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A PlEaMISE LINEAR S'l'ATE VAlUAllLE 'l'EOINIQW

roR REAL 'l'n£ PI{)llULSlOO SYS'l'fl.l Sll-llL.ta'IOO

James R. ~Wmloc..'W National Aeronautics nnd Space Mlliniatration

Lewis Research Center Clevelnnd, Ohio

ADSl'RAl;l.'

Stephen 1). Roth Pratt and Whitney AirCl:aft Group

Govtll:l1l¥:!nt Products Oivision Wost Palm Bench, !~lodda

'l11C CJllilasis on increased aircraft and propulsion control syst<l1\ integration and piloted aim.Jlation has created a need for higher fidelity real till'(\ dynamic prOIJwsion nodcla. A real tin"Ml propulsion s),stan nodelins techniquc which autisfies this need llnd which provides the capabilities needed to evaluate propulsion systCJl\ pcrfoll1lilllCc zmd aircraft system interaction on mannod flight sUlulators Ims been developed at NA&\-kwis lU'ld d<llonst:raced usins flight sinulator facilitios at NASA-hll(w.

A piecewise linear statu variable technique is used. '1111s technique provides th(.~ SystCJll accuracy, stabilit:y and transient response required for integrated aircraft and propulsion control sys too studies. '11\(,\ real tilllC dynamic \1'Ode1 ino1udos t:he detuil and flexibility rcqu.iJ:'ed for the evaluation ot critical cont.TOl l'>llrll1l'Cters and propulsion oolqXXlent lunits OVer a limited flight: envelope. 'llu~ n'OClel conCll:Ul8 apprCXlti.llIlltoly 7.0K l;ytes of in~lino calq)Utationnl code and HI.7K bytes of block da t.:a. 1t hus an 8.9 illS cyclo till¥:! on u Xerox S;tsillll 9 caupute.r.

A Pegasus-Harrier propulsion SystClll was used as a baselino for devoloping the. mathClootical m:xJeling and sil1u1ation tcclmiqua. A hych."a:lIechanicn.l IJl'¥.i watCt' injcction control systan WilS also smulated. '111e \1'Odel has beon pl;ogrr.unncd lor int:ertnc:U"lg \.,ith a Harrier aircraft s.i:nulation at NASA-limos.

Descriptions of the rcal time ll"Mlthodology m~ llt>del capabilities urc presented.

ImroDOOL'IOO

Sil1l.llation, with its inherent flexibility I will play a l{oy role in the. deve.lop1:cnt of integrated aircraft-propulsion control systalls. 'llmse sinulations will provide a cCllrprehensivu source of qualitlltiVG m1d quantitativo informatioo. regarding the characterist:tcs of aircraft oud pl,"Opulsion. systans Ul a dyru:uuic stato, 'll1oy will also senre as tools for the analysis and synthesis of control logic m1d Llii test vehicles fOl" control Software and hardware develOflnent.

Since the advent of piloted sinulators and the grCM.l.ng anphllsis for systems integt:'lltion, there hIlS been an :lncreasu"lg need for higher fidelity real-tilnll Eropulsion systan uooe1s. Propulsion and jllteg-rnted ~ont:rol system evaluation of aircraft on flight emulators require that propulsion syst:al1 sUlUllltions be realistic and include sigt'lificant dynamics as well us important engine pID:ll111etOl:s. Such sinulations p.rovide the capability to evaluate propulsim systans and their :Interaction with aircraft controls. One such trodelinS teclmique to accanplish this cttpz.:\Llity i.s described in (1) and (2). A dynUllic digital1:cul-t:lme trodel of an advanced propulsiOt'\ system based on a piecMsc l:l.near IlXlthodology to meet the SllIlle requirE1llents is described in this puper. 1111s nodel provides the engine-control SYSCClIl accuracy I stability and tra1'l.sient response required for studies which lI1ight u1c1ude the evaluation of criel.cal control parameters, aystCll1 response, SYSCClll envirommtal effects and critical propulsion canpone.nt aerodynamic, 11lechnnicnl and thettl'Od~ynLmiC liJn:l.ts. 'l'he trodel mlly also be used to aoolY/..e propul!:;::'on control fldlure m:>des m"ld Mfects.

A Pegasus II propulsion SystCJll provided the baselirle engine for developll18 the lMthunatical ll'odelinS urd sinulation teclmique. 'rhe engine trodel is a piecewise linear state variable tocpresentHtion. which was derived fran a detailed aerothenrodYl1l11'llic sinulation of n typical Pegasus 11 engine. Dynamics included in the sillulation are eng:lne fan and cOO\)ressor 1,'Oto);'

dynmdcs, engine burner hoot transfer dynamics wld engine cont'rol dYl1\IIlrl.cs. '11\0 rrodel calculates trtUlBient pct'formuncc by r'...rneri~l :lntogtation of tillWol-depcndent diffcrential state equations tlnd contains the dynanics nccessary to sjJlulate aircraft [orcas rCllultil18 frem engine thrust levels. 'l'his higher fidalil), propulsion control nodal providCll scrody state and trllllSirot chnrl1ctcristica for vnrious engine prcss1%es, tarpcraturcu, fl(Xols, ntnll. margins and thrust.

Application of the n~lcl to a sinulated flight program end cVllloocion 'tcsults arc prcsC11ted in (3).

row.. 'l'IME ME'nlOOOI.OO'l

The renl tunc lIv:lthodology it) based on 11 picccwlile linc.o.r st:L\ce variable technique (4). Within this process, the engine nodel uses state vuriablcs and matrix fumulations to represent the engina process at specific operating poinC$. 'Ille effort llrcsented here was directed tCNJard applying this 1I~'!.hodology to nodel !~ross engine transients accurately and efficiently. 'l11e roothodology int.;'udas the following sceps;

(1) apply a nodal £U1Illysis and sensitivity study of tho detailed base modol to select nxxlel states and input controls

(2) optimize state variable nooel selection to accurately define steady state and transient characteristics over a specified rango

(3) generatc accurate noool partial derivative matrices using offset derivative techniques

(4) cormect and scl\£.'Clulc tho stnte varillblo nodel partinl derivatives to represent gross transients

(5) l3lploy a sunple but accurato integration ScilO1Wol for fust cCITpJt:ution

(6) optimize progranming fOJ: real tin..: operation

State Variablo Representation

'1M state variable representation is shown in figure 1. It is charactorized by the follQNing t"WO equations: .

X"'AX+BU

Y ... CX+DU

(1)

(2)

Equation 1 is a linenr constnnt coefficient matrix differential equatitm that represents computBtioo of engine dYMlIlics. Equation 2 is a linear algebraic equation that: renresents coo1pUtation of tho observed engine parameters. X i.s the vector of: state variables, X is' the time derivative of the state variables I U is the cont-rol input vacto)." and Y i.a tho veccor of observed or ouLllut par£ul~ters, A is the plant mntri.)t, Its eleoonts are tho partial dcrivativcs of the time dcrivatives of: each state variable to ench stato variablo. l~lanents of the output matrix C define the effect of each stace variable on each OUt.l)UC vw:iflble. '11le control lIl!ltrix B ;n'\d the direct couple matrix. D define the effect of: ~lch control variable on each state variable ture derivative and each output parameter.

1I~1 Analysis

A Il¥xlal analysis is used to detenninc which states in the nonlinear aerothe.nrodYMllic IOOdel would adequately represCl'lt the oystl.ln within the required control bandwidth. '11'0 nQnl;i.nenr oorothenJooyn81l1ic tooelel is then linearized to obtnin the systan A, B, C and n nlUl::rices. I\n cigenvector-c;i.gcnvaluc analysis of the A IMtrix. ;l.s perfotmCd. 'll1e eigenvectors arc exauined to associate eigenvaluas with states. High frequency states outside the control bandwidth us shown on figure 2 are eliminated. 1\ nxxIe controllability matrix. which defines the effect of c..ontrol inputs on stutes was also generated. States which are uncontrollable by the inputs are el:i.m:i.nated. 'llle result is a set of state variable vectors for the rool time nudel.

t-kxlel Resolution

Modeling gross transir;mt excursions efficiently BOO accurately ;1.1'\ the state variable torm depends on the t'luri;>ar of I100eb selected. Initially, a p:i.ecewise linear fit of th<:: stcl;\dy state operating line is perfonood to define a min:inun resolution. 'loose roodels are then augrrented with additional IllJdels to accurately define transient response through the full

2

pcMOl" range am any Q)(.tranely nonlinem,' arens.

Matrix l'art:l.al Derivative Generation

State vnrlable techniques can be Illlploycd to obtain a lincm: apP1."OXirtV1t:ion of n nonlinear systOl\ by OOl1fJidoring operation in the vicinit:l of a particul.m,· opcrutu18 lX>:Ult. Matrices A, B, (,} BOO D are ChlU,'llctcrized by:

• axt (3) Uij "" 5Xj •

• a~

bij .. ruj' (I.)

aYi cij .. oX

j (5)

OYi ~j .. alij (6)

To generntc finite difference uPP1.'OXimutions of the partinl derivlltives, the strody stRte level of each ~ltntc vm:iable and systm\ illput is indcpcndimtly stepped Ul both n pm>itivc and

.. negative d:i.recticn while bolding the other stato vllrinbles am systll1l inputs fixed. CorrCllponding "lllucs of tha state variable derivLltives llnd syatun outputs arc recorded for ooch step change. Each matrix partinl derivativo is calculatoo by t:l.lldng an uvernge of both s tap v&lues: . .

tl .. a~ RI x..t'Xj+OXi - x:t,~-&Xj "ij bXj 6Xj (7)

'rhis process was autanuted em the decnilcd nonlinear buso 1\'0<101. ]i'irst, each X is pert;urbed one at a time while noldi.rig all other states and control i,iputli constant. ')1U1) al1O¥."8 calculation of the A Ilt1d C mat.-rDt partial derivativC!:l. l!:nch U iii then pel'curbed ana llt II t:iloo while hold:u'l8 all other contl:'Ols rum stat~ constunt. '1his allows cnlc.'Ulntion of tho B and D IIllltriX partinl darivntivcs.

Severnl different levels of pcrt."I.Jrbntions on the stut\lS tum inputs !U"C used. l)erturbntion step size wns m:UUJ1lized to prevent driving n'iXlel parrullotors out of range but \lInde large enough to excite ouch stato vuriable and output pat'runotar. '11\0 best overall agt:eeS\~t bOt.'WCC1l tho lillear baso Il~clol nnd tho nonl:incar m:x.\ols occurs When the SQttes arc pert."I.lt'bcd about 0.5 percctlt and tho inputs about 3.0 percent. '11\0 pm:tinl derivntivcs generutati by the offset derivative technique are reasvnnbly aCClJrate for stoody sC!\to at roch condition for which they ware generated I For largo trru'ls:kl'nts, ha.vever I this is not IIllcossarily truo anU a for.c~ steady atato match \1lI~y be required to ens\'Il'e steady stato accuracy f-ran ono state I variable n~dcl to U\1l next ,along the engine operating lino. To avoid this, ull perturbations ro:'o n'e1lSured l-ran a steady sente operat.::ing line !rodel ~1ich insures str.m.ly state accurucy over tho wholo rlU'l8e.

State Scheduling Parrunecer

'rho state varinble l\'Ddcls nus!: be connected efficiently to provida continuous operation over the entire rl.U'l8o of the ll'OdeL 'rho interpolation is controlled by scheduling the matrix elClllCI')ta with lUl imlQpcndent variable called the s tato scheduling pllrlltlXlt:er (SSP):

SSP .. f(X) (8)

This parlltl~ter is derived fran the state varillbla vector and indicates the relative energy level of the eng:l..nc ayst:an. Application of tlus pl'OC.;ciure rCllults in rech.lcing several lille.ar trodols to 000 nonlinear nDdel as ShONn ill figuro 3.

State Vector Il'\tegrtltion

A slrrq)le self st:ru:ting run reasonably accurate int:egrntioo SChetXl is required to minimize exce\lsive function evnluatioo and callputntion tilna. Euler integration was fOl.nld to meet: these requirBOOnts and was subsequently adopted for this llPdel developront.

3

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P~ OOSCP.IPrICN

Figure 4 conta:lna an overview block diagrBl11 of the irrportant state scheduled parameter Btnte variable m:xlcl logic. . The code contains first pass checks and initialization steps. The initial steady state point is calculated 14'an the state and output operating lines. 1'1la initial time point of any transient run is assuned to be in a steady state condition, which requires that'::

AIlX. + BAU • 0 (9)

Since the state derivatives will not be zero on the first pass if the initial point is not on the operating line, nultiple passcs are made through the state dynamic c.a:Lculation loop until the states settle. The aolution will be a set of steady state values where the AX vector is given by:

(10)

Transient operaticn occurs as fo11ONS. The last time step values for the Btates are used to calculate the SSP. Bracketing m:xl.el points are then identified relative to the SSP level. The SSP is also used to schedule the A, B, C and D matrix elarents. These matrix elarents are stored in a linear equation form (point-slope) which allows for rapid CCJll>utation bctween the discrete m:xlel points. AX's and flU's fran the IOOdel points above and balON the SSP are COOlJUted. A relative distance weighting schane is used to caroine deltas fran the trodel point above with deltas to the 100del point belCM. 'This basepoint srroothing is set by the tooelel point distance fran the current SSP value.

State derivative canputations are perfonned by the matrix nultiplication of the A and B elanents with the AX's and AU'S canputf:d earlier. The derivative vector is then Euler integrated and sll111led. into the storage vector holding the cunulative state level. The CUIllJlative h.X is . passed out of the matrix integrator and vectur St.ITITIeCi with the basepoint values. This final result is then the actual state level to be used for both output and calculating the new SSP.

'Ille basic values for the output parameters are set: tbIQI.!gh tb.e operat:W.g ),ine curves as f1.mctions of the SSP. Base output levels change as the SBP dynamically varies indicating engine energy level changes. In addition to the basic transient respot'lHe, dev;i.ations fran the output operating line IlUSt be calculated. 'Ihie is done thrCAJgh the output equation which uses the sane AX and AU vectors used in the state derivative equation. The (}.Y vector c::atq:IUted then represents the change in oul:p..lt values fran the current tiet of operating line output values. Sunning this value cmd AY elanents gives the required total output response vector.

APPLIC'ATlOO

The toothodology presented here was applied to the Rolls-Royce Pegasus 11 profJulsion system (3). This system is used in the Harrier VSTOL aircraft. '!'he primary purpose was to generate a real time digital trodel for use at the NASA-limes flight sinulator facility to evalw\te integrated control concepts for VSTOL aircraft.

The source for the real time rrodel clevelopnerlt is a canprehensive aerotherroodynamic sinulation of the Pegasus 11. Data fran Rolls-Royce was used to represent mininun engine steady state performance on all. parameters. A band of 3 percent for specific fuel consllllption and 1 percent for thrust and jet pipe temperature was established as . an acceptable matching tolerance. Four differ-mt match points Wf'..rt! chosen covering the full ~r range. As shown in figu'!:e 5, the results obtained at these points on the aerot:hertrodynamic sinulation wr.:re within 1 percent of the m:ininun Pegasus 11 steady state perfonnance.

Transient sinu1 'ation accuracy was obtained by adjusting rotor inertias to specific Pegasus 11 values. The :~~ults are shoWn in figu'ce 6. The apparent diffel'ences in the initial steady state are attributed to differences in the steady state and transient Pegasus 11 data. The transient accuracy WdS canputed by CCllq)8!'Wg the maxinun rates of change. This was within 5 . percent for all engine parameters.

'l11e state variable input alld out?Jt vectors used in the real time m:xlel are shown in figure 7. The state vector was derived fran the trodal analysis. The:input or control vector and output vectors were detennined primarily by fuel control and flight envelope requirements. Fourteen linear mxlels are used to define the Pegasus 11 operating

4

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characteristics fran ground idle ilt 7 pm:ccnt power to !ul.l UIIl>tU1W\ at 109 p&ccnt pt'MCr.

'l'hooo point troOOls aro linked tosetl\l"r by schcWl:f.t~ tho mnb::ix ulmmts At each trodel liS n 1'Uf\Cticn of the SSl'. In this lrodel too SSll was oofined lUI the avoragc of the Ii toody u tate £\101 flows that: woold c.x:cur lit ooch l>articular current stnto vu1ue!

~n \~f

ssp.. .J:. n

1-1

(11)

'l1ll1l1, I3teady stata fuel flow bcC<Xl~ an intCl100dinry utat() schodulil'8 purlJllQtcr derived fl:an tho state vector.

'l'ho rosultiIlS flipJlt s:1nlllator state vadnblc Cfl8U'c nooel is a hiSh fidcli~ prol~laiOl\ nwel which provides 1ttoody state and tranuic:mt: charactcrintics for dcsirt.'!u l!ngU\e pretJsUl."cs, tUl~aturos, nirflO\oN I surSQ JI'lnrSms I tht'\Wtu and rotOt' BllCl.."<hl. 'l1lC c.al.lUc~r Prob't"U1l silrciuting tllO Cl~ine culculut(!1l ooth stem.!), state lllld dyn/.ll1lic cnginu chrn:uctc.ristics thllt nro repr(~scntatj.ve of the PCSllS',w 11 l'll8:i.M.

RESULTS

'Illl:! opcrationnl r001 tUI~ clinical IIndal cOtlt!ili\D llpproxin\ltoly 7.OK by'tes of in .. Uno cu'Wtutlonal code m'ld 1'~. 7K byte.'l or blocl~ data. Progt'mtlrlng techniquoo developed in (1) and (2) were used to optilluZO tllc code £'Ol.' 't'ool t:illlQ operatic:m. 11113 bloc\t daCll C(Xlblins ull of the AUCD IIl1ltr:lx duel! £01.' the 1/1 nweln and supporting £'I.il:lCtions. \~ith lI\1.oor C'hllngcS in the code t the \\urocr of llodcla used could ensily be clw\I>tXl.

For n sinulation time s\~l2p size of S()us the propulsion systCln 110001 O)Ct:~cutcd in 2.llls on l\ Univuc 1110 for a rc.al-to-C)(ccution tinlQ rutio of 21.7. tAl the Xerox Signn 9 C<lnputcr used t\t NASA-limes, the Q..'tccu\"iOt\ tinlO wns 8.9l1'6 for u rutio of 5.6. '11ume CClIFutllcion t:bnes included L\ do wiled hydroo'cchlllliClll cool .. rol LUld uu uircrU£t: forca lU'\d blllru'\CQ sl"Jetton, 'n\l~ Cl'lgUIO portion consuncd about 75 JX!rcent of the tt)Ctl1 execution till~, '11m Inntrix operations ilwelved in thu atata vurir.ble ~T~tivTtS e-~.sl.l~ ItO i,m..-cent of that: cinko.'.

A nuroor 02 sata of trru'lSicnt test cases were 1.'\.lll 1,:0 dcm':ll'lStrnte the l-cnl till\Q digital Posusus 11 110001 llccurney lmd Cilpubil:l ties. Only the engino section. was UiL'<i here. 'l'he tos t C8SCS shCM response to full rm'lgo 3 second ~uel 1:10\01 I'll'll t,;'rru1.Simta. '11'e mmullln to tlllutill1,,!1\ [uol f1CM rLUlF prov:i.dcs a llooeratcly rllpitJ C1:ansicnC disturbm'lCU that allows the full .;~t of engine purtllll dcd.Vll.t::ivcs to be cxcrei~,cd. A test Cllse nt each COt'l'lC1 of the flight envelope as shown in f'igurc 8 is consi<.lercd. An mlditionlll ti;wt (:lUle Ltt S('.ll level static with response to the 8t.I)O. fuel. £10\" input:: but including a waCQ1.· ir\lection stem lit 92 percent fun speed was (lls() run. 1)1t:a fran ooch CllSQ \o/us c{lllptll:~J w.l.tl\ tho ao,rochc:ll,1100Ylllllllic silllllution f1."(l1l \~\icl, it:. ,.,us deriVt.~. All parlUlWJtCtS ~\Le itl PCl."CCl)C of .cull ~'nngOI

'J't:ansient:s Ut"C Ilt .. n in figures 9 Illl1 I 10 for only tho full rlll'l8C llccolerntion at: 5000 fOOL llltitude DUd the Sllll'll case with wlitor 11\.100t:iol\ HI: sOO leval stude.

'11\0 full range ~cceleration alt:it.1.Ide l:Ul\, figure 9. clalonst:rat;cs tlm effoots of nltitude on tho 11'0001. 'l'l:lU1.Sient behavior lllld hi-gh pcMut" stoody stntu levels nrc vel.">, closely llllltched cspccinllyfo>.: the output thrusts. Since tht'\lSC is the pdl1ltu:y OI.IL~t to th~ ail.oct'afc sUlulution, it: is CSSCl'It;illl to have tht"usc rcpresclltntiVtl of eho ronl ayst(lll. Also, al.though not sl)(1Nll here, diffQl.:'Cl1CCS up to G percent \~O cxhibiccd U'I sum intl'.l.1\ul ~il1c puran-ctet's. :41 pllrti<,:ulnr, jet pipe tUIFCrature £11)1;1 £m\ ru'\d cal'q)l."OOsor surge mru:sin, 1\lcsa differences c!ould be significzlnt if theywel.'c us<..'t\ us a sensed parameter :41 u c.ontrol systun,

'1110 test cnse witl1 water injcotiOt', figure 10. showod UllCh the sam lICet.tl'LlCy. 'l1lt::t"C wn.fJ no pnrtiMLlr difficulty in 1I~'Wlil18 the stlWtly uCnte or r.ynullic effects \.,.ith 'WU(;er injection.

l~lts of tho ()th~ test CUSOS I nlth0U8h not shO\ll\' he.re., \o1ore. silnilnr. 1110 effect of (.'Qrbined altitude and speed sh(],ol(.,~ tl\l\t altitude lu,d ",he l''IOst cUccI:. rull rlU1gc trlll'\Sicnt nv.'tchu18 wus C)(col1cnt here witll\1'()st diffe.L"cnces occuri.ng ngnin at lCM paMlr lovcls. Speed clfccts n1000 hud a relutively il'l::d,gniflcnnc effect t,Jl\ tho stnt:e variable llWOl. 1Iigh llnd lCM stoody state level lU1d trtmsic.nt: pl.'Ofilos matched extranaly \'JOll.

11'0 engine 110001 prase.nt(.'<i llel7e was carb1..ned with n dotailed ropr~cntntion of n hydrancchmdcal fuel cont::l;'Ql ru'ld in~rlltod into e aiuullltl.on for tho llarx':l..c.r ait'Craft:. Thu results of that evnl1.llltion ure $ivcn in (I,).

~1NG RflolARKS

A IItnte sched.1led state variable Qlethodolos.v has been wed '0 genet'ate a real emil digital propulsion aye tQll s.iJTlllacion for p;l.1ot~ t; lilulatorB. '1110 Mthodology proved effcctivQ in S(J1()rati.ng a n~l of GKcollent: llccuraCl 0\;(11: n lillrl.ttld oporational range as C<.l'l)(-U:oo to the nororhernooyr.mdc trOdel (TOOl which it: ~lS derived. '1l1U lootOOOOlogy provides a very flexible Ulemlll to aCCQlllUsh a real t:lnle m:x101 in a rcnsonnb1y al'¥lrt t:lnle.

'1.110 !rode1 GKccutoo 5.6 t:lnle8 fnstCl." than rool tin'il usinS an intogration titro step of 5Ql~. Coo-pJl:iltion tin'il nnnJ.ysis .indicnted that uppl."ORimnte1y 30 pcl."COl\t of tho. ~yclQ t:ln~ \o.\UJ C.:OMU1~>d in IMtrix opcrntiOll8. It uppcar:;l that t.his could be a high pt.4yoff arM for future dewl~llmt. Altl¥XJ&h tllt) tot:a1 ~lt of code g£!1lOl.'UCed in this 11Q()el is high. 21.71\. bytOB , tlus is generally not a lilldting factor in real timo Silluilltions,

1. Nihnloew, J .R. and ltatt, C,E.; "Real 'l'in~ Digit:lll Propulsion SystBll Sinuwtioo for Mannc<i night SiJlulators," NASA '!N-78958, National Aeronautics and S~ce Adl1inistrntion ({llevoland, Ohiu) , 1978.

2. HJ.halocw, J .R.; IIA Nonlinear Pro~Bion Systall Sil1ulncion Technique for Piloted S:UlUlators," NASA '11-1-82600, ~llt:ional Ae.ronnutica Md SpIlCc Adninistrlltion (Clcvolmld,OIli.o) I 1981.

3. ~Wlaloew, J.R., Ratll. SiP., llnd Cr~l~rc, R.S.; "A Hettl 'l'inlQ Pagru;us llropulsioo Systel1 ~bdcl for VS'l'OL Piloted S:imJ1ation EvQluntion," NASA 'J}1482770, National Ael1lllautics and Space Adll.inistratiw (Cleveland,Oluo), 1.981.

4. Roth, S.P., Sailgal, R" '1'~mson, D.N., and 7.unt, R.: "l'Ull Authority Digital Electronic O:mo.'ol: Rc.al Time OyMndc Propulsion Hodel For Miln~in··tllo-Loop S:f.nullltol;"S, II l~ FR-91133, Pt-lltt !lOO ",ut:ncy Aircraft Croup (WeB t l'alm Uooch, Florida), Decarocr 1977 I

6

o

CONTROLS

MATRIX OUTPUT EQUATlO~' A Y • C' AX + D' AU

STATES

AX

OUTPUTS

AU '--......,.. ........ B 1 S

MATRIX PLANT EQUATION AX • A • AX + B' AU

Figure 1. - Real time model state variable representation.

MODELING

r--- I---' REAL TIME MODEL L===~=======~"""",",=~ ... ""''''''.J

ENGINE

r------~~------------·-· --------~------, L _______ I AEROTHERMO CONTROLS I ______ J

METAL TEMPERATURE DYNAMICS I ROTOR INERTIA DYNAMICS 1 AEROTHERMO DYNAMICS 1

...

AIRCRAFT

AUTOMATIC FLIGHT CONTROLS ]

VSTOL AIRCRAFT

1 FREQUENCY N HERTZ

10

Figure 2. - Modeling frequency range of Interest.

A MATRIX ELEMENTS B MATRIX ELEMENTS

AII~ BilL' sSP sSP

C MATRIX ELEMENTS 0 MATRIX ELEMENTS

CIlV . DII~ SSP SSP

SSP II ((STATES)

MAX

MIN

MIN

STEADY STATE OPERATING CHARACTERISTIC

\ , A,B,C,D

~MATRICES GENERATED

:~_----- AT EACH

ENGINE POWER

~DEL POiNT

MAX

Figure 3. - State scheduled parameter nonlinear real time model.

--~,

STAlE FEEDBACKS

CONTROl INPUT VECTOR

STAIE S EQUALS SlEADY

~~~ I axm

fNlEGRAlE MID SET r

aXwm STAlE ~ INIITIAl

EQUATION COINDITIONS J allw(J}

)0[0} A (I.,J,LJ B (tJ,U

J .- IDENTIFY Cjij.CULAlE COMPL'lE f " CALCULAlE

SSP AND MODEL(U aX'S

I MAi'RlX

'--'--STAlE BRACKET AND ElEMENTS -'L. SCHEDULING POINT MODEl

aU'S PARAMETER MODELS 0..+1)

~X(l) a{J(J} C U.,J.U D (tJ.U

CAlCUlAlE t-Y-

CHEDULING PARAMETER WEIGHT OUTPUT FUEL flOW RA lE IN

~XAND~U aXw(I) EQUATIONS STAlE. aUwW

Y{LJ)

I COMPUlE OPERATING LINE ~..:tTPUTS

Figure 4. - Slate scheduled state variable engine model.

STATE VECTOR

OUTPUT VECTOR

~~;~. _.;:,,,..~,~_ ~>_~~W"'~~~l~~~~·"~"""""'_·~'"':~"'··~~-=--<e'""'~~-z~··~~~~r::;:.~~t"-''-';~'_e_._~ ....... ".....~---.,..~.,,::'7~~_._. -'~~'~-"*'.~«""""'.~~~~t":~.-«.-- -

o

• PEGASUS 11 AEROTHERMO SIMULATION

! j I I ~ ~ 1m

FUEL FLOW, PERCENT

Figure S • .. Pegasus 11 simulation steady state match.

j , 1

l I ,

! ACCELERAllON DECELERATION )

100 I 100 ~as

PEGASUS 11 I

~ffi 60 .. ,. - - SIMULATION 60 - Z!:)U z!:)u ~~ffi ~~ffi fEo,. 20 -fEQ.,

0 100

100 ~~~ z~ffi 60

< u 60 l.I.Q.,ffi looI.Q.,ffi Vlo,. VlQ., 20 0

l00~ w~~ l~F r ~ffi 60 \ $!:)u o::~u ;gE rd

.... ~~ 2~ I:\;--rw_.e:=t= o~eJ ufE::t!

uj:a.. I

~ 100

UJ~ffi 100 LIJ~as 0.:::1 0..:::1 90 -!;;t 80 -~ Q.,ffiu 0.. U mo..ffi ,

~--- .. -tuUJ~ 80 . ,

o.LU """530. \ "'~Q., 70

0 60

0 ~

1-

2 4 8 2 4 6 8 TIME, sec TIME, sec

Figure 6 •• Peg usus 11 simulation transient match.

CONTROL VEC/1R STATE VECTOR • FUEL FLOW • WATER INJECTION fLOW [

• LOW ROTOR SPEEO ] AX· • HIGH ROTOR SPEEO

• REACTION CONTROL BLEED FLOW • BURNER MEI"AL TEMPERATURE • ENGINE FACE PRESSURE • ENGINE FACE DELTA TEMPERA'(URE • AMBIENT PRESSURE

OUTPUT VECTOR

• CORE NET THRUST • FAN NET THRUST • FAN PHYSICAL • AIRFLOW • HIGH COMPRESSOR PHYSICAL AIRFLOW • FAN AVERAGE PRESSURE RATIO • HIGH COMPRESSOR PRESSURE RATIO • FAN NOZZLE DUCT TEMPERATURE • FAN NOZZLE DUCT PRESSURE • HIGH COMPRESSOR INLET TEMPERATURE • HIGH COMPRESSOR INLET PRESSURE • BURNER INLET TEMPERATURE • BURNER INLET PRESSURE • HIGH TURBINE INLET TEMPERATURE • HIGH TURDINE INLET PRESSURE • LOW TURBINE INLET TEMPERATURE .. LOW iURBiNE iNtEi PRESSURE • CORE NOZZLE DUCT TEMPERATURE • CORE NOZZLE DUCT PRES SURE • THRUST SPECIFIC FUEL CONSUMPTION • FAN SURGE MARGIN* .. CORE SURGE MARGIN~

*AUXILLARY MODEL OUTPUT COMPUTATIONS

FIgure 7, .. Real time mvdel state variable vectors.

ALTITUDE

FULL ENGINE ENVELOPE

05"~~~~---------o .. , 0.2

MACH NUMBEI~

Figure 8 ... Real .. I me moo91 flight envelope,

"

I I 1

.1

.~

I ! " J1 "1 ·1 q

u..,.~" T

3l:u 6

oQ) ri'" :c z-o l-= 4 i=0 uO::: ~~ ~3l: 0::: 2 l=!t=! I Ca) « 3:0::

0

l50x102

~ 0:::

3l: g.2 100 u..:c --1-

~ ti I (b) 1.1.1 3l: 50 ~ (!)

a:i o

llOx102

E r Cl.. ~

N z 90 X

ci 1.1.1 1.1.1 a.. en 70 z

o AEROTHERMO MODEL L:l. STATE VARIABLE MODEL

4 5 6 7

~

1 2 50~_~_~~~~_J_.-l._.J

3

7Oxl02 E e:

..... -~150 ci 1.1.1 1.1.1 a.. 30 en 1.1.1 0::: 0

10""'---u '- L _ -I

l5Oxl02 ..c

u: Z u..

..: 100 en :::> 0::: I (e) ¢= I- 50 1.1.1 Z Z

o~ ~

l5Oxl02 ..c

cS Z u..

..: 100 en :::> 0::: ::t: l-I- 50 1.1.1 Z 1.1.1 0::: 0

0 u 0 1 2 3 4

TIMIE, sec

Figure 9. - Full range fuel transient with watt~r injection at 92 percent fan speed, sea leve! static.

. ~ .. -:;!Ii

5 6 7

~ -:E t-!' Q

i ~ 0:: 3: 0 Ii ~ t; ~ a: ~

~ "-.c -:E

t:tT s:

~ ~ ."", ;::.. 0 Ii cd E2 I.J.J a:: (.!) z L.U

E e-N' z X

1/

Ff1\'f(' r ';;:'J. h,·: , t'"", I,; U ", "~" r , r,~"~ '~F- "'(':" ~ n,';.Jr'" ,I \')" 1~_ ~.J'.,.. ',~

3.986

2. 986 e 70 CI. ...

.-I"

~ 1,986 0" 50

~ VI UJ ~

.986 8 30 (a)

-,014 0

150 150

== c.J 100 EE 100 ~ ...:

VI :::l

50 0 AEROTHERMO MODEL ~ STATE VARIABLE MODEL

0

15Ox102

~~----~----~------~----~

et:

i= t;

50 I. z LI.J oLL I (el ~ 0 u

200xl02

== 150 u.. ..

a: .-.­VI

~ 100 F t; z z L:E 50

o 2 4 6 8 00 2 TIME, sec

Figure 10. - Full range fuel transient at 5000 feet altitude.

4 6 8