved understanding ofihe possible effects on thepilot's describing function of changes in the nmnipulator.d. An extension of the population over which the operator'sdescribing function was measured. (It. would, of course,enlarge confidence in the model if it could be shown that no t

Just a few subjects, buL many, could be described in the sameway.)A natural result cf an experimental program carried out for theise purposeswould b.. a set of definitive data measured over a wide range of frequenciesfor elementary and .imiting-case ,ontrolled elements.

In addition to all the purposes described above, it was hoped thatthe experiment results would yield the information which would permit ja variety of questions concerning human operator performance in trackingtaskr to be answered. Some of the questions which have arisen time aftertime in connection with previous investigations are suggested by thefollowing phrases:

a. Motor responsi models: Few, if any, investigators usingrandom-input describing function techniques have carriedtheir measuremerts to frequencies high enough to reveal any but Ithe crudest facts concerning the motor response (neuromuscular

lags) of the human operator.b. Stationarity during runs: Data reduction performed in

the frequency domain requires fairly long averagingtimes. The describing function which is measured is then anaverage describing function for the period of the test. Itmy be postulated that the operator's performance is far frominvariant over a period of minutes, and that, for example, hisdescribing function might be appreciably different at the endof a long run than at the beginning. If this were the case,there would be good reason to doubt the validity of the mathe-matical description under any circumstances other than as anexpression of the average performance in a particular experimental

-, * ~ c32


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situation. The nature of cross-spectral data reduction techniquesand the forcing function frequencies of interest preclude theexplicit examination of variations in the operator's describingfunction for any period shorter than approximtsly 2 min, butevidence for or against rapid fluctuations in the description ofthe operator could be indicated by considering p values in con-text with other measurements. Also, the question of whether ornot the description would be appreciably different for the first2 min and the last 2 min of the 1O-mmn run could quite easily beexamined.* a. Lulsing control as ar expression of lead: It is a matterof fairly common experience that in control tasks which

are very difficult, either because of potential instability ofthe controlled element or b-ecause of a very wide bandwidth ofthe input forcing function, the operator will tend to exercisecontrol with e. series of discrete pulses. The reasons why hedoes this are not clear, but it may be that hr can in thisfashion arrange more lead or phase advance. This behavior,howevero has some nonlinear properties aad is a possible sourceof remnant.d. Samlini effects: Some authors have Ruggested (Refs. 5,

6 and 60) that the human may belbA-'e as a sampled-datsystem with a reasonably constant sampling rnte for otherwisestationary conditions. Physical evidence for or against thishypothesis is very sparse.. If the human operator f..pproxirwatesa constant-sampling-period, sampled-data subsystem, the ef-'`-ctof sampling would be a directly testable physical source ofthe remnant.e. Nonlinear effects: There is evidence of nonlinearbehavior in both the perceptual and actuationmeohanisms of the operator. For examplep there are ratethresholds below which the perception of motion is notpossible. Further, in attempting to mKke predictions offuture motion at rates above the threshold, the operatorwill overestimate slow rates and underestimate fast ones.Some such nonlinear effects might well be a suitable sourceof the remnant.

* f. Other sources of remnant: In addition to Items a-eou ve a variety of mechanisms might possibly be thesource of the remnant.

It was not thought desirable to attempt to resolve all of tbesequestions within the framework of the present program, but certainresolve some of them. (Those which are treated, to some extent, in this

nlIt!


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report are mrked with an asterisk in the mrgin.) All the questionsand data deficiencies listed were carefully considered in evolving th erinal experimental p2Ans, and where at all possible attempts wereincluded to attack such questions within the main framework of theplanned experiments.

34

ISi


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I IVMf-PUlD ANA=?IU

Pragmatice.lyo, the most comxelling justification of any model is itscapacity to subsume past experimental results and to predict the outcomeof future experiments especially contrived to exercise the model to itslimits. The existing human pilot model, discussed in the last chapter,was cor.structed on the basis of compatibility with past results, so itlerformed the first function noted prior to the initiation of thecurrent program. Its next test was in forecast of extended situations.These predictions were the basis of much of the planning of the experi-ments and choice of particular experimental situations.

Three kinds of predictions are summarized here. The first is,fundamentally, application of the adjustment rules and rationale tothree simple controlled elements. Although one of these, Yc a Kc, hasbeen extensively studied, it is treated again here with the model topnovide the basis for comparative statements between this and the othertwo systems. The second kind of forecast is also concerned with describ-ing function adjustment, but in a rather special way. The intent was tofind special controlled element forms which would tend to tightly con-straiii tie operator's choice of' characteristics. To this end the modelis used to explore possible control situations which will tend to can-firm or deny conjectures about variability. Finally) the third type ofprediction uses the model to nake mean-squared-error estimates for sub-sequent experimental validation. In pursuing this objective, unexpectedresults were obtained which provide the raison d'etre for two adjustmentrules which previously lacked a theoretical basis.

A. M3W=WZ5A COAUIMTM~A series of controlled elements which require the pilot's equalization

selection to range from lag-lead to essentially pure .gain to lead-lag have


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already been diaicuosed in the last chapter. From there it will berecalled that these pilot-adapted forms correspond to Yc(Jw) forms ofKQ, 7e/Ja, and Kc/(JW) 2 , respectively. Detailed analyses of man-machinesystems involving these controlled elements will be summarized below..I. TO - 3r,e

This simplest possiblo controlled element Is also the most extensivelystudied, since it was included as one of the controlled elements in threeof the programs listbd in Table II.' The most definitive investigation wa sthat conducted by E.kInd (Ref. 13). Yet some characteristics remainedill defined (e.g., phase margins) or not too well understood (e.g., regression).

The operator-adapted describing function fur control of tLe pure gaincontrolled element will beyl)(j( 2s7 ..le'T, c + 1)(1

YPW)I

Documents

(McRuer) Human Pilot Dynamics in Compensatory Systems