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- hen governtrnt er other drawvimr, speci-tic,.tiora r other data are used for any purpoeez'thr tbaa In connection vlth a dea•litely relateAwover•-cnt vrocurmaent operatinn, the U. S.Goverment tnexe*y incurr no r1sponsibility, nor anyoo2lA.f-t- whatoeier; and the fact that the Govern-ment xy Lave fomdAted, framished, o0 in any waysupT.L!ed thV said dravings, sp-cifications, or otherdata i nov Zn be regarded by tiplication or other-v-sr- -2s in aW saner 1.,censing the holder or any
.her person cr corporation, or conveying any riotsor, .vrzmsslon tc Yenufacture, use or sell anyatented invention thai xe- in any way be related
IU-Muto.
ASD-TR-61-&3I
PE"*FRMANCF CRiTERIA FOR UNEARCONSTANT-COFFRCIENT SYSTEMS WITH
__ ~DEVERMINISTIC INNMT
TW!PNICtL REPOIT No. ASD-TR414WI
FEBROARY 1WC2
I ~ iL*Grr CONTROL LABORATORYI ~ ~ AAON.AUTICAI STSFmS DIVISoN
C *4PROJECT P&L 3D).* TAM U9 N1 -1go f I'"Of 28 w92
rU LbJ;VT'SIA
(Prvaed uodw CoubWa No. AF 33(8I6).?S41by Syzttm Tedhosk-v. !L.%'
Auffi-: Ju1jn W4ovItc. Pay ?.Cqiaeoo, Duawme dw,D-stogt Criuso. Jobe 4d~owseU)
NOTICES
When Cývcrnment drawings. srocificatio-s, or other data are reed for any purposeother than In connection with a definitely related Government procurement operado. theUnitod States Government thereby incurs no responsibility noreny obligatlio wbstsoe r.and the fact t t he Governtret may have formulatd. furnished, or It un way supp)4e2the eadd d&'wIngs. apeclftcations. or other dee. Is not to be regarded by Implicston orotherwise as in any manner licwncing the ho~der or any other persoo or corporaltUo orconveying any rIgbta or perminieo to manufacture. use. or eL% av patented inventionthat may in any way be related thereto.
QCa•lified requzsters may obtain co•lea of this report from the Armed Services Tech-nical Information Agency. (ASTJIA). Arlington Hell Station, Arlington 12, VirgInti
This report lis been released to the Office of Tac.mtcal Services, U. S. DspartmeWtof Commerce. Washington 35. D. C.. in stock quntitea for sale to the general public.
Copies of ASD Technical Documentrz7 Reports Should not be returned to the Aero-nautical Systems Division =n-,et return is req.ired by .weaurity cor.nsderations. con-tractual oblptiona. or notice on a specific docuwme.
M - May 1t - 3D-13to0 1331
"11.is repart, represents ore phase of an effort directed at the use of per-forrwne criteria as elenentz in flight contrel system opttnizeatio studies.7-e kNieaxc% rported ---s sponsored by the Plight Control Intorstory of theAcrorautical Systrs DIvteion under Proje,' To. 8219. It van conducted atSystzes TC.ennology, Inc. under Contract Nto. AV 5,(616)-7841. The AnD projecternineers were Mr. R. 0. Anderson and IA. L. Schwartz of the Flight Controllahonatory. The princil investigators were Messrs. D. T. Meihier and DunstanGrahan. The principal contributors to this report are listed as authors. Messrs.D. T. MW-uer acAd Dunstan Grshans, Sy-te=s Technology, Inc. teehnicel directors,piwined t.he geieral a.-oach nad contributed many dreails. J. bielkovitch, $21proj.ct engineer, wrote the report and originated the mstcrel of Chap-ers IVa= V. hay ftdall.no originated Appendiceb A and B, and John McDonnell producedthc =r-.p,he "d detailed relatiorxhipr of Chapter lI.
The authors wish to exrress their thanks to v--. V. J. KKcacevich for hisdiligent work in gener•i•g the M2 and 712279 general and standard forms, andto Lt. L. Schwartz for his thorough check of the AE iteri.vation and for evaliable ccments. Acrnowledgment is CrstefelL• made to Mr. J. Taira, and toMr. H. R. Pass who contributed zany of the calculations, and to Messrs. W. E.Ellis and R. N. Nye for their careful work in preparing the final ilnustrationsfor Lhis report.
Permissio, haa been granted by the AIEE to publiab material from theTransactions. Volume 7Z. in this ASD Technical Report.
AsD TR 6-0 5om
ABSM|
Pcrformance measures and ascoclated criterla for lincar constant coef-ficient systena forced by deterministic i r.cut are Inventleateed, vtth particu-lar referer.ce to flight ccntri.d cystens. 1i is shown that the, application ofperforn-ncc ensures i flcilltatcd by substituting for the actual flight con-trol systm " ý.u-ivalcnt lou-order lineariýed system having similar dynamiccharacteristics. A critical survey of current performance neasures is given,and iwv suthods for the as.lytle evaluation of some !ndicial error measuresarc presen:ted.
Th-•.u ,ritorla are examined vith regard to their validity, selectivity,and case of application. Nir:.alized pr-sentations are ýued so that practicallimitations on the tine scale of the response (e.g., due to power/irnrtia re-strictions) cae be taken Into account separately. It is concluded that mini=fl. (intecr.eec tine moment of absolute (.crrr) and ntnimr ITE2 (integratedtine moment of error-squared) have particular nerit. The lIAr criterion yieldssmooth irdicial responses having little oversnoot, but Its analytic descriptionis complicatcd. Of the eriteria-ninina IIE- (integrated error-istarc4), mini-num MT2, l12•-, IT'E2 (Integrated tinrst-, second-, and third tine sarcnte oferror-square.))-Isa ha sIgple analytic fo•rs, but selects poor ind-ci•l• re-sponses IT32 respooses are an duod as those selected by lITA, but IT352 (and'alo iT12) aaljic exoression arc too complicated for general use. I se-lects moderately smooth and veil-damped respons=es (less good than fTAE), butpossesses tractable analytic form-. Thurefare, 1112 is reccmended for analyticirnvesttCtiors, uheress IMAR is prtfe.-r for optii=sation using analeg coaqiters.Some other criteria also anpear proms!rng for use In conjunction 'with digitalcomputers, but require further invrastigt•_on t deternine tics valdity andsel-ettivty.
MBILICATIONFXI
This report IAs been revic'ed and Is approved.
FOR TlX MWANMM~:
Chief, Aerospace-cehtancs BranchFlig-ht Control laboratory
ASD V 61-,o1 Mi
CONT'KS
Cl-apter
f•,IfRODCfON . . . . . . . . . . . . . . 1
I CAWCUTATIOI OF DXNAIC FP EMfCE ...........
II SYSTEM. CHAMC STI'.C3 . . . . . . . . . . . .. . 11
Formulae for Indicia] Reponse Characteristics ....... 21
General Correlation of System Charnteristics ... ...... 30
III INDICIAL ElROR R . . . . . . . . . . . . . . X
Derivation of IE2, ITE2
, and IT.......... 39Evaluation of IsL, ITE
2, and 1T-92? as Perfcsnee Measures. . . 45
.rivwtion of IA•, ITAE, and 1T2AE ........... !A
Other Indicial Error Measures . . . . . . . . . . . . 61
Other Standard Forms . . . . . . . . . . . . . . . 60
IV IWECT OF A P TIM JAG UPO u DICIc L niz MoAS'jpS . . .. 67
Effect uf Time Wan on ITAZ and ITS2
for 3econd-Order7ero-?oGtion-"-rror Systems . . . . . . . . . . . . . 69App-oxtmation to Performance Measures of Higher-OrderOptimal 7.ero-Position-Error Systeo .......... . IEstimation of Delay Time . . . . . . . . . . . . . . 71
Interpretation of q, . . . . . . . . . . . . . . . 75Example of Approximate Calculation of Perfora-nccW-asures for Hieh-Order Systems . . . . . . . . . . . . 81Further Approximationc- to the Delay Tim ......... 81
V FmMANCE WAS=URS FOR GENERAL DIMShISTIC IT .... 89
IS for General Inputs . . . . . . . . . . . . . . . 84
Genernl Rclationc Betveen Certain Mcauurcs ofOpen-l,4p ri Cl•os•-Loop 3Z-Lysc . ...... ........... . 5
SUtWARY AND COWCIUIOU. ........ .............. 103
FannaU ... ..........................
ASD TR 61-5o01 Iv
Apr-tiiver
A AIALYTIC IXALUATION OF .AE, ITAE. AD IAE PW AiCE1nsU FOR WUT-2M43ATOE 23 -ORD S=M .. I. .
Clcu, tic, of the Inm.vLai Error Responzc of =
Unit-Numerator Cecond-Order System ... 11.
Eveluation of lAS .. *.. 1.
Lealuation of ITAS ................ 116
Calculation of IT2AE ............... 113
B A M4HOD FOR EVAIUATIM IAE A•D ITAE FOR THIRD-MER SYSI • 123
A Wthod for Lsullmtlrr IABE and ITAL for Third-Odcr Systems. . 1°5
Ihterm•irt-Cn of tie Transfer Function of a Pall-Vave Rectifier. 125
Characterization of the Input Signal, x(t) ........ 127
Characterization of the Output SigrAl ......... 128
Evaluation Of C(t,m) ............... 130
Numerical Che•k of Epres3slo for je(t)I ........ 15
Evaluation of IAE ................ 136
Calcumletion of IaE for Optimum Third-Order System ..... 153
Evaluation of Iw ................ 15j
C DETAILS OF THE MCAMI4E FIOGHT CCM!ROL SXSYM .• TODMUOMRRTE TH• V EvIS2IT == COCz ........ 157
)ctsils of the Example Flight Control SystcM Used tuDemonstrate the Equivalent System Concept ........ 159
ASS TR 61-o31
I I
I . Open-Loop Bode Diagr of G(Qo) Transfer Function . .... 6
2. Comparison of Exact and Approximate Closed-Loop BodeDiagrtme of O(jwo)j1 + 0(3w) Transfer Function. .* 7
3. Idealized Partitioning of Indicial Response ....... 12
4. Comaerison or Actuol asid Approximate Phase Margin. . ... 18
5. Cc;parison of Actual n Approx1mat Bandvidths ...... 18
6. Compariscn of Actuc. 2-rd Alproximate Settllng Times ..... 22
7. Comparison of Actual and Approximate Equivalent Time Constants . 24
8. Peoparison of Actual and Approximate Delay Times ...... 25
9. Ccmparison of Actual and Approximte Rise Times ...... 26
10. Pesk Overshoot of Unit-Numerator Second-Order System . . . . 28
11. Time-to-Peak of a Unit-Numorator Seccnd-Order System . . . . 29
12. Settling 'Tim- ef - Th~rl-Order System ......... 32
13. Rice Time arc Peak Overshoot of a Third-Order System . . . . 33
1i. Closed-Loop Bode (jc) Diagrams of Stndard For ...... 46-47
15. Indlcial Responses of Standard Form-s .......... 48-49
Is. c-jpA o Ana calandExperimentally Determined10 ., WETsr and o IfAn icrfors o -asi.res ........ 5
17. Cuaparinon of Analytical. and Experimentally Determined
IAE cr4 ITAE for Unlt-Numerator Second-Order Syst. .... 58
18. CcTcerison of Analytical and Experimentally DeterminedIt
2AE for UV.it--Ncx:t--•or .cond-Order Systm ....... 59
19. Poclm. U~ Loain f UiLNsoorSewun-Opljr Stsxulrd For.&. 32
P0. Pole Locations of Unit- -.-erator Thbrd-Order Fo..• ..... 63
21. Pole Locations of Unit-Numerstor Fourth-Order For--m ..... 64
22. AE for a Second-Order Zero-Position-Error Syntcm with a
ime Delay, T.... . ...... ................. 72
ASD TR 61-5o1 vi
23. ITE? for a Second-Or-ier Zero-Position-Error System with a
Time Dela., .................. 73
24. PA-p Appro%-st!=t- to Upt!._l System .......... 74
25. V-azotion or Dcezy Tmc vitn Integrated Errorfor Otimum ITAE Zzrc-r'osjtIon-Error Syste-.s ....... 76
26. Variation of Delay T1e with Integrated Errorfor Opti-um IE2, IE2, 152E', and IT
5 System ...... 71
27. Variation or Delay Time vItn Integrated7,..r -or Butterworth Filters ............ 78
26. Indiclal FLsponaes of iuttcrworth Filt-rs of FirstThrc•ugh Eighth Orderr (reproduced from Ref. 37) ... . 79
29. Effect of a Time Delay on ITAI. for a Unit-lhrerator
Sýcond-Order System ................
30. Ccoparisot. of Measured ean Predicted ITAE ...... ........ 83
31. indicial Responses of ITAE Standard Forms of Second-Throug c ,ig:t.h-Order (reproduced from Ref. 37) .... ....... 84
32. Correlation of Del" TJme with Coefficient of an-I forith-Order Unit-Huerator ITAE Standard Forms . ........ 86
33. Correlation of Delay Tine with System Order for
Sor=ai.ed Unit-Numerator Standard Form .............. 87
34. Rectifier Transfer Char-cter tle.....c ...... ........... 125
35. Inversion Contours .......... ................ 126
.%. Contour of Integration for C(tm) Coefficients .... ....... 130
37. Calculat-d Value of Ie(t)! for Third-Order Optiemi ITAE System
vs RNuber of Harmonies at t . 4.55 Normalized Se.. ...... .. . 7
38. Calculated Value of Ie(t) for Third-Order Optimal ITAE Systemvs Niser :f FHarmonics at t - 3.20 ,ormalized See .... ...... .13
Ny CacuLated VaLue of je(t)i %or Third-Order Opt=l rTS Systemvs NHmuer of Harmonica at t - 6.13 Norm•lized See .... ...... 13
40. Calculated Value of le(t)l for Third-Order Optimal ITAE Systemvs libter of Harm=onlcs at t . 6.41 Normalized See. .... ...... 140
41. Calculated Value of Ic(t)I for Lightly Damped Syste2"vs Number of Harmonics at t - 0.4105 Normalized See ... ..... 1.1
ASD TR 61-w0o vii
42. CMCl~ated va•ue of Ie(t)l t•o- Lightlj Dsped SZys-vs Ihmber of HarmOnics at t o 3.664itomnized See ...... 142
ASD Th. 61-5woii
1 Stability Mesures for Constant-Coefficieot Linear Syste=n. . 9
II-A Single-Loop r:edht•.k System Cr~cteristicnersosfer Function Ktsures ............. .3
II-B Transient Response Characteristic3 ........... ii
III Correlation of Se-ond-Order Unit-1hoeratorSystem Characterlstie ............... 19
IV Indiclal. rror **&sures .............. 37-38
V Lieal. lorms of Integals Required for Ij, ITR2
, rmdITE for Systems of Pfrst- Through Fourth-Order ...... 43-•
VI Standard Forms Eor Unit-Mzcrator Systems•merator is f0O bere n is System Order) .... . . . . 51
VII Freq.ency-Dcsn Charfteristics of Standard Fors ..... 52
VIII Itlic'-l Response Cearscteristics of Optimal Systems ..... 60
AisD TR 61 -501 ix
NOMCIARMZ
A~t) - ýc7 (in' Al-per.Ix B)
[A) )'atrix or oeoffilclnt,' (Cee 1"~bln 7)
a Coet"Cieint -if aperis41v term in tr,-1r responxe
23 Arument of CMux,* function ( j,ý)
skj Azeu.%nt of fGnsrn luictions( 3/2)
;jw=$ýcft of 'ft-- f5C1Ct'.O, (. ',2)
al Coefficient of 21 in charncterlstic equaucn, &(a) o
aj Co-ffIcient of n- i' 'IrenocInAter Cr E(fC)
2 Set Fq -3 nd B30(ApperAix B)
B(z,a0 ) -I'(z)r(C0)/r(z + CO)p the seta function
b. - eg
b Coefficient of e- inj qh
bo Normalizei Coefficient of r,'In deonionator of C,'s)/li(s)
C(s) -/Ct
C(t,() See E4 B-18 (Appendix B)
C%~) /- ~iýC(t,:) -
C2
C3 Paths of integration (s". Fig. a.3)
I --/ -P (in AppendLAB)
e COefficltat Of 4 1- i:q h7
CO r- IIC .,eli'. , - dcv. mirneo of C(a)/Bt(s)
ASD fli61,41 X
C Peak oyerskoot
,(t.) RetsK03e MCe
D "sncfer function dermitor
di Ooefficient of co'in ot~erstor of E(s)
'-(Z) .I ptoi
eA -24O18
-0 Error responst
Gr C(5J).ft(-) G (L)/I + G(S)
G_ E(:)/B(3) 1/1 ~
go Arqg.snt of Beta functI~o
b Ik Ej B-4i4 (Appendix W,
it, See E4 1-83 (AppeodIn B)
b(0) - 14 Pb-, the trenafer eb t~rls^ýcel'~ the full-wave lioear *ectifier
h+ I Tat p~rt or transfer cbaxasteristtz for % positive or !Iegatiw-ph- respectively
IAS Intevml of*bsolute eror, .`*j*It~d-
XE Litegral of irror, jP'0(t)dt
AVID TH61-5M1 xi
XE2
Integral of error-squ.red, P. (~D t) ý't
MTE Integral of timc-elg1toX aLbzolute error, fl'tez d
IT. Integral1 of tiae-velatted erwror f.'~ te(t)dt
r=2 integrali of time-veisbtc-i error.sqTmzv6, J.' t am 't)
n2E2 zuteiru of ti1.e-sqimredl times error-somedre,. t2
r(t) d
IT3E, Integral of time-cubed ýLmes. error-squezed, t3 .'t 2C
rzOE Integral or nth~ time-xmaot of error, f.' t0
(t)dt
fl¶7 ttldt
111 Integral of a functIon of error not involving Usme expluc~tly, foo' uWc
16,(z) YNcdtfie' Aessel function of the first klW~
ja~q) Beacel functio-
K2
I See Eq r-19 jEpl-en'x )
k Coefficient of oscinakrr t~i in error r-24ponse of tnirn-order sfaets(Appeclix B)
Sei Eq B-61 (Appedlx B)
L. aplace transform operator
Inverse laplace transform operator
10 Noiling moet derivative with ri-spect to sideslip
See E-1 -61 (Apperzlix D)
mp Peak mg-ificatlon ratio
a Order of 'Larmr~ct in output signal
l.sD mt 6-501 x~i
N Transfer function nuzrotor
ni Number of ciockucc cn-cimlricrjts of -1 by C(S)
-90 Yawing mncent ae:1'~n-.ve with respect to sideslip
N, Set Pq D-33 (Appendix B)
* S-t-tion or infinite product index for Beta function,
* Sy~teu order
P ihmber of poles vf %s5) in right-,alf Plane
21 Coeffictezit of k1
in the nmaerator of Eq 19
p laplace transfer voriabie it s is ±ndepe:,dsot variable
Pi Colefficient of 11 In the numerator of Eq 23
Qj Coefficier* Of 1." in the densslrntor of Eq 19
q Bq-Aunt (Appendix 2)
qd Coefficient Of I" In the dereanLmtor of Eq 23
q1 Co0efficienlt Or s It, Ctn)/fRs denc~inator
It(s) tr(t)
r SimatILoc Index
r(t) Treput
J.- ~ *5 [5:.st-+a .4a
a laplace trsnsform -ariable
31-1bs sin- b,
T A tine conns-nt
I- Atr :-% cm-~
T, .(ln(a/kj]/y - n. ti~e when uatio of envelo'pes equals oreTe flevtor se.v: -.lze constant
Teg Equdvilenc tin: cmbn.Ltn
ASS 6'-,01 xiii
T1 Closed-loop tdn constant
t Time, or rwmrl.zed time
t1 -firc or n^rum-slt4 time
tdj Delay timle
tp Tiw to peak
ta Rie tine
ts Settling time
U. Set Eq !ý4 (Aprendimnr
V An error function, not involving ttw etplictity
V(W) = ke-nt
w.-) 2- Eq B-69 (Appeo~nz B)I2
d o, + " .- I t*
x Be I..t) 1-o I 2t
x or1(t) !n-¶xt to #?Uf.%v^, liner rertlfler
Y Open-loop tr .isfer function
y . InI:)
y)or 0Ottut of the 'l-Avsve linear rectifieryAt)
. Nlumber of zeroo of i + G(c) in rivptht.lgf plane
Zor::(t) See Eq B-37 (Appenixi' B)
z - X* J) complex Narlable
I Real Pa~r' of c--7ex roota Of tbird-order art,.
Q(S) Vf~mzrator of G(S)
:5 Ieagnrary Par' Of cccplex rootf Of tbird-ordler 3yste*.
19,14i- or (Appendix A or.Ly)
P(S) r1ezom~natar or G(s)72
'Wx Cama f'onticn
71 Real roo0t Of tbird-order syatra tran~sfer functicon denmirntor
Y thplaze trosforo VI~riabio (in abapter III)
I2 Ste Eq B-6! (Append%. B)
-A3 -tan-, 2aa
a2 - 32
S A time coon-tant5e Elevator deflection
£.1 a2 01,?,.. So.:-wr factor
"41'J~) * * + J, rplxvariable
Dampinz ratio
;j Closei-locp d4-ing ratio
ASE -R 61-301 xc
o Angle of pitch
G(t) .Pt + * - ,
x Open-2ocp gain
C, Pqialization gain
KU Gain Of rýrwv tzr plus a.Ulller cucbinstion
K1/•C (,.in mnrgln
Kt/IK GCcialIceII gain UMargi
S •,Open-loop time constant
1 -S4o0
p Wbgnit%-.e of laplace transform variable, a
PC Generalized crossover frequency
t - Re (0)
4 DazpiAg ratio cf c3-5ei.ioop roots
o Real part of colplex variable, a
a, A particular value of a
"I Time delay
o: -*e- T1 e
92 Pixe margin
On Generalized phase margin
E - t ar 7 M, (pta angi)
Normalized fxvqýuAcy
0 imaginary Part Of coXVIeX variabla, a a +4 jw
A~TR 6:.-5oi
0 Laplace transform yarlable vb•o x is the independent virlable
ab Bandvidth
CDC Crossover frequr,-cy
9I Closed-loop rvot natural frequency
e() See Fq D-72 (Appendix B)
1 Undaeepd naturml frequency
1 Peak maniflceation freq7.ncy
1, Frequency of instability
.4 Angle
I I Absolute value
I I~b )Mnitude in decibels
~, Partial dertynktive
Srnatiom
U Prc•uct
ASD . -=01 xvii
Dynamic perfo=3nn-e Is only ore of rany fa.tora to It crtaidered in
assessing the merit of a flight cono:o1 4satem. Coat, ueight, reliability,
schedule, etc., must also be taken into account. The beat choice for ony
given requlrement can be found only by veighting each of these factors accord-
ing to their relative Iaportance. Cost, weight, etc.t are measured directly
&a unequivocally In termr of dollars, pounds, etc., but at the present time
assessment of dynamic perforance depends heavily on intuition. '!is is cot
du- to aso shortage of performance wasuree; many have been proposed. The
Problem Is which measure or combination of seasures to choose, and, baving
made the choice, how to apply the resulting criteria to the system tnder con-
sideration. It is to the solution of this problem that this report is directed.
The work reported here vwa performed under an Air Force contract directed
at providing
1. A foundation for the specification of dynamic performance
criteria for automatic flight control system
2. The methods of analysis required to apply uc-h criteria.
It vas convenient to present the results of the study in two part=. The
present report deals vith Performance criteria appropriatO to deterainistic
inputs; randm inputs and associatvd topics are discussed In a subsequent
repcrfl.
Chapter I consists of a broad discussion or dynamic performance, and out-
lines exact and approximte calculation procedures. It is shoWn thet although
actual flight control systems are descrited by differential equations of high
order, moe tractable equivalent systems of low order can provide a convenient
and sufficiently sacurate basis for analytical optimization techoiques. Qasli-
ties defining the merit of the dynamc perfamance of given systems are out-
lined, and their interpretation into rmuerl.s peerformance Measures is dic-
cussed. Performance criteria a t.en defined in ter=$ of optlrmi ralutc of
xnece measures, and the requirements of validity, selectivity, and ewe of
application tmt a Cood criterion should possess are formulated.
Ibnuacrt released by authors 11 Deceaber 1961 for publication as an ASDR eport.
An mR 61 -5w
Chapters II and 17I survey published measurcs a dysmaic performance.
Chapter 11 deals prlcarliy with criteria rot explicltly involcing Integral
ftuctiors of error (e.g., phase rmrgin, bandwidth, race tli, etc.), and pre-
sents exten¢zve correlatios of tiese criteria for second-order systems.
Chapter III is concered wits ndicinal error masures; i.e., criteria directly
measuring somne tZstgrased !'•nction of the errnr response to a step Input. This
clas: incis-les sore of the rost useful crlttri•, such as IM (integrated time
cer•,t of a.oolutte error), Ie (integrate.d error-squared), etc. Calculation
procedures for these crit-ria are presented; in particular, analytical expres-
sions for ITAE and some other measures not previously expressed in analytical
- -.ir- -•b.hished for the first tire. Chapter III concludes with a survey
of the closed-loop and ccpn-loop charsntriatics of those optimal systtma which
-rinntotc particulvar :rndicial error ressures.
C!Apter IV derives geersi exact for---= expressing the efftct of a pure
time loG on syste. idzicial error measures. By comining these foruslae with
the ob.servastion that the responses of high-order good =dC opt -,4 systess
closely approin•te to the resposes of lover-order systems vwil a tire delay,
a -ctnod of sppro'týizting to perfornAnce reasures of high-order systems is
drvissd. X- ecenple is give. to show how this approximation simpilfies opti-
sltatios prozedurces
Chayter V Cenerai•tes some of the fcr-sIan for iroicial error reasures to
cvnO tate t••o crre.spoming nessures for inputs other than steps. Relationships
be open- sA1 closed-2oc- 'oc-= of certain measures are also discussed.
Chapter VI discsses -easures or sensitivity to parameter changes, and
pover rciree..t, ans prnsste the conclusions reached from the present study
c,. zriteris, for inetcroin'stic inputs.
2
CHMTM 1
MACUIATION OF DMMNC PDWC
The dyr-4-c pertormance of a control system or element is assessed by
comidering the folloving fectors:
1. Stability2. Response to desired inputs3. Response to unwanted inmptsl4. Accuracy5. Insen-tivity to parameter changes6. Power and/or energy demands.
The quantitative specification or dynamic perforsance c-neistt of choosing
measures or tme alove qualitlos (either -inciy or in coblunatlyn), and setting
desirable values or linlt3 upon these measures. Defining a "per foreance meas-
ure" as a quantity characterizin6 dynamic perfor=ance, the term 'perfiemnce
criterion" may be defined as a standard or reference value or a perfcrvance
measure which provIdcs a basis for a rule or test by which soe aspect of
dr-. c perfr---nnce is evsl',ted in forming a ýudqment of system quality.
Pcr a perZormance criterion to be of ue it must be valid, selective,
and readily applicable. Validity implies that the criterion is associated
with desirable performance characterittics for the input crviroent of interest.
Thc -rcquircnn of selectivity decend3 thee the criterion should differentiate
sharply between "good" system and those which are merely "acceptle-." For
the criterion to be readily zpplicable, its exprecssion i. term of system
pa-a-meters should be compact, %Ac convenient procedures for its evaluation
should exist.
In principle, the p:oc-ss of designing a rfstem to meet the specified
performance consists of calculating the dyn-ý-" lcrforance, apyirg the
measure to the system under conxideration, and thenp If necessary, modifying
the system so that the specified pericrsance is attsined or approached as
closely as Practicable. For convaueuisuu in isplemetinUg this stap-by-step
sequence, analytic pzucedurea for the calculation of the parformece mea
should be aval able. However, mas:y flight control sMstes are so eo.plic. don
that their response to any given input can only be described by a differential
equation of high order, or by a large number of siLultaneous differential
equations of lover order. Physical realities thus tel te be o tz.cA !n a
fog of zathe-atics. To avo~d this situation, a :ober of simplifications can
he intriun d 1-. the analytical repmesentatior.s of actuAl flight control
systees. Poreort zmong these Is the ass=mption of small perturbation: ant,
hence, ltneerity. The further a-umption is then cmonly made that changes
in vehicle configuration and cnrirorent occurring during the motion are
ctsli, so tkat the coefficients of thn differertial equaticns are effectively
constant. The resulting linear constant-coefficient equations arc still of
high order. For purposes of calculation, a lover-order system which posatsese
(for a given input) approximately equivalent domnant amoe dyanics can be con-
structed. This artifice is particularly valuable in tVe ci;culation of perform-
ince criteria. It will ibe demonstrated beloo 'bat this =mre tractable equiva-
lent srteu Ja relatively easy to deduce from the open-loop trcnsfer function of
the particular loop that is being stAlcd. later in this report it will also be
shown that the result provides a sufficiently accurte basis for the calculation
of perforvance criteria.
The geze~ul procedure by which ecuivalent systems ar derived is ,oat
¢clarly illmutrated by an example. Cinsider a pitch control system for the
fighter airplace detailed Jn Appendix C. The open-Icop tra-nfer function for
the pitch loop Is
0.060) Trnsers.
Airplane Funcsteon (i)
Controflar Transferunction
The Bode diagmra for G(3() is snown In Fig. 1. The closed-loop system has
three regions of interest defined by
Io(=)I >> 1, over which IG, *.•1
Io;(n)I << 1, over whichjrIf IIIO(J.) I
The form of the closed lcop transfer function, G O) , in this lost
re.ion 4IPnes the "do=inant =- 'c" of the closed-'icp syste dyzs-,ic response
for Impulse and step inputs. In most cases G(jzl/j) - 4(b)] in the regionwhere ;o(3•)I i of thz order of unity can be approximated by a first-, second-,
or vtird-order syste, the modes of which vwlt dctermine the major features of
tin response. The open-loon amplitude asymptotes of an appropriate equiv-cent
system for this exsnp2e are shown in Fig. .
Applying this approximation tc tht prer.nt example yields the closed-loop
(to)) Bode diagram of MS.- 2. 1he B'- diets for the esact closed-loop system
Is also shown for cemparatlie purposes. It wil be observed that the error of
tL. approximAtion is small. If greater accurzacy is required, nore complicated
open-loop equivalent systems can be produced by retaining more of the terms in
the complete open-loop transfer -function.
In the enmople cited, the servo break frequency is of the order of .. Hore
Lypiý•.iy, this frejeLzy will "e >> xv; the effect of the associated high-fre-
quency leads ar. laegs can the,. be approxiza-ted by replacing them In either the
open- or closed-loop transfer function, by a pure time delay term, e-13. A satis-
factory approximation for the time delay in r - -(Tlees - 'lag)hgh frequency.
(A~termativc approxinstloýs ar- discussed in Chapter IV.)
In general, airplane transfer functicn break frequencies and tine constants
nrcspaccd so that C($v) z Um, rsgiw. of crossover can be satisfactorily approxi-
mated by a system of nD. more than fourth order.
The equivalert system concept can be ap;lied to form rnuerical rsurmes of
each of the ss.erts of dynamic performac c acIz-ed at the beginning of thiz
chapter. However, -etcrinascion of stoablity Is us'slly (koly slightly more
21 0
a),
Figuxe I1. Opta-toop Bode Lia,,Or. o G(,ýA) Tra.nsfer PFrnction
*2e .C PrsllutE - 1)vrm~eCoajo
bc~cDý&w-sof -7L ý47 T vvýýer I 0
7a
conplijted fcr .the eomplece linearized system model then for the simplified
equivalc.t. ?ablc I =srriszes techniques for Investigating rtab2Vity. Since
the as W epplication of almost all stsard performance criteria requires
lost W-e system shall ve stable, the use of one of t'he tests listed in Table I
:orc a prerequisite to tibe applicatino of more refined criteria.
Of the retaining aspects of dyrnaic performance listed at the beginning
of this chap•er, this report is principally concerned with Item 2, "response
to desired !nputz, and the main body of the von, presented herc Is concerned
with the !.ecrpratstioa of this quality In numerical terms as a performance
&essure. Investigation of the --•.t'o ceiacity to suppress most unwanted
Inputs requires cons!&lrstion of gustcs noise. etc., which are best described
tn statistiJcal terra, t-Ac then fal~l outsidc the,--: W? thope seeit re:port.
Hovcer, teo= unvanted insuts (e.g., engine faLlure) are vell-describ.- by
deterministic expressions; for these inputs the mezhods of the present report
are quite applicable. inc topics of accurazy, lnsensitivity to parameter
cianges, and power/ener 5 y dcrmnds are discussed In Chapter VI in terms of
.iAl-eI performance measures.
tonventionasy, Lic quantitatire interpretntion of the response to desired
inputs is acieve1 by scarns of me•sures of the motion follso'm s. limited
selection of determinrstic inputs. These inputs are:
1. Impuls (Dirac delta func-tion)2. Step3. RaAPhs. Power srles5. Sine wave.
Besides their conventional nature as test Inruts, the first four of these are
saso representative of a vile variety of flivit contrul system -mmand, dMa-
turbqnce, or initi.l condition inputs.
The well-krnn convolution relationships of linear theory (4-.g.9 Duhnel's
inceg"l), scat sy•p.-i•e.d t-chnricuer dec.ribed in tf. 36 ar•d 72, rnable tU.
response to any •et•r•inistic input to be calculated from a knowledge of the
response to any of the inputs listed above. A comprehensive surrey of perform-
ance measures that have b-ee propoced for detaxr.Instlc Input: is presented in
the follovwn two ct*ptezs.
a-a
?MIZ I
N.C.,t.s f~. M.f -- & Cf.Ara.e1ristle equation, a(s) 0 Pushe of vmxI&,tio. In .lgn
dsAW*$ CO,4±tlo- X0 rOt.TistIc q-tl. 4(-, , 0 71.1.11 v~w e0s oflid) cofficiens;
Oteo.oIft.4 Ru.lsao 1110 HMIf~e 104 at.,1.11c equgtlio,of Signs &M 0 -&W )... Maher CC '"I.Ll000 I..
7.OslbCbmle.rnsti equati~ ~ ~ ±on, a,*&) - 0 Hurul.1110 fotetlt..
r-o,.lloed 710.th-110lt: Mol41110 'etweerl.tlceV.tlont R-.t1. test functionsor C. 0,ItsCriterion &(A) -0 : 4(x) ot10010fo 101 1fild
LEpoAf '1.1115lkrl of oe0.ficlent., ( A]; oe. 5 701U,. r,at.I f.- or Uo.sytm1galoss I h 010~ JAyo.tr10 04111 [P]
(o-[A] [Z]
lakw0115 Crieloor., Also M-s-eotr1.tle equation In f=,E ' O -4.. - A(2 - jYj(cw!) Hoot. of A(.2) on4 b1(.2)
motor of .I.01r01.sto N, or -1. a-5.0401 of %-#o Ps of No,) - 0 in
C.right holf pla-
CwyluatOpcs-Iscp 1001.01. function, .n 11±17*t1
9-ta s0.1. tability
Mob-e of e01000.of -1, -A Cs._b1 of seýs In pC.) 11111 real partsgstlr 1100 o, or daVIsg mt.110 1es-
G3.rllo C rnoclto Op-oo m~e f--Cs 1 etefe 9~.o n
TA=2 I-.tV4ILITl MAU402 Fdt 020270 T-COOF.4102? L02EA 012202
.021 DI= M%=IPLIC&I.Ca 21u2612 A=2 AIDS; leMWX
~~ct ~ ~ LUcoeft3~t ,, ~ U~ ficients of Cass sign - .ce.tc-*3"7, tout not 9odffcient, 00.02024.tfor
rarI.tlonu, of sign of 4t(-), [(t.(.)j.-
0.1.t'so v0200. of als) 001 of~lcts -'-l2Oo.2tions for clot of 6(c) .,o'ýO (.1 > 0), to betopnl- .02 real .oe: - o 10..1.1-ic of u. .. 6 Aj =1. 11o (] add [so. selj
* -c f. -. *ýl In -19 All coef-!tI-t. 1Z Sae sign ano a decessr.ooy uo ot onfflelent, condit0o for allroots to t.an neg~tire real parts LeCa than a .ogned "a'. or -.
?coyide necessary and sufficient condcitions for stablhity; also detec.2.e numer ofP-oth too-t !.-otion; root. In right half h1mum. Routh'i Algorithm i a0 Ssimle aid for deveoopoet of
b1~her-'rder test fi..tions. One test function 1s crttical %iben a pargater variationtauscs a otoog too.ý et.Oflty to insotability. (Rtef. 22)
* R-,.th teo toi... o;i- "lruit. OrcId. necee-ory and o~fficient conditions for MOCt0 to ban real purt. 1ess tha.4oto,.nant. f-, .0.11! el *o.l..d .011... of -c..c:.orcterlstioc.4t0 01
lyotc- 10 .ye-Ptfl.11cs nta10 if the m-rlo J0], ohito Is2 ri
too~iee ~f,,. fro 0toO ~'~[PJ j.P~), satisfying the stri. toto [A~JP [IJEL] -[-(I]hT,ýýtricratrs, [] cl t, of [?] are found fr=the 02. o +. 01)2 tlowltsc.Ots equetito. resultiog from
eximi1di0 the &,stch, eqution tav 0e. The tbecred can al"0 be stated Io terms of theLh.Po-~ff foorolon, Y([o]) -o 3P1]
lyx o4 rlotoeo~ooo ofcoo. oht~tO O fltto of 00121 or intbhilit.,.oooof o(oý) and B(.;-) In001 tolo.rooAo) 0.13() 0oeoop. real, .01 positice;
and tbey alternate in tO. zcqmon#rol A D A D, etc.; usually aplied graphically.
1.02cr of C~loct , -f - 1 .02 .- cco.4- r of n od,Z.I.p4 2 oftI 0(3% -Olnr~bujbet plf anso. tOdLoarily00rof zerns, F, of 0(& 1'r applled graphically -. -"Ir ;act -r G(J. od ý diagram..
3s1,. 00 g1t, "in. riusnpc mees..y for otarl. z-ability.ooro toot~lloyI'ctvol_________
10.00 aargi., n~. T.c~ ~ ored for neutral stabUilty, .202 gainbc! 01 constant.91 )- wor.ooI stability ____
Peak ..ooiZ.stsooo rati., mp reooluces ==tIs= dioed-ioop resonance. Ubually deteraljoed frcý-cm oloop ýplot. (pails,or logerithmic gLoinphase) and clotod-boo, overlays (M otreles or 31000. dr
3000cr If o'olrclemeots of;' and vs C . -r0c of t-r- of 1 (o 0 uft200 be" cool parts pn-ter tO.j, or dapingnot, of Inco X.2() otOro ot o:.10 10 .OrtioLy "plied with G(s) 301. diagras.
tre.e too. I~ orIopo Itoll
Gteroltocid gain margin,
USU.o) L -I GIr zr000e g. eired to, schoser routs .102. specified I wrt
top iý''12ofcoo-op Copoeafn-w I; syte trasfe cberat~ortic~.oc coot. knowedg opcf Inp: ct.
9 2
CHAPTE7. 32
Sv=CMmAURXMa-"M
The term "system charncerlstie" vwill be used to denote all p•eorfance
mensures not expressed as explicit functions of error. For example, phase r
Gin 4 ti.--to-pech arc both system charscterlstics, whereas Mr, is classi-
fied as en "'.i-cial error measure" and is discussed In Chapter IIn. Although
an infinite number of syst= characteristics could be devised., useful performance
rm.asure In practice are obtained only fro. characteristics directly descripu e.
ot the systes transfer function or the inpulsive or In-iciel (step-Lnput) re-
sponse.
Tables If-A and fl-B summarize a bread cross-zection of system character-
istics. All the characteristics that have been found in the references listed
at the end of this report are includcd, vwih one exception (the product of peak
overshoot and tine-to-peak), which is discussed on page 27. The characteristics
listed in Table fl-A are quantitics directly obtainable from G(s), G(cm), or the
closed loop ro ,. . It Is possible to gererU-d-e some of the
G(O) neasres, such as phase nargin and gain malgn, Into analogous quantities
for G(s) for.-. Apart from the error coefficieont:, the remainin~g entries on
Table fl-B deccrihe the response to a step input: and wIll be called "Inilcial
.-- posse characteristics."
Ideally, the indicile response character.rtics weld partition the response
into the reglons indicated in Fig. 3. -Dead tine" in the tine to attain 10 per-
cent of the final value, rise tie in the time to -_o from 10 percent to 90 per-
cent, and decay time Is the tiL-e for tLe transient to decay from 90 percent to
witLin 5 percent of the final value. The sum of the dead tine, rise ti=, and
decay tine Is known as the scttlng tine (*.;tch can he deflned directly as the
tine for the indicial response to rea~h and thereafter remain within 5 percent
of the final value). Unfortunately, only the settling and rise tines are read-
ily related to transfer function characteristics, so the quantities Involved in
the Idealised partitioning of the indicia) r'cponse are replaced by the some-
what orerlapoing measures indicated on Table II-B.
11
105% 6--100% -
95*/- --
90% I
Rise reow wa
fls-e 3. Idealtzed fwtrttoanlo or Thaidl Bep.se
12
c...i.eft uinwitr I i..'u4II,- - .Un~ta _W.*i~ ii60GO) - ~ Ic.p. III~tn~. - - tw
S. t.c - wfltSt =W
a ( .4 im.2 GaAj .
*Q.C.i) t
O...7t...7 ~rr,.Oct.)i
,~~99 -.1W ?
1 usqa. ~ St .80..) - -
us In Is.. ~mammo ý ?pmmm
60.)-
ICA) flR m
?g~s~ 5~ U-S ~ /
Seha.. ,I~. . .. n - 11 .,a
* Ct~lQ ft~ t ~flkaflSWSI)~~~ WC 1 t .JC cl-
SmI~ - t. nS ,. .C:C3
PAOMM~O od I PAZMANC=
Step Fi..i VoIlw Ill. )* C(.)
V-t ofo flMo~l .a-p
tq.lOl--1 t TI- t tfor ltot.. "0000 to
0100~(7 ro 120)~ ~ ~j 0
T- f totloit -opo-o to 0_ ___
T!, too 10010101 00000 to -. . .
Rio. Ti-, ri r ro0 too.2 to W0 I.-.oroof 1.6 oito Mo.1 V.I.
looko~oooo.~ o~ Ib ý 00.10 of lo100001 *v4~0
1''I
IN TIF= -F al"t.LO IN 1ENO Or C1CC=.1O
recpovo toIN7. 5. 700ý fi00t-o000S. 40.10.0 mod. ts -.~o~e ~ - 1. '~ ~zo rig. 6 otoo oo.wo1. of -~t o.1-s with LhuCob 0~OOJ ~ ft, 06 Peoo~o plld to . -..oo.1. *Y.1-.
(tir. 36)
reo~~~oolo -f- ~ - - ~ ( . ~ ed.ee.. Apeo-, ;t - 1 un-t-tbl. for -.. d-0
rtfl-r-,to I 0.7ro_.lof fLi 7ik 0.'~ se FIN.7
I 0?0to 2
20~~~ e~00. h.6. 2-2 * 0-2- 000000 vithO. C30i P-00 for 0.1 <O a
lb. Croo elfflcl.U0 -~ b. Il.epeted 1. Woe. of
goj L (Iwt. 72 .04 72.pur V) 1.C' 0 ~ 1 L.e~. toeb fte711 00
go .01. . 3om.
2
In order to assess the me--it of any given system dynamic performance from
its Indicial respoose, It Is necessary to define "good" dynamic performance in
term of the indical response. It is generally agreed that "good" dynamic
perforcarr iTelies low overshoot, short dcns tin, fast rlse tine, cend good
danping of motionz ibscq'cnt to the decay tLie. (The last requirement implies
low settling time.) Thus a good Isldicial responsc 411 comply with certain
specifications on its "shape,- which can ..e defined in terms of overshoot,
ratics or dead-tine to settling-tine and rise-time, etc. However specifica-
tions of "shape" alone Is insufficient to ensure that the indicial response
will be satisfactory; soe parameter defining the time scale of the response
"ist also be specified. The implications of this last requirement will now be
discussel briefly.
The over-all time scale of the indicial response depends upon the bandwidth
of the system. Practical cornzidcratlons of inertia, powcr demands, etc., result
in increasing penalties in ,eight and complication as bandwidth is increased.
However, In this gensallzrd investigation it is not possible to set these upper
limits upon bandwidth explicitly. For statistically described inputs and for
deterministic inputs (such as rectangular pulses) which ore of finite specified
durations, lover limits on bandwidth can be set at least approximately. For
example, the settling time should not exceed the pulse duration. However, for
impulses and step inputs, upper limits on settling time cannet be specL•ied in
the absence or further information regarding the operating envirornent. Thus
this chapter and Chapter III are essentially limited to a study of those aspects
of dynamic performance which can be represented by the "shape" of the indicial
response. To focus attention on "shar-' rather than time scale, performance
measures such as settling-time, rise-time, etc., are all expressed in nondimen-
sional forms. For example, second-order system characteristics such as rise-
tie, scttling-time, et,:., are normalize•i through nultiplication by a,, where
ah is the system undamped naturml frequency. A more general procedure for nor-
malization wil b,- given at the beginning of Chapter ImI.
The general procedure for obtaining syste. characteristics is to construct
the appropriate transfer function reprecentation or indicial response, and read
the teasure directly. T-hus, crossover frequency, phase and gain margins, a.nd
frequency of instability may be obtoined from the open-loop C(jks) Bode diagram.
Application of sie wuified servo analysis method of Ref. ,3 (hereafter referred
15
I
to as U.S.A.M.) facilitates cwift construction of the closed-locp G(5m) Bode
diagram, inspection of which yields peak maeicflcation ratio and freqzency, and
Uandwidth.
For second-order systems, it is possible to develop -xact fosunlaa for
Many cystem characteristics and approximations for the remainder. Tables 31-A
sod fl-B list these forzulse In terms of open- and closed-loop parameters for
a system having the open-loop transfer tunction.
G(s) K (2)
a&= a closed-loop transfer function,
as2
where % is thc "nde-ned natural frequency, and t is the damping ratio. The
closed- and open-loop parameters are related by the equations
2 (4)
S1 * (5)
(This second-order system corresponds to a wide variety of equivalent systems
encoutered in flight centrol applications). Elementary operations on theor
relntions enable exact fomulae to be found ror all transfer function frequency
doeain characteristics. The resulting formulae are listed in Table fl-A, to-
gether with references where derivatiors can be found. Simpler approxteate
fortnalae may sometimes be rreferred, and two standard approximations are dis-
cussed belov.
Figure 4 Illustrates a lineor approximation to phase margin
TZ37 (6)
Compari•on vith the exact result inLcates acceptable accuracy for ;6 < 50 deg.
Barnwidth may also be found approximately using
+ {I ÷ • .% (7)
'tich is obtained by neglectirn term in ý in the exact expression in Table III.
Equation 7 in cocyured with thi exact bandyi•t- for a second-order system in
Fig. 5.
Pbas': uargin, bandwioth, and peak magnification ratio are widely used for
performance specifJcation. The phase margins of all the standard foms pre-
seated in Chapter III (Table MII) are between 0 a.-d (0 deg. In systems with
a dominant seccd-order nrto, this would be ekoected to yield adcquite (t Z 0.7)
,iaoping. Sixilarly, elimination of the frequency response peak leads to &do-
quate damping of the dominant modes. The connection of these measures with the
transient response is, in general, neither unique nor explicit, except for
---ýn 41sQcunsd it. -ýcnzeon with Tabll M.I
A good syste. for a given application =mst of necessity have phase margin,
banowidth, and peak magnification values which lie %ithia ,vlatively narrov
limits, but a system utich complies with these limits is not necessarily eood.
This follow, or course, froe the fact teat the behavior of the actual transfer
fumctions of interest is defined only in a gowns ==c•o over a rr-" -frequency
band by these particular measures. In eeneral, therefore, none of these
characteristics taken alone yields a valid, selective, wen reliable measure.
Of the indicial response characteristics listed in Table fT-D, exact
formulae exist only for tlme-to-peas and peak overshoot of second-order sytens.
For hieher-order systems, time-to-peak, settling time, and rise time can be
estimated for some special classes of systems by means of the charts of Ref. 13,
17, and 2h. Reference 13 presents rise times for several classes of third-
aiter systems; the resalts are discussed on page 31. The rest of this chapter
is mainly devoted to a discussion of the calculation and interrelation of
second-order system characteristics.
17
100-
Or4z 1146C, deg
60- //cto
40- s(s+M
0 0.2 0.4 06 0.8 LO
?3.gre 4. C=Paris'-n of, Act~a ýA~ Aproxfrate Phase Nshinns
Approximation wib-(l'+e!)wc
3 1 1 Altual
0 0.2 0.4 c 0.6 0.8 1.0
Figure 4. r,.--ario-. o:- kt~a ard A~pr:xi" &-Lwiidtai6
16
lo 2t cot(p
9, tan 1 2t% I
(I - 2C2) sin q_ t o9.%
UE2-
2-M 2•
Vý( ý4 k -Ztan 46 VIFcs % - slne q
-2 - +2 sin
(Approx. 1(i-o)(~7 2)/ I +40.35 tanq.os/ j
/
-61++
2
S"• - 42,
sin q6 taan;~2
2 ._-
YI- -...% - 1/2
2i zi~t n 2 6 an
:i;ý atnn 2 sinY t a in 1/2n2
- + 0.35 1, % - o/2 (14o495I IT
-sin %~ tan 2
!. ~1+: [-HI'o.2q >
_ x + I
T:
CORRELATION OF SECOND-ORDER UNIT-N
-IH
11112
)[31
I -eýrj ---y + eX[,
3 (6-"
+F r i +,
3
TABLE MEATION OF SECOND-ORDER UNIT-NUMERATOR SYSTEM CHARACTERISTICS
- _ _ ___1/__
,p-7 )(• • ' -••'• ,'I[.-• (• )•- .2-b•,]
11/
1 21/ 2X 1i e1/2(~-
4)4
_ _ J _ _ __1__
j '9C41
(ftpprox.) tp
tang 2d36 (%td) 4.8%,- 1)2+I1/
--l 2 q2) (2-%2 - Rtp2)
2.8
2.04't %ta, - 17e /2a~~
(" 2 ____ x2 + I
t 0~i +o-7ý) V, t
* Jtd
1.i&27%(%td -*
1+ -2.O4(matd )
__ _ _ __ _ _ __ _ _ Cp
Lan-1 V~qxr-7,1/2tan- 2 In (c, -1) ]2 In2 (ep -1) + [4+ 2K2 n
2 (cl,-1)+
(2-2ý )[2 -4 tf (Cp- - iniJCx2 + ]I2 (cp _-[1)
1/2 ~~~n~p x2+12(p-1
X2) 2rIn (c. - 1)
x22
cp
K 2+ IM2 (c, - 1.) 11/2
tal, 2 In 2 (cp 11 ' I n2 (cp -) + r + 2x2 1 n2 (Cp - 1) 4 5 In4 (Cp - / 1/2~
J X2 +e -n (1) 1E]In(cp1)
2 4In( [-i-) .2
22 +n (Cp -c 11
+ rl2 + In2
(Cp 1)J
in (cp - I
FRomAsu FOR I=DCIRIL FJM~PEs CHfARKMuMSnc
Exat. f.o-nue n.- not available for some of the indicial response charac-
teristics, end the range of validity of the approflcatiorr quoted In Table 11-B
requires some considcerstio.
The Irndiela response of a closed-loop system having the oper-loop trams-
fer function G(s) - * s
C(t) - . V g e" t sin tain%% _-2
t - 0) (8)
&here V tan - Co-.
and
This equation can be solved e-Actly for dt , yielding tine-to-peak anddtpeak overs'oo .
Settling tine is cononly approy.r•ted by considering only the envelope
of the response
ce(t) 1 71 1 - 7. 711
For an overshooting response c(t . ) - 1.05-. Putting ce(t.) - 1 .05 in Eq
yields
Equation i0 differs from the approximation givcn on pages 22-hi of Ref. 36 .tre
the 4,1 -7 tern is replaced by unity. Figure 6 compares Eq 10 and tee further
approximntion ts - 3 given in Ref- 1 vith t.e settling tine obtained by direct
nescurenent of analog compter rcz;npees. It should be noted Uat approxirations
to settling time (e.g., do not reproduce the sawtooth shape of the ejact
graph.
21
15
C(s) 2
RSZ s+2~C,~s+4.ý
10--
AcIUGl-IsV~Z3
5
_0 0.2 OA 0.6 0.8 1.0
Figure 6. Ccep~rlsof of Actual and Appmuz~ate Settling ie$
22
The Equ'vslc.-t Tim~e Constant ceazure in cost approtriate for c.ystens jith
a dorant first-order =ode. To the exivrt La' .Uz obtsico, Te•;q I/Z, and
Sparticularly simple connection exists betL.- the frequý.,, .1. tiL-e ---rxlrs.
When extended to second-order syste-s, the variation of Equivalent T-n. Constant
"i.tn C In =ore .reguwar thwa th- settilrZd tine, alth=.q the =sasure is t.tzally
inappropriate. For imsance, Fig. 7 illustrates .easured Teq, and the asrroxi-
nation of R-f. 36" ..q - 7c fr n .. eood-ordier syste=, uith tihe cpen-2.oop -rans-fer f.'nction of EL i. Note that the -inism value of To, occurs at. ; - 0.
Delay tine is derlned here as the tine for the ixileial response to .
aThteve ýO percent of its final value. The exa-t value Is •;aredý vith t.e
np.o'--atiun ((iven in Ref. 7-2)
in Fig. 8. Th-Is aprox'l-tion Zs satlsfactor- only in -_2 "cpLal" region 3f0.6 < ý < 0. . •n e-ý trlca.l linear -elation t. - is =m gener
applicable. Again, as vith TeIZ, n-Ini delay tin i xIleved ea . 0.
Equation I1 Is I .xten.tl to hiWoer-order syste=s in Mhapter IV, "here It Is
sho-., that for otc-al systetMS, sinyle arn accurate aFprcxiatic•s to delay
time can be obtained.
Rise ttne is defined as the tine for the i=ndclal -esponse to rise :u
10 to -O percent of its final vr• 2.•. Reference ý6 presents the siu=.pl ap-
proxmnetion tr, I *.-'. This is cocpared wjth tie exact rise time, ,.ti a toreref ied approxiat
1on (Eq 12) in Fig. -
A inL1- rise tire syste- rossesses a 10. 1- Am apprc1iate for=-'a for
rise tine is given in Ref. 2'
; ;
vhere E1 ar-A are tie velocity and acceleration dynanrJe error coefficients
aeflined at the bottom of Table Tl-B. For the second order syste- consldered,
the accuracy of this approdnation is far inferior to Eq 12.
23
__2
R(s) S2+2C 4ns+w.n
Time to reach63 percent of
o05-
0.2 0.4 ý 0.6 Q8 1.0
2"
C(s) Wn2.0-- F(s) S2 +2C.,S+.,
S Approximation, td;2±! f
// I Region of valid
0.5- / approximotion
/7. I
0 I0 0.26 0.4i 0.6 0.8 1.0
Figue 8. Co~prisn of Actaal end A~ppoxirato Delay Tlaes
25
Ic wfor 0=.707
1.3 t 7.040~+0.2
4•_
2 /Actual
0 02 04 06 QS LOC
Figure 9. Co~ariscrn of Actual and Approximate Rise TineL
26
Fi•ured 10 and 1. illustrate tr -,-Y .r-r-hnct and tine-to-peak, respec-
tively, for a unit-numerator second-order system. 7he ninira occur at • - , and
0 0, respectiiely.
It should nov be apparent that a criterion based on a single jndicial re-
sponse tine or transfer function measure is unlikely to be valil for the variz.ty
of systems encountered in flight control optimization. toat of the measures
specify only particular elements of the response, A:,I/or are restricted to spe-
clsl types of G(s) behavior in the region Io(jw)I ' . Miniizsatien of the
measures results iller i, low damping rttiln, !.-6 lase 'rgins, etc., or is
unselective. The enor use of toe indaicisa response end c1i.esoary transfer
function measures is for specificator, purposes; this will be discussed furtherbe!¢'".
One indicial response measure, i.e., settlin.; ti•ne is a possible cxception
to the general statement abovy. Miniims, or nearly mind- settling tins is
often uscd &z a crterion, and because it gives a • - 0.69 for second-order sys-
temz, it it- worth ý==1==&irngI the li_,ht of -h 1wTeir..as foýr "r!-"ia.
Settling time is a possiblo cxzcption to te ntstutme.,t above. Sinco "good"
indicial respone Implies low settling time, it is reasonable to inquire whether
tue caiterrin of low settling time always results in irnlicisi responsea which
are satisfactory in relation to overshoot and the other parameters definirg the
"shape" of the response. Figure 6 shows that for t unit-ntmerator cormalized
second-ordecr syctem, the criterion nr 4-n', -ttling time selects * 0.69.
Howver, the sawtooth graph yields almost equal settling times for 0.43 < C < 0.68,
with sudden Jumps at each end of this range. Similar discontinuous selectivity
characteristics occur with hiahe,-order systas, an shown in Ref. 32 (which pre-
sents an exhastive investigation uf settling times for higher-order systezs,
including standard forms of first- through eighth-erder). These diecontiruities
can be remcved by using approximations to -ettling time (such as t, - 3&,/,, see
Fig. 6), but ruch apprcximtions re difficult to derive for higher-order tyates,
-ie.. don.I•nt nodes can be idsntifled. in general, therefore, minimum settling
time cannot be reccemnended for use as a sole criterion.
Minisization of the product of tine-to-peak and peal overshoot, proposed as
& criterion In Ref. i',ý Is also examined in Ref. 32. Its failure to discriminato
against undershooting respcnses and frequent selection of very poor response mrske
this characteristic unsuttable for use as a porforstane criterion.
27
00 QZ2 04 06 08 10
PI~are 10. Peak 0vemhQt of Unit-Nllýtrtcr Second-0-der Syste=
10-
29
Th.e uce of error :ucfficiento as performance criteria Is discussed in
Chapter V. It is .hown that each error coefficient it" p;;,ortWisl to a Li,,n-
weignted Integral of the error response to an Lu.ei-e input. The error reef-
ficients are thus c-he su of positive end negative errnr integraAs, and fail to
'odicat.e vhether a small value of this sum represents a smal absolute error,
or a e-al difference betwen larpe positive and negative error). It is con-
eluded, therefore, th--t e-rr '-rlcients by themselves do not provide vaid
performance criteria for general systems.
ODURAOL COtEEMATIO2 OF SYhTD CHAACTUWTICS
It is not .urprisirZ ttat nor of the init~ad l response characteristics
considered in this chapter provides an acceptable performarne criterion. these
characteristics are suitable tor performncee pecdflcation, rather than opti-
notation. For example, it may be convenient to s;ecify system performance in
teer= of bandwidth and time-to-peak. It is imprtant that any spemlfication
framed In terms of system characterixslcs shoa be prncically realizable (i.e.,
mutuasly exclcalve values or limits of characteristics nmst be avoided). To
facilitate this process, it is desirable t have complete correlation express-
Ing each system cht.rascrictie in terms of any other dystem characteristic.
Table IlI kwt been prepared for this p•rpo3e. Crossover rrequercy, bandyidth,
phase eLargin, peak frequency, magnificaticon ratlo, time-to-peak, pshk overshoot,
and delay tlire are all expressed in terrs of C, ct, and each other, for % second-
For flght control systems, despite the presence of nmer•vs unalterable
eleorot: ,::r'tte vilti t" veh-cle econfiguration, the variety of possible
systa.3ý Ic n-o gr-at thkt no meczure Of the iniicial response based, u;=n anr
single instant, or meaure of transfer function cheractrietics at a si•gle
frequency, can hope to pro"id wire tVa neceesssry rather ta•n eufficient con-
ditions for system goodness. Recegnitio'r of thi's fact has led many investiga-
tors to prOp3e e'ssures bsed upn integrated f.ntlons of the indicLal error
response. T:ease initstl arrsr meaures axe discussed !n th- next chpter.
The precedi discussion of system ch•racteristIcs has examined the merit
of each characteristic as a perfors-nce crlterion for unlt-nmercetor second-
order systems. Thls type ef sysem v33 selected because nany equivalent flight
control systrei are In this cWStory, and cmsequently- It provides a fair basis
30
for inritii asscsac2:nt. Systen charxateriotlcs that do lot yield natIsfactoly
,erforcance criteria for second-or4er svs*-rx need =o be ec-.-i•ired f.rt.er In
the scarch for a s.tisfactory performance crIterion. The diaeratty of possible
*-1rd. and higher-ordwi fi£j.L- precludes a gencral currelttion of syst4m _har-
acteristicsp as ta" been achieved for second-order systems. Hvwcer, t.ere is
saw value .1n collecting the limited data avallable on synt-. chircterisatLes for
third- ard higber-order syste-, and c=kz'irg trm resuit, wl-v,.sr fr-cticabIt.
Me vell-known charts of Chestr.at and Yer (Ref. 15) reproduced in pef. 36
cornelate time-to-peak and settling time vith frIeq"ncy response ch=,oot.-stic#.
Elgerd (Rcef. 24) presents indiolal response tine histories of a variety of third.
ord.r systems from wotch rise tise, time-to-peak, etc., msy be sesti dixect1ty.
To ezanine the implicatioc.n of his resultt, and to compare then with those ob-
tcaned frm0 other sources, a standardized third-oarer unit-numerator systen Is
considerel.
Thos fo•. In not covs•d by the charts of Ref. 36, tut Is discussed by Clenent
(R-^. ,7), wno presents ricc t4.-, Lettllng timse and peak overshoot for A
normalized system having the following transfer fanetton:
RrO (I( b.-,) 2s
FiL,,rr 1U illustrates Clent's resuitn on settling. tIe for tMe standardized
system of Eq 1i. The points mzrhed doa te wslvus ert-,ed by means of the tran-
sient responses of Eef. 24. The a6reem0't Is Cox4, satzugh l-locance must be
cadi for tle Pact that the saytooth chaps of tin settling time graph has been
s-ooted in ReC. 17.
Wrnett and Shua*se (2tef. 15) Oe-ueice rife tire with peak power for a
variety of -thi-d-order sysleza- TM resulting Talses of rise time and over-
shoo-. for t!". cyste. of Eq 14 are presented in Fig. 13, from which it vill be
seen that the agreenbot vith Ref. .A is 1norally cl•se.
51
R(S)-
C (S) [(3-2 (.5)(.3s+113210-xx
xxx ~Ref ;7*
2- xx Ref 24
0 2 4 6 V
32
0
I• I'T- -0
01 ..- If)-/o E
tv<~+ 0 0Itc
x +
INDICAL E110 3'ISMWE
Toe term "Indicial error measure" in. be used to denote integrated fumc-tionr of the error response to a step Input. Table IV summarIzes all the
indicial error measures that have been suggested as criteria in the references
listed at the end of this report. Most indicial error measures are of the form
00IU - f Udt (iQ)
U Cf tUdt (17)
0
etc.,
where U is a gene.ral functIon of error, cuch as E2, IEI, etc., not involving
tine explicitly.
These measures are meaningful only for zero-position-error systems; cost of
the published investigations of the relative merits of these measures haye been
concerned with their application to systems haring unit-numerator closeO--1-pinput-output trmnsfer functions. Since many flight control oystems reduce to
unit-n'•.•tor equivalent systems, this Is a fair as well as convenient basia
for assecsmcnt of criteri-a. To achieve comact presenttion, It is custom•ry to
nornalize the systems and associated performance ceasurte. Thus, a cysten baring
the transfer -function
R(7) C +09)
where 7 Is the Laplace transform vari-able at this point (eltevwhere a Isempuyned)
can be put in norzal1:ed form by the procedure uf Ref. 37, reproducod below for
35
1. Dcftin a constant % so that
'ý . %(20)
2. Define nev coefficients t.or the numerator ean denominator terns
1 - , 2,-..n (21)
Pj11 p- - , I C, 1, 2, ... =(22,1
3. Divide tbh numerator and dencirator of Eq 19 by Q0, A apnply theaefintions of Eq 20ý 21, and 22. The transfer fu.lction theu becomes
2w _.. _ Pj %n12
72 _ qleL + 4A (23)
,. Introduce a r"v transfer variable so that
2.- (24)no
Then the transfer function reduces finally to the normaized form
C(s) POR (.) ' n + 4n-1 sn'• + -.- + q=; 2*2 + q + 1 (25)
From diL-nsiooral onsiderat'l-n it = S:l.o .. i 4L to convert a nOAlze6d per-
forrAnce measure to its denornalined form, the folloving relations should be used:
i,(n"v o -- •3e r c.,l- ) (26)
Iog I(I O.i) (27)
IT2U (28)
Ibc normalized fora of the ciosed-loop sec.~-.1roer system cusnilac-'i pre;Ic~iiy
36
'TA= IVMNICIAL ZR1f NEASUHM _________________________
C14 C1tlTZ0IO ASSEBST(AMUPONI URI-N1Nam " SIMM) A1802A11 APPrICAfION TECHRIWN ANDUS 1 M
if constrained to conoverehooting second-order syst-e,t - 1..if overhotocts are eacued,ý t - 0 is
telected. Lack tf J.aidity and selectivity eliminatesAs a criterion. (Bef. Y[)
If constrained to noroverzzlootlng second-omo.r system, For impulsive response (not step) each of these measurest . 0. Appiles aeavier ve~xgnt-ing (amd ftoe is PrOpertioula to a particular error coeffilent.e-ic-tivity) " system with. ; > I t;.an tim control (Sec Tnole 33-B And Chapter V)Area criterion. If overthootz ame Allowed. - 0 isselected. Ian" of general validity eliminatos as acriterion. (11-f. 37)
Similar to j, te(t)dt, except Veigits responses for
ý > I even zero.
For normalized second-ordcr system# critcriots selects.o.63. Criterion Is moderately selective on
low-order unit numci etor system; but nonselective onF:rveoiyerror systems And higher-order sytm. Aaltc form given In Chpe 132 ,,,55
Easily.mechanized on analog cosputer. lack ofAdAppen&ix A.
selectivity eliainate* as & criterion. (R~ef. 37)
Criterion is thoroughly explored end is good in nearlySeclcctive, rellable, And easy to apply for all every respect, except complicated nature of sneaiytiesyntens (through tighth-order) investigated. for=s. Analog coaputer-derived standard forms andApproachcs the ideal criterion for routine tresslent, responses exist for zero-dlsplacemeet errorOPt~szfttIcn employing Analog ecmputer"- (Pef. 37) system through eighth-order; zero-velocity and zero-
Acceleration error system through sixth-order.(Ref. 37)
Highly selective, giving -0.6 for a normalized
"send-order unit-numerator system. Analytic fore rtro amc cctoogl xlrdthnMEis very complicated And less easy to apply than rM, Prtronsi s not crh lthe ethra l eomploredtthn ovr MAE.Also less convenient than lITk for analog computer ' Psil o ot h xr opiainoe iRnechanization.
Is the simplest of tns higher-order messures to applyanalytically (,w?. 63), yin:
for(~t- 1 D(rEOd
Se'Lene a vslue of -0.5 for second-order systems. N) isElack of selectivity snd specification of highly which Is a vell-tabulated Integral. (BarF. 44, 58)osct.latory responses for higber-order systems General sod optimsl (standard) form. sod transientmcasn use Go a criterion unsuitable. (lHer. 37) responses through fourth-order exist (Tables T an~d VI)
And allow a fairly sigpie dct-rssinatioo of effect& ofuncertainties In system parameters. (Can be evaluatedfsirly easily by operations on normal design charts.)These two sdvantsges seke its use as a criterion cootonIn ipite of Its relative lack of selectivity sod
Ya ii d it y ._ _
2
TANZ IV
r=2~ Jto(t)dt M.im-o .a-~ nnllomblt y for hlmi*er-ondcr systmn(thnnili fOMath Order)- (atf. 70) Appl-n
anry "toiaig " - laos 10 Anlto7 nritenianIn n. .yt.. *taiine.
Fn,aOOM.dan0. nystas, 0 .65ta,TV, fnY(d X11 l- 0. (FAn. 3?) 3sac.tivlty become
1 1a5 0~~geee f"n ~ n ctnan Incrnsaa. and onitanlon 41,
bncns an. MM MFX to ASawn.
ý2-.toMdf nn (Wn. 73) Res not Ynt t--41. nltndnd for blaon nd.- nYet-s.
at Yd Vel tl -Ay~) m.nnny~ inability tas or thjon mo:Itan.1 tof.a ae . raa d~itfflfht toJ(tnn-qoa
2dt _ont. .an n itb provnot toir, af nrtntnon
Probnbly tI ail ndy nynllonIit7 test.
dittlint to an.
Nit yet enniottd. A i4nl tn.4 fori-o[ -t Ign,)4t_4
0 )]2t ieliee TIna a t-sa alto finit. 5altla error. 0.1in.bern n/t I. - .nIt n.-r a
T. in stbltnay.
Tod -Me inl K vsgn wd sinynt
.In~ t~u- .. tKrogdt yaft-na (%nf. 5%)
t.f..an t (j.a"g i xnt X. **tN4-t~.bt. W In . pitointnnindnontno
18
TAUL. IV
§WhtotnIY sky to apply O-1yttcllr 00 It -n to Ol.±
( 57 M. foAr _ nv 'tty-Iro f.ta "It~lot - o. 0.0. be,.
I(toroob foarth older)- (hR.f 70) A33.00 no..., 0 iscrre as010 a0 poro., in to.. Integration. 000.001vey r~a~e r n del nior citrin ndCM se (-tsNord) f~re. nod transient reapnoen oohrowogh
anlyi t0031 asoin. fourth. 0rd.r .01.1 (Tabl.s 5 and 6) and allow analytic*Ydolmnto of effects of yatm0 ooaoertsotol...
I ~Analytical c ýreno~n beoon,. am. diffti. as a increases, v102
70.?' To seodo rde systees 30.. 0.65 for t6ztt,( 1)o ! 1O"P 1on. nt, o 2.(Mr. 31) 0010041O1y "oo~ 6 K.-tot I .I00 0 ~ Eo.Zo.4
greater 00 0Loon..., and criterion al-. Standnt ort o. and "omitted traonot~t re00100000 to" b~eoboon. no 0o1oo o oo00..1olved f,- n - 1, 2, 3 u.100 rcospors. (3.?. 70)
I.00 (1 ., nno rrn raefn, preoonn'd Il1.1 ta.lStandard000 torn In.010 6. renernfll not norto. t0. extra
00000t104 effort ov. 0 J0
ti2(-)dt.
Yield. *0.867 for -.onod.r4or .yot- wtot... -lue. cn be foent .oal~ttc&UY (Ref2. 73), 01100000It 010 &I fcr o- to (P f. 73) Has0 00t yet bonn -o tega1~. mOn.. are fairly COM10o. Onlortift Of . to
1.1-ot.4 for 0151.0 onder 0yst- svoi~on,7.
A typiral critonim' for. .,.ltoble for &*olotwvia 01 oynnowlPr~rooeltot tecnibns~. This00 oneha boen 0.1.01.0 fro a1.01
AL t0 J2, red splCY 1iionb1ty, toot for the 0001 (r.-7-18, 9, 31, 1.6, 47) because tit reduces Io
00000.06 ctorfloco.,t 03.4,4 rorIdror 11.41105 00000, to criteria 1101 have onrit for .10"1 oottrt;1a ot%. orwaretro, 11). born. Choi- of a 1, nebitr~oy, -oir the onofltn t 03to,
ease001n tLe 000000 tares of reference. e08., .ltnt. J c~dt vhbti. tI 2d-cm ?W001.04. 0.7 for ytexo-od~ 100. (002. 63) 000 notboon evaluated for hi~otr-order 01,0.00 Vill boona so..foonter and itporient se dyno.Io P-rnctrlog o~lo
Probes1 fa1l, rc.Ay tnitoitotltt tot. V[, t, At] iso Cn ooral fortflon if -rror, thie, and syntes,60.8. Choit. of Po Isabitrary. ft"Ig Z- t~ ne ,or-i jn lrs..o Al. p(t) to the. prob,'Ilt-tltmt the1 MI0. Vtln ~ be0
f0,. of Z00 o.0. awe- -doftood end dead. 60300.1. asn aoi1..ooro0oo s otr~to~1n.dIffiO.t to states.
I ot. -t arolonOM. A logcarl fr, for A obonor u0e of 1.40000 fot10. I eotinn nev tog the Integral1100 ml. ysteas Vitto foinit poottmno~n ror. Cho1ce appears 10 nof 3. nbich sago oo..10on to.. caa oithto o
of To t. arbitrary. lapboo 10p0t. Key 0.e" &il~tiotno to speca o dMrootm "nIno
966,9 Not yet .o"oat.4 . taok of flood .01... for bovoceod a00 ̂ F-e-raltood nooforoýo - -or. o Innoobot
T00sake thin asever K-g. 04 tnflt.f 0. Inea Red whn . - 0 of2, ~ t
5. Mttoo11factory response fnr btl..r order 9144 .Plrt..t to Ie33t-ootPto~ loll t foofr fgottoa 1n.Yet-. (hft. 34) eas 02 00 error coefficients or. related to 61f."L0150.
10o13e poono of error (See Chapter V).
5 1 0IOdateOO.to
toot
is
- 1 *(29)
For systems with a nonzero steady-otztc crror folloving a 3tep input, Indicial
error measur-.o= ssL - •plio~iie, nu so. n d asove. Howeer. It is easy to derive
associated - d response measures yielding fin ite criteria values; four
such rnasures are appended to Table IV.
The measures win noi be examined with regard t^ their validity, selectivity,
and ease of applucation.
Au tnted In 7ahbl TV, TR, YPE, and ITnEr ire al: ir.v•,-•ated by thoi
iuability to discriminate býtwzen rms~onocs which are C-od I,) the sense that the
obsolute error is srall or d-cOy3 xstidly, and oscilittory responses which are
lightly damped, and in vhich the large pcsItiv. Asm .-eeti~e errors ýpproximately
caucsl. Similar drawbacks apply to the associated isclUaiie respcnse measures
fo c(t)dt, £otnc(t)dt Therefore, tLv.. measures will sot be Alscxsod fur-
.her in this chapter, except insofar as they reprez:nt limitins valucb of other
measures (e.g., ITAE - IE for sonovershoating resptrce). :fou of the rcaizs.ier
of this chapter will ta devoted to a stuay of •be ý.Zasures IE2, ZTE2, IT
2E , IAM,
ITAE, and IT2AE. Pirticular attention vill be gl .en to the analytic evaluation
of -iese measures, not only b,'c-se of the inSgiii. U1&at acoiYtc esressions
;,.uviue, but also because -kcy enable a cte!:k to t- male on puhdashed valut.s
obtained by mechanizat'on of nnalog comput'r rr. n-t.. As will be shomn,
several errors have been detected.
DERIVATION OF IS2, ITE2, AND 1720
Tables Civing IE2 for noonit numeratr systems of first. throish 3:•,-th-
order in terms of nwerator and dencminatnr ,oefffrleCtg Are given In Ref. ".
Appendix F of Ref. 58 extende these tubler. through tenth-order, and corrc•ts an
error In the soventh-n;i.er intef.i-' ,,f P-r 44. These literal exprecsicns are
lcngthy, wad .i f. 4L notes that t1.. int-grals can be expressed aore -apect•y
in terms of Hurwitz dete-mirnants, a poinc which is discutsed further in Ref. 6
and 4
;. Table- for IT-' (Arnd IS•% a-e S;You ly *WLtL (Ref. 79) for systems
ol first- thrmgh rourth-order. The topzc also been stalled by Xoothe (Ret.
48) and Stons (Ref. 70), using a o..what different approach thSn t's previous
referinceI. StneIs procedure, •Id that of Westeot, is outlined very briefl...
below.
The Laplace traz.sform of the error can be erlreesed as
Fi) - iiil i i - n-i i - + d (30)
. 5 +5+ + +.
Stone first calculates literal expressions for
•i ~ ~ ~ ~ ~ ~ ~ d,- +i i, a,,ni _ii • -Z-- - ' + + d= - --
.. _ -+ .an1 .. 4- an .1.
IE2 say thez be obta~ned by use of the final value theorem
- I Ih f1 E s1 ] 21 1£ d ,n-I + dhan- I + .. + d n11 j 2. .. *a05 + a:..+ei an
rr? LimIa I Ill 4tr Il
,.2 11rsd212
The Pro-tedare is sisple, In principle, once th~e literal fure, for Sq 31 has beenObtained, Hzwever, in practlie, for systemssb -- 4A- -- -4. -Iferm becomes exceedingly long. (Stone prtents such a form for a ononnt runrmr-
.tar 'etb-zd.i zyatem t•w. h o ccpleero nine pages.) a =sy bn
Introduced into the differedniation required to obtain 12
by the apir
neglect of tcrms in s nd show, wichv'crsh s-w as illowed to tend to zeroin the second derivative. liwer-rsIeas, Wletcottl'A rocedure seems to te zcoo-
what briefer, sod is preferred for the pfrpose of ctktnnir.S ITEaM r22-.
40
i*s2I i~oli.ving Per. 44 ezrl ý8, ftnlq.ys PasevsI Is theo'ez. to express
rE -Co 2 +3w 13f. t [.") -2_ . E(S)E(o - 3)ds (3
.bich, tor a t.~b~t, system. bezcmea
Wbea E(s) is, in addtin, a rutio of rationa ply omazI&~ Eq 36 is.o £yetri-
cal. rational funt!m 01- the poles of E~s), being the sum of the rosiduez at
W%"e jyuIee "i ceo therefcoie be expresýeI iP tors of tin coefficents or
E~s) only. The procedure Is well-illustrated by eovsinerl~kg the simplest ease:
L2n dB do (38)2t3 fJ o, (son , =1)I -V~ + aý)
vbich has a single pole at S- -6/s enclond Iy the contour integration which
is an arbtrc.arily large w cictin the left-half plane. Ths residue at this
po~l ix equa to Mand is Six-m by
a5j -C s)-.I a rr So a 1 ] 2a (39)
The proeedure for c-naining m2 Ind.Imicated by Eq 33.
f~i I. t (t)f dt 0 b f h ka~tj. - s)ds tw")
It is not possible to take 'be ow-o 0 mUzang process under the Integral sign
vithrit first pemrfomng tea differentiation. lTo ohcain a ryumtricalk function,
j, as-.uwed to be real and p .:. , the ccocoxr is teken as the aiidole of the
d•splac~w-nt of the E(s) and E(-s) functions, copleted by n large se•=icrcle
occupying the left tLtlf, and per. of thR r4Mt half, ecrpez plane. This is
"-q-tvalnt tao vmply cl-angin the variable a to a + (oa/2), eoýv treting the
contour Integral na an lodlrry Integral. ZT..stlon 36 them beces
01
Td - •Ids
By substituting a =o/2,
. . -o0 l L, I H(s. + o)E(o - .)dr]
Tais my now be solved by making use of the X7 tables. For x le, taking
the .aplest case,
E(s) " do (42)
the Intrgrsn Lecomes2
(uOO # c0C + 4,)(-aow t0. 771
Tht group a, + e.,a ro corresponds to al in Eq 36; Z'ee the tpproc'rte substi-
tutltoo in Eq kO yields
1 22
L. a i o ro kd a. , 0-42(3
By zeans of the procedure, Westcott cbtains literal forms for iC-;. •e process
has been extended by a further -Z/A operuton to obtain in2E I the present
.-cp-rt.
Table V preser•s the resulting literal forms for TIF, TTTý, a-d -IrE2
for
systems of first- throu.. '- rtb-order. ý.h- I=? results havs been rster-ved
1roe eeden,•y of Vestvott's vork, srf4 eheck his valuea exactly. So g=era)
42
TA=I VL-ITXA FUOS C II!&&SM fl=MlS MO iE?, =
2 tam n,
2z
2
POR inilE OF Pi=- MtRQ PO'J17n-UWES
rr -1r
L-A
84 a o.*3 *
.~~4.0 - Sc) * a
W- 41- *03 -Klts (4 - j(j~-
--A - %i4 -
-1-3 W[L) ah3
TADLZ V (Continued)
~~o.
13 - ~~,
23
23.
113 3ee(3i P 44Al(3-02 --
* %s.,4 ,4. -~t
-,-~%:-v34 - a) - "(4 - a4ast 1(ua.c
,. ^ 4- *t 3
"fah4o4- 6"3% t ~.3 5 4A(; ~.2(~
literel valjaec or Tk havt pierlously teen publi3shed; hence, it is not port-ble
to have a .ccp~ctc Irdepcndent check on this =casure, althm-.h, as will be noted
it tnc follovurng section, pflrtisZ cheek 4ots cxist.
L'VaW& mO a it2, LTE2, Mt WJs? As kmuowmc ts=Wsnt
1ndlelisi recpeoues Of nonaliz-ed unit-nmmrator ninirun le?, I?!?, rS,22
=1n 1T3E2 ryrteon= 3e Zgiven !n tic!. 70. Thee ar-c =;reprduced In i. , f=
ease of recferene, sno corresponend dlozed-loop Bode diagran are given In
Fig. 14. The transfer fuLnctiona coefficients that mi' '-'-e these nornelteed n2',
=1!2
, etc., perfornance tcesures are call.-' standard tones, i.e., the partied
4,ariva~ive of the raernsltzcd pcrfoeno~ee nase=re with respect to each coefficiett
of t1- :'olorl-locj- transfer function Is zero uthen the trarafer fjnctlion nas the
appro-priate etanoaco torn Table vi preaentt. such standard forma for a variety
^f ;,erfwr-sne criteria, incluling minit:= M2?, T!E2
, a=d =l2i
2. Thee- ho.ve
her obtaines by differteu..ating tin: general. nnslytlz .=rzrza.=3n of Ts.e.: V
with respect to the tronfer function cocfficiznts. The results -rece with ttose
Or Def. ?0 chich %were obtained by a digital icesputer jrograred to? iterative
Plnleizatiwr. (This provide' a partial cheati On the accuracy of the 112!? gen-
eral ton-s.) flfermcce 70 a]t- pr-snts atandteI force. aor,! undieal risponses
"tor opti-a lI!St? syateces. Theae nave not bceen cntcsed, but are includedL in
Table V1 to ensure torpl~teness. The aralrtic expressLun for IT31 fo Lya'a
of higher tha... Gecccnd-order are very cecplihcat-.l, adI'2' dee =et nertooff-!r advantages over r2or IT20 that wculd heuffi.-o Coresainfo h
ecocputlng efforts Involved In practi-al optinietion cerlulations.
Nbls N7 also lnets the open-loop t.-ansfer function coefficients associated
with the stanlsed fores. As with the etoer!d-lozp coefficients, the sieosti+vity
of te ntorzalined perfocoance neasune to *=dlz changes in these coefficlent:
tuout the -specified values is set- Ver.u-ever, toat the ,.cnormalized per-
forrance ecanure in affected by still changes in open-loop Coefficients tronm the
standard ton.s.
Tne c-iosea-.Loop o acn naxarama tor tee 1 t;2, ia-2
, ±' s?,ao TL
ntcArerd fort arc given in Fig. i1h. (; tc-,.astcs cerrcel to one sIgnificant
f,gur em urhsed to construct these UtvSraaS.) By operatiors on a Nlichols chart,
or by direct fartorinatlon of the lower-order synteun. the nsacciatcd open-loop
=-a'cterlsttes of phase nargir. erousever fr-q1ae-y, etc., Iavc 'een found.
4)
40, 40
j 20.920
\\
O O 0 O 9I 0
oj L 00w0.1 LO 0. 0 w.
, 40, . 40,
0
\ .cJ• \\ IC ~ r- \\
\9 _g >~~ 90
ITAC '~ -80SI'. ..... "-- -I,
Figu. 14. Cloud-Loop Weic (:w) Ditag.-.s. of Stwxldrd F0.=3
-:6
0 40 ~ 401
0 -20.
"" L J
-270
_____134 - z~
Wj 1o v.0Ow 0.. 1.0 W.0w
1-401
-90 n~ System Order
-1 ns 4n=\
0A ~ w -W
_ _ _ III
-t AE Chu S a a w ThC
Fi.xý V,. I Thdicial Responses or' St~ndard Forms
'.8
WE1
4 //i'~*LEG=
Fig~~~rt 15 (onl~
4,9
Tl'ese m_•r resezted IL. Table "I1, together with thp eorrvscr...ln vo1"", for
mlntims, HAE systems and otter standard forms. Table V1I and Fig. 15 isdicate
that the standard fora having the nmoothest indicial responses possesj phase
zag-- t een so ad 70 degrrs.
For a norznlized second-order unit-rsnraur "YstsL having a transfer func-C(n) Ition aescribhd by Eq 29, II *- ,. The general expressLona for !z2,
I132, and 1T2232 sheplify to
le, .~
112E2 (46)
These reasourc• have been eval.ated ard graphed in Fig. i6.
The rapns of Ref. 37, which were obtained by mcnhanization of analog :o_-
puter responses, are also shown in Fig. 16 for comperativc purposes. The dis-
crepancle in these ruces are likely to be attributable to
1. A scaling error of a factor of 10 for r1j2
2. A scaling error of approximately 16 for IT2
3. -A.-f4cr drift for t > 0.9 for all three measures
The con~lusiens of Ref. 37 regarding the value of these measures as 'ritcria for
nor-alized second-order syste--_ Ped not be clanged by the correction of these
errors. M is min•ni-td by ý - 0.5, which is not obviously dedsirable, althsrfih
rather high-t damrpts ratios Qt - 0.7) are usually preferred Zor step respons-s.
The strongest objection to IE n a performance measurc for zecon.--order
systems is Its lack of salcctivity. Vsrfylg ', from 0.2 to 1 ., raises 1ý2 from
its lnridc- vnluc of 1 to only I .g'. qy contrst, 1232, winch has a rninam value
of 0.9 at ý - 0.67, increases "0 percent (to 1.35) at - 0.51 .a ; - 0.9. Fes'
the nerealized third-order system investigated in Ref. 37 with the transfer func-
tion
M C bs - cs - 1
50
TAhU.1 VTOTAIMAi4D P10.43 F014 IJIIIT-NUISQATIOR syI1TFM2
________________________________________ Ctugmxxxro in fu, wHK14ly nis syWEI24 ORDEJR)l
OC~mAll POLYNC1.UAL FACTORP11 (I'falWl-UIr
4 110 Vo P.0mlf)"(lb 02 2(.')Uau :1
aI I .W() 0n 4
n2- r, + .1'p8~2 * 2(.t;,)5)voG 4 0* 42.0) z4(a4.1)E2 + 2(.24.j)(1 .2y'f"OU +
TIMl WrEHIMJO L2,~d
13 ~ ~ ~ o + 11 .. 47wZon P +.4~ a* tA)a 2(. 3P.) (I .227n0)o a (1 .227. o1
13 41 .),2PO 4 ..1.l0 '-'( 2'oC f8it ('e' 4(7I(.9n~ 2~ 0.~,,i~jp*~(i .84 (10)1
THE 4TMINUMPa34 it 0' 0
02 4 ~41 2 2(22)noc3 41 .66:j*8cl" 80 4 7041o L 2 (.1-2')% (191* 6 0-1
*I.6jo'4 2.09141ýn 4 (a.,1n)[2 *2110)2]( '9f2 + 2.35(1 Y35ri
5ie 4 1 o 5476 o +nn
L; 1 *)rjflOC l + f? +,2*~ 2(.43 6) 0I Io5 + I M
64 2. 1.Oj +IC 3.4n 2 2a -7001%[a 0 2(.4.)(.18O)fl (1.752.11103'4 00 2.7ln 4o +~ 2(.318)(1-33'I10-a + (1 .33300)21J[02 2+ .24(TOl
I~ou it C2 +2(.7)gos + '3 . 1.5500, 2
4 .10 4c +a .6610o)[02 + 2(.361) (1 .V50)c + (.2300).
Cl4 + 1.6o0,10 3 - 3.150882 4 ~ 2(grg 2 + 2n(.75)(.691fl0)1 + .9n)jo
a 4 1 no +4 1. 0
4-0a 1 ".040 4 n2 + *o) 2(.5)Pi *
e~~~~~~4~& 2.nn 4p~3 13 * On +. 11 &2(4f~ .*4 2.6fn r3 + 3.4a802 + 2.6Xlg * p (9)nnu g2 (8"
TICE B1NOKCAL
* 2f4 *n (10 no)2
42+2 n 4 ngno'V4l4 G- * xr~ *14an 4 (a3 C4
14A 2+ ian
TANJE~ V It3TANPI)A IAI UM3 FOR 111IT-NUMeRATOR :YIynTii wrffas~ mtim4. i, T
(flUtKHATONt 1.1 IC. U14HEl. R 18 SYSTI-24 OWNr~) - fNlu:'AIi4xi iAi 14. wmirm. 1
F~ACTORED ("OOmyl-la FCW ACP0JU? CIAY:JEP,,
(it
4_04No L2( .:')c'(Iý06 )B
THE Mla.uJI _ ft~c2dt T - ,#nr~rJf
02 2( l.Y)!))v'o * 00 0)i
(f) 4 .6,N)Ea2 , +(a'( (I +~~2 L
. Ž(bI:).6~)c+(.WO )-2, 2(1'+ .9~ go * E* (Id (.2 4 ( .66,Ap))"j V
TILE MNIMNL~2e-t TIM WtKrtIJM/u
2. +
TH IW.MMN4?J ~~s THEKMINdIMUM 4 ~710 A a0'+ 110
52 "2 +(I'.~o * .' 4 1uO
(',70I)V*~.2~ o~ (1.1~,2:10)'l y (', 704'2 0 ) &2 +2.1)(11
THE M!N1MLS-PO ýLcd t THIR MNISIuN4 Pot
112 2(.6)no + n 2( .6)iioc
+ 24 318)(l-'113o~)u + (I.333 0o)2J1. [B* 2(.824)(.7500OO)s ( ___________________+_(1_33300_2]___
4.6)[ +2(6)(.Q)c4 .6)J ( + .66100)(a2*(p 2.b).6910 o)[ (. 2.36I)2] [2Xr + 't'.23-0-0-y)2] & i.2?1la~~0~ 2 4
THE 1UMIWIWORTH THE BLIM
*2 2(.,()Iloe + its42.7%
& 2a ) a 2(.920o1 * a * g [a2 2 f(g
.no) (ano)
(S. + g) 2 (a +*o~
541
VIN~ MINI"IM flu,'.I
' *1. ~ *u(s
* ~ ' ~ .~(c~~I(t )~ (I fl~~ * Awj,U
0); u'wL AO,4nB 1'a.
TJ F
AA). .~ )~ L2 4 C(. 'I) I'A0 (1 u" [a- ;?.J lý(&iv ~.42911O u * (1 hýý,g
THP MfINIUM Atd
n* '+ 1
N +: * b, (J02)(1 .1,211 0 )c + (1 .'#Zu')L 6& 4i.A~ .42)3 + 0IIa)2,]
THF M1141 4UN/'
u + 11 5~ JIa~
4U- 4 2(.,16)(1lj.1or, GO11,~)J ()()I~~di)
* (u1).:;~) (.(..,:1[,P 1.2)I~~AO. (1.:~)J ; I. 1 irn 34 .,2 "1 s
THIfF SMINI J /TIM CRITERO
02 + 2(.'[6)110L 4 s+I !.O
+ . . 661,~ 0 0 [ a2Ž + L'(.J 1,ui)(i P3 0R )n + ( i . M O ) 1 g [ 03 I~)1 C W ~ c2 . 15iu)l * 2.I 4 503o* ~(.~~i)(1(.61,100) * (.92')(1.4,too0)a .5~O~]a~ *Ph
THI4N E1UN MTEROTH CiTR
4~ + ~ I .I9ii. 0 2
& 2(~:(.9l +a * .90) 2 + 2(. 2)(1a4oo) +09 4~o) .0c 2 ~j4
* (a + 00)
(a*~ (B + s(441?o Poo
(g + ,,(0) 3 2(.866)(1.73Mlou +4(
(8 4 fl) 4 2w0)[sl + 2(.707)(l44ý)%(1. 4114'0)n
TAME VII
REQMCY-DQPALI CHARACTMIUSTICS OF STANDAIM FO
CL40SED-MOOP
DESIOFATIN ~ CRITER1ION YS
lp, db
2 1.00o% 0
butteryorth - 3 1.05 0 04 1.0o0% 0
2 1.00 no 0
ITA Minim=a J tleldt 3 1.05% 00 0.95 no 0
2 1.00o% 0
..Mini= SettliP3 Time 3 1.1t, % 0
:4 0.80% 0
2 1.280 1.%
IE2 Mi== 2d 3 1.52%0 1.5
"" 1.67% 2
2 1.16 ni 0.5i t t2dt 3 i.4o % 0
S1.58 0o 0
2 1.00% 0
ITA? Him t2e.dt 3 1.15 % 0"0 i 1.'5 % 0.25
2 1.00% 0MY Mni==-or t3e2dt 32 .5
1 .o % 52
l -2
TABLE VII
M OArNCY-D1CM CHM•-TSTICS OF STANDA2L FOIOE
CLcSED-LOOP OPW-LOOP1ON SYSTE__ ION O I
On Mp, db IC 9q6p deg Gan Margin, db
2 1.00 no0 0 o.67 no 64. ----
3 1.05 ra 0 o.,9 Po 60 134+ 1.00o% 0 0.38 fo 60 8
S2 1.00o 0 o.61 o 67 I I
tleldt 3 1.05 0 o.48 A 66 11.51+ 0.95% 0 0.37% . 60 8.5
2 1.C0o% 0 o.67 % 61----ng Time 3 i.14 % 0 0.50 %n 67 10.2
1+ 0.80 Do o.37 60 8.6
2 1.28 0 1.3 0.77 f 52 ----e
2dt 3 1.52 % 1.5 0.56 no 71 6
14 1.67 % 2 0.45 %D 53 6.5
S2 1 16%0 0.5 0.75 % 56 ....e2dt 3 1.40% 0 0.55 %n 68 8
4 I.581%8 0 o.1o% 60 6.5
2 1.00% 0 0.66% 62 ----t2 e2dt 3 .15 % 0 0.490% 68 9,5
t3e2at 2 1.00% 0 0.64 % 66 ....
3 1.0o %) 0 o.46 % 66 8.1
14 1.00%n 0 0.!o8% 6+8.3
522
1T2E2 -
Io Calculated-- Re' 37
8-
ik TE
.2 \ \\ _ 8 1.2 2.
Figure 16. Ccprrlw% cr in~'cal &-,A Epcri~entaI1l Dt~l.nedrt12E
2w r2erformAnce 34aeaumf
10 lcares "ern :s .! ieti'z ;s chow' in Fie. 11 of lief Y1, for b rangingS
iron 1.Z'5 to 3.0, c can be vrrted fro= 1.5 to 3.O 4ti, 1::: than 10 percent re-
smltier !sts.n Is E2. The judicial respo.!es or Fig. 15 Indicate that antn-
misatoio' of IZ- ýcicnt: oscillatory responses fOr higher-onier systems.
An advantage or XE2 is ta.&- its t.±ljt~ic exprenclomi are simpler than those
roe t-F2
,An )~2
The last es eIs h~ghly s~eletive, but it is d',ubt~il
vhcth-n tie. %dded conpilicat'on of the general forms is worth,& the extra conputa-
tio A!i effort required, ciryared -.o rIi?.
Having stet.-it the advaiu.tages and disadvantoge: of 1Z2,jb
1 o IA S
sn= alternaktIve criteria can no'J be assessed to arrive at a unique selert!len,
if such, a asiection can be -ide. Theý next section uf this chanter ia concerned.
vIlt TAS, ITfS, ar-nd 12 %-t ±t ma:.A be 2inlcadt-g Wo close toae dlc~ussion of
1L' .t7e. etc.,p vithout . ix.,r ligresetor. relat- rg to the suitatnlity of thtescriteria for statistical Inputs.
A strong arp~esft in fsz~r of 102
is Vat, for attatlastical isasts,- the ccr-
responding measure 7(t) a fT m a enthruhyinsl
ganed, ani form the founrdatict 'or an extensive litcrature, Znsl1udlng eny rf
tee =ot valuable and vidsy used contributiors to ol.tinisatimse theory.
Statistical inputs fail sutsate the scope of th'. present re:port. Uilsuever,
it wcould me Unrealistic to ignore the ract that s'LJght control syatera aref 4us-
Jei-ted to both statistical and deteelteso I" nguts, sea it s.U :ý' is ufortumte
if the already considerable gap between -anar end detcreiniatic =nies:oz Ofanalysis was widened y dcdoe ral Pr different eriteuia for ecah type of input.
It Is suaeested then tic possibility ol' developing a os. at4ble criterion nLght
.ell fern the subiect Of a Cutcs'e svrestigatios. Mlwever, for the remninder of
this reps?', statistica:- csn~idernt-c' wll- -~ dlee'gaveed, end assesesnts of
Zndiciai error nca:.a'es will be s=ce with rew,'eni only to their "stutbility for
the task of see-sur'inG nsyten perfomnnrce fol'ovirg j ;ti-p input
.he 3AP, LIA-R, and zT5 eazu~cs- have been J-seostigated in 7cf. 37, 35, and
39. Reference 5o7 includes en extensive study of the propertiets of ITAE obtained
by nsctanlratioo CZ &arlor Conpitcr res.pones for unit--orerator sewo-position-
error aystess of flnt-- Threseh citzth-zrdcr. fiSrst-order nmerat.r sect-velocity-
- -: ..joc... . *v. u-jg u a)fl tat-l'der, x4d zrro-aeceleratIon-error systens
if third- th-rongh, IAith-order. jtxiitsl reopenerz and stamdaxd ferns for 'sob of
t s~e.ycte=n art nrtnented. DSo- ?%tta are als'r given on IKE and IT22AS, but WlAS
c=;. ý-Iz it .-.& I- . n-,w-t've than L%--, end -- teazcaer to apply Liar,
n- As. ( Cl car itVslly sre Wte andteaed en an analog ocqput~er, -t.j.
using tierzple .Iirest describe,' in Ref. 37.)
The pr~sernt rrc~ert ennt-ore; th* I iventigetie, of these ,eaaurss. Exact.
,vrlytie expressinen for MAZE, XTAE, sar. IT'AE of eceonl!-crLvr ys-4=s ere pre-
rented, PM tthe -esadle are exsploeyd to eheeck those cf Ref. 37. As will be showrn,
'mn, -- rvc arL d'tr-ted4. An 5.anar Ic eti-ed rer cfhtairnir W.? TTEAR, CAn I-AE
fo:" hicncr..odcr sreter: Is sinso -n' 1r-4 In vdditllcl: Cbayg,,r .'d~jczilcen
pruceodre for the apprex~nte caleoletic . of lIkE for near-uptime system.
Ti-e precedirt for cale-ttatlrg WI, MEAnd. md 2AE for a se!.-nr-s-
ten. Is detailed in Ajprndlix :7, mut bey 'rie belt..'
,he £tarilg poInt ts Ire ealeusltiun Of' the taplaee rnfr of the Lb-
ac-zt,!~ error tins hicte-y, 4Oce en'; b&- been obtalled, IM, rVAR, and
172AZ ere r:Sily estained by mcans o1 tie' final valuee tttoree., an indicated
1A I- e't)'dx al"r LA(S)(h,
-cj ace a0.' 7
The metbnd for obtaining U4s) afL~ytmr-011 i,- an exteosior of the torh-
nique utrd -or obtaining Uth leplace .r>x*orm of' a re-titled sine wave. The
err-or- time history Is a damped nine wave w',tb a phase lag
e(t) - e-.-j.- ~Aa in t(52)
70
where -a& j
Fro, eKe -rassor, t-hles Of Prr. 16,
!'S 'd. e - ýO." li tA (cotn ~-(4) (~
As show. in A4~psodx A, tL, dasigm factor e.% cto In'1 314 ean be .eceoat.4 for
by replAcing a with a + ts 441 to ohtAin Cieýýntslo i. The Phase lag itote-drsn cileavinneA. b',' tw~e tiedl rewilt it, reeSoeobl, ecm~et.
tns -+ tý S + 2;0:t ("A)
Mrae that tb': ;As* term in Zj 5t Is sitPly E~s).
Qoi prteedze 1 ýe bt&anig an smeslytical expression for Ie(t)I, or E4(s),for ýxccC& higher-order systems is nore ccoii!.csted. For these systems, thethe rawv ckwtnas of' be error time hixtter, are nwt eqnUj opaces. and Eq 53*V1.ts tov Ap;iy. ýt' procedure used in Appemiix B for a third-order system havirgz. crclet pair o: roots ^s to represen.t the error response bj a Fourter-likke
&eries, the coeffl.ients of which arm ti-.iependent. The first result ib quite
eocp~lested. It providesa Abests for fwrther tiheoretiral stýuies, and for cheek-
!cg eooriný,tslly obtatrned MA, et:., but seems too involved for routine opti-
risetiM.; cwlculattons.
7ro a normalized uait-=araror secona-oroer s7:=tzs described ty Eq 2y, toe
IAE sr4 IMA are gives by
erA tV77 ehct(ir;lI ~ (5
16
bIAu t.• +rrr 2 sin-'c .
(!7~~_ +s (56)
T nO=O P onsiO , n a • q AO 0 aprodA- orwrick th..at l erits u e
rented in Fig. 17 and ex, sod are -Ccpred with too var'• •obtained by Graben
s:4 aitr, - .l Ref. 3-.. It ir evident that o sealine error of to eyists In the
-1 al IOo graplh for _ePAA. Ade t on IoE ids exrellefe; some ae or
discrepancies exist iT IA. X1 itinzatiou ancurs at 0.7 instesa of , '0.7,
but this error Is insigoifieartt.
The o eonclusirnt of Grasuan s Iathroe regarding the relative merits of these
criteria am not altered by the present study. Ps would be artclpated, Usi ,
Mre, r4o r9AE exhibit sinilyr characteristics to le'yerIe' sod ITa ,t respec-
tivntly. la, IS Sod T12
A% is too complicated, therefore, eaR is
preferrvd. l•%h 3ha been explored wore thaortghly than any iedicial error
neasuieýe %d the consierable bacrgrosd of knowledge established by Tef. Vi Led
38 permits thir nossure to be used '-ith confidence. For each east lerostigated.
in Ref. 17, vin m T. A lAER yielded "goodr responses. s.-S toe criterio, reale d
highly selective for cystens of third-, fourth-, and fth-order.
Cr--psred with lIE2
, lIAR has the advantages Of giving les. 03CiLiatory In-
dl cilx standirid focus end of bel g rore thoroughly explored, but it is the mare
difficult of the two measures to express &anayticciLly. The choice between thesecriteria nust depend on the circumc-tsnces of thc Individual easts In% which they
are to be applied. Possiblyr ITE2 world be sore convenient for so analytical in-
vestiga~tion, and IER boest for optinizatien using analog co~uters, because of
the ease with which 1z can be mchanized szA scaled.
cliosed-loop Bodec diagram- of the I=A standard forms listed Inx Table VI are
given in Fig. lb. Associated indicial responses are given in Pig. 15. Table VIII
557
II
to
-
/
30-
I Calculated
~5I Ref 37
IT2AE
20-
c* 5
o /w
ti
5-/
04 0. 6 Q8 1.0 .
-IT2E
Ficu.-e ;6. cc~parisoo or Ao.iyticil "n Eztripe1-,tafly Deter=ioedIT'A:- for U it-N=-t'.v Stvc d0-Od~.r System
ZIIDCIAL ESMVFM CXARACEMM~S OF aUM" SY0US=
Time SettlingDelay TI RseTie hoot Costn TI-
SaccxA.U-trderI
maimu sttling 1.63 2.2h 4 60 0.046 1.46 2.96
4ýwtje(t)Idc 1.51 2.56 503 0.023 1.86 3.47
.fo e~t) 24t 1.32 1.65 3-62 02i65 1.56 5.34
f" et)2 14 1.86 3.93 o~o6 1.67 5.26
f 0 ,t2
e(t) 2ýt 1.114 2.00 6.25 00o62 1.72 5.03
f 0 ~tS e(t) 2dt 1.5 2.21 4 .6 0.041 1.81 3.05
Third-Order
M~nim= SattlIng .2 20 .2 ,3 .2 35
f£wtle(t)Ils± 2.11 2.3 6.67 0.029 2.66 3.58
f~cot e(t) 2
dt 1.96 1.93 3.90 '.051 2.:n 6.98
J~t2 e(t) dit 2.0 2.32 h.16 0.063 I 2.30 6.,
fo't3 a(t) 2dt 2.03 2.20 h.%6 0.038 2.37 3.61
Nmaimao Settling~ 2.56 2.17 6.78 0.05 2.88 3.8Ti-
f£ntje(t)Ilt 2.62 2.67 5.66 0.027 3.03 6.23
J~e(t) 2dt 2.40 1.83 7.75 0.141 2.6a 10.32
F .W3e(t) 4d 2.49 2.23.9
I t e(t)' 2dt 2.5a i.16 is.6 i o.-06 2.86 5.0
Irt ajt 6d .2 22 .9 j 0.060 2.9.3 j 3.93
60
cmtwarizes the indicial response characteristics of aU the standard form of
Table VI. Standard tom root locations am shown in Fig. 19, 20, and 21.
0102 UTICIAL L.( ;=
The measures discissod in the previous section4 of this chapter .avi- all
been integrated functions of error. It in pconible to -encr-!4-e-, ' thi measues
by tneludi functions of the ti= derivitlyes of error. Thus, several recent
references (e.g., Ref. 79 an,! ,0) have proposed optimotion procedures based on
the following very general mewiure:
F~e(t), t, vj]ptt)dt
Here F[(et), t, y1] is a general function of error, tiae. sad system parameters
), and pi is the probability tht the output A11l be used. The gemermlity of
this measure is both its #Avantage asd Its dravbadk. It cannot be put into con-
crete, usable form vithout -4n a mzber of arbitrary choices to arrive at F.
aonsideration of the resmaning measures listed in Table IT Illustrates this
point. The criterion
fe (t,)dteo + c e2(t)d, Jon (~r dt
yields, for a orer-lized sccond-order unit-numerator system, - 0.866 for
a co and 0.5 for - 0. (As shown in Ref. 73, the ý selected by the crite-
rion can fall out- -de the 'ae 0.5 to 0.866 for a negative a.) No guarantee is
failsaole that an a suitable for a Aecond-order systen wul yield acceptable re-
spaz-as wlihn the same criterion is applied to a eystem of higher order. Hove-er,
the -riterion cannot be relected on this basis alone without further investigation,
partic;.arly as its rando•n areIn should be easier to usr With statistical i.rut3
thU. the rendo= Oanalog of tine-weighted cr-tteit.
Arbitrary constants also appear in the =,L stic form associated response
zeas-res listed at the bottam of Table IV, and In the very ge.ri%2 nessure
Second Order -2
LEGEND L5-no Measure0 IIAEx IE2
0 rrt2 Iwa hT3E2 x7' PWorrm Setlin Twme+ Butterworth/
C15
L L I -- - J
-2 -1.5 -1 -05 0T~
7101"~ 19. Pole Incations of thit-Rumeratcr Second-Order Steandar Porms
Thir Order -2
LEGM -. 5
ITrAEx 1E2
o3 T ~ 2w0 1T2E2 1* 1T3E2
+'V Wiimin SetflingTi
-05
I 0-2 -L5 -I-Q5 0
Figw~e 20. Pole !Locatforno of Unit-f'o ermt.r Th.ird-Oder Fors
xLLEGEND I 1.5
ymbo, Measure A0> ITA-EX IE2
a rrE2
jW
v Minimum SetkngT•me+ Butterworth
Se x -05
4-o
II I
-2 -15 -I -0.5 0
PFiure 21. Pole locations of Unit-Rumrato,- Pourtb-Oder Por=
f"D dt. n-0 dt •J d
Here the choice or & a2 , ... , an Is arltitrary, so that czy desired weightingof the elecents of the c'literion may be used. Unfortunately, little gaidarce isavailable on the choice of al, a2 ,. .. , an and considerable effort may be required
itf it ti necessary to evaluate the interrals. (However, this m•asure has usuallybeen proposed in conjunction with dynamic programing procedures exploying digital
computers. For ouch applications, the latter disadvantage would be inslgairlcant.)
New maesures are constantly being added to the already vast literature on
this subject." It is perhaps worhwvnile adding a plea for caution in acceptingthese m*atures as valid criteria. Tt Is all too -asy to present a "reo" per-
foroaknce criterion which is valid, eclective, and even readily applicable fcrcome particular systen. The task of foraulating a criterion which retuinc thes.
merits over toe range of syctems encountered in flight control design is much
sore difficult. Of the indlcial error measures diseusa.d in hias chapter, ITT
stands out because of the thoroaghnuss vith which Jt has been investigated. The
large number of examplez given in the studies of Graham and Lathrop lend credence
to thoir assertion that the IARE cr.terion has ex•e-•lonal meril.
Thia point is weli-il1atrated b1 rve. 42, which was received whoLm the presentreport was being typed. In addition to MEB, MT
2, ITAE, I-T, IT?2, And I2AE,
Ref. 42 Investigates the following nanures for normalized unit-aroerator second-and -hird-order systems:
f0Vet dt, ( Jtle(t)I dt, f t11ie(t)I dt,0 -e0
t4lett7, dt. and] _* Ato• , + (e(t)J0
Both digital and analog computers were employed to generate the measures. Theresults are generally in agreement with those of Ref. 70 aed this report, exceptfor errors (in Ref. 42) of apprixiately 10 percent on acme standard feorcoefficients.
65
orimi srAwioti Frop;
Ts* standard !,,r.i; dll ----- d rerln? 1n this chapter have afl been cptlral
the- stuttit ti-ey - U-t In ,AIIW% V61-1, of parthedlar iuldilal error4- r ft-riq could kA used for optiniouniun, asia Aid. Dd pP...Lhs
tro stnndard f.Tors asocniljtd with minimu= settlino tir-.e for systems of see erA-
throne tighth-ordl-r. TIhe necond-, tnhrd-, and fenrth-oraer ioins assA associated
transient respomnse acm r- r',eed- in 7t1.- VT &:,d Fig. 15. Figure 14 illustratesthe T-ode ( %(,I di._--:: f,:s these- systems. Table VI asons linan bin.Aial andflotirusrth ateanudwd rm ThL bnioimial forcs are not optIL&alsIi the a%=- sen"e
as tae stanzdar forms considered previously, being generatcd by the following ox-
!ee S the isystem order.
The Butterwesrtn for=s arc characterized b) the lact tnat their Doles areequtuly spaced arowel % -;ericirole of radius 12 in t-* cosplets plane. The
frequenny-dcezsercrse.ato of th- ?neitiflZ ireestfer functiorin Iý lulusratedin Fig. 11. for seccond-, thirdn-, and foerth-oncdr systems, the indicial responses
are given in Fig. 15. iiatterwnrth farce are naxls=aldy flat, i.e., at jus 0, thefirst I, derivativ-. us en- ssnpistesle..frsqeeer.scy diagv=~ axe sees, b*ut their tran-
sient respolnse: tIave increasinpjy large overnshoot: me the noder of the systen isin=rez-&I. rIt~rr..e $7' iv Leetransient responses of tic k$utterwortn ftlte.rs
for first- throegh eighth-order, cvltch nrc ropreceured In P12. 28.
11- Itleer the Buttersrre, :.Or tIec binoutal standard firms are of direct "ce Infitei., cenre: system' nptizicstione. Hlowever, they do provide a convenient start-
ing point for devising, otter stastard terms.- The ITAE standard forms of Grahan
and Intisrop were, In fact, generated b., nystesatically modifying Buttelrworih feces-
of the -pprclpriate order.
(A,
({AFTM IV
MECT 0? A TM'~ TIM4E 'AG UF07 r.=CAL MMOR =SL=i
This chapter presents vnnnrol vxect forziulae expressing the effect of
system time Is on indicial error =ea3urC3. These forn i•n eay be used to
obtaln approxinate relatiors expressing performance measures for some high-
order systcr= in terms of the performance of "eqvJivale-t" lc-'er-sr-er systemns
Possessing time las.-
As is usual, the analysis -4ii be restricted to systei, having zero
s•e•dy-stat* error In response to u step-input, to obviate infinite values of
Indiceal error measures. Three classes of indicla2 error measuras vill be
UoDsiderod, desiGnated It in tq 16, 17, 13 of Chapter Ills and repeated below
for convenience:
0
z1J - f tUP 07)0
tf. (18)
It r-l1 bn sioc. that when a pire tiýe d Ilay, s Introduced, the measores
can bu expressed in terms of the meatures appropriate to the undlayed syite-c
1,7 =-%z "fth !ollowing formulae:
Iu.U d . (50)
ITV - Uo 'I!,. l lo (5")
nI" - u. l2IuI2,.o+ + l- lJ + l I"l,..o (6a)
wiire S(3.
67
proý
proof:
IUI, 1IU, IT2, etc., can be caleula.ed by application of the final value
tbeorem. Thus,
*u .lt . • . 1 U(s) (61)
s-.o0 i d.;62-IV .Ji " d• U(S) (62)
110 l. d (63)
For tOe syltem vi-th 1g,uo
U() - ! (I -e--) . e-" u (s) (64)
By the fizal vaue theorem, expnding e-'sr • -Ii + (is)2
-
iiif W .-. 0 " " ) - 0 u(S) (65)0
r"vo ÷ 1M .(4) (66)
"- • ÷u0 .0 (67)
vhIc. checks Eq 58.
171 Is evaluated sizlsrly. From Eq 62 and (4,
/£tU(t) . -~J(.) I....e [L,. o -- j -U (68)
e r... U O,1 02+ .o(S) - - + ! (69)
68
Applyingq Me final va~ue theorem
CDtUft)dt "a' e7's I U_ (a+ 2ia Te.u
ii.U0.1's Tj l-t (70)S~ r, 2
[ 1 ITEUI. + +T[I] +o 2. %0 7
v h ic h e k a " k E q 5 9 -
So evaluate ITJ q 63 am ý ame tc±omied o give
LtU~t) .U(g) -- 1e-' #U (a) -e" -2- Te' u.(s)-
-13 A.. u s.e a2- U00
-e da T- _d` 0,.0(2) ~-~ -t-.e
ST "-t d - t-'s -0
-se - (73)
After trte rzluction, thit yicldý
2I[=] [r2]. -[~ (74a)
Vhic~ htk C!caq 60.
DV=2~ C TDO L-- Or, ITAE ArD Ir P S0 D- C8ýSE2
12te effects Of A time lag., to on soe erformance measures for a second-
order zero-potitioo-errur f.yst= are ow considered. Apart froa their jotrivaie
valt.*, tbc results also provide approImsaticas to performance mesures of nigber-
order systems. The transfer functice of a second-order systen with lag applied
directly to the input is
69
2
It ij c.nvcz,.iet to work 1i ncn,•flcd (non..-dl.craL) time so that
C3) 0" (76
De,wr=MIZatiC% of perfor=anee measures ic easily a=cn=spUed by use of
Eq 26, 27, crn ;.8 of Cl-z-zr MI.
For the System of Eq 7> vith a Step input
E(.) - s 2t (77)s2+ 2ýs +
Appendix A hc-3 that for this systca
Va.0 T,- [,t" ~ 1 2ý (78)
e~.0
and
,T 7 2o21j, t2 24 If 2 4F;1
+ 4t2 (79)
For t > I
[,TA].O- - , (•)
[ *TJy],.o _[],.. q2 (81)
70
SubtItut•.io of z.hese expressioi,a into Eq V9 yieldz the -*,edte qiaphed in
Fig. ZZ. TVP effect of tI m lag cn IT2
has been caculated similarly. F-meCiaptrr 1II.
ne -r ~~ ]2d,- C2 (4.5)
h.- c-ffct of yarrouz tlre delays on IT2
is snorn in Fig. 23p, hich Las
bet-n constructed by co•blnotre Eq l, , seud o.
,PP1IYA.7ION1 TO PEOMOPJMCE WMA.2MS CF HIGr*R-CFMM
Pig"re 1 -ilusttratts the indicial responce time hictoriec cf ipt;-i TlrAi',
le2 and rT2E
2 :.rO-position-error systens. There criterin ray all be regarded
a- r=c-hil vnl' d &-.d aectIxce at. lossi Au, the system orcers shown. Ineach case, It vwil be rnted that the optiým nth-ordor systen response (where
n > 2) ca.. be fa'rly well ap I~ leated by adding a tl= I=-. to *.he opti=ýmsccond-order renpo=e.. Thin tme lag is conveniently chosen as equal to theeifference betveen the delay time of the stcuni c-tel and the delay time Ofa zero time lag secona-order systeL. (Note ttat the delay time is defined &ath- t'C. f.r the -=:o:--•c tu ac-i;c 0 pOe x.t of Au finsi value.)
A pessltle procedire for r .roxicating to the perfur-m.zce measures ofr.ig%-order systezs now becomes ar•,arent. The actual system say bp replaced by-, equivalent second-order system •ith time lag, and the performance measurecilc.•-lat-cd f.r. the latter syste= by =eans of Eq 58, •9, and 60. Thin pro-cedure is in fact quaite aimp.Le, and of reasonable accuracy, as vill be demon-strated for te IMAE performance measure. Hoverer, It does demand knowledge ofi-he delay time, and a method for ertinating this parameter vill nowr be 4c-
EST1HATION 0• DELAY ME)
Each of the optiC-s responses shown in Fig. 15 can oe roughly approli-=ated by a delayed reap function ter-minatei .,c=r L•c rirt c(t) zerv crozzind,
as sa "tched In Pig. 22.. If the delay times of the actual wd approximnted
response are ra•de equal, ther
30- tyST-fT-40
25 -T50T3
20 T =20
2015
10-2,T-
5-
0 02G4 06 08 LO 12 L4 L6Dompirg Rfijo,
n1re2. TTAF fro a Second-Ord.r Zero-Poslt~on-Ermr Syste= vith a T1w Deylay T
72
I 0I ~~r6) 2 +s4
RT~s)
T.1.
TO
02o 04 06 0.8 10 L2 1.4 1.6 18 zo 2a2Dompig Ratio,C
kigure 23. M fl or a SecorA-Order Zero-Posltion-Error System 11tb a Time Delay, T
71,
TotdTI
Figuz 24. ha~ Approimtion to OptL-al Systo=
t * - To+ coty (7)'
To and y cay be ftrt-e: -4•latcd to the actual recy•nce by deanding th.at the
integrated error of t•e aetual and appivflmate reoporena shall be enyenl. Thia
kmplies that(To . cot 7) ' To
q, - - (8a)
where q1 is the coefficient of a In the tyiten tra.sfer function. (Note: ql - IE
for a urit-numerstcr system) (See Chapter V) If the ramp approximation I- valid
(as It is for the mlnim= fl!2
, iT291, IT31?, and ITAE systems of Fia. i,ý. r.q 83
,411 be losely satisfied, and can be colined with Eq P2 to &ive
q, - td (84)
Figures 25 and 26 show that Eq 84 is closely satisfied for the minimum ITAE,
T-172, Tl42, and fl r+-s. It also predicts the delay time for some nopti-
mal systcms with good accuracy. For example, F1i. 27 Illustrates the delay
time/q1 rolationship for the Puttenwrn * iere v! first- tnrcotgn el.o.*order.
"The indicial responses of these filters are graphed In Fig. 28. It will be
observed tat even lor tne eighti-order filter, the delay tlie exceeds q1 by
orl) 6.8 percent. It Is therefore coclskl that Eq. 84 predicts the delay time
vit good accuracy for optimum I['E, ITE2
, 114V, %-V -yte, and for systems
such an Puttervorth filters which have responses approaching ttese optima.
IUIM11IEZATICR OF q,
The coet•i:Ie".t q, is equal to the IE for a unit-numerator systen, since
•n-i sn-2
- a l i m .L ( ) . a n - :, + " ' " + q _a o q4! sts*'t + ... + qa I
ql (which in the generalized inverse velocity constant of Ref. 72) can also be
interpreted in I-rrz of the open-loop system characteristics, as follovc:
rut C 8) - n 1 * 1 (An)
R .,n. + qls +1 1I
75
6-n System Order
/ n:8/ °5 /on=7
5'4
/n
/n :
NI I sonI I n
0 I 2 3 4 5Nord.mensional T'rne
Delay Tine (Time To 50% FI%' Value)
FJgure 25. variation of Delay T,2e ywth Integmted rrrorfor Optim IAer Zero-Positio-ErTor Systems
76
6-
/5/
SI LEGEND
I / 0 11202
I /x 3- L OE2
2kn=
o?1/ x
OL I - - -IL- - - -
0 1 2 3 4 5
Nondimensional Time
Deloy Time (rime To 50% Fnc: Volum)
Figure 26. Variation of Dehy Time vItk I1te-ral& errorfor optimus &E, iTE2, fleE2, sod iT-. system
77
na System Order
onn=8• --=7V
o /on=5o /0
0"0=4
b 2-
C n/
//b 2
v I ! I1I0 2 3 4 5
Nondimensaonol me
Delay Time (Time To 50% Finol Value)
Fgr 27/. Varatz•u of ueZly Ti wi•th Intpeted.Enhr for Btterwrot Filters
78
3
1.0
00.
5 10 15NONDIMENSIONAL TIME
Figue 28. Indic1l Responses of Btte vorth Fi±t-..s of First Throw=_Eighth Orders (reproduced from Ref. 3?)
79
""I , q1,(X,. 1)(. , 1) ... N-16 + (87)
unerFe - --t- c. etc. are the open-lwn. z+,ot.s
rfasue for a rormolized mro.-p•,•tion-error systen, ]r/(s) is alvays of tme
rout= 9(X, * )(G I 1) .)-.. (+,a.:a + I),
q, (88)
vnere K is '.. realized gair. (or Inverse velocity constant).
A further Interpretation can be obtalned by applying forculae expressing the
coefficients of a polynceal It- *.er-s of its roots (e.g., Ref. 71U).
Let
an qr..1 sn * .<..qls. - ( %- , )(- a2) .. v.(sa-o) (89)
v ere a,. a2. anr a te ctLtlose.-o,,, poles.
Nov -.. • a," .-.- . (90)
and
Divi -- -. Eq 91 by Eq 9
-,J (02)
Hence, for 01'2 ... n all real, q, is sl--ply eqVlal to the Sun or the &ýStem
tine constants. (A second-order factor yields tao 'tine constants' uSznlg to
the approprinte 2t/%)
Some furth-r approximations to delay tL.te are discunsed later In tnis
clA1,tr.
Having obtained a good aeproxinaticn to d&lay time, and interpreted its
-nI&-if±-'.c, the result =ay nwov e applied to the calculation of perfox-r=ce
measures.
unTza OF APPRMIMATh CALCULTION OF pvPy-TgMMECAWSVft FMR !CH-W101 215To
* ~~~~To -omoKne*1trat the -petitol proposed, It 1-is bcc .,Tlin~t ' tý "eigte
oZ flAB for nonmalized wnit-noueratcr systezmw of third- tL~rough eighth-Ord-er.
The opt!=u response of each of tesse syatpee has been approxlested by a delayed
secornd-order syst-m vitt a &.-ping ratio of 0.7. FIigre 29 above how the Ml42
of suer. a system varies wita the tinuc lig. Ti-c nessnered sand predicted optiac
flVt are cakparsd In FigE. 30. The- agreement is seen to be good, espacisily for
systems of third- through sixth-order. Mhe slight falfloff In accuracy obtained
with seventh- and oighth-ordsr syatees Is dun to the deparL re of these systezre
from tbe approrimatirng fors. Figurz, 51 illustrates this well; the egregious
behavior of the seventh- an! eighth-order time tiatories is ratened by a cor-
respc-viigly unn. a-td distrinution- of poke (Flg. 21 of list. N). There due.snout seen to be any erpisestion or this phena-zeson. in %.is connection, the do-parture of the delay tine from the predicted value for the Deveoth- andi eighth-
order systems shown in Pig. 25 should also be noted. Fur htg-oi-rer systems, it
is important that the effective z should be estimated as closelZ. as possible.
Thus, whorerer onse-t vzalues of the delay tize are availab)le, they should býe used
in preference to the approxneste xg ; 4. insis sLpproxisation Is setisfaetory for
systems of mhini- through sixth-order, hut undecesticates IMTAX for the eighth-
order sycten by 20 percent. Hceoevvr, in vin-Y of 11w large Ovexohocts ti cfuea
"opt" -AEh seventh- and .i~htbt-oedrT systn''.: it -. "- 1--. t%- .ƒ
discreperoyf will he significant in practissi optimization problems.
naM"= APPSCIOXDIOfMi TO TEE DUAY TM
In the course of neveloping the spproxintinn described above, a zneter ot
alternatives we-re Lnvestigated. It is folt that a brief discussion of two 02f
th1se wcald be of interest.
7w~, ýý- MAX values for tot -sixth-, seventh-, swee igbth-onter systemswere ',btoincd by applying Sicpscons rule and the trapezoidlftL-grt ruletc the cptinje r~sponsea shown in Fig. V, *icd by then averaging the reoults.Keasured MES volues for optin,. second- through flth-order system are givenin zAnC. Y,7.
1 - J i~24 MR___$14____ ___ ___ ___ ___
*161
1o
4-
0 1 2 3 4 5
Twne Delay
P1iure 29. Efect 0: a 21z~e De~ay ou VIME for A thet-Nuera-zSecd-ordor System
Time Log - Delay rime-Delay Time For SecondOrdsr System
2- Damping Ratio, CS0.7 For SecondOrder System
22
20
16
14 o0,n=n
lo-/
t n-5
c 6-
I0 12 14 16 18 20
Measured I T.AE. For Optimum System
F1gre 30. C=;riamn of Xeased and Predicted rVA!
L23
.75
Plý4ure 31. Indicial Dc~p~-c of ITAE Stzndarld Forms of Second- Tw~ouighElg5hth-Order (repradotrd from Ref. 37)
Pimt, delay time was plotted against th- coefficient of sn'1
for varous
ntila sy3tes as shown In Fig. 32. The relation Is not lintar, excelpt for Ut.
binomisl filters w4ere thc coCfficients of s and san- are identical.
As show =n ha. 53o, a faLiry litear relation exists for ttc cpti- 'ff.E
systes between delay tine and system order; this linearity is retained for
lnosial filters although the slope changes. Liuearity for the blnmiul tiltera
would toc anticipated since the coecfiCient of s1
"1 (- coefficient of a) -. equal
to Lhe system order. P.r the optin IT!?, ITE2
, s)d 1l3
E2
tystems, the cot-
relation with zjnt.- so isn cot as good.
It Is concluded that bol, or these approximtions yield lower accuracy than
the reli-t•hip eoxressed by Eq 8li.
85
6 +
5+ 0
4 +
3[ +
c 0 ITE2
o / A 1E2+ O ITAE
+ Binomial
El
I A A A
_0 I 2 3 4 5Delay Time Nondiinesir.'aIl
Time
Figure 32. CDrrelation of Delay iTee vIth Coctr~cinL or' sn fornth-Order Unit-N*merator ITAH StwrAard Fhras
86
co+ 4- 1>Nc
WCJ 4 01>
+ (> o
Inc
0Y
-+ (m 00
oun.LAoia
873
CHAP'ih V
PEaMru(AC Mwdzu- Aft GMMPL rnMOwMnc IWMr
Mte lisp inpIt has become stanlasrided as the Input required1 to generate
the responses from which most perforns~ne measures are obtained. Ibdicsl3
'eR,.,-' ere. easily generated on an analog computer, sl provide a ,co,ve'ctut
means to calculate te .esponae to -e..een lnatt. -h., theb use of
- 1' -- tegra, sr epivent time-zerle3 procedures. It would be extr.eely
convenient if perftorncc r-asures appropriate to inputs other than steps could
be calculated in a si=ilarly direct fashion. (ID general, it is necessary first
to calculate the response, and then to appiy toe seasure to the respoow). in
this chapter it is shown how for ace measures this procedure can be -short-
clrcultd- i.y calcujlating the response Assure frrA sasures applied to the
Input and to the synte indictal or Impulsive response. Thus, instead of ap-
plying anesut.- to a in.lice trsnsforr of high order, it my be calculated by
algebraic operations on measures mp'lled to two Laplace transforms of lower
order. It is also shown how certain perforzance measures of closed-loop systems
my sWI1iarly be callelated directly frum perfw.-unce ,",,res for the open-
loop system.
,hese procedures constitute the first step towrs the establishment of a
"calculus or perfor•ance measures," i.e., a method of expressing performance meas-
ures for complicated system in terms of (tore readily computed) measures for
simpler sysLte. This chapter concludes with a brief discussion of error coef-
ficients.
UE FOR GEnALI iMn
The response to a unit ipnlae, or Dirae 8 function, will be referred to
here as the "Lpu.lsive admi4ttance." It is also known as the "weighting function"
and memory functlon." A system having an impulsive adaittance vhich is nero at
t - cm will now be considered. By the final value theorem
1 . Efi) - o (93)
whr E- is the syste= transfer function.
89
W4 arca enclosed by the i-*plsive edmittance and the t-axis is
-cf k:td lim si1~(%B -- so
Us
The area ealosed by the Input and the t-axis is, silarly,
R S(s) - inut
Tie area -closc by the error response to a general input r(t), satisfying the
conditicn that r(t) O 0, is
s- -0o " 1$ R(s) - 1 (9-)
Tis ara is tLi it-grated ereor response produced by the actual inpet. De-
notirc this area by IE yields the simple formula
I" - "ad-itta.ce x i' pt (98)
The implications of this formula for systen optimization using the 19 criterion
are wortbh of comment. To opt'ization proeedzre is assured to consist of t•e
atjustzert of aystem paramters to that the TR 'a ,*ddin.d:,*the irxt being
fixed. Minimization of the i3admttance vwil thus result in the minimzation
of the it for any speolfl.' flenltt inpt.
Note that the conditions that have been Imposed on the admittance arn on
the Input result in a firite (or zero) iE.
As noted In Table rv, constraints must be imposed to avoid selecting C 0 for
minima it.
90
Intciated Error (19) tor Step Inputs
There is ,o ese-iti, difficulty in extending the analysis to deal with
atep Inl•ts to tero-position-, veioc.ty-, or a~celcrati"--crrc- Cynte=s, but the
straichttorvard geo-metric interpretation of Eq 98 1i lczt. For exasple, con-
sider a zerc.pozlcim-errer cystem hAving the trensfer function
aC 1 ... 1 qe (99)
lik +n G n1 n , ... . all + 1
For a step Input
S1 an %.1 +.. + ql. I
0 + a. ;.s n• a0
+ ..' + 1 + 1(101)
- q,
To apply Eq 98 here, it is necessary to replace the actual systom by a
substitute which b"n an Impulive edmittance identical to the IndLicial respone
of the orieinal systen. For s-eh a systez, the transfer function is
* ""e t q, (102)
Rs a n. + + a +
91
The I•adttta.cc is
un I sn + £ -1sn-' +. qs6-0 + qu " ' + - . 1
_0 ... + y'-i + +
The Uinput is unity for a Dirac function. Fence Eq 98 r-duccs to the trivial
form
L . Ta•dmittance of substitute system 1
Integrated Tim moment of Entr (mT)
With the same restrictine =n the input asd asoittance as in the previou
sections,
iE-, E a -- o s • -; • -t [-iEs)J d1o.s)
c (So - I t a0 ).i (106)
-(. [If1lf<.>..) (t --• B.s))
• (.) <, R(S)•• •<,, MO=
/ 8-0 d. aR FEj (107)
92
But .iR -./t t (•i) s . I E,- )
f ((-1E•ct C ~~t - t 6s(19
are_ tl..- t Etc1 CD , d
im a I Ld~s 010EC- da M34~ i(s) aITo~ittsce(1)
Hence, Eq 10 can be interpreted as
1 - adnittance x TlEinlut * r admittanee x EEi-nput (11)
An example of the eczlelation of Uý r1E by the itse of this formula now follows.
Tk, implicatioas of Eq 111 with regard to optimization are discussed sub=-
ouettly.
Ezaple of the Calculation of MTE for a Second-Order Syst
The fufloving exampleMs 1~atratoa the proxced.rc for calulting =E =ingZq 111, and checks the result against that obtalued by direct calculation.
The transrer function of the systen considered is
- (112)
The input is assumed to be describe4 by P(t) - e•ct - e d. Evaluating each
-iar.tity on the right side of Eq 111
•Lt ftI (eCat- e-bt)dt (113)'Esamittance * t_•co f.
I 1 1 1
a 1
1 1 -(116)
U ia I at t bt
.tt.ace -t (te - te )dt (117)
limo 1 dL. I a d 1, (18
Similarly, minput (120)
Inserting these results in the general for--la, Eq 111 yields
S.. . (121)
= - z, .
a.2 b bj b2c ac db2
Conventionally, this formula can be derived by direct application of the Laplace
transao-m
1i - IA '
914
IE - !! [ 4 (I , na - i - c
s-0 Gn - a)(z - e)2 Cs + e)(s + a), (s _ a)(s _+,
(rn- d(Z -a)' (S -b) (s -e)2
(n'_ tUs .b-)2
( s - W(s + d)2 -s )(s - sa 125
aL_ ,.c .. ... -- bed)
whien in identical with the result ottatr-e' freom t'se fornuha
GMRl~AT. EEIATIý_ BETWEEN CERTAIN MEASURE OFOPEN-LDOP AND CiCMfl-iCOF oSYS-I
The. relations 01 tie runlonc= section have been 1ýlelcieLly expressed in
uf zls,-Il.--., pa-ameters. Squivalent. svr.-lolp exprennior: Are now
derived. These exprvsniont enable the cloned loop Sb end ITE to be calculated
withost factorinjg (or even -'riting) the eloned-loop transfer function.
Iziegs-ated Error
Denoting tin. open-Icop transfer function by G(s), the eloned-loop transfer
function, with unity feedhac-le milating cerrr to input is
R& *h 1_ws (127)
Pro; the Fýevio~s section,
915
I~lsdlo ad~wc 1+ s ~ s ea *--.. T (0) (129)
1~,as zc~in
so
3 ý IEnPut 1 0 3
and
=tlo.ed loop 5ad IonCme I TVjopen loop adnttttw,co
Me'e~r~t-d Time-vtiment of Ekr~o
For a unit.y feedbacK closed-loop Ssytem,
E(x) (I * C(sa] - (s) (132)
Differentiating vith' respec- to s,
A-G-- Go(9,, Ed s R *1 (133)
7UAIdng the limit as s -. O, and. uking use of the results previously jbtaioed
for 13 and M1, Eq 133 caa be rewritten as
-13 .113 -tIE(134)
96
flea a -1x1i
from Aich I .~i-at - Z o x Penlo. , a-'mittancc 0*5uopmloop c- ttznee
As outed it. Qiapner li, -e nn..-;- A- optirni syttcusu *za'l and
TAEA and 1hZ Can be approxtated e!tv. ioot accuracy t7 =i --- it.en-,thn toe
"-.-ts of validity of thim approxiation, the formulae developed in the present
chapter for it and .E with general Inputs, and for open-loop nit closed-loop
forms of in and IE, can also be applied to ITA and IME. Corrsponding re-
lationsaips for IZ9, 1i72, 1. ad enact formulae for TUEA "n 1AE have not yet
bce. Obtained.
The time--mighftcd -e r: of the L-," e re.s.ase, referred to asn) "lttance I., this cr:apter, are simply related to the drraL error otef-
ficients discussed briefly in GCapter In. The relationship veLl now be damon-
strnetd (follo-Inr Ref. 11 and 72), and its implications tudied.
Error Coefficient.
The generalfie4 dynamic crror Coefficieut are defined in Ref. 72 as suc-
cessive coefficients of a power s-lres expansion of
-().E E0 .3 1 - B2 2? .. (136)-Uls)
wniere X, Fj, ... are the dynamic error coefficients.
2s) -s) F;() ;:r) t~%() -:. (137)
For an impulsive Input R(s) - 1.
z(s) . +o . (138)
e(t)dt i 1 1 E(s) - (130)
cW te(t)dt • • [E(.]o._ (- ( ) . .z. (A,)
97
a de
%: m 'f ve LflcFICI"' ore Airectiy proportional W tdAiointeGrals of tr" 1ni. delve response.
For a secord-erder "nit-ntroerator sy.tem having the transfer function
Ea (14?)
d E 3l + 2t.,. * 2')(2. * 2~C.) (- 2 * 2gwns)(2s *2t..) (144)a.- 4 (.2 +(5
Taigthe Uni.t s: -O0
-l -725P -
SuSJamr'r. !t cnr be shc'e. that
E2 (11.6)
None of' those error coefficients results In & satisfactory criterlon for en
izpualive input t. toe second-order system cotsidercd. As noted in Chapter II,error cuefftivints fail s. =rei:s =ms of their inibility to -distingu.-sh
between positive &nM negative error contribjt-4l-.. f )e(t)vd*fctgqLt, etc.
98
CHAPMT VI
AcMxmwY, cý=-vIT, ANlD roMIn/lacr ThmM
In Chapter I, it vas stated that the dynamic perfouasnce of a control
systen or element is assessed by considering stability, response to desired
inputs, resp•nse to .ra.nto- inputs, accuracy, Insensitivity to pgrreter
,hanres, and pacrenerr, dcmw--z. T'hI. rujczL is pri'n-ipj concvrned with
the rcsponse to desired inputs. Az previcutly noted, the topic of response to
undesired Inputs requires cornsideration of ztsti.sticsily-cc=rtb'- inputsp, and
falls outside the scope of the pesent report. Stab•lity s-o-ures have been
otoarized In tabie I; the r-ealnLng chsra-eristllcn of ac-urac)s, in:r.itivlty
"to parameter cstnges, and power/energy demands are greatly dependent upon the
cctalaj =ectnlzation u,1 aseruynseic properties of the particular flight cma-
trol cynic. being azsesbed, and do not lend themselves readily to generalized
.Lulies. This chapter tnresore prrconto only a brief discussion of each or
these tile-e topics from a hturistIc viewpoint.
Accurac•
a. accuracy is enacutial. the suppression of error, it can be studied
by the direct or t.ntegratcd error s.asres discumsed in previous chapters. Rest
cf these neasures have beer, related to zero-positton-error systems. Fcivalcnt
syste=s are nornafly of this form, althooi the linearized mdel of the actual
eos ky siossesc a •-•al position error. ThIs ateady-statc error is explained
belov.
A unity feedback syste wilt be considered, vith Open-loop transfer func-
tion G(s).
Ef I~Gs)(i7
For a step inp•t
E(t) lis I I 1t--w s--o a " " 7 "3 (108)
99
Thus, the system is a trie zero-position-error system oly for G(O) - Go. In
deri•nR equtd lept oystems, this lnaccuracy In genera3y neglected, because for
a ogood system, the open-loop asplitude ratio at 1-v ercqr4n' is usually
high. The. equivai-rnt 9 y ndetolctA '_n Chupter I i sutrmtd zhis veil1 !he
resI8-A&I error Is - 0.6db for the eXnd system; U in error ~a ý_Zet~ed In
forming the equivalent syztes. T general, it i3 believed tiat this neglect is
justfined; noveyer, circumtances may arise vteiv steady-state errors of this
magnitude ecoud be significant factors in perheemnce assessmet. In such cases,
the system would either be modilf-d by addin integration to the equalization
to remove trim errors, or a steady-state error would bo permitted, and V- PAri-
tde considerd together with other criteria in judging the over-all performance.
lnsensitivitt to Parameter Ch-npes
lI in desircble that the dynamic performance Cf Z given syste shall not
be degraded by c?--,en In parameters cc=-rrine either as a 'evult of a change In
fligtt conlitions or because of xcspsies !betren prect:.- d =t• oki cc-
ponent characteristics. In flight co•r.al sstmy the latter "r ^% my become
particularly acute, because of the well-known uncerteainties in derivative eotLma-
tion. Important derivatives such as II and lp are often made up of eeoponents of
approxmately equal magnitude and opposite sign; the relative magnitude of the
over-= derivative is thus very sensitive to =all changes in any of its aero-
dkamic components.
Foruslation of the performsnce measure L-ing employcd In analytic term is
the firt step iowards =%stering this .; . . . .... _ _' _"ti
Process is carried only to the stage of defining the performance masure i terms
of the transfer function pole a zero locations. The further step required is
to define these root locations In term of the aeredycs.ic derivative•, eP solo
pilot gains anA time constents. Referer.ce 4 describes the approximate factoriza-
tion of conventional aircraft =t=sfer 1xlactions in such Literal terms. To apply
the procedines outlirnv in thý ror it is desirable to study (by the techniques
outlined in Ref. 4) the sensitivity of zhe equivalent system prmetems to uncer-
taint,et in the basic aerodynamic and autopilot charactexistics. A rood under-
standing of the effects Cf sea.l chnge In pern••etcrzo s=y also be Obtainel
throueh the ti-- rector method or Rf. 12 and 21. Further analyses of sersitivity
ar$ given in 1,ex. 3; and 57, which investigate the changes in the roots of tLe
100
characteristic equation restating ftoo :=all changes In Its coefflaieeur, and
In Ref. 75, vhich uses root-locus techn.i-es to Study the sennltlvlty of the
closed-loop roots to open-loOD pamreter changes.
Pwer/Energy Detwnds
As noted in Ref. 67, "if the gain in a physLcal system ir mdc large enough,
a point is reached at which the peak acceleration of the output response exhlbited
In toe liear rxodel excced that wbich -y he phyrically obtiined from the power
e.ctuator c2 the act-ml systen. At trte point, th linear rodel my cease to be a
,ailid basic for design. Either a nc-llaý.-r cteoticsl model mat be ceployed,
ir the design procedure =zt be modified so that, although loaed no linear theory,
"z!-x. po=' 11ity of saturati.n is recotnized.-
It is therefore neceo-sery to have o=ne c1eck as to whether or not saturation
is occurring. Frequently, this my b" acceplished by ezacing the onnit3wei
of the peak overshoot of the output rmsp==. "the %pprop.-lat. systems charge-
terietics gaphsa •iven In Chapter In, and in Ref. 13 and 24, wiu be f-.-d •u•ef
for this purpose.
Poucr/enerey d-maods my also be of interest as direct perfcr-ance measures.
In space vehicles particularly, stringent limitations zst be enforced on those
ncators. Under these conditions, dynamic performance optiamiation Is achieved by
=eans of comined or con.-ninsed criteria. Co1birwA criteria am tiypicailY of
the for•
indicial error measurce plus a constant times total energy measz.=]L ~~~equals a ils.j
Constrained criteria are merely single criteria of any of the classes discussed
earlier in this report with l•..•tlctns upon z power, torque, or total
energy.
Generalized assessc.ent of combined criteria is very difficult because, In
the absence of any specific application, selection of the comstant becams cm-
pletely nrbitrary. However, such criteria say be very valuable for a given ap-
plication. It 1c hoped that a discussion of tihe use of these criteria say be
included In a subsequent report.
101
Procedure' employed for calculating certain judicial error measures say also
be employed to calculate energy requirements vr specified syst-oS. .4fr---ence •4
cortiderr a-n Lnerti•-wheel attitasd cortrol cystem, aer shows how the crplex
convoltion ethod ct•ojed to -0slclate 1R2
c be ate',-.i vlth littt.. -
tion .to calculat- tic cncr9Y eXpMres in sBtalilmling the respav3e to & step in-
put. Because the tine htstory of the power required Is a damped sinusoid, the
IAE calculation methods or the present report could al.o t0b cplted to ttiL caze
to cclculste flPIdt (w.hich equals the total enrCy when no provision is made for
recovering the kinetic cner- of tIe inertia wheel).
S&M(&M AND COU• BOIO
I. Perfomnce -- t-"" -p4 ,sse:lst4 'ritcy-t, 1-" 11 ...r 'cstant cr
fioeut systea with deterministic inputs b•a.e been Investigated, with particular
reference to flight control systems. The application of performance measures has
been fsc'iltsted by substituting for the actual flight cotiol system an "equiva-
lent" low-order linearized system having similar dyrAnic ch-racteristic. This
cqulvaIent systen was ronstructei by dividing tte actual sycten transfer function
into reicios of ir.crczt defined by
IG(u)i 1, eves which 1
I3,(:,-,)I <i, ove.......n __-_--_" -" o,)
Th, for- of In this last regnt t.efljnes the dc-.---t mode.. of tie
clcsed-loop nysten response, and can usually be closely approxi-ated by a s -tte=
Of fIrst-, scconrd-, or third order.
?. At critical and exhamustive survey of currentt performance measures baabe - cn-snct". Analytic fe•rn f r IT-AE, •A•, etc., are presented, And a nob""
of errors ir previously 1-itlishe4 mesures have tben corrected. A complete cor-
reletion hat been given of crossover flcsyncy, bandr•dtz, phase margin, peak
frcqi-ency, regnifiestion ratio, tire-to-esk. peek overshoot, a-6 a-lay i-4
second-order unlt-numerator systems. Normalized presentations are used so that
yjrsici•ies limitations on the time scalC of the response (e.g., due to power/
inertia restrictions) nay be taken Into account separately. It is concluded that
sininu= IMkE and minima ti2
t•es satisfy the co-ined requir-eents of vsliity,
selectiv.1ty, and case of application. The IM criterion Ir.cld- stooth indicisl
respts•rcs havin; little evers"t. but its analytic description is complicated.
Of the other Ln-icial error =cosu'es exasnied, mlnim I!E has sisplz analytic
fo.se. ba it lcts pOor in.-••nc rsponss IT-E- responsei areý m good as
thoat, selIetead by ITls, but flT'T (a.'d Qsos IT2E) ,nalytic expressionss are t=o
coplicstcd for general use. M2 selects "-dertoly smooth ald vell-damped re-
sponses (less good than I-M), but it porsesses tractable analytic form. There-
Core, Mho .s r-tcnwded for analytic investigations, uherccz ITAE is preferred
f tr nit•atlc. a.An analog ccoputers.
3. The inteaated errcr-vti sc sal lte6rated tk-.we 0 -,-Ac error re'
sponac of closed-leop rystems to a general deterministic input have been related
to the corresponding meairec of the response to the impulsive input, ahich in
turn have been expressed in term of open-loop pmramotcrc.
lo0s
1 Nehrd orArt Piloted Aircraft ; nfl t Control
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3. Aigrain, P. L. and L. . Wifli.me, -Design of Optium Transient Reeponv*Amplfiers,- Proc. Inst. Rado MIrs., Vol. 37, August 1949, pp. 873-879.
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10O
15. Chestnut, H. and R. W. Mayer, Servoaecham and ating SYtes Des1is.Vol. 1, John Wiley and Sorns, Inc., new Nork, 1951.
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'08
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.10
APPDEDIX A
AAICATic EVAWIATON OF IAR, MEA, AND IIVAE FEm)Fl2EICE MEArtVSl
FM wlt-TNw*MtAlw SEWIW-ORERf 516136
C&IflhITflN OF ME ICICIL EMOR SEPOIOP A URI-NLGt)OMt sXCuD-wrZ SYSmC(
The transfer fnot tico of the system ccfdxx:&
R (A-2)
wbr a step input
E(s)-
c(t) - 4,-'-1 -ICaht . _k2 +(tq (A-4)
9-tan a in _ -1 ,
let2
L~b(tfl e s. Ii[%, r-7 t Iit
if -t,--+ - - '
1132
Ljb~tI e-. lsin 'ol Ir e s~ ~Id
fe/ (P-6)
*jh.--tcr vint egm, (v1ban beenn eitted- for brev.ity) 1:
f~~~ ~ STsiiflot w'O
e- nl - 2 2 2 (A-7)
With the approlriate limits, the results of the Integration are
. 2
f 3Tm.+ -x/
0 0
- ml -2.s~/0. .~t
F 0 --54t/m -e 5,/0&2 +0
11lk
B(S) I ex/ -2so +et/ A8
&2 _ _ _ _ _ _ _(4
EA- - BIs
where EA(z) is the !*jI.ee transform of the absolute error. Naklnig the substitution,
IAh _ '] _I(A-11)
c-t -'[ C O 1 (A-12)
Notirz Clint + .~2
eo
43 Nri7
XAE -[Coth(A-1 3)
EVALUATION OF ITAE
ITAE is defined as
I2Az f flemt)4t - l~temitIt]
- 1..'' -!L[tmoct]) - a~ 0- -Lia -11
where EAC:) is tht Laplace transform of tJ'e absolute error (Eq A-Il) I EACs)
In eale-1ahted as follows;: let
Then Eq A-11 becomes
EAWs - c1 a Cti.]+Gt% (A-I-,)
dA Es)do dE (A-l6)
and - -li
[('ot xm
11 -111 02m
116
Since m
ITAE ct C.X-
+ 1 1 - } (A-19)
Vaking the substitutio-
(lT o .)2,, cot - 2 in-'
Both the IAU an HIAF have been calculated &ýd cc.i.red vith the values obtained
by Glraben an lAthzvp (Rlef. 3I7) (l'r-a -1 coIutpr w.-c~h-mlt•on) in Fig. 17.
Th analytically obtane results agree well vith those of Ref. 37 for LJE an~d
show rdr discrepancies on ME.
A t ý1 i . er expression (sq A-20) heemes indetez-irmte, but ••
Aldditlopaillyý the liesmaxles may be written in terja of the norumlized maue
by a~plicaiton of Eqo 26 and 27 of thle mai tex of this r-eport.
".A(q,%% . u(v i)1.•
• ,( - _ __CO (A-.)
Ji -
Sa .•
11, I. d2EA(s)
-12Ak -(A-21)
As sh1own previouS)',
A B C D E A-~
The differentiation rsy be simplified by grotning the terma as shovn by the
br"ckci., and operating on X gxOk~y at a Ine.
Then !A- -A(lBi + E (A-25)
d2EA(.) A C D A E
and - + (A26)ds
2 (A.+B C D + LE)
Mhe calculation of ca= group is out.l]Acd on the follocrmi rases.
118
II =.A ----% % (A-27)
a (A-29)
2 2 2( A - 2 8)
(A-3)
.V, -t2(A-52)
C C&th~ 112
t2 (A-331
2w 2rn
ii CV
I1,
(. ý. , -- -
D- - T-, 2- - (c (A-38).Vt We
2fv O. (coth 8.4 1)
E' ~ a 2 C9) 2 -, C (A-40)
2ce2
, -,,0)-
&Iio N -. ( - 1) (A-41)
--- [2 '- - t a I (A-42)
vher
2I2
2(1 - 2t2((I.~ _ I _____ A-.
ý71 W,
exrW Ag oC.~J ~
0- - 2 "
21 0.,2 -2(,
The last tez= of Eq A-48 is the IT2
L,
n
rle -4r-&alizcd IT2AE w.1 -22E ane graoed in Fig. 18, and coparmd vith the
values Obtainee oj Ortham san Ifthrop (Ref. YO). A scaling error of 2 is
detected In the latter curve.
122
APPFJDHX B
A METHO FOR EEVAILATI3 TAE FED flAE IF3R OPDE-M Sr2SM
a10
A )•TOD FOR EVAULATIMP IA& JED MAE FOR %!UR=-OCMFR SYXIMt
T•is appendix presents an analytic prx.elure for evaluating IA and ITAE
for third-order systems. The zeros of the error tirt history for these systees
are not equally spaced, and the procedure In Appcndla A cnrrnt be used. Ii. the
-vrk presented here, the error time history is d(scrlbed by a Yourier-llke
series, the coefficlents of vAich are time-depenment. A isimlar technique was
emp.oyed in Ref. 20 to describe the output of a linear full-vave r-tirfler sub-
J-cted to r. d-,--ed---. 3snicidt i:ut (second-order error response). This appendix
extends the procedure of Ref. 20 to third-order responses consisttig of . deaped
sinusoid pluss an .-ý ,itita! tam, ai u•.•s now the resuitine expression ror
le(t)l my te Integrated to give IAM. The ,rocedure to, obtaining I•TA is also
outlined.
ro examine the aceuracy of the method (i.e., the -.=bar of hsxmnics
required to give acceptable rcpr-xentations of the actu-ol functions), several
examoles or the calculation of Ie(t)j are given. It is shown that only three or
four harennics need be taLem In mont cases to achieve an accurmcy of within 2 or
3 percent. The cnevergunce of the IAE Geri.n is even more rapid.
DW UIRNATION OF H • •IAJSFiR J!CfId OF A FULL-WAVZ k=TIFIM
A full-wave linesr rectifier has a trausfe. characteidstic shown In Fig. 34.
.yh(x)
x
Fi~ure 34. licotitiur ---*=far Characterintic
.aer,- . is the input signal.
h,(x) .
C when x > 0
h _ ) - x I w e n x <C
Then y h(X) .h.(:) +h_(x) (B-2)
h,(x) arA bj(x) have unilateral Lapl.ace trar'sforms, f,(cs) uArjn). re.1jectiveiy,
where
f,(mn) -Jn,(x) e-" a.w
f -( o h-(A &("x N
The Output of thp fi11-urmc linear rectifier Is gieen by t~e following lIn-
verse Laplace transform (Fig. 35):
h 2x) - *m)e s.J fj.(m) edx (B-i.). c_ JCD-C-jco
Where > 0
wzu+jv plane
,77 - -
Figure 35. lr-vrýnIon C-e-...
126
T-.e trensforma f,(n) and f..(c) can be evariated as follows:
f.(=c) x -dx . -
f-(W) - (-x)e-dx - d
and h(x) in Eq B-2 teccues
which can be condensea to give
hCx O C J ()D (B-6)
CHARALMUZA2ICII OF Cm 1IP1. SIGNAIL, x(t)
The input signal is the Indi.tFs1 -rror response of a nors~lized untt-
m-erstor thi4rd-order aystem
for Il(s) I /s
E(s) g2- b~s + C. s2 +bos +co (B-8)
where (.42 - ft2)7-
1ý 2 + 7n
127
from flieb
e(t) ae-7t +tkeCt
elmm (-,t + 0-9)
vhere a
$ taz &-I tar1
f~t
c('.) m~y be Vritt~ln
xt(t) ae(t) aA(t) + Vlt) ecos 0(t) (B-10)
vhere A(t) a -7
V(t) a ca
O(t) ti-
CLAPAafZKLtIZOlC OF 7MOIJWW SIGNAL
The output my be obtained by substitutLng 4~t)(Eq B-10) in Eq B-6.
. CKA, v *cise) -c(A4+Vcosa0)y(t) . .- edeea (B-11)
The erponrtth± terz =my be expanded usaing the followinhg '.iforml? con-
vergent s-flee(3tco'hl-Ar.Cer for'wAU, 3t-. 52, p. i8 snd Hoef. 20, p. ;62).
4,us -S 1.(z) Cosq (33-12)
vhare Z. 2 (aa-l, 2, ,
I..z it the vnti11e. '.I? on. sft '%c tr
128
Naking use or the reiaLImnstap X,,(-z) - (.i )ni(z),
At) m co me+ 3)
Uev Mt
"a V(L)lo Cos so 5B+3o Em(OPrL.+ (-I)em.-b]dt
y(tm) fB-Ja,
yc)-V(t) l C b(tm) +cot 0(~
OnJe
E %(tc) co V (t) I -lb(+ ) +o co 0(tj (B-17)
129
EVAWLASIcf OF C(t,U)
r(t,U) -, ~ ,) ~(-8
To evaluate the coaffie .ents C(t,u), first c=Lnder the integrel of C(ttt)amunnA the contrur hm in r±g. 36.
C2
+J0~j plane
-jB .maC4
Figure 36. Contour of Integration for !(t,e) Coefficients
130
Witning1CX, x2,iC. X4, as follo~s:
0+3
30 (2.19)
X3 -f C4t,09~,-JA
r,-f c(t,.:)d , C = t-,0
K1Then as 0 -. • -. C(t,m). Because
) L+ (ý + ((B )- u.1 2(3+1)(m+ 2, J'
"the Bessel unetion I.(z) my be qlwroxiated by sm for ml values oa .
The discassion is initially restricted to Lt. cue a > 21 •hen the SA9u-
larity of C(t,C) at the origin vanIshes, old C(tAj) becon amalytic insidp, and
on the contour of Fig. 36. It the-efore folLws fro ucy's theorem that
KI + K֥- 4 - 0 (B-21)
2hbe integrals K2 + 1h will ow- be conside,+ed. An araptotic expa•sion for
a(z) for lsame values of 121 is
1(z) - - ~2..(6z)2 -~ . Ba
131
z
INobec, for large values off 0
- eb14'O) (.lUebt~~]d~(15-24)
IX215IS [ebe(a) (B-26)
'v JV (.1 .e(b)B) + c.~ 5 aoie(1 b)) -3.0 as o-w(D .?
Ths 2-3 0 and, sixljsluz, 94 -300 as p -. . Bence,
K1 - X3 -J C(t,t)dt 03-8)
- C(t,t).d(329
Note that
K.C(ta)
132
On the i-,iflry axis
K3 - - -dq
because
r.0 - ?JOT(-I) (-
for as even
2 0~ ''~ . ~ (-~
for a l
Y3 2- J-(ii) a- bqd, J- J3 ,) min b-. q (-5
because
(D-36)sin (-b.1) -- sin bqi
These integral~s ame Fourier cosine and Fourier sine transforwa, reepect~f.ely
(Rtef. 26). They ame
Coes (u.~ia O s (m )aT b l < I
a even In b> I
(B-57)
133
c si ms t omW - (j sin bos. b 1S-•r•e(r 1 ()--
U -odd -(-os jvb + mb>
* (• -1)? r. 1
where a - sin-lb
.- O - IF 2
*B - b +479-7
Applying L'Boepltal's rule to the second terms or z and W it can be shorn that
b>1
SIm.1 b >_ I
Becase Z (foe a even) and W (for a odd) .r,- single-valued analytic func-
tions of m for a> 0, the theory of analytic continuation (Ref. 16) permits the
rcntrictlon a > 2 to be rrmoved. The output signal can ow be found by cotbln-
tug Di B-15, =B-37, and B3-.
F a-fl xe-t sin (P~te - e(t) b>1I
Ifjbs + cjt1(K + r) sin (Pt *+)I
y~~t) ~ -C Y oa a(0i a oeven} b5 < Ii'
Lv V in xOt sodd
134.
vhere b -
V It -a"t
n sin lb
8- -- b2
mc coo n + b sin ma=(n2 - 1)
som sin ms - b con rasg - 1)
MILCAL camCW MTSIU PM Ie(t)I
Equation B-41 expresses I e(t) for a third.-oriter system indicial response
as a Fourier-like series id.h tine-dependent coefficients. As wil be shovn,
ZME and ME can ba derived from this series by a straightforward procedure.
Ile uatfuinens or Eq B-J.1 depends upon the rapidity of the convergence of
tie series, i.e., how many haronics must be Included to obtain acceptable ccu-
7-. ." "-.,y of cLc-- g this poilut would be to evaluate UAK and compare it
with the published values in Ref. 37, or with wnlues obtalned by direct Integra-
tic-. uowver, as shou in Fig. I I of Ref. 5ý, IAE Is very 143ensitive to
par---ter coArs-. It would appear therefore that, with a sU number of
examples, a mteb -I.- sensitive and thorough check can be obtaiacd by eowting
lei at selected instasts. Integration produces a msoothing effect so the number
of harmonics required to represent ZAU accurately will be less than the onuber
required for e6(')1-
Two responses are considered: the I= standard form ani a lightIy damped
responsp. For the standard fore, the parame.r in Eq r-39 are
a - 1.2
k - 0.72T57 - 0.707
- 0.52
0 - 1.07
$- -O.279 rador -16deg
135
and therefore e(t), Eq B-9, bec=es
e(t) - 1.2 e"'C TO7t - 0.7?,e`Z'•" sin (1 .-0it - 0.9) (P-p)
For the lightly dampped system the folloving parameter values vere selocted:
a - 0.426
k - O.798
7 - 1.1142
CL 0.03112
9-0.93.5$ - x/2 - 0.7611
and therefore e(t), Eq B-9, becomes
e(t) - O.'264e""2t O ' 4 .790e Sin (0.935t . ./2 - 0.7611) (n--13)
The Standar form le,"" wa exained at t - 3.20, 4.55, 6.13, -A 6.41
nondiluetlnnal see, there values being chosen for convenience of calculation.
Tre IiOj4 oa&peo response wes examined at t - 0.4105 and 3.66 normalized 'ec.
100 ro.t atsre illustrated in Fig. 37 through 112, vhich show that the use of
only the first five basonics fYe$ valuer vithin -1ercent of the true Ie(t)I.By taking 6 or 7 hararcics accurscy of vithln 2 or 3 percent can be obtained.
ZFAWAIC O• lAB
he integral of the absolute error can be found from Eq D-41 for y (t)
f °inE- yltldt - No E = Us (B'lA)
NOh f T, e()dt
O,"•i U f Ie(t)Idt
t=455
10(0
0-
Numb~r Of Harmonies
FIgure 37. CaIuuated valt of io(t) I for 7hird-Order OPtlual ITAR ~rstemvf. gambr of Haxbcoics at t - 4.55 Wormioea Sec
137
W s3+bs4CS+l
t=32O
100
50 5 0!
1 2 3 4 5 6 7 8 9 10
OL - I
Number Of Harmonics
F1•are "S. Calculated Value of le(t) I for Stird-Order Optiml IfTA Systemva Number of Hav•=lcs at. t - 3.20 Normal3zed See
1371 S
R(-)
tz6I3
500 -.-- - - -
1 2 3 4 5 6 7 C, 9 10
Nwxnber Of Hormonics
FP1.~m 39. Calculatt.I Va3.ve of Ie(t)! rai Whrd-Order Optimal ITAB Gyýeve It\wber of At~c at. t m 6.13 Ntoimalled See
R(s) I
t=644
100
,,50
S 2 3 4 5 9 10
Number Of Harmon-cs
'Vxr1r 40. Cacuak2ted Value of le(t) I for lMrx1-CTder Optime ZTA. systen.as hlmer of Hermouica at t - 6.4l s•malOized Sec
140
1 50.- 1
6 4 5 7 8 9 10
Num~ber Of flormonitcs
Fium~ 41 opcat Value of ;e(ti for udoikv au-' sy*fT-- Yer ar Afiz.ones 4t t - o.doo5 Rormuiedt s.ee
a-6 I
2 34 3 3 7 8 9 101
Nmnber Of H1a~mxucs
Fgm-2. Cal1..fttd Vnlrue of Ie(t)I ter Lwk-kv Syo-t-v- Ls~brr : : .tmc At t . rejr=I5ze S-c
T, in k
ANt Yric%'. tent of So a be evaluated in a ctraigtf-iw~rd anner.
t a7
The procedure for integrating the! Um frmes is more cump~licated, and wlll cow be
described in detail.
Evaluation of U0
7cr= Eq B Li and Eq S iii,
*o X (bz +c)catdt (B-46)
let t - 2-1
Note tbat m Is not repl-ced by p In b. s, ox c. ETc resulting expreossio can
be likened to laplacc tranaftita. Other. advantagv will sccreni vhen ITO isevsl'.nate4.
2.U0 - AO (¶'i13, -cO&'-dT (B-47)
where
b, - -
11i3-
L10 . fo. t f('Zid -0(1 / C~er'dT
Integrating. the f~rst teim by parta
A0 [i4..L ] p+~ f0~he d-: J . 2.j-g
(1 -ý')'Ie-dl 29 (Bet. 26) (B-DC)
Eq B-4 1-n
-, B( 2,/2) 2(2-, 3
r(z)r(g0 )ihere BCr, so) - fr--e' is the Beta function
P- vbe--e z - x + jy Is a general complex variable
Eya2'fation Of U,
rmF4e 26A-I4 arA 3-~4
U,- C (bc + a) *n(Pt +J vieCt~t(-2
lett +T
p a
f (blc1 . at) sin (gn + * * fl 1 )e'PTde (B-53JAO 0
let
- b1 1 + 'I
W )- LP. - f (bP,1 * n,)e]
* , U 1 . / w,(p - j13) - W 1(p' J0)\ ) v( + Sin + W,(p+ JAO 2.1 2
(B-54)vhere
4 - v •
The exwesaial tor W, (p) is ;sinr tc that for U10, the differ•nc•c be!g the
replacenent of p by p - g in oa t.rm, and by p + S in the otner term.
111(p) 2p (B-")/222p
The Beta function with ane covlei argumnt can, be written as either an
idfinitý series; ar an infinite product (Ref. 52). The Actual calculation of lAX
for spclific para-eter wTaen In simpler vith the Infinite product, but the infi-
"-dtc zries allows a simple derivation of ITMY, as wll he demonstrated.
The beta function with an. cclcx arrant can be vritten
CO (-'i)n 90(6 - 1)(90 - 2) ... (go " n;/ \
- Ed . ' (D!)
withL g, real and positive and z copoex
D~. - Y~, 90 ) -D(z+ 3Y, 90) .(0-1(, 2 ,g -1)
i2n)
B(X -jr, B(Z 4 •'yo go) OD -1) o(go •o : o( . - 1)(go - 2)...(go "n)
" 2 p 2 "
0 air% cog t- . "p]"g- --a
(Iaoa- 1aOo ) - .(so -( -
(B-58)
2o, ts that tbe first eoeeficieuz.• of esc:, •ri~S IF rn.t,Ot• g i•. On oo~binin6
Eq 2-5•, 2-••, aa B-57 U1 be/c
i ii s+ �# ".o01n;ilmm < -o I (• •+•;+•
"""-i l . ((-)% (o-i)(o" 2)---(so"- n))
II BE r-o L. ;-o--
ii 2 J=-E
An BLets function vith Ont OU owleftg~ent can also be viitt,-n Cuer.
;-ge 2)
3(x +jy, gfJ g)(x_)
'go jy/,n
I
X +% 4M
vherec - Eulers constant - 0.'577
.. (x.'Y, 90 - 3(1.%,) 17 n-~(-0n. 1*[ JL B-
When Eq B-55 snd B-60 are substituted into Eq B->ý4, -o becomsXA0
I~ . X(p Cos 9 psin 9) + in . n0+'J(-1ro 2(p
2.
2 +~
.bere
S- 0~ (t----147
k2 B_ .0 D g.29a(
n.0\
N2
'\2 .2M 2 - s W 00t~' ¾~ sn r r r
&I- 3/2
ao-1/2
The eviasluton oZ ;41. k2 , A, ead 1s an be performed by opertions or. a Bode
'Usgeco- becauase N1 &adM ceesist of sn alteruttirg sct tt poles sod zeros If .10is cegaried as tbc lApin" variables9. O'ly those term with breskpoints less
than a few times ft need be inclWAed, beeaust each pair of poles snd zeros wre
qu-te oboze together.
Lrsiostlon of U
Fros Eq B-41 snd B4t
V2 ' A f oz(t) ro. :)(fw + qie'u~t (B-62)
t -¶T 1
p CL
1- 12. co s + b, sin 2as Co. 2(p, +~ n 1 )e'pTd ("-3)
-3' r l- b2,)3/2 .0. 2CmT + * + (3-Oh)?T
* (I - b2)/
5/2 n
FCobmn Eq B-65 an -6 r zn q35,V ~oe
290 .. 0 as)n a~t )~t (-8
kt (P -2p +4
Ln~usion f U , x 3, odd
Co - : V.('r) si ZO .+ C)epd (D-69)0 u-
wn)- sin MB1 b,~ twno
W,(p) - aw
2 1=~ wo I(J w
Z - S)I : n7 (B-70)
a oda sn0
co e and sin me, Chi be expanded by the roilowing iormuue (iter. e3):
-0 -~n2r>3 rdL 0
2b
s aidR-71)
With th11.c a tres-Ciatn for C~sM and sin "s, 1.4) bemms
A~(P) I -.u br1 )K ( 3 s-a
1(-2) 3'
-3( . (aJ - 20e ,) - . 2r +4a)w 2g
U. jS ýUbtIled by Combining Eq B-72. B-57 asBf-IC. The renulting e.ni
1. tkrcer cospmitted. andl is not given here.
_vIujtt~en ofU 4,i
q B-4 wan B-44&
U5 -4tt~ f-'sc- coS . b csos n(pt + V)e_"tlt (B-73)
m even
let t +. tT
2 sOD -Um ~ - )~ r I vII) cos m(8¶,O~_ý*- (B-74)
ms even
.(r[) MCI Cos Mal + b s ma,
); -a 210;w( 435j
2 m 4 Vp -j wn(*p + 3n0))
m -yven - 23 i:j
Cos ab, i RL. SU. lca e expared by I:w foIn oing .
ý06 ms _ -' ~'r'•: OD m I)!Cl=• 0,---
sin A, O- .1)2 -r b"
>2' r0O
m even (B-
Because ' (P) is the IaplaL.e tranrfcrM Of um(.C
W((e) - (mc Cos MI b" sin me *" (P-78)
m even
,lr~~r~~e-r ( ) ) (B-79)r 0
2 even
'abere
- ~ ~ ~ ~ m/2+1 ~ e8n2
(P P) .~ *(1) (a-20 1T
(-L (+ - 2r)B~ M~ - r)32,
By coabining iEq B-7,.57, and B--, the expression for U. can be found.
Again the exrreunion Is ~cmplicau.d, arAu2.1 e-ut, ixcL 1- l= e.
CArnATTON I? In: PMi Cv'TwWrnca Y 4
.he jn.at.t' .cquisvd fur tie C&lcu!&tico Or tLM for the thi i-ore sys-
tern are g-ceor just prior to So B-42. Thu vuluts of the first. few cocponer~t or!AS are
I- 1.913
U,. - 0.285
V,- -3.109
Total -2.6
The value tf Vkr ecaled o,. .,; F2ig. 11 of Recf. 57 Is spprnedmat" 2.05, and
the calculated eaite of ISA! ±s tbercftre oidtbIL £Wjrcx±.tsý $ý ur 6 percent or
the experimenzafl obtained value. 'foe irctation of murm components of IA! would
Lzmp-nr the Gccc'racy.
The ±'tfnnie product representation (S4 t-61) for tne Bets function was used
fczthe nuzncal mcmlulatxon of IA!. only the first foeu, raýt-3r "re 'n-' -
tee accuracy was Judged to be acceptable.
rnmnIcu OF ITEA
1raz Is defined by
ITAR f c t!e(t)idt (-'0
Because the expression nee-d fr lei is In two p..rtm:
r fT'tedt IfC tlelat (B-82)
It Is convenient to derine
Hl .- T tedt , -1K tjejdt 'a-83)
15,3
i-.e eunds ML correcpsod~s to late m'tI erap~c.cent of the contrfl.'tion of the second
7-s* xrruiNs ur 1; 1141.1c tos tev. Lcn-i -re , in~volved ss.J IengsLhy.
llt,.rictl checks usill tot h-u attempted, '-tII sr.I No .0.11 be evaluated in lit-
ormal.s :Lr 'asu Lwte cd cuisticsn .2f ti-rP too expressions Is the nsat difficusit
part if cvalusting MrA.
ýrro Eq 0-: 1 and P-83,
III tit - a tc-ystt dt t sin (Pit . t)ot0dt (sBa64)
The first of these integrals ic quitc simlM,# but the secoad is rather In-!-j ., ca w, eritten as %integrating &he tinrt terv)
. 1. _ .-,T,(,(sirsUtvos v coL fss in+)cet at
- "t sIn (Pat + )euidt (B-851
Thc tt.and te m or Eq 3-O) !s Usc. laplate taannforua ox t. oin 0t and t c04 alt
with aL repiacing the Laplace -'sarI-bi- 5. The tntir. term car' be snily Z-vslvated.
by mRsking a chu'er nt nisWbl. 'Ir + T,. Upon integrating the s-stond te~,m L11
becomez
- -e I(i - YT-)) hi sin (* )
-ke-'1 f (r * T.fl(sin If" coo i * -, PT si'&,il d ý-6
oo
£yu.amticn ot •
"- T t(b. * c)e'd. (3-88)
11, 1 31" +
Wet.
tan-
tcalýU in or i ex1eee esU;t Uoe0r00eep h
t"aetr (i + T.) fr. th fr.tfe~tandx. -3ince tkle eX'-:sion can be ]lkened to a
Laplace t~raceft-n'I 55 - T(
(10 s gi-n by Eq B-51, but tir deribative with icepezt to p offtrs .=a
difficultj. Reprusentation of the Bets. ft-z-ticn by atn Infinite senfez (24 3-56)
cieqi2 tfiles tbz aif-ferentiat' *a crmnA,.rat!Zy r.sez occurn Only in the- ter=
The dcrivatlve with -expect to p or I,' + ut is. -aey a/). ' withL a
rwItiplicative constant. Accordingly,
-c 't I i/) + ( )- 1'a 0 (a0 ) - i)(to - 2) .. o.A0 2(p - ) n-0 +~gnt( 1 4 ~
on (0 +)0 (:- )C n)?
The expressions for N I > 1, vii be oiSiii-r to Eq B-89 ard B-9U
The desiratt:lity af the ne~es r natefom the Beta. fwiotn is Isi
dent. Note that tle terms; of k, should eunrexqe smre mrapil tUan tb~eofU
because there is S=emeslly &n anditiorat -acto of the torn 1 /(z + n) in each ct-with -rente tU n.
n~6
APPUWI c
DEXAIS OF TRM EXLAfZ flXCM? CalMEL'SYsM. MW To DEC-3flj&
TIM EZnVArn, STDM C=WT&
157
DUM!Ib 0-' T2{k LKAWL* PL'rhr WrW2p1. M-i-M MW
ý-on I of Cha.pttr I Li~ Is.& "n bcd-7 of the mtpcrt cxprcscst4.( f--
'01 t -:Ater fuw~tirn rtiati.,t pf.t.2. &Ltitudc tý elc-.tar derztetionr. or tee
fighter ixrnekee !ct~iled In Tabli. I11-1 .ýf Ref. 2. iTbe altitude It Pole=~ ft,
t!hc -b Z C,:4ý0 It, ýLZ ýr6C M:.i.PeCl 4b 660 rt/Cee (I4~rh NO- C) The
ai.1TiSne tiazfe func.tion Is Sb quoted it. Htr. 2,
_2 2 0c-i)
The servomotor ;lux, s:pllficr tronefer function is estimated as
(C-2)MI7 a- ~
Inc equeliation Ut deacribec. by ic, (y, 1). Combining this Vith the product
xr C-1 and C-2 yields the open-loop trannfcr function of thc cooplete system.
