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NASA Technical Paper 1118

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Eigenvalue/Eigenvector

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Assignment

S. Srinathkumar

PEB~UARY 978

NASA


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TECH LIBRARY KAFB,NWIllllllH1lullll11111illIlllIIIIll0134489NASA Technical Paper 1118

Eigenvalue/Eigenvector AssignmentUsing Output Feedback

S. SrinathkumarLangley Research CenterHampton, Virginia

National Aeronauticsand Space AdministrationScientific and TechnicalInformation Office1978


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SUMMARYThe problem of eigen value a ssignment i n a l i ne a r ti m e- i nvar i an t sy st emusing output feedback i s cons idered . Ne w s u f f i c i e n t c o n d i t i o n s a re de r i ved t o

a s s i g n a n a l m o s t a r b i t r a r y s e t of minimum (n,m + r - 1) d i s t i n c t e ig e nv a lu e swhere n, m , and r are the number of s t a t e s , i npu t s , and ou t pu t s , r e spec -t i v e l y . These c o n di t io n s p r e c i s e l y i d e n t i f y , t h e c la ss of systems where such anassignment i s imposs ib le . The syn the s i s t echnique a l s o h i gh l ig h t s the freedomi n se l ec t i on o f c lo sed - loop e i genvec t o r s under ou t pu t feedback . The u t i l i t y o fe igenva lue /e igenvec tor ass ignment in t r an s i en t response shap ing i s i l l u s t r a t e dby the design of a c o n t r o ll e r f o r t h e l a t e r a l dynamics of an a i r c r a f t .INTRODUCTION

Control system design based on eigenvalue o r pole ass ignment has rece iveda g r e a t d e al o f a t t e n t i o n i n t h e l i t e r a t u r e . I t i s w e l l known that for a con-t r o l l a b l e s ys te m, i f s t a t e v a r i a b l e f ee db ac k i s employed, a l l e igenva lues can b eassigned ( r e f . 1 ) . Also i t i s known t h a t fo r mul t i - input sys tems, the feedbackl a w as s i gn i ng a given s e t of e igenva lues i s no t uni que and t ha t d i f f e re n t con-t r o l laws can y i e l d i d e n t i c a l e ig e n va l u es w h il e y i e l d i n g r a d i c a l l y d i f f e r e n te igenvec tors . S ince the e igenvec tors de termine the in f lu ence of each e igenva lueon each s t a t e v a r i a b l e re s po n se , f a i l u r e t o use the mul t i - inpu t design freedomf u l l y may r e su l t i n undes i r ab l e mode coup li ng and o t he r poor t r an s i en t behav io r .For n-state feedback s y s t e m s , i t has been shown ( r e f . 2 ) t h a t w i t h m i n p u t s ,i n add i t i on t o t he a s si gnm ent of a l l n eigen value s, up t o m e n t r i e s i n eache igenvec tor can be a r b i t r a r i l y ass igned . However, t he prob lem of e igenva lueassignment using output feedback ins t ead of s t a t e feedback h as not yet been com-p le te ly resolv ed. The problem of determining con di t ion s under which a l l eigen-va lues of a sy s tem can be a r b i t r a r i l y as s i gned t o a system under output feedbackhas been i nves t i ga t ed i n r e fe rence s 3 and 4 . Bounds on th e number of s t a t e s , i nterms of number o f i npu t s , ou t pu t s , co n t ro l l ab i l i t y , and obse r vab i l i t y i nd i c e sare es t ab l i shed fo r comple te po l e a s s i gn ab i l i t y . R eference 5 shows tha t fo r asystem with r ou t pu t s , i f m r 2 n then th e system i s pole-ass ignable p rov idedthe feedback gain elements a re a l lowed t o be complex numbers.7 add res s th e converse problem: give n a co nt r o l l ab le , observable sys t em, howmany e igenva lues can be a r b i t r a r i l y ass igned to the syst em. In gen era l , i t i sconcluded ( r e f . 7 ) t h a t minimum (n,m + r - 1) eig env alu es can ' Ialmost1? alwaysbe ass igned t o the sys t em us ing ou tpu t feedback. The qu a l i f i c a t io n l la lmost" wasi n t roduced t o cover c l a ss es o f sys tems where such an ass ignment i s impossible .I n e f f e c t , t h e a n a l y s i s i n r e f e r en c e 7 does no t p r ec i se l y de termine the condi-t ions under which ( m + r - 1) e igenva lues cannot be as s i gned t o t he sys tem.

References 6 and

This rep or t co nsi der s the problem of d etermining the number of e igenv aluesBy formula t ing an e igenva lue /e igenve c tor ass ign-s s i g n a bl e t o a given system.ment problem, s u f f ic ie n t con di t ion s requ i re d f o r th e assignment of minimum(n,m + r - 1 ) e igenva lues are d e r iv e d . T he se c o n d i t io n s p r e c i s e l y i d e n t i f y


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t h e c lass of sys tems which can be assigned only d e igenva lues , where maximum( m , r ) 5 d < ( m + r - 1 ) .a lgo r i c hm to a s s ign ( m + r - 1 ) e igenva lue s . I n a dd i t i o n , ( r - 1) e igenvec torsca n be p a r t i a l l y a ss ig ne d w i t h , a t most, m e n t r i e s i n each v ec to r a r b i t r a r i l ychosen . In the event n > ( m + r - I ) , v a r i o u s s y n t h e s is a l t e r n a t i v e s t o s t a b i -l i z e t h e system are a l s o i n v e st i g a t ed s i n c e i n t h i s case a l l sys tem e igenvaluescannot be ass igned . The counter example of re ference 7 i s used t o demonstra tet h e u t i l i t y of t h e new s u f f i c i e n t c o n d i t i o n s i n i d e n t i f y i n g s ys te ms which cannotbe ass igned ( m + r - 1) e igenva lues . F i na l l y , the advantage of both e igenva lueand e igenvec tor a s s ignments i n re sponse shaping i s i l l u s t r a t e d by d es ig n in g ac o n t r o l l e r t o meet the l a t e r a l h an dl in g q u a l i t i e s s p e c i f i c a t i o n s f o r an a i r c r a f t .

The new formulation pe rm i t s t h e development of an

SYMBOLSValues are g i v e n i n S I and U.S. Customary Un it s. Ca lc ul at io ns were made i n

U.S. Customary Un it s.A sys tem matr ixA E Rnxn n x n r e a l matr ix A

l a t e r a l a c c e l e r a t i o n , m/sec2 ( f t / s e c 2 )aYB i npu t m a t r i xC mea s u r ement matr ixC measurement vector

ED ( k )d , i , j , k , t i n d i c e s

ma t r ix used i n equa t ion ( 4 5 )matrices def in ed by equat io n (B4)

v e c t o r d e fi n e d i n n o t a t i o n (1) i n a ppe nd ix Bmatr ix def ined by equation (9)vec tor de f ined by s t e p 3(a ) of appendix B

matr ix d e f in ed by equation (IO)vec tor de f ined by s t e p 3( a > o f a ppe ndix Bmatrices used i n equa t ion (23) and de r ived by su bs t i tu t in g equa-

t i o n ( 6 ) i n t o e q ua ti o n ( 7 )ve c to r def ined by s te p 3(a ) of appendix B( i nde x ) h o r d er i d e n t i t y m a tr i x


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j E {~(k> j i s an element of index set A ( k )K feedback m a t r i xLi matrices used i n equa t i on (23 ) and de r i ved by s ub s t i t u t i n g equa-t i o n ( 6 ) i n t o e q u at io n ( 7 )R feedback vec t or def ined by equa t io n (17)M(k) ,M(k-l ,M (m number of inputs

matr ices def ined by no ta t ion ( 6 ) i n appendi x B

max maximum v a lu em in minimum v a lu eN mat r ix def ine d by equ a t ion ( 1 3 )n number o f s t a t e sP m at r i x used i n equa t i on ( 4 6 ) and defined immediately afterwardP roll r a t e , deg/secQ(k-1) ,Q( i )Q ( 0 )9 feedback ve cto r def ined by equ at io n (18)

r number of outputsS mat r ix def ined by equa t io n ( 8 )S vec tor def ined immedia tely pr io r t o equa t ion (20)TO,*1U i npu t vec tor def ined by equa t ion (2 )V modal mat r ix (mat r ix o f e ig env ec to rs )v ( r > m at r ix u sed i n equa t i on (12 )V , V i e i g e n v e c t o r sW vec tor used i n equa tion ( 6 ) ; a p a r t i t i o n o f vX s ta te vec tor def ined by equa t io ns ( 1 )x E Rn n x 1 rea l v e c t o r x

matrices def ined by equat io n ( B 3 )matrix defined by no tat ion (3 ) i n appendi x B

t r ans fo rm a t i on matrices de f i ned where u sed

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Y outp ut vecto r defined by equat io ns (1Z ( r > z ( t ) matrices c o n ta i n in g t h e v e c t o r s z i ; i = 1 t o r and 1 t o t ,

r e s p e c t i v e l yz vec t o r u sed i n equa t i on ( 6 ) ; a p a r t i t i o n o f vB s i d e s l i p , d egA ( 1 , t ( 1 , r ( 1 , ( k)&a a i l e r o n a n g u la r d e f l e c t i o n , deg&k& r rudde r angu l a r de f l ec t i on , deg&Xk,&Zk

se t of i nd i ce s de f i ned i n append ix B

de t e rm i nant de f i ned i n s t e p 1 of appendix B

p er tu rb ed q u a n t i t i e s d e fi n ed i n s t e p 1 and s t ep 4 of appendix B ,r e s p e c t i v e l ye vec tor def ined by no ta t ion ( 6 ) i n appendi x B

A r 9 % diagonal e igenva lue matrices used i n equa t ions (12) and ( B 6 ) ,r e s p e c t i v e l yX,Xi,Xk eigenvaluesC , W I 91-12k scalar defined by notat ion ( 4 ) i n appendi x B@ bank angle, deg

vec t ors def ined by equa t ions (21)

9 yaw r a t e , deg/secSupe r sc r i p t s :-1 m at r i x i nve r seA -, ,-,* t ransformed quant i ty

Upper case l e t t e r s of t he a l phabe t i nd i ca t e matrices; m a t ri x s u b s c r i p t si n d i c a t e p a r t i t i o n e d q u a n t i t i e s . Dot o ve r a quan t i t y deno t e s de r i va t i ve w i t hr e s p e c t t o t i m e . Prime denotes t ranspose.

EIGENVALUE/EIGENVECTOR ASSIGNMENT FORMULATIONConsider a l i n e a r , time i n v a r i a n t , m u l t i v a r i a b l e , c o n t r o l l a b l e , o b s er v ab le

system

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X = AX + BUy = cx 1

where x E R n , u E R m 9 y E R r , and B and C are f u l l r an k ; and f o r at r i v i a l p robl em fo rm u l a t i on , assume m , r > 1 and m , r < n. The problemf i n d a c o n t r o l law of t h e formu = Ky

i n o r de r t o a s s i g n a r b i t r a r y ei ge n va l ue s f o r t h e closed- loop system. To

( 1 )

non-i s t o

(2 )i n d i -cate c l e a r l y t h e f reedom ava i l ab le in t h e se l ec t i on o f c l o sed -l oop e i genva l uesand eig env ect ors under output feedback, th e measurement matr ix C i s assumedt o be i n a sp ec ia l canonica l form:7= [Cl : CZ]

c1 = [.I; .Ic2 = [. 1 .Ic = p I . . . 1 1I ( 3 )Here, t = r - 1 , C 1 E R r x t , c E R l x n - t , and I t denotes a t th o r d e r i d e n t i t y

matrix. Appendix A d e t a i l s a procedure f o r reducing any system ( C , A , B ) t o t h i ss p e c i a l f orm .The closed-loop system matr ix ( A + BKC) a f t e r applying feedback l a w ( 2 )s a t s i e s

( A + BKC)vi = Xivi ( i = 1 , 2 , . . ., n) (4)where X i i s t h e i t h eigenvalu e and v i i s t h e correspo nding eige nvec tor . Theeigenvalue/eigenvector assignment problem i s t o de te rmi ne t h e number of eigen-va l ues i n equat i on ( 4 ) t h a t can b e a r b i t r a r i l y a s si g ne d a nd t o d e te rm in e t h ef re ed om a v a i l a b l e i n t h e s e l e c t i o n o f t h e a s s o c i a t e d e i g e n v e c t o r s .

I n o r d er t o see what f reedom ex i s t s i n t he cho i ce o f e i genve c t o r s , wri teequat ion ( 4 ) i n p a r t i t i o n e d form as


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where A11,Bl E Rm x m and B1 i s nons ingula r . S ince B i s f u l l r ank , t he non-s i n g u l a r i t y o f B1 can be assured, i f ne c e s s a ry , by r e o r de r ing t h e s t a t e v a r i -a b l e s i n e q u at i on s ( 1 ) . Complet ing the mul t ip l ic a t i on of the p a r t i t i o n e dmatrices and some algebraic ope r a t i ons ( r e f . 8) permi t s equa t ion ( 5 ) t o beexpressed as a se t of c ons t r a in t s on t he s e l e c t i on o f e i ge nve cto r s . For c l a r i t yo f p r e se n t a t i o n, t h e s e r e l a t i o n s are d e t a i l e d o n l y f o r r e a l eigenvalues . Exten-s io n t o complex conjuga te pa i r s i n quas i -d iagona l form yie ld i ng rea l e ige nve c to rp a i r s i s s t r a igh t f o r w a r d ( r e f . 8 ) .

For rea l e igenva lues[XI,_, - F]w = [G +[A I + B I K C I V = XZ

where i s t he e ige nva lue , v ' = [z' : w']; v i s t he e ige nve c to r w i thz E Rm , and

S = B2Bl-l

A 1 = [All : A121Equation ( 6 ) rep re se nt s an underdetermined sys tem of n - m e q u at i o ns i n

( 6 )

( 7 )

( 1 1 )nunknowns. Thus m e igenve c to r e n t r i e s c o r r es pond ing t o t h e z-vector can be

c hose n a r b i t r a r i l y p r ov ide d doe s no t c o inc ide w i th t he s pe ct rum o f F.Examination of equation ( 7 ) r e v e a l s t h a t a t l e a s t r eigenvalues and r eigen-v e c to r s s a t i s f y i n g e q u at io n ( 6 ) ca n be a s si g ne d t o t h e s y s t e m i n e q u a t i o n s ( 1 )by th e feedback mat rix

where A, i s t he d i a gona l m a t r i x of r eigenvalues and Z ( r ) an d V ( r )have the form T ( r ) = [ t i : t 2 : . . . : tr] (where t i are ve c to r s ) . Thes o l u t i o n t o e q u at i on ( 1 2) i s .guaranteed provided th e eigenvalues/eigenvectorsare c ho sen t o i n s u r e t h e n o n s i n g u l a r it y o f I t should be n o t e d t h a ti n t he case of s t a t e var iab le feedback ( C = I ) a l l n e ige nval ues can ben . 'ass igned t o th e sys tem provided th e modal matr i x ( m a t r i x o f e i ge nve c to r s )V = [VI : v2 : . . . : vn] i s nons ingula r . An a lgor i thm which con s t r uc t s sucha nons ingula r V i s d e t a i l e d i n r e fe r en c e 8 . A pendix B e x t en d s t h i s a l go -r i t hm t o guar a n te e t he nons ingu l a r i t y o f

[dr)].

[CV(r)f .N oti c e t h a t by c a r r y ing ou t t h i s a na l ys i s on t h e dua l syst em ( B 1 , A ' , C ' ) ,i t can be shown t h a t m eigenvalues can be ass igned t o the sys tem. This ana ly-sis yie lds the fol lowing wel l -known r e su l t ( r e f . 9 ) .

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Lemma 1 : For sys tem ( C , A , B ) , max(m,r) e ige nva lues can beass igned us ing output f eedback .

The ana lys i s so f a r i n d i c a t e s t h a t o n ly max(m,r) e igenvalues can bea s s igne d t o t h e s yst em us ing ou tpu t f eedback . However, by s a c r i f i c i n g somedegree of freedom i n t he s e l e c t i o n o f t h e a s s oc i a t e d e ige nve c to r s i t i s p o s s i b l et o e xt en d th e number of e igenva lu es t h a t can be a ss igned t oas is shown i n the fo l lowing sec t io n . min(n,m + r - 1)

ALGORITHM TO A S S I G N M I N I M UM (n,m + r - 1 ) EIGENVALUESThe basic a pproa ch i n t he developme nt o f t h i s a l go r i t hm i s t o 2 o n st r u ct t h eoutput feedback l a w i n e q ua t io n ( 2 ) as a sum of two feedbacks (K + K).f i r s t feedback (E) a s s i g n saddi t ional min(m,n - t ) e i g en v a l u es w h il e e n s u ri n g t h e p r o t e c t i o n o f t h e te ige nva lue s a l r e a dy a s signe d . The c ons t r uc t i o n p roc edure y i e l d s a s e t o f s u f -

f i c i e n t c o n d i t i o n s f o r a s s i g n i n g min(n,m + r - 1 ) eige nva lues . These condi-t i o n s a l s o h el p c h a r a c t e r i ze the c las s of sys tems which cannot be ass ignedmin(n,m + r - 1 ) eige nva lues . Fi na l l y, some des ign f reedom s t i l l e x i s t s t op a r t i a l l y a ss i gn ( r - 1 ) e i g e n v e c t o r s .

Thet e igenva lu es , and the second feedback a s s i gn s

S t e p 1 :Assign t e ige nva lue s and c o r r es pond ing e ige nve c to r s t o t he s yst e m i ne qua t i ons ( 1 ) and form t h e matr ix

N = [.N! ] = [V I : v2 : . . . : V t ]N 2

( 1 3 )

w i t h N1 E R t x t and no ns in gu lar . Appendix B d e t a i l s a p ro ce du r e t o c o n s t r u c tN . Let K" be the nonunique feedback (eq . ( B 6 ) ) cor re sponding to t h i s a s s ign -ment. Then th e closed- loop matr ix i s& = A + BffC ( 1 4 )

Ste p 2:Apply a c oor d ina t e t r a ns f o r m a t ion

( C , $ , B ) -+ ( C T ? T ~ - I ~ T ,T ~ - ~ B >where

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The transformed system h a s the form

_I ( 1 6 )where At E R t X t i s t h e d i a g o n a l m a t r i x of e i ge n v al u e s a s si g n ed i n s t e p 1 . I tshould be noted th a t C2 i s i nvar i an t under the t r ans fo rmat ion TI.Step 3:

I n o r d e r t o a s s i g n a d d i t i o n a l e i g e n v a l u e s t o t h e sy st e m o f e q u a t i o n s ( 16 2w h i l e p r o t e c t i n gi s r e s t r i c t e d t o b e o f u n i t y r a n k o f th e form K = qRc with q E Rm , R E R r ,and i s chosen so t h a t

t e igenva lues (A , ) a l r ead y ass igned , th e second f eedback (K)

R ' P 1 : c2] = [o : c] (17)Now q must be chosen so t h a t

i s assigned min(m,n - t ) e igenva lues .S ince ( i2 2 g2) i s c o n t r o l l a b l e , the f o ll o w in g r e s u l t h o ld s :

/ . 4Theorem 1 :assigned min(m,n - t ) e igenva lues i f and only i f ( a ) (c,A22)i s observable and (b ) 62 i s f u l l r an k.The s in g le outpu t subsystem (c ,A22,B2) can be ~

Theorem 1 f ol l ow s d i r e c t l y f rom lemma 1 . F u r t h e r , c o n d i t i o n s ( a ) and ( b )r e s t r i c t t h e admiss ib le s e t o f e i g e n v a l u e/ e i g e n v e ct o r a s s ig n me n ts i n s t e p 1 .For conceptual convenience, these p a r a m e t r i c r e s t r i c t i o n s are formulated interms o f c o n t r o l l a b i l i t y c o nd i ti o n s o f a f i c t i t i o u s dynamic sy s te m i n t h e s t a t ev a r i a b l e r e p r e s e n t a t i on .After matr;x op er at io ns i n equ ati ons (14) and (16) have been performed,

th e submatr ix A22 can be wr i t te n as

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Equation (20) can now b e looked upon a s a system matri x deriv ed by apply ingfeedback t o t he dynamic systemi = Ai25 + Ai21-11 + c'1-12

(211n = $5 + Bi1-11 1i th 6 E Rn- t , q E Rm , 1-11 E R t , 1-12 E R ' , 1-11 = f i& , 1-12 = s l q , an dN2 = -N2N1'1.From equations (21) i t can be shown th a t co ndi t io ns ( a)_an d (b) of theorem 1a re e q u iv a l e n t t o t h e f o l l o w in g c o n d i t io n s : ( c ) & = B 2 + N2B1 and i s f u l l

rank; ( d ) [[A22 + A { ~ G ; ] , C ~ i s c o n t r o l l a b l e . C o nd it i on s ( c ) and ( d ) c l e a r l yin d i c a t e t h e r e s t r i c t i o n on s e l e c t i o n o f t h e e ig en v e cto r p a ra m ete rs i ns t e p 1 . T h is y i e l d s t h e f o l l ow i n g s u f f i c i e n t c o n di t i o n s :

Theorem 2 : The s y s t e m ( C , A , B ) can be ass ignedmin(n,m + r - 1 ) e ig e nv al u es a r b i t r a r i l y c l o se t o t h ed e s i r e d s e t i f t h e f i r s t t eigen value s and e igenv ector si n s t e p 1 are chosen so tha tI N 1 i s nonsingular

nlI1 62 = B 2 + N2B1 and i s f u l l rankI11 { [A;2 + A i & ] ,c l} i s c o n t r o l l a b l e

" A r b i t r a r i l y c l os e r 1 i n t heo re m 2 i n d i c a t e s t h a t s l i g h t p e r t u r b a t i o n s i ne igenva lue s pec i f ic a t i on s may be n ee ded i n t h e f o l l ow in g s i t u a t i o n s :

( i )Assigned e i genva lues coinc ide with the spectrum of F (appendix B ) .(ii)An e x a c t c om bina ti on o f e ig e n v alu e / e ig en v e c to r s p e c i f i c a t i o n s i ns t e p 1 may not yield a nonsingular N1 (appendix B ) .

(iii)Coinc iden t spec t rum s i tua t ion s imi la r t o s i t u a t i o n ( i) x i s t s f o rth e subsystem eigen value assignment of theorem 1 .Condition I i s r e q u i r e d t o g u a r a n te e t h e e x i s t e n c e o f t h e t r a n s f o r m a t io nT I i n s t e p 2 . T h is c o nd i t i o n can be e x p l i c i t l y i n cl u de d i n t h e s y n t h e s i s p ro -

cedure as d e t a i l e d i n a pp en di x B. Condition I11 i s obta ined f rom the p roper tyth a t f o r sy st em ( e q s . ( 21 11 , t h e c l a s s of feedback from input 1-11 should ber e s t r i c t e d so t h a t t h e c o n t r o l l a b i l i t y o f t h e f e ed ba ck s ys te m w it h r e s p e c t t ot h e i n p u t 1-12 i s preserved. Condi t ions I1 and I11 yi e l d nonl inear_ a lg ebr a icc o n s t r a i n t s for the e lements o f th e e igenvec tor pa ramete r matrix and thus ,i n g e n e r a l , c an o nly be used as t e s t c o n d i t i o n s f o r e ac h as sign me nt i n s t e p 1 .However, example 1 of th e se ct io n e n t i t l e d l lNumerical Examples1! shows how th es econdi t ions can be e x p l i c i t l y ch eck ed .

N2

9


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Consider the

-A =

B =

-c = 10

Reduce the system

NUMERICAL EXAMPLESExample 1

sys tem desc r ibed i n r e f e r e n c e 7 where1000

0-001-00

-0 0

0 1 LO 0

O 0 11

(25) t o t h e s p e c i a l fo rm of e qua t i ons ( 1 ) by o r de r ing s t a t e-v a r i a b l e s as ( X J , X ~ , X ~ , X ~ )o makein g t he c oo r d ina t e t r a ns f o r m a t ion (C,A,B) + (ET0 ,To-lZTo ,To-IG) where

sa- no ns in gu lar (appen dix A) and by apply -

0 0 1 0: : : :0 0 1 (26 )

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t o y i el d

A =

B =

c =

0 0 10 0 00 0 00 0 0

10

0 01 1 O 1,

From equation ( 6 )

where th e eigenv alue

can be shown t h a t the closed-loop eig env ect ors s a t i s f y

i s X and the e igenvector i s

-o f g e n e r a l i t x . Then, N2 =c h o i c e s o f N2 s ince B i =/

From theorem 2, condit ion I , n1 # 0 ; th e re fo r e , choose n1 = 1 without l o s s( 4 2 -9 -n4) I . Condit ion I1 i s met f o r all0. Condit ion I11 i m p l i e s t h a t

0 011-12 -n3 -n4

0 0l!should be c o n t r o l l a b l e . For t h e c o n t r o l l a b i l i t y m a t ri x i n e q ua t io n ( 30 ) t o beo f f u l l r a n k, t h e e i g e n v e c to r pa r am e te r s i n e q u a ti o n ( 2 8 ) m u s t s a t i s f y

n2n3 - n4 + n32 + n3n4 # oBy d i r e c t su b s t i t u t i o n e q u a t i o n (31) i s s e en t o b e v i o l a t e d by a l l admis-s i b l e s e l e c t i o n s i n e qu a ti on ( 2 8 ) .( m + r - 1 ) = 3 eigen value s . Indeed, only two e igen value s can be a s s ig n e d t o

(311Thus, th e system cannot b e ass igned

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t h i s s y s t e m ( r e f . 7 ) . However, th e pre vio us ana lys es ( r e f s . 3 and 7 ) do notp r e c i s e l y l ead t o t h i s c on cl us io n.

Example 2Consider the system descr ibed by ( r e f . 7 )

100

-l01010I.

I t i s r e q u i r e d t o a s s i g n e ig e n v a lu e s c lo s e t o - 1 , -2, and -5. The system (32)can be reduced t o t h e form of eq uat ion s ( 1 ) by fol l owin g the procedure i n appen-d i x A as

Step 1 : Measurement matrix i s i n t h e d e s i r e d f orm w i th ea nons ingu la r .Step 2: Apply coord inate t rans forma t ion

( C , A , B ) + ~ C T ~ , T ~ - ~ ~ T ~ , T ~ - ~ B ~where

1 0 00 1 10 0 1

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y i e l d i n g

A =

B =

c =

1 10 10 0"1

10 1 1

O "1The system (34) i s i n t he r equ i r ed canon i ca l fo rm o f equa t i ons ( I ) . The closed-l oop e i genvec to r cons t r a i n t s i n equat i ons ( 3 4 ) can be der ived by id en t i f y i ng ther e s p e c t i v e matrices i n e q u a t i o n s ( 8 ) t o ( 1 0 ) as0

-1B1 = [;s = ( 1 -1)F = O

G = ( 0 -1)and equation (6) y i e l d s

> can be

x w = [A : -(1 + A)] 1.": .1L z 2 1

Equation (35) i n d i ca t e s t h a t z1 and

(35)

r b i t r a r i l y ch os en , p ro vi de dX # 0 .t he fo l lowing syn th es i s sequence.S tep 1 :

Now, app ly i ng t he a l go r i t hm t o a s s i gn ( m + r - 1 ) e i genva l ues y i e l d s

Assign A = -1 . From appendix B , case 11, A t ( 1 ) = {l}, and t h i s i m p l i e sz1 # 0 i n e q u a t i o n (35). (Condi t ion I , theorem 2.) One acceptable assignmenti s14


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N = ( 1 1 1 ) '

jL

(36)

- 1 0

. . . .-1 -10 1-

corresponding to this assignment from equa-(B6) is

f equation ( 1 4 ) is

2:Transform the system to the canonical form of equations (16) using

TI

1 . 0. . . .

1 . 11 . 0

0-1OI1 . 0. . . . . . .

0 . - 1 00 . 0 0

e = [ ; ; ;j15


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From equations (21) we can form the dynamic system

wi th

Since feedback from I-rl does not a f f e c t t h e c o n t r o l l a b i l i t y of system (41 )w i th r e s p e c t t o ~ 2 ,he subsystem i s pole ass ignab le f rom condi t ion I11 o ftheorem 2 .i s nonsi ngula r. Thus theorem 2 holds .F u r th e r , t h e a ss ig nm en t i n s t e p 1 s a t i s f i e s condi t ion I1 s i n c e 82S te p 3:

Choose R' = (-2 1 ) . T h i s c h o i 2 e p r o t e c t s X = -1 a s si gn e d i n s t e p 1 .I t now remains t o choose q so t h a t K = qR' a s s ig n s t h e e ig e n v a lu e s X2 = -2and A3 = -5.Step 4:

Assign A2 = -2, A3 = -5 t o t h e s i n g l e o u t p ut s ub sy st em i n e q u at i o n ( 1 8 )where

c = E1 13

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J

5:The feedback l a w of the form of equat ion (22) i s given by

-1, -2, and -5 t o t h e sy st em ( e q . ( 3 2 ) ) .A I R C R A F T LATERAL, CONTROL DESIGN

The advantage of combined c on tr ol o f closed-loop eig env alu es and eigenvec-r s u si ng s t a t e va r i a b le f eedback has been in ves t iga ted i n r e fe ren ce IO. Thet i l i t y o f t h e o u t pu t f ee db ac k e x t en s i o n s d ev el op ed i n t h i s r e p o r t w i l l now b el u s t r a t e d t hr ou gh t h e d e s i gn o f a l a t e r a l c o n t ro l l e r f o r an a i r c r a f t .The l i n e a r pe r tu r ba t i on model f o r the l a t e r a l m ot io ns o f a n a i r c r a f t ca n

e modeled as 1: = Ax + Buy = Ex + Du ( 4 5 )x i s t h e s t a t e v e c t o r o f r o l l r a t e p, yaw rate 9, s i d e s l i p 6, and0, esp ec t i ve l y . The co n t r o l vec to r o f a i l e ro n 6, and rudder

, a n g u l a r d e f l e c t i o n s i s u. Rol l r a t e p, yaw r a t e Q, and l a t e r a l accelera-ay c o n s t i t u t e t h e o u t pu t v e c t o r y . A l l a n g l e s a re i n degrees , r a t e s i na n d a c c e l e r a t i o n i n m/sec2 ( f t / s e c 2 ) .

The respect ive matrices i n e q u at i on s ( 4 5 ) f o r a f i g h t e r a i r c r a f t a t an a l t i -6096 m (20 000 f t ) , a Mach number of 0.67, and an ang le o f a t t ac k o fare given by

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r - 3 * 7 9-0.14A = II 0.06

25

0.010

L-0.13

11.03where the elements of

0.04-0.36

-10.06

0.050

01

-0.06

-52 04.24 0

-0.27 0.050 0

0 00 0

-3.42 0

the matr-ces are approx-aated t o two s i g n i f i c a n t d i g t s .v e c to r i n e q u a t i o n s ( 4 5 ) i s der ived as a l inear combina-in c e t h e o u tp u t '

t i o n of both s t a t e var iab les and con t r o l inp u t s , the c losed- loop sys tem af te rapplying feedback u = K*y t ak es t h e form

x = ( s i + BKC)X + GPu( 4 6 )

y = (e + ~ K ( ? ) x + DPuwhere P = [ Im - K*D]-' and K = PK* i s the equ iva len t ou tpu t feedback matrixobta ined by se t t in g = 0. Thus, the a lgor i thms developed e a r l i e r f o r s y s t e m swi th 5 = 0P e x i s t s . a re a p p l i c a b l e t o s y s te m s o f t h e f orm of e q u a t i o n s (451, provided

Then the feedback gain K* i s computed by using t he r e la t i o nK* = K[Ik + fiK1-l (47)

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Direct matr ix manipula t i ons show t h a t t h e i n v e r s e i n e qu at io n ( 47 ) e x i s t s i f Pe x i s t s .

The h an dl in g q u a l i t i e s s p e c i f i c a t i o n s ( r e f . 1 1 ) imply t h a t t h e l a t e r a l a i r -c r a f t dynamics should be composed of two weakly cou ple d sub sys tem s. R o ll r a t ea nd bank a n g l e c o n s t i t u t e t h e f i r s t subsystem and disp lay predominant ly t he rollsubs idence and s p i r a l modes. The second subsystem i s c h a r a c t e r i z e d b y a w e l l -damped Dutch r o l l mode de fi ni ng t h e yaw rate and s i d e s l i p m o ti on s. These s p e c i -f i ca t io ns can now be fo rmula ted as an e igenvalue/e igenvector ass ignment problem.

Table I summarizes t h e modal c ha ra c te r i s t i c s o f t h e f ree a i r c r a f t . Fromt h e t ab le i t i s seen t h a t t h e Dutch roll mode i s very l i g h t l y damped and appe arsdominant ly i n the response o f t h e roll v a r i a b l e s p an d 9 as evidenced byd omin ant e n t r i e s i n t h e c o rr e sp o nd i ng e ig e n v ec t o r p a i r . Thus, the e igenva lue /e ig e n v e c to r m o d i f i c a t i o n r e q u i r e s t h a t t h e closed- loop system have t h e modesand mode-var iable as so c i a t io ns o f t a b l e 11. ( I n t a b l e s I t o 111, j = n.

TABLE 1.- MODAL CHARACTERISTICS OF FREE AIRCRAFT.igenvectorcomponentsPiJB

9

ModeRol l subs idenceDutch rollS p i r a l

Eigenvalue of --3.70( R o l l s u b s id e n c e )

-0.964

-.041

-.002.261

-0.35 5 j2.66(Dutch roll)-0.403 0.829

-.096 -.I31

-069 -.034

TABLE 11.- DESIRED MODAL SPECIFICATIONSEigenvalue

-6-1.0 - j0.2

-0.01

I0.03 .( S p i r a l1

-0.032

.002 i

.998 1

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The ou tpu t f eedback an a ly s i s i n the main t ex t shows t h a t a l l system eigen-values can be a s s i g ne d s i n c e n = m + r - 1 and only two eigenvectors ( t ) canbe assigned w i th a t most two ( m ) e n t r i e s i n each v e ct o r a r b i t r a r i l y c ho sen. The igenvec to r f reedom av a i l ab l e was u sed t o c o n t r o l t h e s t r u c t u r e o f t h e eigenvect o r p a i r c or re sp on d in g t o t h e Dutch r o l l mode t o e f f e c t the des i red yaw rateand s i d e s l i p dominance . The modal coupling matrices D ( k ) (eq. (B4)) a i d i nth e se l ec t ion o f the Dutch roll mode t o yi el d t h e appropr i a t e e igenvec to r fo rm sThe f o u r e ig e nv e ct o r e n t r i e s t h a t were f r e e ly chosen corresponded t o t h e rollrate and yaw rate components of the rea l e i g e n v ec t o r p a i r a s s o c i a t e d w i t h t h eDutch roll mode. S ince ga in m agni tude co ns t r a in t s cannot be ex p l i c i t ly inc ludedi n t o t h e s y n t h e s i s a l g or i th m , t h e desig n parameters have t o be i t e r a t i v e l y modi-f i e d t o meet ga in l i m i t requi rements . After som e des ign i t e ra t ions , a compro-mise des ign y ie lded the modal ch a rac t e r i s t i c s summarized i n t a b l e 111. Forexample, Z t was noted t h a t t h e Dutch roll mode damping could no t be reduce d ( t oimprove $ and B r e sponses ) w i thou t v io la t ing feedback g a i n l i m i t s which weres e t a t u n i t y f o r t h i s a n a l y s i s . K*(eq . (47)) as The des ign y ie lded a feedback gain matr ix

-. 65

1-0.16 0.19 -0.6 1

-.05 .05I

TABLE 111.- MODAL CHARACTERISTICS OF FEEDBACK AUGMENTED AIRCRAFT7igenvectorcomponents

Eigenvalue of --6(Roll subs idence)

0.986

.009

-. 07

-1.0 + j0.20(Dutcg roll)0.01 0.02

35 - 53.2 5 .72

-0.01 1( S p i r a l

-0.013

-.031

- 082-.996

- .

Table I11 i l l u s t r a t e s t h a t t h e closed-loop eig env ect ors have approachedt h e d e s i r e d mode-decoupled s t r uc tu re . I n pa r t ic u l ar , t h e des i r ed m odi f i ca t ionachieved i n the e igenvec to r pa i r co rresponding t o t h e Dutch roll mode should benoted. The improvement i n t r a n s i e n t r es po n se c h a r a c t e r i s t i c s u s i n g t he f eedbackc o n t r o l l e r i s i l l u s t r a t e d i n f ig ur e 1 . The response curves demonstrate t h a t t h ecr os s coupl ing between t he roll ax is (p ,@ ) and yaw a x is ( Q , B ) has b e e n s i g n i f i -c a n t l y reduced.20


23/33

CONCLUDING REMARKSN ew su f f i c i e n t cond i t i ons t o a s s ign minimum (n ,m + r - 1 ) eigenvalues by

eans of output feedback have been der ive d. In ge ne ra l , i n ad di t io n t o t h e+ r - 1) e igenva lues , ( r - 1) e igenvec to r s can b er t i a l l y a s s ig n ed w it h a t mosthe u t i l i t y o f a s s i g n i n g b ot h e i g e n v al u e s and e i g e n v e c t o r s fo r r e sponse m odi f i -i s i l l u s t r a t e d by d e s ig n i ng a feedback c o n t r o l l e r f o r t he l a t e r a l dynam-c s o f an a i r c r a f t . a l i n e a r s ys te m o f e q u a t io n s .

m e n t r i e s i n e ach v e ct or a r b i t r a r i l y c hosen.

The s y n t h e s i s a l g o r i t h m i s computa t ional ly s imple and

Research Centerl Aeron autics and Space Administra t ionVA 2366519, 1977

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APPENDIX A

REDUCTION OF (C,A,B) TO CANONICAL FORMConsider t h e given sys tem of t h e form. -

y = ezS t e p 1 :

( A

Reorder th e s t a t e v a r i a b l e s ( i f necessa ry ) s o t h a t the measurement matrixi s o f th e form-[ca : cb] (A

where Ea E RrXr and nonsingular . I t should be noted t h a t the d imensionso f t h e p a r t i t i o n e d matrices i n e q u a t i o n ( A 2 1 are d i f f e r e n t f r o m t h o s e i nequa t ions ( 31.Step 2:

Apply a coord ina te t r ans fo rm at ion-x = TOX

where

l o 1n-r Jand e, E Rrxn-r and Es = w i th c as d e f i n e d i n e q u a t i o n s ( 3 ) . Thetransformed system

ihas t h e des i r ed form of equa t ions ( I ) .

22


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APPENDIX B

EIGENVALUEVEIGENVECTOR SELECTION PROCEDUREFor com ple teness o f p resen ta t ion , an e x i s t i n g a l g o r i t h m ( r e f . 8 ) developed

r e igenva lue /e igenvec to r a s s ignm ent us ing s t a t e var iable feedback i s d e t a i l e dt h i s appendix . Extens ions are made t o a do p t th e a lgor i thm for t h e o u t p u tcases d i s c u s se d i n t h e main t e x t .

SPECTRAL SYNTHESIS ALGORITHMA d i r e c t way of construct ing a nonsingular modal matr ix i s t o g e ne r at e t h ehich s a t i s f y equa t ion ( 6 ) s e q u e n t i a l l y a nd i n s u r e t h a t t h e y dot l i e i n t he e igenspace genera ted by t h e vec to r s a l r ead y syn thes ized . Thei thm presented accompl ishes t h i s c o n s t r u c t i o n w h i l e c o n st a nt l y t e s t i n g t ot h a t t h e s e t of e i g e n v e c t o r s i s a l i n e a r l y i n d e p e n d e n t se t t o a degreea numer ica l to lerance parameter se t by t h e des igne r . For c l a r i t y

t h e a lgor i thm i s d e t a i l e d f o r r e a l eigenvalue ass ignments . Theare used throughout t h e a l g o r it h m p r e s e n t a t i o n :( 1 ) z e ro .( 2 ) v; [z; : w;] i s t h e k t h eigenvector , where Z k i s des igner

( 3 ) V k ( k - l ) = Q(k - l )Vk , where Q ( O ) = In and Q ( i ) , i # 0 , i s d e f i n e d i n

( 4 ) V - (k-1) i s t h e j t h e n t r y o f Vk(k- l ) and f or convenience of n ota-( 5 ) A ( 1 ) i s t h e s e t o f i n d i c e s ( 1 , 2 , . . ., n ) .( 6 ) M ( k ) , an e lementary upper t r ia ng ul ar matr ix of order n and index 1 ,

e j i s an n-vector with j t h e n t r y e q u a l t o 1 and a l l o th er e n t r i e s

) which fo l low s .Jki s denoted Ok .

the form

A ( k ) i s a s u b s e t o f A ( 1 ) ( d e f i n e d i n ( 5 ) ) c o n t a i n i n g t h e i n d i c e s n o ts ed i n t h e c o n s t r u c t i o n o f t h e matricescan be cons t ruc ted i f an d only i f Ok # 0 ( r e f . 8 ) .M ( l ) , M ( 2 ) , . . . M(k-1).

( 7 ) Q(k-1) = M(k-1) , ~ ( k - 2 ) - . 9 M ( 1 ) ( B 3 )23


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APPENDIX BThe algori thm is now accomplished by c omp leti ng s t e p s 1 t o 4 f o r

k 1 , 2 , . . ., n and by computing the feedback gains from equa t ron (12)using C = In .S t e p 1 :

??Or A = Ak, COmpvte 6, = d e t [Akin-, - F]( a ) i f 6k = 0, pe rt ur b Ak t o [Ak + &Ak] and repeat the ca lcula t ion of

6k*( b ) i f 6k f 0 , p ro ce ed t o s t e p 2 .

S t e p 2:Compute

S t e p 3:For some j E {A(k)},( a ) Compute [gjk)] = f j ( k - 1 ) + hj(k-l )D(k) where [f j ( k- l) : h j ( k - l ) ]i s t h e j t h row of t he t r ans fo rm at ion m a t r ix Q ( k - l ) - g j k ) E Rm andh . ( k - l ) E Rlx(n-m).J(b) Compute

( i) f ak # 0, compute w = D(k)zk and ~ ( k ) from e q. (B2)) and

(ii) f a k = 0, s e l e c t ano the r j E {~(k> and r e t u r n t o s t e p 3 ( a )r e t u r n t o s t e p 1 f o r t h e ne xt value of k .

(iii) f a k = o f o r a l l j E { ~ ( k > , go t o s t ep 4 .S t e p 4 :

For some j E {A(k>,( a ) if g j ( k ) 2 0, p e r t u r b

(b ) if g j ( k ) = 0 , s e l ec t ano the r j E and r e t u r n t o s t e p 4 ( a ) .( c ) if , j ( k ) o f o r a l l j E { ~ ( k i ) , p er t u rb Ak t o (Xk + EAk) and

Zk t o (Zk + &zk) t o make ak # 0, computeWk and M ( k ) , and r e t u r n t o S t e p 1 f o r t h e n e xt v a l ue of k .

r e t u r n t o s t e p 1 .

24


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APPENDIX Be fol lo wing obs erv at i ons can be made regarding the algori thm as o u t l i n e d .

( i ) xtens ion of th e a lgor i thm t o ass ig n complex e igenva lues i s s t r a i g h t -( r e f . 8 ) .(ii) f some of th e eig env alu es of A c o i n c id e w i t h t h o s e o f F , e i t h e rp e r t u r b a t i o n can be made i n t h e e i g e n v a l u e s p e c i f i c a t i o n a s i n s t e p 1 , o r

e i g e n ve c t o r s t r u c t u r e s ca n b e d er i ve d by n o t i n g t h a t e q u a t io n ( 6 ) hass o lu t io n f o r z i = 0 ( r e f . 8 ) .(iii)The i t e r a t i v e p ro ce d ur e i n p a r t ii of s t e p 3 (b ) a t t e m pt s t o meetc .t e igenvalue /e igenvec to r spe c i f i c a t i on s . In s t ep 4 (b ) an a t t em pt i s mademeet e x a c t xk s p e c i f i c a t i o n s w i th s l i g h t l y r e l a x ed Zk s p e c i f i c a t i o n s .t e s t i n s t e p 4 ( c ) i n d i c a t e s t h a t th e xk s p e c i f i c a t i o n i mp li ed t h a t t h eres pon din g Vk l i e s i n th e e igenspace a l r ead y genera ted . The t e s t t h u san e igenva lue pe r tu rba t ion .

( i v ) T he a k provide a good measure of th e l i n e a r independence betweeni f a l l v e c t o r s are normalized to a s t a n d ar d b a s i s . S i n ce t h e

inant of th e modal matr i x V i s g iven by th e p roduc t o f t he ak,1 , 2 , . . ., n ) , t h e n u m e r i c a l ill cond i t ion ing o f V f o r i n v e r s i o n c anc t i v e l y c o n t r ol l e d by s e t t i n g a t o l e rance on the a k .

OUTPUT FEEDBACK EXTENSIONSS i nc e t h e d i s t r i b u t i o n m a tr i x C i s a l r e ad y i n t h e s p e c i a l c a no n ic a l form,i n e a r l y i nd ep e nd e nt v e c t o r s t o g u a r a n t e e e x i s t e n c e o f f ee db ac kn e q u a t io n s (12) and ( 1 4 ) i s s im ply ach ieved by r e s t r i c t i n g the admis-b l e p iv o t a l i n di c es A ( k ) ( e q . ( B2 )) i n t h e s p e c t r a l s y n t h e s i s al g or i th m as

Case I : Computing E, Equation (12)In equa t ion ( 5 ) , l e t A r ( l ) be the s e t of r column indices correspond-

t h e r l i n ea r l y independent columns of C ( I r ) . Then apply the spec-s y n t h e s i s a l g o r it h m u s i n g i n s t e a d o f A ( 1 ) ( n o t a t i o n ( 5 ) ) f o r1 , 2 , . . . r . T hi s a p pl i c a ti o n g u a ra nt ee s t h e i n v e r t i b i l i t y o f C V ( r )u a ti o n ( 1 2 ) .Case 11: Computing E, Equation ( 1 4 )

I n equa t ion (51, l e t A t ( 1 ) b e t h e se t of t column ind ic es no t c onta in -c ( e q s . ( 3 ) ) . Then a p p ly t h e s p e c t r a l s y n t h e s i s a l g o r it h m u s i n g. i n s t e a d o f A ( 1 ) ( n o t a t i o no n g u a r a n t e es t h e n o n s i n g u l a r i t y

CN i s rank t , the feedbackl a ionEC N = Bl-I[Z(t)At .- AlN]

( 5 ) ) f o r k = 1 , 2 , . . ., t . This app l i ca -of N 1 l e q . ( 1 3 ) ) . F u r t h er , s i n c e i n t h i sm a t r ix K ( eq . ( 1 4 1 1 , computed using the

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APPENDIX Bt o a s s i g n t Zigenvalues , i s no t uniq ue. However, a minimum norm least-squares o l u t i o n for K can be computed using t h e pseudo inverse of CN.

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REFERENCES1 . Wonham, W. M.: On Pole Assignment i n M u lt i -I n pu t C o n t r o l l a b l e LinearSystems. IEEE Trans. Autom. C on tr ol , v o l . AC-12, no. 6 , Dec. 1967,

pp. 660-665.2 . Sr inathkumar, S.; and Rhoten, R . P . : Eigenvalue/Eigenvector Assignment f o rM u l t i v a r i a b l e S ys te ms . E l e c t r o n i c s L e t t . , vol . 11, no. 6 , Mar. 1975,pp. 124-125.3 . Kimura, Hid eno ri: Po le Assignment by Gain Output Feedback. I E E E Trans .Autom. Co nt ro l, vo l . AC-20, no. 4 , Aug. 1975, pp. 509-516.4 . Kimura, Hidenori: A F u r th e r R e s u l t on the Problem of Pole Assignment byOutput Feedback. I E E E Trans . Autom. Co nt ro l, vo l . AC-22, no. 3, June1 9 7 7 , pp. 458-463.5. Hermann, Robert; and Martin, Clyde F.: Ap pli cat ion s of Algebraic Geometry

t o Systems Theory - Part I . I E E E Trans. Autom. Co nt ro l, v o l . AC-22,no. 1 , Feb. 1977, pp. 19-25.6 . Topaloglu, Toros; and Seborg, Dale E . : An Alg or it hm for Pole Assignment

U s i n g Output Feedback. Proceedings: 1974 : Joint Automatic Control Con-ference, American Inst . Chem. Eng., c.1974, pp. 309-312.7. Davison, E . J . ; and Wang, S. H . : On Pole Assignment i n L in e a r M u l t i v a r i a b l eSystems Using Output Feedback. I E E E Trans. Autom. C on tr ol , vo l . AC-20,

no. 4 , Aug. 1975 , pp . 516-518.8. Srinathkumar, S.: S p e c t r a l C h a r a c t e r i z a t i o n of Multi-Input Dynamic Systems.

Ph. D . D i s s . , Oklahoma State Univ., 1976 .9. Davison, E . J . ; and Cha t te r jee , R . : A Note on Pole Assignment i n Linear

Systems With Incomplete S t a t e Feedback. I E E E Trans. Autom. Control,v o l . AC-16, n o. 1 , Feb. 1971, pp. 98-99.O . Srinathkumar, S.; and Rhoten, R . P.: Eigenvalue/Eigenvector Control V i a

Sp ec tr a l Char act er i zat ion : An Appl icat i on t o Hel i copt er Hover Dynamics.Ninth Annual Asilomar Conference on Circuits, Systems, and Computers,Shu-Park Chan, ed. , Western P er io di ca l Co., ( N . Hollywood, Calif .) , Nov.1975, pp . 605-609.1 . Hartmann, Gary L . ; Hauge, James A . ; and Hendrick, R u s s e l l C . : F-8C D i g i t a lCCV F l i g h t C o n t r o l L a w s . NASA CR-2629, 1976.

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Il1 Ill I1

10I-10 -

.5--An

--.5I

-.25 L

tda

-3

Time, sec( a > p ( 0 ) = 10 deg/sec.

Figure 1 .- Comparison of f r e e a i r c r a f t and augmented a i r c r a f t r e s p o n s e t o rollra te and s i d e s l i p s t e p d i s tu r b a nc e s .and B corres ponds t o augmented a i r c r a f t response .A i n d i c a t e s f ree a i r c r a f t response

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- ,-

A B

100I-100I

Q 50r

1 - 26 8 10I4Time, sec

12I0(b) f3(0) = IOo.

Figure 1 .- Concluded.

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1. Report No.NASA TP-11184. Title and SubtitleEIGENVALUE/EIGENVECTOR A S S I GNMENT USINGOUTPUT FEEDBACK

. Government Accession No. 3. Recipient's Catalog No.5. R e p o r t Date

7. Author(s)S. S r i n a t h k u m a r

9. Performing Organization Name and AddressNASA L a n g l e y Research C e n t e rHampton, VA 23665

8. Performing Organization Report NoI L-11869512-51 -02-03

11. Contract or Grant No.

-

I 13 . Type of Report and Period Covered2. Sponsoring Agency Name and Address T e c h n i c a l P a p e rN a t i o n a l A e r o n a u t i c s a n d S p a c e A d m i n i s t r a t i o nW a s h i n g t o n , DC 205465. Supplementary Notes

6. Abstract T he p ro b le m o f e i g e n v a l u e a s s i g n m e n t i n a l i n e a r t im e - i n v a r i a n t s y st em u s i n gm ,

o u t p u t f e e d b a c k i s c o n s i d e r e d . New s u f f i c i e n t c o n d i t i o n s are d e r iv e d t o a s s i g n a na lm o s t a r b i t r a r y se t of minimum (n,m + r - 1 ) d i s t i n c t e i g e n v a l u es wherean d r are the number of s t a t e s , i n p u t s , a nd o u t p u t s , r e s p e c t i v e l y . These c o n d i -t i o n s p r e c i s e l y i d e n t i f y t h e class o f s y s t e m s where s u c h a n a s s i g n m e n t i s i m p o s s i -b l e ;l o o p e i g e n v e c t o r s u nd e r o u t p u t f e ed b a ck .a ss ig n m en t i n t r a n s i e n t r e s p o ns e s h a p in g i s i l l u s t r a t e d b y t h e d e s i g n o f a c o n t r o l -l e r f o r th e l a t e ra l d y n a m i c s of a n a i r c r a f t .

n ,The s y n t h e s i s t e c h n iq u e a l s o h i g h l i g h t s t h e freedom i n s e l e c t i o n of c l o s e d -The u t i l i t y o f e i g en v a lu e / e i g e n ve c t o r

r . Key Words (Suggested by Author(s))E i g e n v a l u e sE i g e n v e c t o r sP o l e sAircraft c o n t r o l s y s t e m sL i n e a r s y s t e m s~-

. .18. Distribution StatementU n c l a s s i f i e d - U n l i m i t e d

S u b j e c t C a t eg o r y 6 3

~- - 1 29 ~ -U n c l a s s i f i e d-~U n c l a s s i f i e d* F o r s a le by t he Nat iona l Techn ica l I n format ion Serv ice . Spr ingf ie ld , V i rg in ia 22161 NASA-Langley, 1978


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