eigenvalue assignment

7/27/2019 Eigenvalue assignment

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TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32. NO. 5, MAY 1987 417.20421891&1.20421891&01 -. 7067847eCOl.47067847e+01.55962903e+00 -.48656740e+01

.48656710~+01-. 4935756ei-01 .18923245et-O0-.44935756&1 -.18923245e+OO-.3535747141 .10549139e+01-.35357471e+01 -.10549139&1-.23336170e+01 .17320967e+Ol-.23336170et01 -.17320967e+Ol-. 3501639et00 .34502561et01-.43501639etOO -.34502561e+01.27751362e+Ol .97235727&0.27751362e+Ol -.97235727et00.S4061450e+oO -.27631868e+Ol.8406145Oe+DO .27681868&01

-.96503305e+00 .1836795Oe+Ol-.9650330540 -.18867950&1.14b41641eH)l .14546167&1.36284301e+OO.1464164141 -.14546167&01.12146926&01.36284301e+00 -.12146926&1

. 5 5 9 6 2 9 o w n

Remark: Note that du e to the possible ill-conditioned nature of thethe computed first row do notell with the corresponding ones of the original first row; however,oop matrix ar e almost the sam e as those ofis an example where the problem of computingthe eigenvalue problem is quite well-

ACKNOWLEDGMENTThe author is thankful to M. Arnold, a graduate assistant, foralgorithm and numerical computations. Thanks a re also

REFERENCESD. Boley, A perturbation bound for the linear control problem, SIAM J . Alg .G. . Golub and C . Va nb an , MatrixComputations. Baltimore, M D: Th eDiscrete Methods, vol. 6, pp. 66-72, 1985.Johns Hopkins University Press, 1983.G. Miminis and C. C. Paige, An algorithm for pole assignment of time invariantP. Murdoch and S . Shriba, Eigenvalueassignment in upper Hessenberg m atrix,linear system, Inr. J . Contr . , vol. 35, pp. 34-354, 1981.

R. V. Pate1 and P. Misra. Numerical algorithms for eigenvalue assignment byInt . J . Contr . , vol. 41. pp. 493-497, 1985.state feedback, Proc. IEEE, vol. 17, pp. 1755-1764, 1984.C . C. Tsui, A n algorithm for computing state feedback in multiinput linearW . M. Wonham, Linear Multivariable Control:AGeometricApproach.systems, IEEE Trans. Automat. Contr.. vol. AC-31, pp. 243-246, 1986.New York: Springer-Verlag. 1979.J . H. Wilkinson. The AlgebraicEigenvahe ProbIem. Oxford: ClarendonPress, 1965.M. Arnold and B. N. Datta. An algorithm to assign eigenvalues in a Hessenbergmatrix: Multi-input case, SIAM Cony. on Linear Algebra in Signals, Sysf,Contr . , Boston. MA, Aug. 1986.B.N. Datta and K. Datta, On eigenvalue and canonical form assignments,submitted for publication.-. Efficient parallel algorithms for controllability and eigenvalue assignmentproblems, Proc. 25th IEEE Conf. Decision Contr., Athens, Greece. Dec.1986, pp. 1611-1616.B. h. Dam and Y. S a d , Solution of large Sylvester equation and an associatedY. Saad, A projection method for partial pole assignment in linear statealgorithm for the partial pole assignment problem, in preparation.J . Kautsky. h.K. Nichols. and P. Van Dooren , Robust pole assignment in linearfeedback, submitted for publication.

N. K . Nichols, On computational algorithms for pole assignment, IEEEfeedback, Int . J . Contr . , vol. 41, pp. 1129-1155, 1985.Trans. Autom at. Contr., vol. AC-31, pp. 643-615, 1986.S . P. Bhattacharyya and E. DeSouza, Pole assignment via Sylvesters equa-tions, Syst. Contr . Lett . , vol. 1, pp, 261-283, 1982.G . S . Miminis and C. C. Paige, An algorithm for pole assignment of time invariantmulti-input systems. Proc. 2lst IEEE Conf. Decision Con tr., FL, 1982, pp. 62-67 .P. Hr. Petkov. N. D. Christov and M. M. Konstantinov, A computationalalgorithm for pole assignment of linear multiinput systems, IEEE Trans.Automat. Contr., vol. AC-31, pp. 1044-1047, Nov. 1986.S . Barnett, C. L. Cox, and C. R. Johnson, Assignment of the characteristicpolynomial ofaHessenbergmaviu,Int. J. Contr . , vol. 44, no. 1, pp. 245-24 9,1986.8. Smith. 3. Boyle, J. Dongarra, B. Yarbow. Y. Ikubi, V. Kleema, and C. Moler,EISPACKGuide, vol. 6, se con de d. New York: Springer-Verlag. 1977.

Eigenvalue-Generalized Eigenvector A ssignment byOutput FeedbackB O N G - H W A N KWON AK D M Y U N G - J O O N G Y O U N

Abslmc r-This note generalizes the previous results of the closed-loopeigenstructure assignment via output feedback in linear multivariablesystems. Necessary and sufficient conditions fo r the closed-loop eigen-structure assignment by output feedback are presented. Some knownresults on entire eigenstructure assignment are deduced from this result.

I. INTRODUCTIONOne of the popular methods of modifying the dynamic response of alinear multivariable system is the placement, via linear state or outputfeedback, of the closed-loop eigenvalues at arbitrarily prescribed points inthe complex plane. Since Wonham presented the fundamental result [ l]on eigenvalue assignment in linear time-invariant systems, this problemhas generated a considerable amount of literature. Won hams result statesthat the closed-loop eigenvalues of any controllable system may bearbitrarily assigned by state feedback. However, in most practicalsituations the state is not available directly. It is desirab le to ind thecondition under which the system is eigenvalue assignable with incom-

plete state observation.The problem of simultaneous assignment of eigenvalues and eigenvec-tors (eigenstructure assignment) has received considerable attention [2]-[101. Most of the previous results for the eigenstructure assignment havesome limitations in the sense that eigenvalues of the closed-loop systemare distinct or different from eigenvalues of the open-loop system or thereis requirement of state feedback. To avoid a condition which eigenvaluesof the closed-loop system are distinct, Klein and Moore 121 havegeneralized the eigenstructure assignment in [3]. Fahmy and Tantawy [4]have generalized the previous results [ 5 ] , [6] to accommodate the casewhere the set of closed-loop eigenvalues and the set of open-loopeigenvalues have elements in common. Howeve r. these results cannot beused in the caseof output feedback. Kimura [7] has generalized hisprevious result in [SI. This result can also be utilized only in the casewhere eigenvalues of the closed-loop system are distinct and differentfrom any eigenvalues of the open-loop system. A lthough som e necessaryand sufficient conditions for eigenstructure assignment are available in[9], [IO], these results also handle the case where eigenvalues of theclosed-loop system a re distinct.In this note, a generalization of eigenstructure assignment by outputfeedback for linear time-invariant multivariable systems is presentedwithout using assumptions that eigenvalues of the closed-loop system aredistinct or different from any eigenvalues of the open-loop system. Thewhole procedure is attractively simple and provides more insight into theeigenstructure assignment. The presented method is illustrated bydesigning an output feedback regulator for a fourth-order two-input two-output continuous system.

11. EIGENSTRUCTURESSIGNMENTConsider a controllable and observable linear time-invariant system

i ( I ) = A x ( f ) + B u ( f ) (1 a)v(r)=W O , (1 b)

where x , u, y ar e n , nr , r-vectors, respectively, and A , 8, are realconstant matrices of appropriate dimensions with B and Cof full rank. If aconstant real output feedbacku ( f ) = K , ( t ) (2)

hlanuscript received July 14, 1986: revised Nobember 25. 1986.The authors are with the Department of Electrical Engineering. Korea AdvancedIEEE Log Number 8613531.institute of Science and Technology, Seoul, Korsa.

0018-9286/87/0500-0417$01.00 0 987 IEEE


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418 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO . 5 , MA Y 1987is applied to ( l ) , the closed-loop system becomes

i ( t ) = ( A + B K C ) x ( t ) . ( 3 )A set of complex numbers A is called symmetric if every nonrealelement of A is accompanied by its conjugate. Let A = {A, , . . . X,} be asymmetric set of complex numbers and let {dil i = 1 , .. ,s; s 5 n }be ase t of positive integer satisfying Cf= ,di = n. In [11] it is shown that if theclosed-loop system has s blocks of order dl , * .. d,, in its Jordan

canonical form, then therere s corresponding generalized righteigenvector chains defined by( A + B K C - A ~ I , ) U ~ ~ = O4 4

( A + B K C - A , Z , ) U , ~ = U , ~ _ , ,= 2 , ... d i . (4-b)Then, s corresponding generalized left eigenvector chains are defined asfollows:

t , > i ( A + B K C - A i Z n ) = O ( 5 4r , ; ( A + B K C - A , I , ) = t , ; , , ,= 1 , ..., d,- l . (5-b)

In the following, J , J r , an d J n - r re n x n, r x r , and (n - r ) x ( n -r ) Jordan cahonical matrices, respectively. The prime denotes thetranspose and given a comp lex vector or scalar u , 6 denotes the conjugateof u . A matrix V is defined as

v=rv,, 2, ... K1 (6)in which V , s an n X d; submatrix of the form

V ;=[u i l , ut23 " ' 9 h d i ]and similarly columns of matrices T , W ,Z are also composed of to, w " ,z& = 1, . . .s; j = 1, . -, i).Th e following theorem gives necessary and sufficient conditions for theexistence of K which yields prescribed eigenvalues and eigenvectors.Theorem i: The re exists a real matrix K such that for i = 1, ..

( A + B K C - ~ , I , ) V , , = U ~ ~ - ~ ,= 2 , - . e , d, (7-b)if and only if the following conditions are satisfied.independent in C" an d Ai = Xk mplies U, = ck j (j = 1, .. d,).that

1) The vectors in {v,jli = I , .. s; j = 1, . , d,} are linearly2) There ex ists a set of vectors {wi,li = 1 , . e , s; = 1 , .. d i ) such

where uio = 0.3) There exists a matrix 2 in C'"" such thatV - ' B W = Z ' C V . (9)

Proof (Sufficiency): From the conditions 2) and 3), on e can obtain. the following form:

A V - V J = - B W

Choose an output feedback gain K by

K =( C V ) ' [ C V ( C V ) ' ] - ' . (1 1 )From the condition l) , ua = Gkj implies w , ~ Wkj, which verifies that theoutput feedback gain (1 1) is a r e a l matrix. Her e, it can be shown using (9)an d (IO) hat the output feedback gain (11) satisfies ( 7 ) . Indeed, for K

given by (1 1)V - ' B K C V = V - ' B W ( C V ) ' [ C V ( C V ) '] - I C V

=z'cv(cv) ' [cv(cv) ' ] -~cv=Z ' C V= J - V - I A V .

Therefore, (12) shows that( A +B K C ) V = VJ ( 1 3 )

as required.eigenvectors given in [ I 11. Equation (7) ca n be written as(Necessity): The condition 1) follows from the property of generalized

which shows that the second condition is also satisfied. If there exists areal matrix K satisfying (7), then there exist generalized left eigenvectorchains

t : d i ( A + B K C - A i Z n ) = O (15-a)t G ( A + B K C - A , I , ) = t , ; + , ,= 1 , ..., d,-1 ( 15-b)

such thatT ' V = I n .

Equation (15) can be written equivalently as

wherez, = K ' B ' t , and = 0. hen, ( 1 4 ) and (17) give rise to thefollowing equations:A V -J =B W ( 1 8 )

T ' A - T' =- ' C . (19)Multiplying (18) on the left by T' and (19) on the right by V gives

T ' A V - T'V J= - 'B W (20)T ' A V - J T ' V =Z ' C V .2 1 )

Therefore, one can obtain from (16), (20), and (21) the follow ing relation:

Thus, the proof has been completed.1 is not required , and hence o ne can obtain the following result in [ 2 ] .a rea matrix K such that fori = 1, ..a, s,

In the case of state feedback, i.e ., C =Zn,he condition 3) of TheoremCorollary 1: Let the matrix Cbe the identity matrix. Then, there exists

if and only if the following conditions are satisfied.independent in C"an d A; = implies u,, = 17& = 1 , . . di).that

1) The vectors in {u,,li = 1, *..,s; = 1 , .. - di] are linearly2) There ex ists a set of vectors {wij(i= 1, * . ., ; = 1, .. di] such

where uio = 0.


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IEEE TRANSACTIONS O N AUTOMATIC CONTROL. VOL.AC-32, NO. 5. MAY 1987 419

[ A ' - A i I n ,' ]t , , + ~IB/such that

T V = I ,f , ~ k J , U,J= 61.1 if X,=

where u j 0 = = 0.Corollary 2 follows from the proof of Theorem 1. A special case ofCorollary 2 is proved by Kimura [7] under the superfluous additionalhypothesis that eigenvalues of the closed-loop system m atrix (A +BKC)are distinct and d o not include any eigenva lues of the open-loop system A .In the following. matrices Vo, Wo,To, and ZO re defined as

V o = [ VI , V2,pIW o = [ W , , w2, . . . I Wp1

T o = [ T p - 1 9 T p + z I .., K lzo=[zp+l,p-2, .. , Zs l

where V , s an n X d; submatrix of the formV , = [ U ; I , U~Z. . e t uidil

and similarly matrices c, Wi ,Z, have the same forms.Theorem 2: Le t A = A , U A 2 such that A, = { X I , . . p ) and A2 ={A,, . , A,} are symm etric with d; = r a n d If,,, d, = n - r.If A = A , U A2 exists such that there exist vectors u i j , fo r i = 1, . . ; j= 1, . . e , d ,and to , fo r i =p + 1, .-., ; j = 1, - . . , d, satisfying1) r; v, = 02) CVo s of full rank an d h, = x, implies u,, = lkj here

with u io = t jd i+ = 0, then there exists a real matrix K such that a set ofeigenvalues of the closed-loop system (A +BKC) is A, an d u,(i = 1,- . * , p ; = 1, ..., d,) an d t U ( i = p + 1, e . . , s; j = 1 , ..., d;)constitute corresponding generalized right eigenvectors and left eigenvec-tors, respectively.Proof of Theorem 2: From (23) and (24), we can obtain thefollowing equations:AVO- VoJ,= -BWO (25)

T i A - J , _ , T ; = - Z i C .Then, an output feedback gain

K = WO(CVo)-'satisfies the following equation:

T,'AVo-T,'VoJ,=T,'EWo, (29)T ; A V o - J , - , T ; V o =Z i C V o . (30)

Using the condition l) , one can obtain the following equation:T iB Wo=ZiC V o .

From (27) and (31), one can show thatTi ( A+ E K C ) = T , ' A +T , ' E W O ( C V O ) - ~ C

=T i A +ZdC=J n - r T i . (32)

Equations (28) and (32) show that the closed-loop system matrix (A +B K C ) ha s A as a set of eigenvalues and columns of Vo and To constitutecorresponding generalized right eigenvectors and left eigenvectors.A design procedure based on Theorem 2 for finding a desired feedbackmatrix K is given in the following.Step I : Find maximal rank m atrices

fo r i = 1, . *, ; k = p + 1 , .. s atisfying the following relation:[ A - L I " , Bl [SI, . ] = [ I n , 01 (33-a)

[ A ' - h k I n , c'] s k , Ek]=[Iny 01 (33-b)where Ni E C ( n rm ) xm ,, E C(n+m)xn ,k E CCncrJxrnd s k EC ( n + r ) x n ,

Step 2: Form the generalized right eigenvectors and left eigenvectorsf o r i = 1, . * - , p ; = p + 1 , - . . , s a s f o l l o w s :U ~ , = S ~ , U , ~ ~ ~ + N ~ , ~ ~ ~ ,= 1 , ... d, (34-a)

t k j=S]k tk j+ l+NI kPk~~, j = 1 , dk (34-b)whereu,o = t k d k + ,= OandvectorspJi = 1, . . - , s ; j = 1, . . * , d i ) a r eselected to satisfy conditions 1) and 2) of Theorem 2.

Step 3: Calculate vector chains as follows:W , = S ~ , U , ~ _ ~ + N ~ ~ ~ , , ,= l , . . . , p ; = 1 , ...,d , . (35)

Srep 4: Calculate the output feedback gainK = W o(C V O)- ' .

R e m a r k I : If the matrix C is the idenity matrix (in case of statefeedback), condition 1) of Theorem 2 is not required. The eigenvalue-assignability follows readily for this case. Assum e that A does not includeany open-loop eigenvalues. T hen, the following matrices:N , , = ( A , I , - A ) - ' B

N2, I,SI,= ( A J - A ) - I

s2j= 0satisfy (33). Equations (34-a) and (35) can be written as

V,I = N I , P ; I= ( A i I n - A ) - l E p , I

~ , ~ = S : , ' N l ; p i l +.. + N I ; p , ,=(-I)'-'(X,I"-A)-JBpil+..~+(h;z.-A)-'Bp,j,= 2 , ...,di

( A + E K C ) Vo=OJ,. (28) w V = p u , j = 1 , ..-, ,.


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420 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO. 5 , MAY 1987Noting that [6]- ( x , ) = ( - l ) X ( k ! ) ( X i l ~ - A ) ~ ( k i i B ,k = l , 2,d kd Xkwhere S(X) = (XI, - A ) - IB and using ( 36 ) , he following relationshipis obtained:

for i = l , . . . , s ; j = 1 , ..., d,.This is corresponding to the previous result [4] obtained using Brogansapproach [121.

m. NUhlERICAL EXAMPLELet us now consider an example not covered by the theory develop ed inConsider a system given bythe previous work.

0 1 0 01 00 1 00 0

A = O O O 1 B = 0 0 1 0 1 0 0 1

O O 010 1 0 0 Let a set of the desired eigenvalues be 11 = {- , - ) such that dl = 2an d d2=2. Then, elements of the maximal rank matrices can be found asfollows:

0 - 11 0 0 010 0 0 0

NII 0 0 1 0SI,=- 1 - 91 9 0 0 0 0

17 -300 0 1 - 2- 30 0 0 0

Nu= , s 2 2 =- 2 0 0 0 1 1 - 1 0 0 0 0

The generalized eigenvectors can be represented asU I I =NlIPII

~ I ~ = S I I N I I P I I + N I I P I ~f22 =NIZP22

t2I=SI2N12P22+~12P21.To select eigenvectors satisfying the conditions 1) and 2) of Theorem 2 ,we first check the condition 1) in the following. From the above equation,one can obtain

If pII an d pu are selected as p I 1= [O 11 and p22 = [O I], the aboveequation becomes zero. Then,t & u 1 2 = - I S + [ - 3 Olp,,

Therefore, p12 is selected aspi2 = [- 01 such that the above equationbecomes zero. Sincef;,U~~= 13+p;,[13 O],

similarly pzI = [- 1 01. Then, one can show thatf ; , u 1 2 = o .

Since Ti V, = 0, the condition 1) of Theorem 2 is satisfied. T hus, thegeneralized right eigenvectors and left eigenvectors selected as above

~ l l = [ - l 1 - 9 91V I Z = [ 0 - 1 - 4 -51r , l = [ - 1 7 1 1 -11 t = = [ - 3 0 - 3 2 - I ]

also satisfy the condition 2) of Theorem 2. Then,

so that the output feedback matrix determined by (36) isKI1-1419 -1 861

and therefore0 1 0 0

-14 - 6 1 0( A + B K C ) =-18 -1 8 1 0

which has the Jordan canonical form- 1 1 0 0

0 - 1 0 00 0 - 2 10 0 0 - 2

together with the generalized left eigenvectors v II, u 1 2 for A I = - 1, andthe generalized right eigenvectors f , , , f2 , for X2 = -2 , as required.N . CONCLUSION

In this note, a generalization of entire eigenstructure assignm ent byoutput feedback for linear time-invariant multivariable systems has beenpresented without using assumptions that eigenvalues of the closed-loopsystem are distinct or different from any eigenvalues of the open-loopsystem. Necessary and sufficient conditions show that the closed-loopeigenstructure assignment by output feedback is constrained by therequirement that the generalized right eigenvectors and left eigenvectorslie in certain subspaces. The presented method has been illustrated bydesigning an output feedback regulator for a fourth-order two-input two-output continuous system.

REFERENCES[ I ] W . M. Wonham, Onpole ssignment in multi-input ontrollable syste ms,[2 ] G . Klein and B. C. M oore, Eigenvalue-generalizedeigenvecto r assignment withIEEE Trans. A u t o m a t . Contr., vol. AC-1 2. pp. 660-665 , Dec. 1967.state feedbac k, IEEE Trans . Au to mat . Con t r . , vol. AC -22. pp. 140-141, Feb.1917.[3] B. C. Moore. Onhe flexibility offered by stateeedback in multivariablesystems beyond closed-loop eigenvalue assignment. IEEE Trans . Au tom at .[4] M . M. ahmyand H . S . Tantawy,Eigenstructureassignmentvia inear tate-Contr., vol. AC-21, pp. 689-692. O c t . 1976.

[SI M. . Fahmy and J . OReilly. Eigensuu cture assignment in linear multivariablefeedback cuntrul, I n t . J . Contr., vol. 40 . no. I . pp. 161-178. 1984.


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TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32, NO. 5. MAY 1987 421systems-a parametric solution , IEEE Trans. Automa t. Contr., vol. AC-28, pp.990-994. Aug. 1983.161 -. On eigenstructure assignment in linear multivariable systems, IEEETrans. Automa t. Contr., vol. AC-27. pp. 690-693, June 1982.[7] H . Kimura, A further result on the problem of pole assignment by outputfeedback, IEEE Trans. Automat.Contr. , vol. AC-22, pp. 458463, June1917.[SI -. Pole assignment by gain output feedback. IEEE Trans. Automat.

[9] L. R. Fletcher. An intermediate algorithmfor pole placement by output feedbackCont r . , vol. AC-20. pp. 509-516. Aug. 1975.in linear multivariable control systems. Int. J. Contr. , vol. 31, no. 6, pp. 1121-1136, 1980.-, Some necessary and sufficient conditions fo r eigenstructure assignment,Ini. J . Conir. , vol. 42 . no. 6. pp. 1457-1468, 1985.I I ] C. T . Chen. Introduction to LinearSysrem Theory. New York: Holt. Rinehartand Winston, 1970.

W . L. Brogan. Applications of a determinant identity to pole-placement andobserver problem. IEEE Trans. Automat. Contr. , vol. AC-19, pp. 612-614,O c t . 1974.

Jordan Pair Assignment Via State FeedbackT A L A L M . BAKRI

Abstract-This nole considers the assignment of the eigenstructure ofThis assignment is achieved bygning the Jord an pair of the denominator matrix polynomial of thee MIM O system via state feedback. It turns out that forropertake into account the almost o rgotten nfinite

of the system, even though they are invariant underof the eigenvectors.

I. INTRODUCTIONThe notion of eigenvector assignment is recent [2]-[4]. Based upon the

(A , B)-invariant subspaces, one can design the

on e can assignpair of the denom inator atrix polynomial [ 5 ] .The approach isinof eigenstructure assignment. The method is motivated by the[ 5 ] .Thispair (r, J ) to the

II. P R E L I M I N A R E SConsider the regular m X m matrix polynomial +(X) given below

d9 0 ) = 9rhP (2.1)P = O

d is degree of the polynom ial. I f the leading matrix Coefficient#d istheQ0 s the identity matrix, then the matrix polynomial iscom onic. If the matrix coefficie nt &has full rank, then the degree@(X) is md where det (.) is determinant of (.)I an d

+(X) are finite. On the other hand, if $d is+(X) is n, hen @(X) will have n -md - n infinite eigenvalues (see [ 5] for more

Manuscript received October 20, 1986; revised January 8, 1987.The author is with the University of Petroleum and Minerals, Dhahran, Saudi Arabia.IEEE Log Number 8613846.

In order to determin e a matrix polynomial 4 (h ) uniquely one has toconsider together the finite and the infinite Jordan pairs. The finite andinfinite Jordan pairs satisfy a certain relationship and constitute adecomposable pair (see [5 ] for definition of decomposable pairs). Thefollowing lemma chara cterizes the finite and infinite eigenvalues of 40,).Lemma 2.I is/: Let $(X) be a regular matrix polynomial, and let ( I F ,J F )and rm,, ) be it s finite and infinite Jordan pairs, respectively. Then( [rF,], JF o J ,) is a decomposable pair of $(X).

ID. PROBLEM FORWJLATIONND SOLUTIONConsider the following state equation:i ( t ) = A x ( t ) + B u ( t ) ,s o (3.1)

where A is an n x n real matrix, B is an n X m real matrix. Th e pair ( A ,B ) s assumed to be controllable w ith controllability index d. Th e transferfunction of the above system is given byT(X)=(XI-A)-B=N(X)D-(X) , (3 2 )

where(3.3)

The matrix polynomial D(A ) s the denominator matrix polynomial of thematrix fraction description of he system (3.1). The degree of thepolynomial det D(h) is equal to n, theorderof the system. Theeigenvalues of D(X)are the p o l e s of the ystem (3.1). If he controllabilityindexes of the system (3.1) d l , d,, . . , dm are thesame, hen hecoefficient matrix D d is nonsingular and this implies that the order of thesystem n s equal to md and also alleigenvalues of D(X)are finite. On theother hand, if the controllability indexes are distinct (at least one of theindexes is different from the others) hen n of theeigenvalues of D(X)arefinite and (md - n ) eigenvalues are infinite.Suppose it is desired to use state feedback to relocate the poles of thesystem at certain desired values, the form of the feedback control is:u ( t ) = F x ( t ) + u ( t ) (3.4)

where F is the matrix feedback gain that assigns the desired poles to bedetermined.With feedback law (3.4), the state equation of the resulting system isi ( t ) = ( A + B F ) x ( t ) + B u ( t ) (3.5)

and the transfer function of the resulting system isT,(X)=(hl -A-BF)- B=N(X)[D(X) -FN(X)~-I . (3.6)

From (3.6), the denominator matrix polynomial isd - I$ (X )=D(X) -FN(X )=C ( D ; - F N ; ) A i + D d X d . (3.7),= O

From (3 .7 , it is clear that state feedback does not affect the mat r i xleading coefficient Dd of the resulting denominator matrix polynomial,this in turn implies that state feedback can only relocate the finiteeigenvalues of the denominator matrix polynomial. Suppose it is desiredto assign the finite Jordan pair (r, ) o the resulting denominator matrixpolynomial #(A) of (3.7), then from Section II the pairs (r, ) an d r-,J,) have to satisfy the following two conditions:i) The pair [(r, -), ( J @ .)] is a decom posable pair

ii) $ ( r J ) = CD , - F N , ) I J + D r d J d = O ,d - I,= Od - 1 ( D ; - F N , ) r J ~ - - + D r d = O .i - 0

0018-9286 /87 /0500-021$01.~0 98 7 IEEE

eigenvalue assignment

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