Control optimization, stabilization and computer algorithms for aircraft applications

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7 i (NASA-CF-145b62) CCNl F CL CilIMIZAIICN, N76-13113 STAE1117ATUN Phr C(?[Ul? r ALGC?IIHMS ELc AI I ;C• Fl AFFICATICNS Status r,- Crt, I Jun.

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25 F HC $3.5 1 '-1C G ?/,. d C EC I C 5 J 4 W "riLir, 6 7

, , , n(- I , 1975 will be :r niscarch cluring this tine period Janxictry 1, to I 4 0n D) tinder the fol1.,),v- ;n(7 topics (:-,ea Sect- summt- -sterrs r,!liability Optjm* tiuti 1) S I riiluro Detection Algorithms 2) 3) Ar, -tly--,i-s of I;onl.inc.-ir Fitters Time Cclay.,i •) ro!;ign oC -.ti rn ttion for Systems- .

X1 Echrjos 7) CcsI ',ja C-ntroL1,2c fir -1-D rrobl,-ms 1-ind C, With Cc.nsLrai!iLs I-o.—"n.,;,atux TM,d%,n for t:L:- Lv ; l,n)v 1;,,,.merica).

and r,irjei-c 12) PetL • irl),ttion M.- , t1iods in

_.;- A.

" y su'Prort receivc-A partial salnr ,rarinq t-his time reri.c2 the iin,1er this yr,int ac ,t 1 .

rtofessor M. Athans Professor 'r.I,. .Iohn,,,.)n Prufuesor S.K. Idi,_ter rrof.cssor r. Varaiya rr ofo.snr A.S. :eill_.ky Hr. 'I. Safanov r. B. T.loretrs M Mr. T. AL-it.ty Mr. P. Fw,-^nq t'r. 1).

_ . I'.`". :.onq r,.,^r,^Ll- ^tr. h eC1'ort to the Y3..

t'...

In aO,1iticn , t"me 1'-'-rii<.l.

. U1 'Urt CM - in') till ; , rocci`: _ 1 no 1' b it _oct Sal l; / P • ;oarc h St.-1 `f ,1:1t <lit:l N.R. Izandell Dr. S.B. Ccrshwin Dr. K.-P. Curtn t11C? ,"nts K. Tong P. Lee A. "ota: edi R. "a E'. I..tx S. E. (-how !'a r.:1t s L,^bano`f R. T'ueno 1'. : " c;. Rare its J. Eterno

this t im-^ l:criod between M.I.T. pursoncnel i nq il:o major interaction clai , . , ^:t Workshop the M,.T.T.- NA`'A Ar l d,iri.r.g aarc:h str.fr o^currc and NA.?./Ames r p ie ctim , held at Aircraft. I,llis r o m Ful:ability Is:3ue,; for F'_,tucc on sv-,t a nutrber of p c-eminent researchers, from Auqust 17-20, brougltf toyct:, T.T.

of :cluiinq many from 'LW-.SA/Ar.!e3 and M.I.T., t.o discu:% a number of their :or,: t.liat impact on tl.a question c- - c,Aial-le aircraft design. '-'one of ti-,t2 work cutlin^d in Soctiun D -- specifically that relatad to rc.linl,ili.ty u1 failure decoction -- wa > r. ;i or;.e l at t`.lat

P. -chni(- . ) rise I.:inn w :trtn^ F:'1 -i - tcloil I . inhi? i ty runt i To date, research by !'r. roull a, Birc:woll arts )?t - ofet -,_()r Mi%o ;'lt han s ep of , has 1;t2,?n cot;corneti WiLll '::1" v re.--nation sy: >to.n-3 for llS ` in the ';I:1vIy hol,tj}lt O _- s •l stores rel i.ibil ity an,! oirtimiz.)tion. A theor-.ti(^al r rar,cty >: k is t • iich 13 zp'no al erow,a to r,,,nsi for 1'l.1ny . i i r01,1.:,'s, `t . I °ferent ty! .'.- of ( is Specific cn(-ugh to allow oi.timal design of rei;..;ble 537st-21as. The `r.wo-ork currently bein l accnrnts 1''rr all discrote lur p s of a s y su , .n and a'L!,-ms's for opt`mi^ lt'( n a :I-t G)_ :;ys tem , }1 Coll i-j ".lration:, whic a critr?ri "n. Gi.von a F7 L ": 1- j :'7' ;._, h s^ i:nl.:; to wor:'! wr..11 and h.?:; J1'IQn good r; .ult,.i ;'lc v ^bl('"1, ::111:; a l for t:at? prn t ` L:_:i o' !;on: G1 - .lc)catil^n for a 1 ' ! .i' ;t.1T^, wlier" failure?

of ccwo,itittc.., 1,)hs 7C ('xe set sonC,Ors. Tl - oF i^:I i`.rO.. 1: 11:;0 seeP15 to he tl.i,l i :a: la to otl.er :ic,?as>, s,ucil a:; the of l.ineari: at:ion , f,)r v.,e 7oriL,'C!`i for . _.-z, rvat:ion :.n.: CCntrei uC :ii•jni ican ` Iy n<r I inoar in ti - .e :L^4%c d- r and toms. Soma i`roblc 1"; with the aj.proach incl u1,?

t' . I 1 .l COITj'1.'_':i:} o nt.ou:;tC- , r''1 in of _i:e motho ' '. '' loxit_y r.,`.y be reduced 1.1 tl.Q future With C.0 aat:l; -i. n UL theoretical t;.: ...

y found, to Zppl1,cabl1? to ?ar.c;,, sets of problems. Ways may lil COtti'lltll t :l^'[ldl COT^,) r .A:tlt.;l thr o ug h he car!?f:ll U;C of ai';)roxln.1tionS reduc thy?

ssiblo T-.rides of failure.

invol.vi11-4 tho Sot ':ontai.ling all i:0 : ? v;<r: k in •. 1 es A brae :h of r '?search vr' i.. h hi; been is rluc "1 by the tl:?; of rnu:.tiple rode1 .:i31'ntiv .^ con -rolof a nanl.inear .:;:;*_e:^. A new roach h:i; been uutlir the -t- of lin' Iri.zatic'ns of they System functi<; 1 to be a finite-s':ate >1>ac _, problcm Leing

-6- S)l:,tt`11 U:il"; sorr:-: I • 1 fF N-JAC it t (•`:.t::!)1 follows: invest1'flted 1R a'; i nearizations ub.;et: of avail," Ible l with :0ro .

is .1Fproxill.ztinq the sy:;t, ;c :alat sul) 't of 5 at some time t, how "hould the control law cllou: 1 ine rizationi; _;' to u.,r at ';o>'1e future time t'? c)n,= al-t.ma( i1 to the tl-,o "(Ustance" betwt•en a syst:.Ji and a tlivcn rt)hlen is to estimztt^ i^_ i h Or ' S' tI';1:1 J a variatL;tl of tho n,?ilr(1:;1.-ilcl li Ctd',G: ; : .:ticn, and liiieari: a:; the norm of tt,,e vec:tor difference is .11(Jo • ithn, ir .,ty l ina 1ri2ati'.n, t.lifi ap p roach xid a poo r t O` :;yitt:m state 1)t'twr,Cn the howetf'r, tilt?ra may 1:,l a yielrj a locally o ptimal method for chor, ,ilvi talc! est:imat.c, but which i-i a lar.(le di::tance from ljr•eavi-..ttion xiiich It :•J:^t:l(i be to) ;ystem.

the , !..^r. :t riatics of the qu: r , close a t3lobai-ly '• „};i - st.'(1 . ' sure of .'.1.:tlr cE] L:itlt' .1 1' tllce i - ` 'W C^ 7 e.x, t_7, 1`; a qu. • ion or 1.'3t Li:16 07t1[ ,- 11- C:loic(l of S' .

still to be an,:.,-red.

r_a,a^ o.' a -:,.ore goner- 1 1 .area of prcblerts.

-aci` c ' i5 research , e tchre•sen- co.ltlt^120 ^1:i :St• - ito spac ^lYCht'C il.:,i d :iV:-itC:a with a the r^'ti „ ^111'1^J.i ri If he stat,2 1;1)acf-.

? co,': t .act, sl&, g et of the s iiiLere:AC(l in SO.'. , tation .ind 4 links t i sp ac e, wita aJ a 11n LL : 1 1 .0 f.)! this J ub`-',2t ccnsid , ^ 73 tI t,.tvo in q t?!.emt'nt: in the = path from o ne exit;tt nco of a l,,.twr(,n s t atef. r(:1'res..:lting th,i coverinq to a point wit:-,:r, another nnit;hborhoo.1, in L:hat L:.ty flocs opti:liza- j state spice Corr,_'.>L.ond to optimization ever the tt - . .; fin 'to - tiorl o •.,•r coutin,.l(: • ls slat:`? Sorh corn,}(.tinn:; with previous work apfear in the for _'.igital i:urposes. Future ai.plic it_ions sp ace di::cretization ')f a ('rutP. ,oc,:, .°ast Ele n .1 inq tl;i.s vi •w . p :nt to ',g irls :rh as

._ )- ::imat ir:ns to th! .:olut ion of ohti.mal cuntrol 1 r-1 1a•tr.; .tv l rcl iahil Lty .

01 ' .'1:.It lull probl!'.ms Concerned with Cnntill`.lous lallur'.. I".'.la'S lel s;y:;tcnl!;.

lur ^_'tec t-ic,n '+iclorithm!i 2. ii P. Dunn, Mr. R. Chew, K. - - shwill, Willsky, ''r. S.D. Col Dr .

F'rof. A.S.

'tom rIC Si<Jn cl• • '.•!(: t icn .,f l of fd t i'.1CC continued the `;tud R. Bacno hx c and hir.

time p> riod, Ta tlli.; iress report.

f methods, as outlin_,d in the 1Tr.ncedi.n(i pro d,C^!•:tirn c.ltl:<''. , y of a number of fail , WillskY comrie!ted a surv^ t,.- offs i.rl Ind tll•! va' i 1i.nind tl:E!ir !,trengt.1v: ar:l ou t raper or tll'.

Cie s'ir*.

metho.l j n- , , the t:F.l?

ra c d.lil.

th" ut!signs.

i t i^..tl l.y, ^; ' tt '.!t_ ,il.

itous progress ieport) has 1.^`!n ,tudied in ., - twQ—u i m^'GS LGRa1 le)ilJl 1 tis•! .;1 L; 11! :• St!1k1y, . . ':1 ; ^t:itCP.5:1 _ .;imul.ttion etocl, arl`1 a roport COn - ,p j 1'•:%':1 .,r the tL:.11:1.11 .a) I ._ - c:^7T'Y a C t , _, 11 •!

)r of t ... C' . ` ; ^^.eTr, f .....i . 1. -. 1'. ^ r L;ln!J the ;ill.11 1 In this re ort we al o '•!:,crib:!

deltection tradelolft:, etc. , i s t'ort.':cominn .

11p#!Ei.

_in+.:in e l to be E1.lv al an l in. c r m l tlat has 1, . • - •.lotaile.d anal;• tical ha,:' . -a Oi cor"'i Ct .1.`_te.ctiC)n, ics 3 to CalCli1at`3 prc::3billt on • ac'.eatjo alit'.::; This 1` .,Mail lair^ in (xtcd !

.1 of failu:_ .;, ar.d the P' q falls! 3141nt.is-o?a: : :;ii r'-Ocaircra* r- J;robl• , and 1)lit^d then techIlLquns to the h,tv,! a r %Ct:Ct10R.

•'rforn!anCe bet_:J(."R tl !S^ a sevcrjll of. the r^:l,-.vunt tradeoffs det` in tho co: i.nfl . . , developmont of tl.esce mcthokls will b a don rurt l we r erar titers.

, tha ^llowillg issues are to be con ,idernd in the r,ionths. Sper-ificall.

ir.,edLate future: of Failures Dct—t' Lon tit!:t;le'..., f")r i;E: ; l(Jll Sur'p2y of p,-!: Ames Workshop on Sy:;t+-!m i.i cA " hresentod at the M.I.T.- t i tits ' I: sues for future Aircraft,"'.ugust 1`175; to l.e suhnitted for 1 r'^li tion.

Of the d,--te:t%;r vi.x Sensiti%- itf :.:c. t. • tinztiun of th e (1) rel•-.-ant acrd coeMcivnt:;.

1 1 a ror:; in the aircraft wit-.

of the F-8C dttection sy.item and a :;tuly of its (^) Gt_vt!lopm,:nt of a sin,i)lifi ?

atiOIIS.

, ince via analyL;is ani simul t-orforin of "ail.ttr.e sx)(1e5 ;oprtent of detection r•(:tl:o<ls for other (3) th:n those presently being explored.

t i e " ' • inl ` t i c , ir FL ' l rs 1t1Y;I` p tt 1 in t.!'(' ;lAt •c u , ; r l' .:1 an(l Mr. ..T.

he work of Prof. A.S.

T "ill ten up in :several Paper ; (.:t,c tht!

• .rt hau b2c-n we ,)u r, research relx :':). 111-x ' rl the -tre y 4 f r.r'n1 t)I' this rot tr . i:l cod r tnrc:; at' t:1C y • vinq lines: r.as c<,Ilti:tu<<c' along the foll chast; ': !

;`t 1I°.^i noell.ioar exter- tun Ul t t .t are "t'.' l'( .!;t 2 it ing :ho We i • - l to a an(l W'Ll! A/ t n,::tlic; . it her ^.l- ..

rl^. iL^rS f-r Ot Oht.. 1 .. .. `•_ ^. 1 '' system:;.

, ,-IO based dt_.:l ,:ed ,^ .,.1:;^ri ir411 ._..tir'Zt • ion tcchni Ti hi-; tl e::t 2 , ...'cus boinq aphlicd to design • ; • . s. :''.;is t;.c:l:ni it:e i on :1.1-mcnic .tr.ai .j.;t:j.at:ors for attitude e_'timation system:.

t!.rt "ll-Irr..cnic an it/si.s banofl are f .: I .^.q .dill ;'<y and 'Ic. V. Kle 'ternllNe C, T` 3 'r t7 V Ly 1dL11;':y Lin , !lt )rcus d. ?Velo,`&I lytical measure; of their l-crforr..anc of C,c roblem ., rr.o are cone; y and 11 c. J. rt i • rof. -Illsk - here is .I( lm dul ition of t , i ,:usoidal t..

opt i.

f :^:: z d i'iil lsk%.

;cries ..Y'.

from t, -.e F ourier quite dif.fetent

'Ehi:i h."! a; 1 • YOaIC}1 iRb'U1'vt • S ti'. it f cun":ide lal::t,rl of t he 50—C. ► }lta-1 "r•^t-resentaltion theorem" exr-rr.c;:;ion for the cunrlitional proLab lity density. We are presently 0oriving at-proximate exl:ressior.:; for the exlrctations needed to evaluate the cien ;Ity.

This approach r^ - ty preve to tie oxtrer^ely useful in d ride variety of nonlinear estimation applications.

5. 11:;in r 1 the sar-! Volterra r• !'

a ; t,ar. j!%pl()rr,1 uy M,lrcu ProF. :: 11s y i.; titr , lyinq ti: !

of !h.ick ry :' :.rs cunt-i4ni.r:+; randortl.y varfintl T;- _ Pola ys rr(.f.

WillAy and Mr. A.

^rtak: n a st:cat :y of :!:<^ c r •• , n or ,: t-cr . .

:atoi s th•tt i., '_.ire O valays. .OrlSider the li .ir t7 •n ;t) :;t) . ru(t) (t) (t) _ (7M (t) ) 1.,aY1`itraly. tC WQ 1+ ? tri • :t Outs'!', 5 t0 i:.:eir r .ct't..ick ia:J.; u( t) _ , :(t) (?)

the Le:;t we can }:ape for : •; jr. r:,; m^•ntial decay of the output. E{o::•^ver, if ::c allow time delays in t-he c•)n: r -1 u(t) = K 1 x(t) + 1 , 2 x(t- , r ) ( a) ono can )ften t;ri .t the output to Tern (:n9 4•'Ap it '::st -e) in finite timo.

In hi; 3, has^or.^.'_' to A develo;rren*. of s;lch a des: ,.l in n and ...•,.ly;is of t'onlincar stcc` i:;ti 0: ^ctri(:.tl Fncuincering ind Computer Sci•_ ty 1

.in.l ro t •t^•ri;ini: 1 1.

I I. i•a • i , k a ; •,>cc I k ^ i ;.111 t.asO j:l f th ' . I- ..;ic least cuatroll .z` ', ; , , i ^: of dimension a, ;i (^, ) : l in a brief :oirrvary haler.

rt orted i.; ults of this 3tuc?y -.truly are: this Tht^ next stop:; in i.e.

of "deackteat ua^erv• rs" -- p:-1-1 0m The consider-li en of the dual (1) of the form synt-m rlt-sign a (A,C) is ul,:;cr.vak)l,., asr•u:*:intl ^^^.-i.)) (^) kt) - K 2 - (y(t

"1(:(t)-^f(ti)

i.`(t) + nu(t) 4 th•1t r.he e: -irit:.cn or-or ,.,^ ^,• , ., - ir.it•, +-ire.

nits to th•^ nultild r- c•.:r_l,rtt CO .r.

( 1j l vut:.:t.

d •, r., dc -I' l Corn !^) 11-rrs ( !) If C' ,. ' ult) = K Ix01 + 1c 2 (t-h) time!.

i system outrt!t is null. '• i:i *i.: }_l -;^ . bo . - • e study inherent tir.)

:,*,.;tc.:)S 1`.3'.

il 1 • h - Ac - act` rany =rrs l;.d in uur ALili*.y t " t'. - • • ,. indi:. ^ ::,::..

our overall iicl in for tharn.

;c effective con` :-ul [: f ?l.ty :. ^ ` :1 ^: Cril$,M .^- .t.. ..:1 v 3 t r ,- ,lc Str t 2ctu , a t .

sc iertce, M. T.

^riru; aia•3 Cor E::: _1^_:r ical ,^• • Tk: si^, Dept. of pt. 1075.

Delay Via ,)r Control Zero .:rr ( "Finite- %IIMP ll'^n., , p • ,,1 r1.ti. k l :ELL Conf. nn D4:cisiun and ^ontrol, 1'.ousron., DoI c. 1'375.

E . ," 1!75

-11- 5.

Ci ;ita' c'orni >:^•, ,tc,r fk^^.Yn Prof, Will-;k-,- and Mr. G.K. Poberts hive ce,n?ALnec.d th.ir efforts in atte:mptinei to n l t;r and the fundamental limitat.ic.n of Ugi o, _e2 ii+tFac:^.^ •a- . .,onLrol :;y.:toms (:ec.

tat ion of the preecedlnq progres- report) . 1110 basic m,tivation for this study i:; to take a firat stele in dev.-lnnina1 a control rl^ :ie;n metho•:olo(jy that diLoctl y _rxorpor:ates comi.uter tradaa'_`fs. Succit i- cslly, ue have been coneia•le_ric. q ;.he design of digital cc:ntrollers for lin..ar ,n qu,,•lrati.c p:r. f^rmarce ncasures. i 71r stu :y 1 included c:onsLraint of a Finite ti.n'! to pQ,.'•,tm a t: ,l al':i1,I ie::at n, •onct ntL,atOd on fir.. i • u1 'cw- ur ler exan•7Ies for Jr,pler, t., imat" control la::s e:utperform the slower r ?" -,troal, r t r'uch slower sa mpl inq rat, • . The _ :=int..-il • .zi ; study a'o[.

u c. -,oncovt:, 7, ' CL'nC or L c ile? v} }'. , arge — l wir ini ► :ial i_' 7 will be described in detail in ;Ir. i Aber ^' is.

• cu : n ;,° S'.ejnals f. Mule? GL' t,.

:ed r V i i _la dCIJ(r<l SOUS. nilr!':est i -1 i.'; Y(t) _ 'i l (:.(L)) + '12tx(L--C)) + N(t) . 1 •aa t: i ,,i taz1 to c^ recovered, N is r0ceiv y r rc, s and t is d ti.

,;G :ay -- ••i the e- k:aown or t,:akno•.%n.

Prof. Willsky any'. Mr. ?. 1' , -,g have :tuclio l ;ti^7atic .: of x given such a rec.!lvcd s,"gr...-il (1ct .7:11 a inure q •'n•!ral one ,: .ag t:v! possibility of distributed Bela; .) ..:.en Oeelaj is k::,)wn, u7d the

-i2-

:ear case

A ;c(t—T + W(L) (3)

v W C

xit-T

iwt

-inii*ty of the ill Of (3).

(1; C^^ I (Itaity in C t r

i_ 3 - 7. !.o: -^).-^^---r—r .c n . ttor Cosign J.

The efforts of Prof. 'r.r,. JrAinf;on, Mr. F. Tor r :, tir. T .1cren:', am!

' .x.

MntaZedi cun%inue to ::ocus cm thf, design o^ lo g ::-order dyn ~,i.c com,.,en- sittcjrs for t'iultivariable 1-near a%;terrti. The interplay 1,etw " g in for:uilZt:inn of the o; tilaal cntrol pro!-Alon, existence of solutions, and .Aructtlr , 2 of r Vie optitaaL cont ol la:^ is the focusi of_ thi s research.

In a forthco-dru: I:"L, T( ?ch,:ical "cr.orandut ,.e .All presort t!1e solution *-ho reduced-o (lor (:u >cd ei • til ^al cr:-tpen:,ator design pr c ,hl^^m for stanu.l y d ruaclriltic t?t'`t;r"P.?t •? lnc l ox. Reduced-order ob:,orver-based C n ':Ors, (i.e., tho;c I )aL.i;d on 1 Tm(_nbEr,J3r ob.`:t'1v2r : • lilich estimates , , tr corbinati , :n, of .;tare ; r"liirtA ,ter t!1' ot ti%:!l input .;1 ;nals) n !, i by t 'ra: ^h and Pearson, .'czL.-i, rx. a.vl li11i ;c,n, Pa t hs ,hill, J OCi'.^ rs i'! ^.1''iimal tl] ^., ;iC:? of ;;ucli compensators is =t7 , )f (-1) ".hc: GI ., .' Ji.litJ ,.:.(it^x Of (A,C) MiMl:i 011-2, i - :.U'_h.-r i r i put.; . , n order of plant, and m= o_ - - c . r or,'. , , - l ir._:1r comp,.:.. -ttors ::hick stabilize t'r. t.laut y c ::..: b.:ieed on ^rvc_r thc_ory. The al)ove role:-E:_ It men" apprr.:cil a; c;.l:.^,ed to :?intersizing li? 'tic 'om...nce tnlex. We , a that the of ` l mal s t,;,rll(,t,i re sh(; .. n r :.ic:1 estiaa0.(_3 %'Fx) :!:ere F are the Kalman r • u :1. , :, .anal Clive in explicit t:rr. , '..(: _ `c.- Pr.alc , ;_ts r ,ssul'_.. :;ere proved by ^) 6 :1')73) 7 _ t` _ (n oiler obs,ervor.

vid(' ;vin rwthrc]3 fcr a.v/ ord_r tl.'n[1 to

-III- w,:! have u:,,A initial- ;tate iw( 2ra•.lin,j over a ran-.!(;n initi.il h1 % 1Ctt st".ato to det.•ruiLne t?: , ? opt i-1 i initial sta g , of such cot 1 t_n::ators. 'Aii • , methc(l i:; and the opt_iral comt:ensat:or dynani(:.; in fact not e_ntir,,ly s3tisf"a(.toly, laar-le covari.:nce del-end on the initial !:tat_e cov.aria vice in this case. • T.

i mpl ies fast,.?r uptiral dynam ics. T acre appear'; to 1%c? an alternative, and Iiat more appealix; i:'t( ?rh ret.ttiG•1 of our necessary con litionS; all somev. , and the st o cll,'.;tic eCil11V.11, I1 e relation i'" t '•JCCII the ranc1G11 Lail:lal state cCm..p-_-nsator.

t;t a ?y .it l ` : L' l rL^l)ll ;`.i can he (.:;tnhli-flho i to y t , 1t' (I:-al) or(:• ,r 4.0 al.i:`':t.'.0 CIC7 SIIM', plant and come-en- ` .'a the St: )chastic ste ady slat(? l:robl' ..1 ltor :;t:i_ucWr^, except th it t;- >lal:t is st r iven by •.t ran:lori (white noise) l we !;,:ek to min j :Az,^ the q c er tatinn :knit :nd i i11 ,,tic ;'.y r:t: of t':e into <jr,in(: of t:ht? L.;Ual l .?ra _..t; 1`C,r ft....:ultCe illdIX, :vhiC h i:, tilr.e- i t ions are %1.= ,,L p ed. "el:-- [ally, the rol : ? of the i. r.V,tr L. nt. Perfect ob.;erV • p layed Lh lntetl`:l.ty of .'As iww !l1tial Sta r o cov arl :;,ce i.:3 1)y , ! 1tL •'.lt nui::(? : ...J1C.^.1.

t Tht? pro: , ! of .,'tl`; tl!'i•.y ana.ly j is xid '_inplt:;. .. :.iLi3a of such C__iigns ,ire still unit?C i[1'lest'1gatio n . int^_id to titiv :op a r:?'.1c,C(1-C)Y(?,'r d(2si(Jn i !low).

for a i:-.tlistic ai.:( • raft r^c(!,1 (set-, SOC'e 111ti:r,r. 1 3 h<.vo claii:,,.l th_iL ub.;ervez-based desicins zi,^ s -I.:i.tive to modellinJ error and a.l.,o ao ,,corl- , , in StO-C. :,t,-:te error'; ill the 7- of Lnc:ll2,lin,? fonjard -1!) n p i.nLegrat'ioll in a iacc dti-iturba.'nces. Thcy !at?`er ad-hoc manner o:ilich nimics conventional vet.hods. We ; vLty may be a pr-bl,.tn, but expect that Un tha absence rant that model `'t3n5;1t i n,tl Control of t?-.e 1,-. . _'c.1 ^ ;t21:t`.Gr ',i.;inq it • 'r, .A., "Spocl • .lc Cl tl 11-:- nle: Observer," Irt. 3. Control, 1)73, .,Ol. Ji, 2L^.1, i .p. 139-179.

t'•' • llain, P., "op t1:^al Com .. ator. Structure :oL Line:(r 'Tiin,?- In variant 7..'!

, .: Report F. may 1974.

i t-% inacco.:,; i ble mate:-", r,, L

-15 •- !-( should , r cns iCn cDor-t—atol: lim ltc l i!

tiff? f;trl:ctilre of e,.o of n i c lul he sufficiently q,!ner11 to allr,-.v for fo^ vard loop integzation to achic%a zero steady state error, when requircd by the quadr .ltic: 1),^rformlancc irl,' x.

of the problem formulation.

g neralization -1 ` ; i t jilt demon5trat ion r eq,1 4 re:; l Thi3 atcr. ('e are con::idcri.nq such , above to allow exog(nmus inl:uts to the compen.

problems.

; :;, or tht' free continue to ?.nve;i Igat_e tllt. ,'fLPt ,-,ti.on of uniquenc' I `: We <'W;.ltOr.

etcr:3 of th_ cptim_:1 comJ; gn parar • nr ^^^• •4 - ^^ ^' I :^i..^,itinn T',•-; Contr,^l.l^.r F• fl.

f^,1:; b i l ity of >t.i•;at_, file :,:nt th1 sis by 1'. I,.tx we h,:v in , In a r., foilo%.iinq , c o2.( :;lIi,':i ..''. us tr.li •,•incl control. lase t0 ..Jill • do!-, r j a tixr -- v '. Y Lgrllrl of 3 l : th'-! ,^eE,',`Ilt ^;ll.;;e C'IIlway` Plal: ^e: arrlv11 ti77 .

.ICCIIratt? c""`:rot OP a^ l f 11e ;'or 7-i.r.^.<}ri.':i,l"I th y, r.c...inF•ar air ,--LL,' 1i r 3 ),lt. 7h:3 is L',o "i^m CC1rrJ.ed Ol:t in a wa}' COIm L; t.erit w ii:h tile' }::lOP:I1, bums i (,' .! i", :rel.l- 'ha'e cf fl.i, ht is u:;ually tvl-liti •n, 1e;J; .rl ,.:lil.osoEhy". III 'r si. ,nincJ tluf3 Lire-varyin,) I.lc i by gai:1 ^-,che<lulinrj models, rat!'—r than , :;ava (-arricd cut all these orr. ;.',.

imA trains as csp,u.;f; i b, "tile" ti: Louing things in a consistent t.• ..nn„r for tha 1cncjitu.1i.:.11 c'.yl,amics or the was , rol ter time-vas, :ng 1 ; i,o trizcd con V t3 proach to rog • i.n Airnort .

:III 1 l , i-llot-tested sirtulation ar- i on ti - .c nonlivi'ar hcrR".orc 1[r., ler.•.ntc'd mean-wi.,:d conditi,.,:,s cn +:,,.! ESL cockpit sirrllab-')r. The rrisults ,:r v:lri.OUS forth,.: ing LSL Report.

•-rte quite encc : Ir tging an(i will t., r,.r :r..1d in a •^ r (1 l t rt^'i I r-, ) iI:C^ 1 .'11 1 1 Y:: I t ^l ^'nt ,lO: .i Inf1nl = ^ 1 ' - .atv•; 1, :t1 wit i C t The y.o recently has bcen ccnsiuerablt; progress in n-,n-linear programminq, 'I the n .=tical and cort,,,u tat i-nal. It is rc-co , Iniz ,-3 that dt!ality plays a

-1G- c^ntral role and (1u.il ^t.taoe;s I .iy turn cut t , . he the rv):-,t ii.,.por t ..hnt coml.ut.a-• ti r Thor u ,e of mally. augmented L vjv r in r lian5 (1) is c:xi;ected to play it key role.

Oren loop o,>ti.mal control problems are i..cl mathom.atic.il l:rograma n ,ing problems. liowov,ir, they have , con,;idcral,l , ^ structure (n. , . 11, t:}:(' time structure) t,ust be exploiter:. Vv' • : ; nq them as infinite dimen:;ional mathemltical Frogra r "- , `ln r T prnblc.rt_; hii.; Lhe3 a.1vanLa i . . one can I- ocus on issues which I";. , Pub not: he evidoiit ott.-.Lwi.su. iL hAs loi:q hi- , ,2n felt that a of m e'.1:. i3tical. progr,-iI:.iny an l ortimal control t•:ould be :a im,,ortc^nt dev(.l- of ^.;en i'i:rtharr:icre, Lt :l.so Laen felt that Ole so-c :tte;? direct r^..tihcdc (Rlylr'ic;il - Itit':), wh,'. .^^: t)e'c.[1 largely P.•_•r'_^Utc 1 in compuLitional work For optimal 'Oat'YO1 i%rc , .)Ie Should allot: ur; t-) ;e matiic:r:atir • :1 i r--;ca7.^inq in m oHic:ier.l' way (t'mt ic, the tir • ^2 ;truc:ture).

Ii, r.c c..rit. work ( [2)-(5) ) , .;e] f . 1 wo have t-ikt , n rinportal.t sLep:; in i.s cli.recticn. ?ior` l reci -tAy, 4,c: have gi • :,,n a reasonably Iu.aliLy ,ory ar c:^nvex of ti:z_.1 , on±-ro1 1> ,blems and also sho.;n i:c;.^ the rh_i.il rrc^blem n Lt. e i f i ci _,ntly using the i:it2 methcrl.

in , )rk is - —w progressing .several litections: w,, similar i ie zis ar<: the fc llowin , i linear problem: : imize (cc'::(t) + ))dt cIII U O mubject to Att):c(t) + B(t)u(t) a (L) 0 , K x(t) < 0.

J^

-1.7 t;F:e condition and with fev;i ALity .1 :'1 , ,L, Conj_:cturo t'...t wit;.,-)ut Lhis lE,.;tllt rah'_('Ins that: t1wro is no duality gap.

and dual F , .

of t:h( Ctnd :ill have great comhu- %'] till ' t nc..cll e all earlier re::lllt of L(zvin;on ( le, in Dyn_lllic rise, f^^r cxzmt ) blems of this; type a , cancc. P rc tati.cr.11 si,nifi .

network:,.

1ull-ic , j ti "-' of c0r. T cu!7tain problems 10(lols 01 - -A in Leontief ^ 'tht(ls j finite el_ :,ent.

dual 1 cbleln usin c . atiol, Of I:) After discreti: the ' l!i y:: d rat is I n t t , rocJ rc.t iinQ math- .' l t:i.cal u l th a wo 1 ro l.: ft howover ! ^.. T* 3 { . j=' trS : lncj I ) r'J}.l,' lli.th d j'_l:. r.'"1t JC rro(trztl_Ir C_I; ' a re 1": '_ ? :ae '101Veti u SiI19 not problems ':C.:11 be, c proc,rai,Tciinq c'uadraL !

that these ()f the m .O t i:,^ola- t' A(' ..`'1:':tl1rQ are lnvc , :;tLgatlnq t.. leot y. j :'-' col1.I'le!

; 2 _'iC-i ".t r'_ ..1111 tal-in .

, :;l to c - ^dl l Vi r^:'tL^t i.11I.7 [)!'Oi l _ ti. , :al `or t i!t; Lr ::olll' _'. t110C1`i) I r 1:•3' l.t.iCIl (E.c 2' l,tt <: c +:CO, I 11ity ;dent l T, •l,r l-,%, Multi-lier Fur.c' `)I1s and l. . 'tar, j i I) r?).2t'8-285.

12 (1 ^,;tl( , ° ST:. I J. on Jt --it ions to Di'::crete Ar , rc -ti: nc . _' .

Conver go ► , ;t , "Rates of ► rtm.nt:, Ltathcratics G • :L , 1 i:.D. ;iusi: , Theor y," Control 1974.

u:u1 Convex ,:ality Th , !orl for D , r, "Lagrar.

Hagar am l . S.

Control.

. I. J. o n ear in L,Ii' ol Pro1I 11-us," to Cont- , A and Control. Con.:' rained i for ` t :' '_t" Metho , "The Fit T e ` A .mllysi n.

caner 't - 11 LIl ^1•.:! J. on '. ' appe ar 'O1 I'r(.I,'. .," t0 • i ,)ns to Y '_r_ e1 ; rn::im t _. - r Dic. • "Fatt^3 of Con%-crgence 1 J.

t:.

l L . :1 S ..':J. on ", to appe ,ir jai.• t ^^l ,:ot. .ol Fro' . , t t . L J .

,-.x Control I'r r )'Aclls of -raints jr. cc:.. , ".'3tato Con .'r. rockafellar, , lt) (.L912) , pp. a {1'7l5• on t^ =gin • rn l SIA!' J.

l...l," T rogra*^inq i r^.t)' '^ nuom-^ ? L I.,rvin.7nn, "A Class of Cor:ti 7.

pp.

1^^, (lc)Gt), 13- ipr'icat tc^, ^, ?rt?i^'sl:; and )'. 1 *.h,

-18- C" ^^" CRfSatOY lh: ^i(l11 inv: stig:ltinq hav,- keen ,1. F;thans •, and Pro} .

Mr. M. S.l:,ul(

Sy,-t• ;7 ar,d Control of for deci(3n l ,1:;E ,( ct:; t under large par,unet:cr variations.

Titer der;i^lns that ar e ` stoms,

of dyIlor l! cal sy

;n till de:-.Ion Of comFem;at ,, .;; for the -:().•.troI it is common civi i n.eering i rac't.i c:e to intrc)(Iu :a the as.,Lmj)t Lon that the .ynamical system to he c ontroll.-id c,m be lcl-(jaately P.rdeled by a fihite :,(2t , ; ac:;urri)ti(•n, thn of ordinary linear di' - ferenti,ll e tuitions. t'ndor CA easy, synth(-, sis^ a linear :3tetc-fe(:d *1.-1 "k Com- engineer c;L:,n, :rith w:li.Ch wUl m -nimi7e a quadr-ltic per` , ^rm, n--o ir:.ic:<.

+-.ht , r.oc' :1

.>a'mr foc

proce(lure have

3 ;1 i'r.c'.uced of c _ a:en..a*.or. 1 '.:p is ?c.,i

_ a11y ::nu:n as linear.--- ,u,ldratic- (I.Q) controllers. 'tile research

1)c (3 F .`i!. L.'.l.'

'.+111 bk

iy .... ch J.t

'ria

cNlt it; .11"Cd .lt d-t,^LinIning c ' _ 3rrl..

t^` yL('1'1 can i.0 e:{I'tC

prodecure

:° ;1 nriovi. wti`Al l,^r tt11', (?!:,':lgl1 t) )1.;d ' 5 7Stc m <1Y (1 ; 'rv:^ ":.Ca an y F : ^' :Ytl1i '. , :s fc)" Cie validity of chi, on 11 .=s reportt, <_1 in t;:e lit , ratur.e to (':-.te co: ,cen 11r,:.lr l i:i. ted to a pcst'^riu •:=i tests i ro ^ll.lr(^ helve f :rj t.(1 1lli:'_;'1tf Sl r. i F:Ci Lurbat i.^ns in th'3 Lvlty O1' t'.^ contrul!i1'•1 ^7 Yst little is known

ry

Hc; fi ver, doCi-^-r 11)inq thn sysL-?'^..

s parar it:>rs

` •aricu of L'1 contro'_lt)r:;. GI notable: :ti`:g el' about the validity of I-h^, Lc;e

,t aat Lt:hive to :r. '.:; have bA(^:1 found for th-^!

n i ; t' .a i T` _^)ri

Of S1C(Jle-LnF it^;1ng1c.-outlalt llr:^'dY ';f C3::;

F^ila- <3rcYin

:F".i ^^.1Lr1

F.hase margin-) tiro j71n C1:1Y'^iil (!,tit IV)`.

'.h" PCCCnt'.'I , ]1 1:Q COnt''(J1iNrS.

to include.

by P.K. Uor',q found for Ln con,:roll

-lq- multi-loot, linear feedback sy_,tems.

W- , are presently c.xamininq analytical to_chniques which are expected to make possible more widely applicable quantitate statements about the validity of the use of LQ controllers. Currently, efforts are being directed to the generalizaticn of the aforemrntione notationso` a priori gain and phase margin to LQ controllers applies to nonlinear systems with multiple feedback loops.

The immediate objectives are to (1) provide a theoretical foundation for the empirical fact that LQ controllers tend to be insensitive to modelin,7 error, (2) provide en(lineers with means for determining q,:-.ntitative ly the magnitud(. of me<ieling error which can be tolerated before it is necessary to cuestion the reliability of a system with an LQ controll_r, and (3) Ponnit better understanding of how such effects as phase shift, tire-inlay, and saturatic;n can Lo Pxpu,:ted to affect the performance of multi-loop LQ control systems.

Prelimina rl resu!.-s indicate that a quantitate bound on the arour.t of nonlinearity and phase shift which can be tolerated in the feedback loops determined a priori for entire class cf systems with LQ controllers cin he of systems with LQ controllers.

also provide valuable insight, The results for robust Ln controllers can the design of Kalman filters for which the covariancQ of the fake design variable. In the long r l :n we noise can play the role of a hope Lo obtain a unified quantitative design procedure for the ovcrall

-20- LQG robust design concept.

Faculty M. Athans Research Assistants P. K. Won(; M.r. Safonov On the Interaction Structure of Feedback Control P o forence: P.K. Wong " Systems", S.M. Thesis, M.I.T., August 1975; also reput ESL-R-625, September 1975.

, incv rguation 11. Numerical ':ethrAs for the I •.^ p The importance of the algebraic Lyapunov equation in l:roblems of - invariant linear systems is steady state estimation and control of time well - known. Mr. T. Athay an3 Prof. N.R. Sandell Jr. have initiated a research effort aim-2d at development and careful numerical analysis of algorithms for Lyzpunov equation sulution. The research will be documented ,thay [1]. The following in the forthcoming Engineer ' s Theis of 7.

preliminary results have been obtained.

The notion of conditioning is basic to nu:.erical analy s is. Ile have f conditieninq for the Ly a punov equation. We obtained a precise notion o have been able to demonstrate that the often hoard statement that systems conditioned Lyapunov equations is with widely spread eigenvalues have ill - correct in a precise sense, and moreover is a property of the equation and not the solutio n technique.

g Lyai urov equation solutio.,i cannot Althou h different algorithms for affect the hrobl-n n' s inherent con ,iitionin(;, they certainly will have different

-21- orror pro! ertie : an,! operation counts. We have analyze-.!. a nur •ber of the mor` popular alclorithms with respect to these prr.,pertivs.

Finally, motivated by the blocr. coupled structures commonly found in large scale engineering systems, we have developed certain a1(3orithms that explicitly exploit these structure. Analy: • is and numerical test of there algorit;;m i.-, presently underway.

eferences: P 1. T. At :ay, Numerical Analysis of the Iyapunov Fquation with Applicaticn to Interconnected Power Systems.

12. if- rturbation Methods in Lintvir Filtorin , T and Control Large scale aerospace and other engineering syster.s are characterized by interactions of smaller subsystems (1). These interactions may be 2o that an al--proxi:^ate analysis of the decoupled subsystems may be weak eml-Ioyed, or they may be so strong that cortain subsyster.is can be aggregated into an 1:qtrivalent suicsystr—i with fewer state variable:;. The mathematical tools of herturbatior. theory are an appropr.iatc means of analysis for such situations.

Mr. D. Teneketzis ana Prof. N.R. Sandell Jr. have initiated research aimed at extending the existing tt:eory of perturbation mothods for deter- ministic optimal control problems to stochastic filtering and control problems. The research will be document^d in the forthcoming (-taster's '.e sis of D. Tenekctzis [2). The folLc-dLng results have been obtained: For the weak couplin cz I.roblem, the analysis proceeds via regular (non- sin,7ular) E !rturbations. Conditions have been obtained for the stability of filters designed for the perturbed (deccupled) subsys t ems but applied to

-22-

ti-,e actual, coupled system and computable bound, have been obtained for the

resulting degradation in performance. Analojous results have been obtained for the stochastic control problem.

For the strong coupling problem, the analysis proceeds via singular perturbation theory. An interesting and nonintuitive result characterizing the behavior of the optimal filter as the coupling becomes infinitely strong has been obtained, an! the analysis has been extended to systems with three levels of couplino.

N.R. Far.anll Jr., "Information Flow in Decentralized Systems," [11 Con`er••r.c(-: on Di,• t-ctions in I- ntraliznd Control, Min .-Person s, -o)t. 197^, Ca:^bridgc, :?ass.

n, ,',7 in i rn ar.; er -`'-a1 Control, M.S. Thesis, 21 D. Toneketzis, Pert u rbation. ^thods in CtochaFtic forthcoming.

-23- x E. I':ANC T AL STATUS i As of December 1, 190, approximately 30e of the grant funds ($70,000) were exFanded corresponding to 25% of the grant period.

No consideration for cost sharing is included in the above figures.

-24-- F. F1 74T. T CAT I ONS The publications below describe research reported in this or previous ress re} orts and supported all or in part by NASI. grant T:GL-22-0,)9-124.

g pro The references cited represent new :material generated or I-ublished since the la:=t progress i port.

Copies have been transmitted ro the NASA Technical Information Center and to Mr. Brian F. Doolin at NASA/.'1ITS Research Center.

l er:; Vrl3 r the NA:'A Grant Fal "n the Stochastic Stability A.S. %•::: lsky, F.I. Marcus, D.N. Marrs • , " 1.

r ? " .1til licative Noise", 'FrE Tr.in-;.

of Lin -ar Svstc-s Cor.tainine Colore Control, Vol. AC-20, Oce. 1975, pp. 711-713.

Aut.-natic on enSional S.T. Marcus and A.S. t:illsky, "AlmFhriic Structure and Finite Dits 2.

.1<: E'tiir^ll.^. .`^_^•:-t '?_i^._ T he ory, Fro^. C:."ItO S 1 U1:, T Nnn1 1 near F.^+t `_!' :tlo:i," Udine, Ital; • , June 1975.

Surv•_-y of P-:isgn Methods for the Detection of Failures 3. A.S.

Ames Workshop on Systt:m in 1-ynamic Syst, r r:," s resented at the ' .I.T.- N I ASA liability Issues for Future Aircraft, ,ugu5t 1975; to be suLmitted for fi( , "liblication.

Systems, Nonlinear "Estimation for Certain 4. S.I. Marcus and A.S.

Sominaires IRIA, Focgaencourt, France.

lctural Properties end Estimation. of Delay System:,," "Str , 5. R.H.-S. Kwong, Thesis, P.ept. of Elec. Fnq. and Corp. Sci., M.I.T., Sept. 1975.

I:..D.

7.,_ro Error Control for Linear Time, E . R.H.-S. Fwcz,. and A.S. 1-.illsky, "Finite- Continuous Time Systems Via Delay Fc^dback," 1975 IEEE Coll:. on Decision and Control, fiouston, Texas, Dec. 10-12, 1975.

Ar.plied Self-Test A.S. aillsky, J.J. Deyst, and B.S. CraiJord, "T,;o 7.

V01. 12, No. 7, I F r`ckf-t S, 7m, " .T, c . 1C r, - ra " t ^.'

y stcln : : o bl to an Triortlal S I .Tuly 1975, pp. 434-437.

trol C ans ,;ian Con John:. on, "i+ I.:^." dT-!^U.l't^ ^!

L.Y. Platz:-.ar. •an1 8.

ons-T'ecc:throuyii :'olution", 3ul,nitted to ILEF. Trans.

Prcblori with I::n 1)75.

Auto. Contr.,

-25- sign vid Simulation of a Des-cent rontroller for Strate-rie 9. F.M. !.ax, "!).- , %!. Thesis, Sept., 1975.

4-Disrensionai Aircraft Havi(;ation", .L. Johnson, "Roduced-order Observer Based Compuiisato^r:s", K. Tone and T 10.

ESL Technical Memorandum, to appear.

F.K. Won<1, "On the ?r.teraction Structure of Feedback Control Systems," 11.

Auqust 1975; also Report EEL-R-625, Sept. 1975.

S.M. Thesis, ".

12. T. Athay, Numerical Analysis of the Lyapunov Equation with Application to Interconnected Power Systems.

Dccentral ized Sy: tc;`s, ":^uf.

%.R. Sandell, Jr., "Tnforration Flow in 13.

on Directions in Decentralize,! Control, !:any-Person Cptimization, and stems, Sept. 1975, Carrb., Mass.

Large-Scale S y M.S. Thesis F rt .irl,atior. Metho,?s in Stocha?tic Control, 1-;. D. T-rneketzis, forthcominq.

'I".Zeory for Convex Control Ha—r ,ind S.K. 'Sitter, "Lagranq^ Duality 15. :.::.

SIA". ? on control.

Problem_ ," appear in 16. S.K. Mit • -, draft of nono ,traph; Part I, ESL-P-602, "Rnpresontation and Qualitat:ve Properties of infinite Diirensional Lin%ar Systems," June 1915.

11 •timal Control T.I, EEL-P-693, l T . S.K. Mitter, draft of morooraph, Part of Linear System.3 with a cuadratic cost Criterion," .:ui.e 1975.

1S. S.K. 'Sitter, draft of monograph, Part III, ESL-P-604, "Applications," June 1975.