8/2/2019 Poles, Zeros, And Feedback State

1/7

Poles, Zeros, and Feedback: StateSpace Interpretation

ROGER W . BROCKE?"T, MEMBER, IEEEAbstract-This paper is concerned with the relationships between

timeand requency domaindescriptions of linear, ime-invariantsys tem s and with the evaluation of the effects of feedback on suchsystems. A new expression for the transfer functionof a system de-scribed by a set of first-order differential equat ions is given; th isexpression not only relates the poles and zeros to the eigenvaluesofmatrices but also makes it possible to compute the transfer functionwithout matr ix inversion. Th e effects of state variable feedback oncontrollability,observability, and pole-zeroconfigurations are dis-cussed and the effects of feeding back the output and its derivativesare considered. Th e application of these ideas to an optimal controlproblem is sketched and methods of extending them to the multi-input, multi-output case are examined.

IKTRODCCTIONT T AN NOT be disputed that state space methods

an d x is the state vector. Its assumed that u and y arevectors of the same dimens ion, say nz., nd that x is avector of dimension n. Particular mphas is will beplaced o n the case 7% =1; boldface notation mill not beused for u , y and D in equations which are valid onlyin thiscontext,an d B, C , nd D will bewrittenaslo\\-er-case let ters .

Transforming (1) into hefrequencydomain, t be-comes

(IS- A ) i = BO + ~ ( 0 ) ( 3 )where the circumflex is used todenote ransformedvariables. By solving (3) for 2 and using ( 2 ) , it is seenthathave contributed greatly to modern control theory;

however, a t present the majority f design problems y = C(Is - A)-'BO + D& + C(Is- A)-'x(O). (4)are solvedusing frequencydomainmethods,and t I n th eevent hat x(0) iszero, he requencydomainseems doubtful hat his situat ion will change in the relationships bet\\;een 6 an d 2 and y simplify onear future. The result is that systemngineers must becapable of thinking and communicating both in termsof first-orderectorifferentialquationepresentations 7 = (C(Is- A)-'B + D ) O . (6)

-~

2 = ( I S- A)-lBO (5)and in termsof transfer functions and their related con-cepts. In he ast few yearsconsiderableprogresshasbeen made i n clarifsring the connection between hesetwo points of view. The basic questions associated withthe const ruction of first-order representations have beentreated a t length yKalman [ l ] , Gilbert [ 2 ] , andZadehandDesoer [ 3 ] ; n addition,moresubtleques-tions relating to multivariable and time-varsring systemshave received some attent ion. However, i n spite of therapidly xpanding iterature on the tate pace p-proach? a number of issues fundamental to the contro lproblem have been neglected. The object of t h i s paperis to f i l l in some of these gaps.

PKELIMIN-4KIES AXD NOTL4TIOSConsider the class of linear, ime-invarian t systems

which can be described b): the pai r of vector equatio nsx=Ax+Bu (1)y = Cx + Du. (4

Here u an d y are the input and output, respectively,Manuscript received August 10, 1961; revised Jan uar y 1, 965.

Space Administration under Contract So . NsG-496 with the 1 , f . I . T .This research was supported in part by the Kational Aeronautics andCenter for Space Research, Cambridge, YIass., and in pa rt by theOffice of Xaval Research under Contract NOSR 1141(12), SS-stemsResearch Center, Case Institute f Technology, Cleveland, Ohio.The author is with the Dept. of Electrical Engineering. XIassa-chusetts Institute of Technology, Cambridge, Mass.

This places in evidence the f act that the trans fer m atrixrelating the input to the state vectors (Is-A)- 'B, andthat the transfer matrix relating the input to the outputis C(Is-A)-'B +D.Since the inverse of (Is- A ) can be expressed as theadjoint of ( I s -A) times the reciprocal of the dete rmi-nant of ( I s - A ) , and since the adjoin t of ( I s - A ) hasno poles, it ol lo m ha t he poles of these ransfermatrices must be zeros of d et ( I s - A ) , i.e., the eigen-values of A. Tha t is not to say that every eigenvaluefA is necessarily a pole; unless further assumptions aremade, there s no guarantee that somef the eigenvalueswill not be "canceled o u t . " The additional assumptionswhich ar e require d are that the system should be bothcontrollable and observable. That is,t is necessary thatth e tzXnnz matrix (B, B , . . . A"-'B), xvhose columnsare the columns of the matrices B? B , through AnP1B,should be of rank n, and i t is necessary that the mn X nmatrix1 (C; A ; . . . CA"-l) n-hose ro\\-s are the rowsof thematrices C, CA, through CAnP1 houldbe ofrank n. Th e role of these assumptions has een discusseda t length elsewhere [11- [4].

I t is of some interest to note thatt is possible to givean lterna tive haracte rizatio n of controllabilitl;n

( M ;N) o denote what Lvould ordinarily be written as (W , T j r1 Thenotation used here is not st andar d; hoLvever, the use ofdoes make things neater andeems to be ustified.129

8/2/2019 Poles, Zeros, And Feedback State

2/7

130 IEEE TRANSACTIONS ON AUTOMATICONTROLprilterms of the in put-to-state transfer matrix. I t is shownin [ 4 ] t ha t a necessary nd sufficient condition or(B, B , . . . An-IB) to be of ra nk n is that ther e sho uldbe o onsta nt nonzero row vector N such thatH(Is-A) - lB is zero. Itmay lso e hownhat( C ; C A ; . . . CA"-') is of rank n if and only if thereexists no constant nonzero column vector H, such thatC(Is-AA)-'H is zero. Asimilarobservationhasbeenmade by Butman and Sivian [SI.

Although there are many ways of represen ting sys-tems witha given matrix transfer function in state vari-able form, Kalman [ l ] has shown that all controllableand observable representations of a given matrix trans-fer function are of t he sam e dimension. In the single-inpu t, sing le-ou tput case, this is the degree of the de-nominator of the transfer function after common factorshav e been canceled. Th e following lemma is an immedi-at e consequence of this result.L e m m a 1If y an d u are scalars related by (1) an d (2), and if

these equations are controllable and observable, henevery eigenvalue of A is a pole of c ( Is -A) - lb+d , an dthe ord er of eachpoleequals hemult iplic ity of theassociatedeigenvalue.Proo f : Since the denominatorof the transfer functionis the determin ant of I s - A , it follows that poles mustbe eigenvalues of A and that no pole can be of higherorder than the multiplicity of its associated eigenvalue.If some pole is of lower order than the multi plici ty ofits associated eigenvalue, then the denominator can beof a t most degree n- 1. This means that the originalsystem has an n- dimensional realization which con-tradicts the assumption that the system s controllableand observable.

THE NVERSE QUATIOXTh e problem of obtaining a convenient expression for

the zeros of the transfer functionc ( Is -A) - lb+d in thecase where u and y are scalars leads directly to that ofderiving a first-order representation for a systemwhosetransfer function is the reciprocal f the given one. Th issystemhasbeenstudiedpreviously [6] inconnectionwith the controllability propertie s of dy namic syst emsand is closely related to the inverse system discussed byZadeh and Desoer [S I .By repeated differen tiation of (2) one can , bs; using(1) to eliminate x , arr iv e at the following set of equa-tions (y") = d i y / d t i ) .

y'0' = cx + dZC'0'= cAx + cbu(O)+ dztc')

> I ( ? ) = cA2x + c f i ~ ( 0 ) cbu(')+ d ~ ( 2 ). . . . . . . . . . . . . . . . . .

y ( n ) = C A " . ~ cAn-'bu(O)+ cAn-?bzC(1)+ . . cbu(n-1) + dZG("). ( 7 )

Consider hesequ ence of scalars d , c b , - cAn-'b. Ingeneral, it cannot be assumed that is nonzero, or thatcb is nonzero, etc. However, if all the term s in this se-quence are zero then clearlyu has not effect on y, and,hence, the tra nsfer funct ion s zero. Assu ming t hat thi sis not the case,efine the numbers y an d p as

cAi-lb # 0, and cAjb = 0 (8)f o r O < j < i - - l

4-1 = i f a = O(,A", if a # 0. (9)

I t will be shown that bothand q have a simple inter-pretation in terms of the tran sfer fu nctio n. In fa ct,a isthe difference n the degree of the nu mera to r an d thedenominator polynominals and

q - I = lim s aG ( s ) .S -We will call a he relative order of the sys tem ; it occursfrequently in the sequel.

In ter ms of this notation it is seen that u is given byq y ( " ) pcAax. Substituting his nto ( l ) , thepair ofequations

are obta ined . This pair defines wha t will be called theinverse system. Since the input to the inverse system isY ( ~ ) ather than y itself, this is a s lightly different termi-nology than tha t used by Zadeh and Desoer [SI.

By aking he ransform of (10) andby using thetransform of (11) we arr ive at thefrequencydomainequation

Since the zeros of the tra nsf er fun ction/,Q re the polesof the ran sfe rfunction z2,/9, and since the atter areeigenvalues of A - bqcAa, we see th at th ezeros of theoriginal system must be eigenvalues of A - bqcA". Thisleads a t once to the identityc(I$- A)-% + D

= 4-l det (Is- A + bqcAu),'su de t (Is- A) . (13)4lthough thisexpression may appea r to be co mplic atei t is, in fact, an extremely convenient formula for thetransfer function, because it does not require matrix iversion. Since the characteris tic polynomial of an n x nmatrix isof degree n, this identityplaces in evidence th

8/2/2019 Poles, Zeros, And Feedback State

3/7

I965 Brockett: Poles, Zeros, and Feedback 13 1fact that the numerator and denominato r polynomialsare of order n- r and n , espectively. From (12 ) we seetha t q has he nterpretation given above.

-4 second important point is that a t le ast one term nthe sequence dcb, . . . cAn-'b must be nonzero if thetransfer function is to be nonzero. Otherwise stated,t isnecessary that ( d , cb , cab, . . . cA"-'b) shouldbe ofrank 1. This condition also arises in the study of o utp utcontrollability ( 7 .Thus far it has been shown that a zero must be aneigenvalue of A - qca". I n order to say more, thats, i norder to estab lish some so rt of converse sta tem ent , i t isnecessary to restr ict the discussion to controlla ble andobservab le systems. In this case the following theoremdescribes the pole-zeroconfiguration.The ore m 1

Suppose that the system efined by (1) and (2 ) is con-trollable and observable and that ZI and y are scalars sothat he ransfer unction c ( I s -A ) - l b+ d is also ascalar. Then the poles of the trans fer func tion are theeigenvalues of A , each pole being of the same order asthe muItiplicit>F of thecorrespondingeigenvalue. Thezeros of the rans fer unct ionare heeigenvalues ofA-bqca", the zeros not at the origin being of t he sa meorder as the multiplicity of the associated eigenvalue,and the zero at th e origin being of or de rj -a wh er ej isthe multipli city of the eigenval ue0.

P roo f : The statement about the poles is simply a re-sta tem ent of the lemma. The result relat ing to the zerosfollows from (13) and the f act that since the originalys-tem sbothcontrollableandobservable, hedenom-inator and numerator have no common factors.n orderto get a first-order representation for the inverse systemwe had to "multiply" the transfer function by sa. Thisexplains why the multiplicity of the eigenvalue at theorigin is a more than the orderof the zero at th eorigin.

-4 second result on the inverse system which is of in -terest is given by the following theorem which assertsthat the controll ability properties of the inverse sy stemare the same as those of the original system. (Keep inmind that y ( " ) is being regarded as the input to the in-verse system and not y. )Theorem 2

The inverse system is control lable if and only if theoriginal system is controllable.P r o o f : I n this proof, and in several of thos e to follow,some use is made of the f act t hat a n n x m matrix is ofrank n if and only if there exists no nonzero t z dimen-sional row vector H , such that when H i s premultipliedby the matrix the result is zero. An equivalent state-ment holds for column vectors and postmultiplication.For a discussion of these results see a bookon inearalgebra such as Halmos [8].

Returning now t o the proof, assume th at th e originalsystem s controllable so tha t ( b ,A b . . . A"-'b) is of

rank 12. T o show that his implies that hematrix(qb(A- bqca")qb, . . . ( A - bqca")"-'gb is of rank n ,notice tha t if this latt er m atrix were not of ra nk n thenthere wouldexist anonzero owvector h such hath(qb, ( A- bqca")qb,. . . ( A - bqca")"-'qb)=0. By ex-panding this i t may be seen that this can hold only ifh ( b , a b , . . an-'b) O . (Recall that p is a scalar andthus can be factored out.) This last statement contra-dicts the assumption that the original system was con-trollable and thus establishes the "if" p art of the the-orem.

The c onverse is more direct; all t ha t is required is toshow that f the original system is not controllable, thenthe inverse system is not either. If the original systemis not controllable, then there exists a nonzero row vec-tor h such thath ( b ,a b , . . . An-'b) is zero. This cle arlyimplies that h(qb, (A-bqcAQ)qb, . . ( A - bpcaa)n-lqb= O and hence that the inve rse system is not controlla-ble.

I t should be noted that n o such result holds for ob-servability. T o see his consider he following counterexample. Consider a system of the form

which has a transfer function c(1s -A j-'b = s,+2),:'s ( s+ l ) . Th e associated inverse system is

A quick check shows that the original system is bothcontrollable and observable but that the inverse systemis notobservable.The difficulty stem s from the actthat theoriginal system has a pole at th eorigin. I f thereare no such poles, i.e., i f A is nonsingular, then under theassumptions given the inverse system will be observa-ble, although th is will not be proven here.

LINEARTATE ARIABLEEEDBACKIf all the stat e variab les can be measured, and if the

plan t is completely known, then an optimal controllercan frequently be realized as an inst antaneous functiongenerators whose inputs are the state variables. For thisand other reasons i t is of int erest to explore the effectsof state va riab le fe edba ck n various syst em properti es.Th e discussior, will be limited to linear time-invariantfeedback.

8/2/2019 Poles, Zeros, And Feedback State

4/7

132 I E E E T R A N S A C T I O A r S ON A U T O M A T I CO N T R O Lp r i lTh e effect of linear state variable feedback on sys-

tems of the form given by (1) and (2) is to replace u byu-Kx where K is an n x n matrix of constants , the j t helement being the gain between the jt h s ta te variableand the ith i nput. Th e closed -loop eq uations of motionare, herefore,

X = ( A - 3 K ) x + 3 u ; y = CX+ Du. (16)I t is well known th at feedback can alter pole oca-

tions; i n fact i t will be shown t ha t if the system is con-trollable, henstatevariablefeedbackcanbe used toachieve a n y desired pole configuration consistent withthe dimension of t he s!-stem. The following theorem as-sert s that in spite of th is linear state variable, feedbackdoes not affect the controllabi1it)- of the sys tem.Th eo r em 3

If linear state variable feedback is applied to the sys-tem (1 ) - (Z ) , then the closed-loop system (16) is control-lable if and onlyf the open-loop system was controllable.P r o o f : If the open-loop system s controllable then foreach initial state x0 and each desired final s ta te x,-, thereis a n inpu t which drives the s)-stem from x. to xi. Letu' denote this input and let' denote the correspondingresponse.For heclosed-loopsystems, hesamestatetransfer can be obtained using the input u =u'+kx'.Thu s the closed-loop system is controllable if the open-loop system is. AIoreover, if the open-loop q-stem s notcontrollable hen heclosed-loop ystemcannotbeeither; any transfer executed b>* the closed-loop systemcan be accomplished by the open-loop system simply bychanging the input to u' -kx' .To see that state v ariable feedback can affect ob serv-ability consider a single-input, single-output s>-stem ofthe form

3 = 11[;;I .This system is both controllable and observable and hasa ransfer unction of theform(2~-33)/(s-l)(s-2).If state variable feedback is applied through a k of theform (0-0.5) then the equat ions of motion become

For the closed-loop system we have

1 1( c ; ca> = [1 ,] (19)and , hus, he s!-stem is not observable. The ransferfunction associated with (18) is 2(2 ~-3 3)/ (~- 11) (2~ -3)the reason observability was lost is that the additionoffeedback moved a pole to cancel a zero.

Sotice that the cancell ation of a zero by using feed-back is not possible if just the output is ed back; this isclear from the propertiesf the root ocus. It is appare nt,the refo re, that one has considerably more flexibility infixing the closed-loop poles if the ent ire setof state vari-ables can be fed back. Questions which suggest them-selves are: "Can zero locations be moved by the appliction of state variable feedba ck?" and, "are here anyrestrictionswhatever on the pole configurationwhichcan be achieved using state variable feedback?" In thespecialcasewhere u and y arescalars he ollowingtheorem resolves hese questions.

Theorem 4Suppose that (1) and (2 ) are controllable and observ-able and that u and y are scalars. In his case, inears ta te variable feedback can be used to obt ain any de-siredpoleconfigurationwhich sconsistentwith hedimension of th e system. Any number of the open-loopzeros may be cancelled bu t no new zeros can be intro-duced.

Proof : I t has been shown [ l ] hat by a change of co-ordinates any single-input, single-output system whichis both controllable and observable can be put in th eform

3 =

+

Suppose that the characteristic equation associated wthe desired pole configuration is s~+T , - I s " - ~ . - r l s f r o .If k is taken to be ( Y O - P o , Y 1 - f i 1 , * * . rn - l - f l n -1 ) thenan easy calculation shows that the closed-loop sys temwill have the desiredpole configuration.

From this i t is obvious tha t an y or all of the existingzeros can be cancelled. To see th at no new zeros can beintroduced is more difficult. Fi rs t it will be shown th at

8/2/2019 Poles, Zeros, And Feedback State

5/7

1965 B roc k e t t : Poles, Zeros, and Feedback 13 3the relative order of a system is not affected by feed-back, To see this consider the sequence, c b , c ( A - b k ) b. . . c ( A - b k ) n - l b which determines the relative orderof the system with feedback. f the relati ve o rderof theopen-loop system is zero or one, then it is clear that therelative order of the closed-loop syste m is the same; ifthe relat ive order of t he open-loop system is a > 1, thencAib=O fo r O < . i < o r - l . Since ( A - b k ) i canbe ex-pandeds Ai-Ai- 'bk-bkA'- '+ . . . A ( b k )5 (bk) i - 'AT ( b k ) , i t is clear th at c ( A- k );b vanishesfo rexactly hesame indexes as cAib doesand that

The nversesystem associatedwith he closed-loopc ( A - b k ) " b = c A " b .system is, therefore, given by

= ( A - bk - bqc(A - bk)")x+ bgy'"' (21)u = - pc(A - bk)% + q I J ( " ) . ( 22 )

Fromou rprevious emarks t s seen tha t he dif-ferential equation can be simplified to

X = ( A - bqcila)x + bpy'"' (23)and, hus, heeigenvalues of the closed-loop nverseequat ion a re the same a s those f the open-loop systemand no new zeros can be introduced.

I t should be pointed out that niorga n [9]-[lo] hasobtained a result similar to Theorem 4 using somewhatdifferent methods.

DERIVATIVEEEDBACKInmostsituations t s mpossible omeasure he

state variables directly; instead they must be calculatedfrom the terminal variables u and y . If the system hasno zeros then y and its first n- derivatives constitutean admissible set of st ate vari ables , and eve n though itmay be difficult to me asure these accura telyn practice,i t is clear that the derivative s actual ly exist providedtha t is at eastbounde d nd piecewise continuous.Wha t is more important, even f the pole locations shift,the indicated derivatives still exist, and y and its firstn- 1 derivatives summarize the state of the system.

Contrast thiswith hecasewhere hesystemha szeros, th at is, the case where a#%- 1. From (7) we seethat in this case y ( i ) will not necessari ly exist if ;>orunless some differentiability conditions are imposed onu. In this case exactly elementsof the state vector canbe obtained fromy and its first a- derivatives, the re-maindermustbegeneratedbydifferentiating inearcombinations of u, , and the derivatives of ZL an d y .Unfortunately, if changesoccur in the pole-zerocon-figuration, the linear combinationsf variables that oneis attempting to measure may no longer be state vari-ables, and, hence, their derivatives will not necessarilyexist. Thus, large errors may be introduced if one at-tempts to feed back more than the first - derivatives

of y unless the equat ionsof the pla nt are nown exactlv.In view of the acute sensitivity problems associated

with the generation of the complete set of s tat e va ria -bles, it is of some intere st to explore th e effects of feed-ing back just those state variables wh ich can be gen-eratedaccuratelyeven if smallchangesoccur n heplant, i.e., y an d its first a- 1 derivatives. Apparentlynot a great deal s known about the effects of such feed-back on the pole locations. It seems clear that the type sof configurations which canbeachieved will becon-stra ine d but it s difficult to see exactly how. The sit ua-tion with respect o zero cancellation is quite simple,however, and differs from th at described by Theorem 4in that no cancellation can take place. This is an imme-dia te consequence of the following theorem which assertsth at if only the first or-1 derivatives are fed back, theclosed-loop system is both controllable and observable.Th eo r em 5

Suppose that (1) and (2) are controllable and observ-able and hat 21 and y are scalars. If a oop s closedabou t th is sy stem by eeding back a linear combinationof y and its first a - 1 derivatives, then the closed-loopsystem will be controllable and observable .

P r o o f : From (7) i t is clear that y and its first a - 1derivatives can be generated from he state vector x ,and, thus, the controllabi1it~- f the closed-loop systemis an immediate consequence of Theorem 3 .

Supposeha theeedbackignal is aoy+aly(l)+ . . - so thathe closed-loop equa tions ofmotionare

= Fx + bzt;F = A - aabc - albcA - . . al;-lbCA"-l (24y = cx + dl.1. (25)

I f this system were not observable, hen here wouldexist a nonzero column vector Nsuch that ( c ; c F ; .cFn-l)h=0.Since cAib is zero for 0 < i < k , it followsthat this implies that ch = cAh= - - ' cA"-lh = 0, and,hence, it would follow th at th e original system was notcontrollable.This contradiction establishes the theorem.

A CANONICALORXOR OPTIMALCONTROLLERSRecently herehasbeenconsiderable nterest in avariant of the time optimal roblem in which the object

is to take the out put to some point, say zero, and okeep it there for some period of time, or for all futuretime. This differs from the usual problem in t ha t it isnot necessary to hi t a specific point in t he state spa cebut instead one may hit any point in state space forwhich y is zero, provided only th at it s possible to keepy a t zero for the desired period of time. A little thoughtis all tha t is necessary to convince the read er tha t f the

8/2/2019 Poles, Zeros, And Feedback State

6/7

134 IEEE T RA N S A C TIOLYS OLV A U T O M A T I C C O N T R O L A p r i lsystem is linear and constant and if ZL is amplitu de lim-ited, hen hisproblem sequivalent o hestandardtime optimal problem prozlided th e sy s t e m has n o zeros.A discussion of th is can be found in [11]-[13].If zeros are present the solu tion f this problem is sig-nificantly different in tha t the valu e of th e in pu t z1 for tlarger than the response time t* is not zero and can notbe generated by an ideal relay. I t can, hon-ever, be gen-erated as a homogeneous solution of the inverse equa-tion. This last point is fairly clear if the inverse s>-stem(10) is examined with t h i s problem i n mind. In fact, i f

_ -

and only if the system was output-controlla ble, i.e., if(cb, cAb, . . . cAn-'b) was of rank 1. In the multivari-able case the situation is more complicated. In the caseD=O the necessar>- and sufficient cond itions or hetransfermatrix t o benonsingular equ ivalent o heconditions under which the inverse s\.stem exists) havebeen given previously [4]. A trivial modification of thederivationgiven hereestablishes hat ngeneral henecessarJ- and sufficient conditions for the inverse equa-tion to exist is that the m a + m by 2nrn+vz dimensionalmatrix

. . . . . . . . . . .10 0 0 . . . D

we require y to be identically zero then ( ") must vanishalso, and, hence , we see th at for some choice of x(t*) th eoptim um value of ~ ( f ) or t>t* is given by

= ( A- bqcAa)x (26)u = - qcAax. (27)

Thus,after the end points reached the optim um inpu tcan be generated bs; the inverse equation. Before thistime, of course, the input can be gene rated by an idealrelay driven by the adjoint system. The general optimalcontroller,shown i n Fig.1, herefore,cons ists of th eadjointsystem in the eedback loop plzrs the nversesystem operating in a feedforward mode to gener ate thevalues of u ( t ) for t > t * . *Additional materialand ex-amples may be found i n [6].

I N VERSESYSTEM

I ADJOINTSYSTEMFig. 1. -4 canonical form for optimal controllers.

~'IULTIVARI.L\BLESYSTEMS-Although some of the results and methods of the pre-

vious sections carry over o multi-input multi-outputsystemswith ittlemore hanachange of notation,othersdonot. I t seemsworthwile to ndi cate brieflyhere wh at he difficulties eem to be. Only hecasewhere the numb er of inpu ts equa l the numberf outputswill be considered here.

Th e first question which arises relates o th eexistenceof the inverse equation. In the single-input, single-out-put case i t was shown t ha t th e inverse system existed if

. . . . . . . . . . . .CB . . . CAn-'B

should be of ra nk mn+nz.,Assuming that hiscon dit ion is fulfilled, i t will be

possible to calculate the inverse system for any specificsystem although giving a general formula seems to beimpossible. Th e difficulty here stem s f rom the fact tha tin themultivariablecase,solving (7 ) for zz is compli-cated by the fact that the first matrix in the sequenceD, B, CAB, . . . CAn-'B which snonzero will no tnecessarily be nonsingular. If it is not, then the numbercy loses much of i ts significance unless onegives it amatrix nterpretation.

In the multivariable case the eigenvalues of the in-verse equationdo not correspond to theeros of particu-lar transfer functions. but ratherhe>- correspond o th ezeros of the determ inant of the transfer matrix. It hasbeen shown tha t t hese zeros play a fu ndam ental role inleast-squaresoptimization heory 14],and hat heyare impor tant n determining f a plant can be decoupledby state variable feedbackl j ] . iotice , however, th at itwill be impossible to decouple a multivariable plant bya n y means unless he matrix M defined by (28) is ofrank mn+m and that this is also a necessary conditionformostmultivariable rackingproblems to haveasolution. COSCLVSIOXS

The object of this paper has been to consider severalproblems related to the relationship between state vari-ableand ransferfunctionrepresentations of systemsand to examine theffects of feedback on certa in systemproperties. X convenient expression has been given forthe zeros of a transfer function and the effects of s ta tevariable feedback on controllability, observability, andthe pole-zero configura tion has been examined. An in-verse s!-stem has been defined and someof its propertieswere examined; a possible applic ation in the imp lemen-tati on of optimal control laws was indicated. In the inalsection the extension of these results o the multivariablecase was briefly examined.

8/2/2019 Poles, Zeros, And Feedback State

7/7

1965 IEEE TRANSACTI0:YS OLYA U T O M A T I CO N T R O L 135ACKNOWLEDGMENT

The aut ho r would like to thank Prof . AI. D. Alesaro-vic of Case Inst itu te of Technology and Prof.31. -4thansof the AIassach use tts Inst itut e of Technology for theirhelpfulsuggestions. He would also ike to han kDr.B. S. AIorgan, Jr . of t he Xir Force Offce of ScientificResearch for providing him with a cop). of his thesis.

REFERENCES[ l ] Kalman, R. ., The mathematica l description of linear d5mam-ical systems, SIa41f . 012 Con t ro l , vol 1, 1963, pp 152-192.[2 ] Gilbert, E. G., Controllability and obsena bilitl -n multivariablecontrol systems. SI ;2X J . o n Control. vol 1. 1963. DD 128-151.[3]Zadeh, L . -4. andC. .A. Desoer, Lil-zear System i l ; eov~ , . SewYork: lIcGra\\--Hill, 1963.[4] Brockett, K. \V. an d \ I . D . llesaro vic, The reproducibi lity ofmultivariable sl-stems, PTeprinls Join t 4 u t o m a t i c Control Con-ference. Stanford, Calif. , Tun 1964 (to appear in Xath. A n a l y s i s- _ .aud A p p l i c a t i o n s ) .[j ]Butman, S., and R . Swan, On cancellations, controllabi1it:- andobservabilitv. IEEE T r a n s . o n Automat ic Control. 1-01 XC-9.J u l 1964, pp 3177318.161 Brockett, R. \. , The n\-ertabi lity of dynamicsystemswith

application o ontrol,Ph.D.dissertation,Case nstitute of[7] Kreindler, E., an d P. E. Sarachick, On the concepts of controll-Technology, Cleveland, Ohio, 1963.abilityandobsen-abili ty of linear ystems, IEEE Trans . O R[8] Halmos, P.R., i d e Di?nensional Vecto? Spaces , Princeton,Azrtonmtic Control, vol .IC-9, Jan 1964, pp 129-136.[9] Morgan, B. S., he synthesis of linear multivariable systems byX. J. :Van Sostrand, 1958.stateariableeedback,h.D.issertation,University ofMichigan, .Ann .Arbor, 1963.[lo] klorgan, B. S., The spthesisof single variable sg-stems by statevariable eedback, Proc. Al le r ton Conj . o n Ciycnit a n d System

[ l l ] Lee, E. B., On the time-optimal controlof plants with numeratorTheory . I-niversity of Illinois, L-rbana. Xov 1963.dgamics , IR E Trans . o x dnttowzatic Coxtrod, vol AC-6, Sep 1961,pp 351-352.[12] Athanassiades, h.1.: and P. Falb, Time optimal control for plantswith numeratordynamics, IR E T r a m . o n Automatic Control ,[13] Harveg-, C. Determinmghewitchingriterionorime-optimal control. J . o j X u t h A n a l y s i s a n d .4pplications, vol 5,[14] Brockett, I