Design of a multivariable integrated control for a supersonic propulsion system

DESIGN OF A MULTIVARIABLE INTEGRATED CONTROL FOR A SUPERSONIC PROPULSION SYSTEM Edward C. B e a t t i e P r a t t & W h i t n e y A i r c r a f t Group Commercial Products Division SUMMARY A s t u d y was c o n d u c t e do f an i n l e t / e n g i n e / n o z z l ei n t e g r a t e dc o n t r o l mode f o rt h ep r o p u l s i o ns y s t e mo f an a d v a n c e ds u p e r s o n i cc o m m e r c i a la i r c r a f t .T h i s s t u d y showed t h a ti n t e g r a t i o no ft h e s ec o n t r o lf u n c t i o n s can r e s u l t i n b o t h o p e r a t i o n a l and p e r f o r m a n c eb e n e f i t sf o rt h ep r o p u l s i o ns y s t e m .F o re x a m p l e , t h i s i n t e g r a t e d c o n t r o l mode may make i t p o s s i b l e t o m i n i m i z e t h e u s e o f i n l e t bypass doors f o r shock p o s i t i o nc o n t r o l .T h i s may be o f b e n e f i t t o t h e a i r - c r a f t as a r e s u l to fm i n i m i z i n g : ( 1 ) b y p a s sb l e e dd r a ge f f e c t s ; ( 2 ) p e r t u r b a - t i o n st ot h ea i r c r a f tr e s u l t i n gf r o mt h es i d et h r u s te f f e c to ft h eb y p a s s bleeds; and, ( 3 ) p o t e n t i a1 u n s t a r t so ft h ei n l e t . A c o n c e p t u a li n t e g r a t e dc o n - t r o l mode was developedwhich makes use o f many cross-couplingpathsbetween i n l e t and e n g i n ec o n t r o lv a r i a b l e s and i n l e t and e n g i n es e n s e dv a r i a b l e s . A m u l t i v a r i a b l ec o n t r o ld e s i g nt e c h n i q u e b a s e du p o nL i n e a rQ u a d r a t i cR e g u l a t o r ( L Q R )t h e o r y was a p p l i e dt od e s i g n i n gt h ef e e d b a c kg a i n sf o rt h i sc o n t r o lt o a l l o w a s i m u l a t i o n e v a l u a t i o n o f t h e b e n e f i t s o f t h e i n t e g r a t e d c o n t r o l mode.

INTRODUCTION The N a t i o n a lA e r o n a u t i c s and Space A d m i n i s t r a t i o n (NASA) i s engaged i n s t u d i e s and advanced technology programs f o r f u t u r es u p e r s o n i cc o m m e r c i a la i r - c r a f t ,w i t h emphasis on improving environmental and p e r f o r m a n c ec h a r a c t e r i s - t i c s . As p a r t o f t h i s o v e r a l l program, P r a t t & W h i t n e yA i r c r a f t (P&WA) i s con- ductingadvancedpropulsiontechnologyprograms.

The t i m ef r a m ef o rt h e s ep r o g r a m si sc o n s i s t e n tw i t ha d v a n c e dt e c h n o l o g y p r o j e c t i o n st h a tw o u l dp e r m i t a U. S. e n t r yi n t ot h ec o m m e r c i a ls u p e r s o n i ca i r - c r a f tm a r k e tb yt h el a t e1 9 8 0 ' so re a r l y1 9 9 0 ' s .

The workpresented i n t h i s paper was a c c o m p l i s h e dd u r i n g a b r i e f s t u d y as p a r t o f a NASA-sponsored s t u d yc o n d u c t e db yt h eL o c k h e e d - C a l i f o r n i a Company, w i t h P&WA Commerci a1 P r o d u c t s D i v i s i o n as s u b - c o n t r a c t o r . (1) VARIABLE STREAM CONTROL E N G I N E R e s u l t sf r o mb r o a dp a r a m e t r i cs t u d i e s and r e f i n e di n t e g r a t i o ns t u d i e si n - d i c a t e t h a t t h e V a r i a b l e StreamContro En i ne (VSCE) has t h eg r e a t e s tp o t e n - VSCE c o n c e p te m p l o y sv a r i - t i a l f o r f u t u r e s u p e r s o n i c t r a n s p o r t s . l 2 ~ 3 ! T h i s able components and a u n i q u et h r o t t l es c h e d u l ef o ri n d e p e n d e n tc o n t r o lo ft w o flowstreams t op r o v i d er e d u c e dj e tn o i s ea tt a k e - o f f and h i g hp e r f o r m a n c ea t 3 5 b o t hs u b s o n i c and s u p e r s o n i cc r u i s e .F i g u r e 1 shows t h eb a s i ca r r a n g e m e n to f themajorenginecomponents i n a t w i n s p o o l c o n f i g u r a t i o n s i m i 1 a r t o a c o n v e n t i o n a lt u r b o f a ne n g i n e . The l o ws p o o lc o n s i s t s o f an advanced tech- n o l o g y ,m u l t i - s t a g e ,v a r i a b l eg e o m e t r yf a n and a l o wp r e s s u r et u r b i n e . A v a r i - ablegeometrycompressordrivenby an advanced s i n g l e - s t a g e h i g h t e m p e r a t u r e t u r b i n e makes up t h eh i g hs p o o l . The p r i m a r yb u r n e r and t h ed u c tb u r n e rr e - q u i r el o we m i s s i o n s ,h i g he f f i c i e n c yc o m b u s t o r s . A t w os t r e a m ,c o n c e n t r i c , a n n u l a r( c o - a n n u l a r )n o z z l ed e s i g nw i t hv a r i a b l et h r o a ta r e a si nb o t hs t r e a m s and an e j e c t o r / r e v e r s e r make up theexhaustsystem.

CORE EXHAUST COMPRESSOR VANES NOZZLE AREA D U C T E X H A U S T NOZZLE AREA

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DUCT B l FAN VANES FUEL FI EJECTOR F i g u r e 1 P r o p u l s i o n System, I n c o r p o r a t i n g a V a r i a b l e S t r e a m C o n t r o l Engine (VSCE), f o r an Advanced Supersonic.

S u p e r s o n i cI n l e t The s u p e r s o n i ci n l e tf o rt h e VSCE will be e i t h e r an a x i s y m m e t r i cc o n f i g u r - a t i o n w i t h a t r a n s l a t i n go rc o l l a p s i b l ec e n t e r b o d y ,o r a two-dimensional de- s i g nw i t hv a r i a b l ew a l l s .A u x i l i a r yi n l e td o o r s and bypassdoorsareincluded t o s a t i s f y o f f - d e s i g n and t r a n s i e n to p e r a t i n gc o n d i t i o n s .D u r i n gs u p e r s o n i c o p e r a t i o n ,t h ep r i m a r yc o n t r o lr e q u i r e m e n tf o rt h ei n l e ti st o f i x t h e shock p o s i t i o na t a l o c a t i o n downstream o f t h et h r o a t .V a r y i n gt h ei n t e r n a l geome- t r y , such as t r a n s l a t i n gt h ec e n t e r b o d yp o s i t i o n ,v a r y i n gt h eb y p a s sd o o r s and m a t c h i n gt h ee n g i n ea i r f l o ww i t ht h ei n l e tf l o wr a t er e q u i r e sc o o r d i n a t i o n .

T h i s will a l l o wo p t i m u mp o s i t i o n i n go ft h es h o c kf o r maximum p r e s s u r er e c o v e r y w h i l e m i n i m i z i n g i n l e t s p i l l a g e and bypass f l o w and p r e v e n t i n g i n s t a b i l i t y such as u n s t a r t and buzz.

3 6 Engine M o d u l a t i n ge n g i n ea i r f l o wt om a t c hi n l e ta i r f l o wi si m p o r t a n tf o ro p t i m i z - i n gi n s t a l l e dp e r f o r m a n c e .S e l e c t e dr a t i n gp a r a m e t e r s ,s u c h as r o t o r speeds a n d / o re n g i n ep r e s s u r er a t i o ,a r e programmed i n t ot h ec o n t r o ls y s t e mt op r o - v i d et h es p e c i f i ct h r u s t ,a i r f l o w , and t e m p e r a t u r er a t i n g sa tc r i t i c a lo p e r a t - i n gc o n d i t i o n st h a tr e s u l ti nt h ed e s i r e dp e r f o r m a n c e and environmentalbene- f i t s .

The VSCE f a ni n c o r p o r a t e sv a r ia b l e camber i n l e t and e x i t g u i d e vanes. The compressor has s e v e r a lr o w so fv a r i a b l es t a t o r s .A c c u r a t ec o n t r o lo ft h e s e variablegeometrycomponents i sr e q u i r e dt oo p t i m i z ep e r f o r m a n c eo v e rt h e f l i g h t e n v e l o p e w h i l e m a i n t a i n i n g s t a b i l i t y m a r g i n s .

The advancedmainburner and d u c tb u r n e r havestagedcombustionsystems w h i c hr e q u i r ea c c u r a t e and i n d e p e n d e n tc o n t r o lo ff u e lf l o wt o each stage t o o b t a i n t h e e f f i c i e n c y and e m i s s i o n sb e n e f i t sa s s o c i a t e dw i t ht h e s eb u r n e r de- s i g n s . The c o n t r o ls y s t e mm u s ta l so p r o v i d e smooth l i g h t - o f f ,s t a g e - t o - s t a g e t r a n s f e rd u r i n gt r a n s i e n to p e r a t i o n , and m o d u l a t e dt o t a lf u e lf l o wi n each b u r n e rs t a g et oo b t a i nt h ed e s i r e d power s e t t i n g s .

Nozzle/Reverser Continuous and i n d e p e n d e n tm o d u l a t i o no fb o t ht h ep r i m a r y and d u c ts t r e a m n o z z l ea r e a si sr e q u i r e di nc o n j u n c t i o nw i t ht h ee n g i n ec o n t r o lv a r i a b l e st o p r o v i d et h ed e s i r e de n g i n e and n o z z l eo p e r a t i n gc h a r a c t e r i s t i c s .C o n t r o lo f t h ea c t u a t e de j e c t o rd o o r s and t h et h r u s tr e v e r s e rm u s ta l s ob ep r o v i d e d .

I N T E G R A T I O N O p e r a t i o n and p e r f o r m a n c eo ft h e VSCE p r o p u l s i o ns y s t e mi s a f u n c t i o n o f t h ei n t e r a c t i o n sb e t w e e nt h ei n l e t ,e n g i n e , and n o z z l e .B a s i ci n t e r a c t i o ne f - f e c t sa r er e p r e s e n t e di nf i g u r e 2, and i n d i v i d u a lp e r f o r m a n c ef a c t o r sf o rt h e i n l e t ,e n g i n e , and n o z z l ea r e shown i n f i g u r e 3. S i n c et h ei n t e g r a t e dp r o p u l - s i o ns y s t e mi sa f f e c t e db ya l lo ft h e s ei n t e r a c t i o n s and p e r f o r m a n c ef a c t o r s , it i s a p p a r e n tt h a t an i n t e g r a t e dc o n t r o ls y s t e mi sr e q u i r e dn o to n l yt oo p t i - m i z ei n d i v i d u a l componentperformance,but a1 so t o t r a d e betweenengine compo- nents.

An i n t e g r a t e dc o n t r o lc a n a1 l o wc l o s e ro p e r a t i o nt oc o m p r e s s o rs u r g e l i m i t s t o i m p r o v e c o m p r e s s o re f f i c i e n c y and p r e s s u r er a t i od u r i n gs t e a d ys t a t e o p e r a t i o n , and u t i l i z e r e s e t l o g i c t o accommodate i n l e t d i s t o r t i o n e f f e c t s o r e n g i n et r a n s i e n t s .A n o t h e ri n t e g r a t i o na p p r o a c hi st ou s ee n g j n ev a r i a b l e st o c o n t r o l t h e i n l e t shock p o s i t i o n , and t h e r e b ym i n i m i z et h eu s eo fd r a g - i n d u c - i ngbypassdoors.

I n t e g r a t i o nm u s ta l s ob ep r o v i d e db e t w e e na l lf o u rp r o p u l s i o ns y s t e m s and b e t w e e nt h ea i r c r a f tc o n t r o ls y s t e m .T h i si sr e q u i r e dt op r o v i d eo p t i m u mo v e r - a1 1 a i r c r a f t p e r f o r m a n c e and t o p r o v i d e o p e r a t i o n a l r e l i a b i 1 i t y and s a f e t y b y m i n i m i z i n g t h e p o s s i b i l i t y of i n l e t u n s t a r t s as a r e s u l t o f a i r c r a f t maneu- vers. I na d d i t i o n , i f an i n l e ts h o u l du n s t a r t ,t h ei m p a c t on a i r c r a f t con- t r o l l a b i l i t y wouldbeminimized.Therefore, a c o n t r o ls y s t e mi sr e q u i r e dw h i c h n o to n l yp r o v i d e st h ep r o p u l s i o ns y s t e mc o n t r o lf u n c t i o n ,b u t can a l s op r o v i d e t h e s ei n t e g r a t i o nf u n c t i o n s .

: Inlet control settings : 0

: impact engine restart : 0 1 " : Engine power setting

b : induced airflow change i

Atmospheric effect i Bypass door bleed induced i b b

airflow change and 0 Induceti a i r t l o r 0

: distortion : Control variable induced

disturbances 0 .

1 " i -: airflow change

: Geometry for optimum

i d

: C F impactsmatch

Wing leading edge i Buzz or unstart

flow separation induced distortion i Airflow change induced i d

b

induced distortion I -: unstart and ACv 0

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i Bypass bleed and unstart : 0

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Aircraftmotion induced : induced aircraftmotions : b .

airflow change and .

: 0 0 .

0 b distortion .

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0 0 F i g u r e 2 Propu 1s i on System I n t e r a c t i o n s

-

NOISE AIRFLOW AIRFLOW OVERALL FUEL CONSUMPTION SHOCK POSITION EXTERNAL PROPULSION POWERSETTING RAM RECOVERY

+

+ DRAG

S Y S T E M SPILLAGE DRAG SURGE MARGINS INTERNAL ENGINE OPERATING PERFORMANCE BYPASS DRAG PERFORMANCI NOISE L I M I T S i

-

I I I I I I I I I F i g u r e 3 P r o p u l s i on System Performance Factors I n t e g r a t i o nb e n e f i t s and i n l e t / e n g i n e / n o z z l ec o n t r o lf u n c t i o ni n t e g r a t i o n a p p r o a c h e sw e r ee v a l u a t e du n d e rt h ec o n c e p t u a li n t e g r a t e dc o n t r o ls t u d y , (1) d i s c u s s e dp r e v i o u s l y . The i n t e g r a t i o n b e n e f i t s i d e n t i f i e d i n t h i s s t u d y a r e s u m a r i z e d i n t a b l e 1.

TABLE I - CONTROL INTEGRATION BENEFITS

M a x i m i z es t e a d ys t a t e and t r a n s i e n tp e r f o r m a n c e M i n i m i z e i n l e t u n s t a r t s andenginesurgeduringmaneuvers M i n i m i z eo c c u r r e n c eo fb u z z M i n i m i z eu s eo fd r a g - i n d u c i n gi n l e tb y p a s sd o o r s Improve a i r c r a f t hand1 i ng qual i ti es M a x i m i z eo p e r a t i o n a ls a f e t y INTEGRATED CONTROL MODE G i v e nt h ei n d i v i d u a lc o n t r o lr e q u i r e m e n t sf o rt h ei n l e t and VSCE, and i n - t e g r a t i o r rr e q u i r e m e n t s and approaches, a c o n c e p t u a li n t e g r a t e dc o n t r o l mode was developed. The r e s u l t i n gc o n t r o l mode, shown i nb l o c kd i a g r a mf o r mi n fi- g u r e 4, r e p r e s e n t s a f u l l y i n t e g r a t e d mode i n t h a t a l l a n t i c i p a t e d s i g n i f i c a n t c r o s s - c o u p l i n gl o o p s ,b o t hw i t h i nt h ee n g i n e and between engine and i n l e t , h a v eb e e ni n c l u d e d .F u l la u t h o r i t yi n t e g r a t o r sw e r es e l e c t e df o rm a i nb u r n e r f u e lf l o w (WFE), compressor bleeds, and bypass doors. T r i m i n t e g r a t o r s , whose o u t p u t add t os t e a d ys t a t er e f e r e n c eo rc o r r e l a t i o ns c h e d u l e s ,w e r es e l e c t e d f o rf a ni n l e tg u i d e vane (FIGVA), compressor s t a t o r vanes (CSVA) , c o r en o z z l e area (AJE), and d u c tn o z z l ea r e a( A J D ) . The use o fi n t e g r a t o r s on each c o n t r o l v a r i a b l e was s e l e c t e dt op r o v i d ea c c u r a t ec o n t r o lt ot h ed e s i r e dp r o p u l s i o n sys tern r a t i ngs .

Design and e v a l u a t i o n o f c o n t r o ll o o pg a i n s and dynamic compensation for such a c o n t r o l mode r e q u i r e dd e v e l o p m e n to f a d y n a m i cs i m u l a t i o no ft h e VSCE engine and t h es u p e r s o n i ci n l e t . The e n g i n es i m u l a t i o nc o n s i s t e do fd e t a i l e d n o n l i n e a rd y n a m i cr e p r e s e n t a t i o n so fe a c he n g i n e component a v a i l a b l ef r o m P&WA's s i m u l a t i o ns y s t e m . The i n l e ts i m u l a t i o ns e l e c t e d was based upon a simu- l a t i o nt e c h n i q u ed e v e l o p e da tt h e NASA Lewis Research Center, as d e s c r i b e di n Reference 4. T h i ss i m u l a t i o nt e c h n i q u ei sb a s e d upon a l i n e a r i z e dm a t h e m a t i c a l a n a l y s i so fi n l e t dynamics and, as such, i no n l yv a l i df o rs m a l lt r a n s i e n t p e r t u r b a t i o n sa b o u tt h eo p e r a t i n gp o i n t . However, t h i sl i m i t a t i o ni sa c c e p t - a b l e f o r a n a l y s i s o f i n t e g r a t e d c o n t r o l r e s p o n s e s i nce ( 1 ) e n g i n e o p e r a t i on a t s u p e r s o n i cc o n d i t i o n si sl i m i t e dt o a f a i r l yl i n e a rr a n g e and, ( 2 ) it i s de- s i r a b l e t o m a i n t a i n a c c u r a t e c o n t r o l o f shock p o s i t i o n ( i .e. ¶ o n l y a l l o w s m a l l v a r i a t i o n sf r o mt h ed e s i r e d shock p o s i t i o n ) so t h a t i n l e t o p e r a t i o n w i l l a l s o be l i m i t e d t o a f a i r l y l i n e a r range.

3 9 t REFERENCE * GAINS& CI C O Y E N U I T l O l l cc ”” LOW ROTOR SPEED ”- REFERENCE SENSED -c DUCT NOZZLE AREA FAN GUIDE VANES CWPRESSOR STATOR VANES COMPRESSOR BLEEDS BYPASS DOOR INTEGRATOR AREA AND DUCT BURNERPUEL DUCT BURNER SCHEDULES FUEL FLOIYS REFERENCE TRIM CENTERBODY POSITION CENTERBODY POSITION REFERENCE Figure 4 Conceptual Integrated Control Mode A schematic of an ideal mixed-compression i n l e t i s shown infigure 5. The cross-sectionalareavariation o f t h e i n l e t i s approximated by constantarea sections t o minimize thecomplexity of theresultingsimulation. For eachduct sectionchosen,theconstantareaapproximation and a linearanalysis o f the compressibleflowequationsresultinone-dimensional wave equationsrepre- senting t h a t section. These wave equationsare used torepresent boththe supersonic and subsonicflowregions. The supersonic and subsonicflowsec- tionsarethencoupled by linearizedequations which r e l a t e normal shock, position t o adjacentparameters. A linearizedequationisalso developed f o r bypassflow,assuming choked f 1 ow t h r o u g h thebypassdoor.Finallythe 1 i ne- arizedinletsimulationis mated withthenonlinearenginesimulationto pro- videtheexitconditions of the i n l e t .

+ < J - , / : T NORMAL SHOCK -OVERBOARD BYPASS

- ENGINE

STATION 0 0‘ 1 2 3 Figure 5 Idealized Mixed Compression I n l e t I INTEtiRATED CONTROL DESIGN APPROACH A m u l t i v a r i a b l ec o n t r o ld e s i g nt e c h n i q u e ,b a s e do nL i n e a rQ u a d r a t i c Re- g u l a t o r( L Q R )t h e o r y , was a p p l i e dt ot h ed e s i g no ft h ei n t e g r a t e dc o n t r o l mode f o rt h ei n l e t / e n g i n e .T h i st e c h n i q u ep r o v i d e s a s y s t e m a t i cp r o c e d u r ef o r de- s i g n i n g a l l c r o s s - c o u p l e dl o o p st h a ta r ee m p l o y e d i n an i n t e r a t e d c o n t r o l mode and assuresadvantageoususe o f t h e s e c r o s s - c o u p l i n g e f 7 e c t s . S i n c e t h e LQR m u l t i v a r i a b l ec o n t r o ld e s i g nt e c h n i q u ei s a l i n e a rt e c h n i q u e ,t h e non- l i n e a re q u a t i o n sr e p r e s e n t i n gt h ee n g i n em u s tb el i n e a r i z e d and combined w i t h t h el i n e a re q u a t i o n sr e p r e s e n t i n gt h ei n l e t .A c c o m p l i s h i n gt h i sr e q u i r e d de- f i n i t i o n o f t h e s t a t e , c o n t r o l and o u t p u tv a r i a b l e sf o rt h ee n g i n e and i n l e t .

G e n e r a l l y , it i sn o td e s i r a b l et oi n c l u d ee v e r ys t a t ev a r i a b l ei nt h ee n g i n e s i n c et h i s can r e s u l t i n an u n n e c e s s a r i l yc o m p l e xc o n t r o ls y s t e m ;i . e . ,t h e LQR t e c h n i q u ed e t e r m i n e sc o n t r o lf e e d b a c kg a i n sf r o me v e r ys t a t ev a r i a b l e s e l e c t e dt or e p r e s e n tt h es y s t e m . A more e f f e c t i v ea p p r o a c hi st or e c o g n i z e t h ef r e q u e n c yr a n g eo v e rw h i c ha c t i v ec o n t r o li sr e a l l yd e s i r e d ,o rp o s s i b l e , and simp1 i f y t h e s t a t e v a r i a b l e r e p r e s e n t a t i o n t o i n c l u d e o n l y t h o s e s t a t e s a s s o c i a t e dw i t he n g i n ed y n a m i c s i nt h i sf r e q u e n c yr a n g e .

Based on s u c hc o n s i d e r a t i o n s ,t h es t a t e ,c o n t r o l and o u t p u tv a r i a b l e s shown i n t a b l e I 1 w e r es e l e c t e df o rt h ei n l e t /e n g i n er e p r e s e n t a t i o n . Even

TABLE I 1 - STATE, CONTROL, AND OUTPUT VARIABLES

X - STATE VARIABLES U - CONTROL VARIABLES Y - OUTPUT VARIABLES

“ _ ~ I

X I - L O W R O T O R SPEED U 1 - M A I N BURNER FUEL Y 1 - L O W ROTOR SPEED

FLOW

X2 - H I G H ROTOR SPEED U2 - CORE EXHAUST Y2 - H I G H ROTOR SPEED

NOZZLE AREA

X3 - M A I N BURNER PRESS. U3 - DUCT EXHAUST Y3 - E N G I N E PRESS. R A T I O

NOZZLE AREA

X4 - CORE STREAM U4 - COMPRESSOR STATOR Y4 - NORMAL SHOCK

EXHAUST PRESS. VANES P O S I T I O N

X5 - DUCT STREAM PRESS. U5 - FAN STATOR VANES Y5 - FAN PRESS. R A T I O

X6 - NORMAL SHOCK U6 - INLET BYPASS DOOR Y6 - COMPRESSOR PRESS.

POSIT I O N AREA R A T I O

X7 - INLET SUBSONIC Y7 - INLET SUBSONIC

SECTION TEMP.

SECTION PRESS.

X8 - INLET SUBSONIC

Y8 - H I G H TURBINE

SECTION PRESS. INLET TEMP.

X9 - INLET SUBSONIC Y9 - THRUST

SECTION AIRFLOW though a l l o f t h e i n l e t s t a t e v a r i a b l e s a r e a s s o c i a t e d w i t h h i g h f r e q u e n c y dynamics, it i s necessary t oi n c l u d es e v e r a lo f t h e ms i n c ec o n t r o lo ft h e shock p o s i t i o nr e q u i r e sr e l a t i v e l yh i g hf r e q u e n c yr e s p o n s ec o n t r o ll o o p s . The i n l e ts t a t ev a r i a b l e sa s s o c i a t e dw i t ht h es u p e r s o n i cf l o ws e c t i o nw e r ee l i - m i n a t e ds i n c e it was f o u n dt h a tf e e d b a c ko ft h e s ev a r i a b l e sd i dn o tc o n t r i b u t e s i g n i f i c a n t l y t o e f f e c t i v e c o n t r o l a c t i o n . The f i r s t s i x o u t p u t v a r i a b l e s were s e l e c t e dt ob ec o n s i s t e n tw i t ht h er e f e r e n c ev a r i a b l e s shown i n t h e c o n c e p t u a l c o n t r o l mode i n f i g u r e 4.

U s i n gt h e s es t a t e ,c o n t r o l and o u t p u tv a r i a b l e st h ei n l e t /e n g i n es i m u l a - t i o n was l i n e a r i z e d a t a s u p e r s o n i cf l i g h tc o n d i t i o nc o r r e s p o n d i n gt o an a l t i - t u d eo f 16,800 m (55000 ft) and a Mach number o f 2.3. T h i sl i n e a r i z a t i o nr e - s u l t e d i n a s t a t e v a r i a b l e r e p r e s e n t a t i o n o f t h e s y s t e m c o n s i s t i n g o f t h e f o l - l o w i n gt w om a t r i xe q u a t i o n s : 6 X = A 6 X + B 6 U 6Y = C 6 X + D 6 U The n e x ts t e pi nt h e LQR c o n t r o ls y n t h e s i sp r o c e d u r ei st od e f i n e a p e r - formance index as a measure o ft h e goodness o ft h ec o n t r o le f f e c t i v e n e s s . A q u a d r a t i cp e r f o r m a n c ei n d e xo ft h ef o l l o w i n gf o r mi sr e q u i r e df o rt h e LQR syn- t h e s i st e c h n i q u et os o l v et h e" o u t p u tr e g u l a t o r "p r o b l e m .

T T Performance Index = J ( 6 U) = 6Y Q S Y + 6 U R 6 U d t M i n i m i z a t i o no ft h i sp e r f o r m a n c ei n d e xr e s u l t si n" o p t i m a lt r a n s i e n tp e r - formance" as d e t e r m i n e db yt h es e l e c t e dv a l u e s i n t h e Q and R w e i g h t i n gm a t r i - ces on t h eo u t p u t and c o n t r o lv a r i a b l e s ,r e s p e c t i v e l y .F o re x a m p l e ,p l a c i n g a h i g hw e i g h t i n g on shock p o s i t i o n will i m p r o v ec o n t r o lr e g u l a t i o no f shock p o s i t i o n .W i t ht h ep e r f o r m a n c ei n d e xd e f i n e d ,t h e" o u t p u tr e g u l a t o rp r o b l e m " i s s o l v e d by s o l v i n gt h em a t r i xR i c c a t ie q u a t i o nf o rt h es t e a d ys t a t ev a l u e of K .

A

-I? = KA + ATK - GTRG + CTQC

A where R = R + DTQD A and G = R - 1 (DTQC + BTK) The m a t r i x I; i st h em a t r i xo ff e e d b a c kg a i n sf r o m each s t a t e v a r i a b l e t o each c o n t r o lv a r i a b l e .

R e f e r r i n g back t o f i g u r e 4, i t can be seen t h a ti n t e g r a t o r sa r ed e s i r e d on e a c hc o n t r o lv a r i a b l et om a i n t a i nz e r oe r r o r sb e t w e e nr e f e r e n c e and sensed en- g i n ev a r i a b l e sd u r i n gs t e a d ys t a t eo p e r a t i o n .N o t et h a tt h er e f e r e n c ev a r i a - b l e s f o r f a n match,compressormatch and shock p o s i t i o n w e r er e p l a c e dw i t hf a n p r e s s u r er a t i o ,c o m p r e s s o rp r e s s u r er a t i o and a c t u a ln o r m a ls h o c kp o s i t i o nf o r t h i ss t u d y .T h e s ei n t e g r a t o r sw e r e accommodated b yi n c l u d i n gt h e m as a d d i t i o n - a ls t a t ev a r i a b l e sa l o n gw i t ht h ei n l e t / e n g i n es t a t ev a r i a b l e s , and s o l v i n g t h em a t r i xR i c c a t ie q u a t i o nf o rt h ec o n t r o lf e e d b a c kg a i n sf r o mt h ec o m p l e t e s e to fs t a t e s .T h i sa p p r o a c hr e s u l t si n a s o l u t i o n f o r t h e G m a t r i xw h i c hc a n bebroken down i n t o a G1 m a t r i x f o r t h e i n l e t / e n g i n e s t a t e s and a 62 m a t r i x f o r t h e c o n t r o l i n t e g r a t o r s as shown i n f i g u r e 6.

19x61 19x11 u - 1 6 u ' + x 6 x B C 5 I S 19x91 GZ I 16x61 I A 19x91 L I N E A R I N L E T I E N G I N E I F i g u r e 6 S o l u t i o n o f t h e M a t r i x R i c c a t i E q u a t i o n D e t e r m i n e s t h e G 1 and 6 2 F e e d b a c kG a i nM a t r i c e sf r o mt h eI n l e t / E n g i ne and C o n t r o l S t a t e Var i ab1 es The r e s u l t i n g c o n t r o l mode s t r u c t u r ei sn o te q u i v a l e n tt ot h a t shown i n f i g u r e 4. To o b t a i n t h i s s t r u c t u r e r e q u i r e s a t r a n s f o r m a t i o no ft h ec o n t r o l g a i nm a t r i c e s G1 and G2 t o t h e new m a t r i c e s H, L1 and L2 o p e r a t i n g on t h eo u t - p u t v a r i a b l e s Y . D e f i n i n gt h ed i f f e r e n t i a l s 6 U and 6 Y as

6 U = U - U r e f

6Y = Y - Y r e f

a l l o w si m p l e m e n t a t i o n o f t h ec o n t r o l system, as shown i n f i g u r e 7, on a non- l i near i n l e t / e n g ine s i m u l a t ion f o r e v a l u a t i on o f s m a l l p e r t u r b a t i o n r e s p o n s e a tt h es e l e c t e do p e r a t i n gp o i n t .

The L2 g a i n m a t r i x i s r e q u i r e d i f t h e number o f s t a t e v a r i a b l e s i s l a r g e r t h a nt h e number o fc o n t r o lv a r i a b l e s .T h i s can be seen more c l e a r l yb yc o n s i - d e r i n gt h e summary o ft h em a n i p u l a t i o n sd i s c u s s e d above. F i r s t ,t h ec o n t r o l designproceduredetermines a c o n t r o lf e e d b a c kg a i nf r o me v e r ys t a t ev a r i a b l e t oe v e r yc o n t r o lv a r i a b l e ; i.e., t h e G m a t r i xo rt h e G1, and 62 m a t r i c e s . Then a s e to fi n d e p e n d e n to u t p u to ro b s e r v e dv a r i a b l e s( w h i c hc a nb es e n s e d ) ,e q u a l i n number t o t h e number o fs t a t ev a r i a b l e s ,i ss e l e c t e dt or e p l a c et h es t a t e v a r i a b l e s ; i. e., t h es e to fs t a t ev a r i a b l e s ,s e l e c t e df o rc o n v e n i e n c eo f a n a l y s i s , may n o t a l l b ee a s i l ym e a s u r e do r may n o tb ee q u a lt ot h er e f e r e n c e v a r i a b l e sd e s i r e df o rc l o s i n gt h ei n t e g r a lc o n t r o l 1 oops. In t h i s i n t e g r a t e d c o n t r o l mode, s i xr e f e r e n c ev a r i a b l e sa r es e l e c t e df o rd r i v i n gt h ec o n t r o li n - t e g r a t o r st oo b t a i nt h ed e s i r e ds t e a d ys t a t eo p e r a t i n gp o i n t . Thus, t h e f i r s t s i xo u t p u tv a r i a b l e sm u s t be t h e same as t h es i xr e f e r e n c ev a r i a b l e s .T h i si n t u r n a l l o w s t h e m a n i p u l a t i o n o f t h e c o n t r o l g a i n m a t r i c e s i n t o t h e s t r u c t u r e shown i n f i g u r e 7 w i t h t h e L1 and H m a t r i c e sa c t i n g on t h e f i r s t s i x o u t p u t e r r o rt e r m s . The L2 m a t r i xt h e no p e r a t e so nt h el e f t o v e ro u t p u tv a r i a b l e s .

4 3 I F i g u r e 7 T r a n s f o r m a t i o n o f t h e C o n t r o l Mode S t r u c t u r e t o I n t e g r a l and P r o p o r t i o n a l C o n t r o l P a t h s If t h ek e yi n l e t / e n g i n ev a r i a b l e s have been chosen f o r t h e f i r s t s i x c l o s - ed l o o pc o n t r o lp a t h s ,t h e n many o ft h er e m a i n i n gp a t h sw o r k i n gt h r o u g ht h e L2 m a t r i x will p r o b a b l yb ei n s i g n i f i c a n t and be a b l et ob ei g n o r e d . If a l l o f t h e s ep a t h sc a nb ei g n o r e d ,t h e nt h ec o n t r o l mode s t r u c t u r er e d u c e sc o m p l e t e l y t o t h a t d e s i r e d i n f i g u r e 4. T h i sc o m p l e t ep r o c e s so f mode s t r u c t u r em o d i f i c a - t i o n and e l i m i n a t i o no fi n s i g n i f i c a n tg a i nt e r m s was n o tc a r r i e do u td u r i n g t h i s b r i e f s t u d y . A p a r t i a lt r a n s f o r m a t i o no ft h eg a i nm a t r i c e s was made, as shown i n f i g u r e 8, w h i c h f e e d s b a c k t h e f i r s t s i x o u t p u t v a r i a b l e s f o r f o r m i n g t h ei n t e g r a t o re r r o rt e r m s ,b u tr e t a i n st h er e m a i n d e r of t h ef e e d b a c k sf r o m t h ei n l e t / e n g i n es t a t ev a r i a b l e s . A l l s i m u l a t i o nr u n sw e r et h e n made w i t h a l l e l e m e n t so ft h eg a i nm a t r i c e sr e t a i n e d .

INTEGRATED CONTROL TRANSIENT PERFORMANCE The LQR c o n t r o ld e s i g nt e c h n i q u e was used t od e f i n et h ef e e d b a c kc o n t r o l g a i n s ,p r e v i o u s l yd i s c u s s e d ,a tt h e 16800 m (55000 f t ) a l t i t u d e , 2.3 Mach num- b e r f l i g h t c o n d i t i o n f o r t h e f u l l y i n t e g r a t e d c o n t r o l mode. These gains were t h e ni m p l e m e n t e do nt h en o n l i n e a ri n l e t / e n g i n ed y n a m i cs i m u l a t i o n , as i n d i c a t - ed i n f i g u r e 8, t oe v a l u a t es m a l lp e r t u r b a t i o nt r a n s i e n t sa b o u tt h es t e a d y s t a t eo p e r a t i n gp o i n t . A n o n - i n t e g r a t e dc o n t r o l mode was a l s od e s i g n e df o r c o m p a r i s o nw i t ht h ei n t e g r a t e dc o n t r o l mode i no r d e rt oe v a l u a t eo p e r a t i o n a l b e n e f i t sa s s o c i a t e dw i t ht h ei n t e g r a t e dc o n c e p t .T h i sn o n - i n t e g r a t e dc o n t r o l was d e v e l o p e db ya p p l y i n gt h e LQR c o n t r o ld e s i g nt e c h n i q u et od e t e r m i n et h e f e e d b a c kc o n t r o lg a i n sf o rt h ee n g i n eb yi t s e l f . Then a s i n g l e - i n p u t ,s i n g l e - o u t p u tc o n t r o ll o o p was d e s i g n e d f o r t h e i n l e t t o c o n t r o l shock p o s i t i o n w i t h i n l e t bypassdoors.

F i g u r e 8 P a r t i a l l y T r a n s f o r m e d C o n t r o l Mode S t r u c t u r e Used f o r T r a n s i e n t Eva1 u a t i ons Two t y p e so fs m a l lp e r t u r b a t i o nt r a n s i e n t s wereevaluated on t h e dynamic s i m u l a t i o nw i t ht h ei n t e g r a t e d and n o n - i n t e g r a t e dc o n t r o l modes. The f i r s t c o n s i s t e do f a 1 p e r c e n tp u l s e i n ambientpressureof 0.04 second d u r a t i o n t o simulate an externaldisturbancesuch as a windgust. The second c o n s i s t e d o f a s t e p change i n d u c tb u r n e rf u e lf l o wt os i m u l a t e a d u c tb u r n e rl i g h t - o f f .

F o rt h i ss t u d y , it was a l s o assumed t h a ta l ls t a t ev a r i a b l e si n c l u d i n g shock p o s i t i o n were d i r e c t l y measurable.

T r a n s i e n tp l o t so f shock p o s i t i o n and i n l e t bypassdoorarea f o rt h ep r e s - s u r ep e r t u r b a t i o nt r a n s i e n t sf o rb o t hc o n t r o l modes are shown i n f i g u r e 9.

F o rb o t ht h ei n t e g r a t e d and n o n - i n t e g r a t e dc o n t r o l modes t h ed e v i a t i o ni n shock p o s i t i o nt o w a r d su n s t a r t was approximatelythe same. The i m p l i c a t i o n i s t h a tt h ei n t e g r a t e dc o n t r o l mode i s n o t p r o v i d i n g any b e t t e rc o n t r o l of shock p o s i t i o nt h a nt h en o n - in t e g r a t e dc o n t r o l . In f a c t , t h e i n t e g r a t e d c o n t r o l r e - s u l t s show bypassdoorareamoving more than i n t h e n o n - i n t e g r a t e d c o n t r o l case t o r e s u l t i n t h e same q u a l i t y of shock p o s i t i o n c o n t r o l . T h i s i s t h e o r i z - ed t o r e s u l t f r o m t h e manner i n whichtheengine i s b e i n g c o n t r o l l e d i n b o t h cases.

-

13.4

-

68.2 Inlet normal 68.0 shock position

-

67.8 - I N .

-

67.6 1 I 67.4,

- Integrated

- - - Non-integrated

-

1.5

-

-

1.0

A Inlet bypass

-

door area

- IN.2

-

.5

-

/- I I l l l

O D ' I 0.1 0.2 0.3 0.4 0.5 0.6

Time - seconds

F i g u r e 9 T r a n s i e n t Response t o an Ambient Pressure Pulse R e f e r r i n g back t o f i g u r e 4, i t i s seen thatthereferenceparametersfor theengine,i.e.,the f i r s ts i xo u t p u tv a r i a b l e s ,a r es u c ht h a tr e g u l a t i n gt o t h e s ev a r i a b l e sr e s u l t s i n a c c u r a t ec o n t r o lo fe n g i n ec o r r e c t e da i r f l o w . Thus, t h ee n g i n ec o n t r o lp o r t i o no fb o t hc o n t r o l modes r e s p o n d sr a p i d l yt o changes i n a m b i e n tp r e s s u r es i n c et h i s has an immediate e f f e c t on theenginereference v a r i a b l e s . The r e s u l t i s r a p i d movement o fe n g i n ec o n t r o lv a r i a b l e st or e s t o r e c o r r e c t e da i r f l o wo p e r a t i o n .T h i s , i n t u r n , c o n t r i b u t e s d i r e c t l y t o m i n i m i z i n g shock p o s i ti on movement.

The f a c t t h a t t h e i n t e g r a t e d c o n t r o l mode made moreuse o f t h e bypassdoor a r e aw o u l di m p l yt h a tt h ee n g i n ec o n t r o lp o r t i o no ft h ei n t e g r a t e d mode was n o t as welltuned as t h ee n g i n ec o n t r o lf o rt h en o n - i n t e g r a t e dc o n t r o l mode.

I n o t h e r words, t h ew e i g h t i n gg a i n s i n t h e performanceindexwould have t o be changed i nt h ed e s i g np r o c e d u r ef o rt h ei n t e g r a t e dc o n t r o l mode t o reduce i t s dependence on bypass doors. These i t e r a t i o n so ft h ec o n t r o ld e s i g n were n o t c a r r i e d o u t d u r i n g t h i s s t u d y .

R e s u l t so ft h ed u c tb u r n e rl i g h t - o f ft r a n s i e n t sf o rb o t hc o n t r o l modes are shown i n f i g u r e 10. The i n t e g r a t e dc o n t r o l mode r e s u l t s i n l e s s movement o f shock p o s i t i o nw i t hl e s s use o f bypass doors than does t h en o n - i n t e g r a t e d mode. These r e s u l t si n d i c a t et h a tt h e r ei s a p o t e n t i a lb e n e f i to f an i n t e g r a t - ed c o n t r o l mode i n terms o fm i n i m i z i n g use o f t h e i n l e t bypassdoors f o r shock p o s i t i o n c o n t r o l .

To e v a l u a t et h i sp o t e n t i a lb e n e f i tf u r t h e rw o u l dr e q u i r ea d d i t i o n a la n a l y - s i s o fb o t ht h ei n t e g r a t e d and n o n - i n t e g r a t e dc o n t r o l modes.

#”””

68.0 r - \

/-

Inlet normal shock position - in.

- integrated

I’ 1

67.8 1

I /

--- Won-integrated

I /

L

’ /

‘”’

/”\ \ I \ I \

A Inlet bypass

1.0 \ I \ door area \ ’.‘i I \ 0 . 5 0 0.1 0.2 0.3 0.4 0.5 0.6 F i g u r e 10 T r a n s i e n t Response t o a Duct Burner Light-Off 4 6 CONCLUSIONS The conceptualintegratedcontrol mode, for an Advanced SupersonicTrans- portpropulsionsystem,evaluatedin thisstudy, makes use of severalcross- coupling p a t h s between i n l e t and engine controlvariables and i n l e t and engine sensedvari ab1 es. Designof thecontrol loop gains and dynamic compensation f o r such a control mode can be effectively accomplished utilizing a multivari- ablecontroldesigntechnique based on LinearQuadraticRegulator Theory. Such i ntegratedcontrol modes may provideoperational and performance benefits such as minimizing the use of i n l e t bypass doors for shock positioncontrol.

R E F E R E N C E S NASA CR-145285, "SupersonicCruiseVehicle Technology Assessment Study o f an Over/UnderEngine Concept" , Lockheed-Cal if orni a Company.

NASA CR-135148, "Advanced SupersonicPropulsion Study - Phase 111 Final

Report", December, 1976, P r a t t & Whitney Aircraft.

NASA CR-135273, "Advanced Supersonic Propul sion Study - Phase IV Final

Report", September, 1977, P r a t t & Whitney Aircraft.

NASA TN-D-7839, Gary Cole, Ross Willoh,"Analysis of the Dynamic Response of a Supersonic Inlet t o Flow FieldPerturbations Upstream o f the Normal Shock", January, 1975.