Decoupling and Pole Assignmentin Linear Multivariable Systems:

A Geometric Approach

W. M. WONHAMAND A. S. MORSE

IN the theory of isolated dynamical systems (flows on mani­folds, etc.) the notion of an invariantset serves a crucial role forclassifying the nature of a system.A subset of the state space issaid to be invariantif startingin it, the trajectorymustremainin itfor all time. For control systems, this property is not particularlyuseful. For example, if a system is controllable, in the sense thatany initial statecan be drivento any terminal state, then there areobviouslyno non-trivial invariantsets in this sense. However, ifone modifies"must remain" into "can remain" in this definition,then a very usefulnotion is obtained.This is the key observationthat underlies the notion of invariance introduced in the paperby Wonhamand Morse (see also [1]).

Wonham and Morse consider finite dimensional linear sys­tems. The linear subspace V of the state space X of 1tx =Ax +Bu is said to be controlledinvariant[or (A, B)- invariant,as they call it] if for any Xo E X there exists an input u(.) suchthat the resulting solutionx(·) with x(O) = Xo satisfiesx(t) E Vfor all t E IR. It turns out that this condition is equivalent to(A, B)-invariance (AV S; V + im(B)) and to the possibility ofmaking the subspace invariantby state feedback (3F such that(A + BF)V S; V).

This concept, together with the related notion of controlla­bility subspace and the dual notions of conditional invariance,opened the road to the elegant "geometric" theory of linear sys­tems and a wide range of applications,especially to problemsofdisturbancedecoupling,non-interactingcontrol, and regulation.Indeed, the decouplingproblem was the main motivationfor thedevelopmentof the geometricapproach.The early contributionsto the study of this problem are reviewedin Wonham's seminalbook [20, Sec. 9.10]. The problem of disturbance decouplingwith output feedback was completely solved in the linear casein [16] and, incorporatingstability, in [18].

Soon after the appearance of the notion of controlled invari­ance for finitedimensional linear systems, it was generalized inmany directions,notably to infinitedimensional systems [4], to"almost" versions of these notions [19] (that made it possibleto treat approximate decoupling problems and high gain feed­back), and to nonlinear systems. It also sparked a totally newapproach to discrete event systems; see, e.g., [15].

In the late 70s and in the early 80s the notion of controlled in­variance was extended to nonlinear systems, and this extensionopened the way to the systematic developmentof methods forthe design of nonlinear feedback laws. In the context of nonlin­ear systems, the notion of invariantsubspacegeneralizes in twodistinctways:invariantsubmanifoldsandinvariantdistributions.Givena vectorfieldf, an f -invariantsubmanifoldis a "surface"with the property that any integral curve of f which intersectsthis surfaceis entirelycontainedin it, whilean f -invariantdistri­bution is, essentially, a partition of the state space into a "familyof surfaces" with the property that the flowof f carries surfacesinto surfaces (if, in particular,one of the surfacesof this familycontains an equilibrium of f, then this surface is an invariantsubmanifold,otherwiseit is not). In lightof this, the extensionofthe concept of controlled invarianceto nonlinear systems leadsto the notions of controlledinvariantsubmanifolds/distributions.as objects that can be rendered invariant by means of suitablefeedback laws. Thus, for instance, in a control system

ddt x = f(x) + g(x)u

with state x E lRn and u E R, a submanifold/distribution is con­trolled invariant if there exists a feedback law u = a(x) whichrenders it invariantfor the "closed-loop" system

ddt x = f(x) + g(x)a(x)

In the nonlinearcase it is thenotionof controlledinvariantdis­tribution (in contrast to invariantlinear manifold, around whichthe linear theory is centered)that plays a fundamentalrole in theanalysis and solutionof the problems of disturbancedecouplingand noninteracting control. Decoupling a fixed output from afixed disturbance input is achieved by imposing, by feedback,the existence of an invariant distribution that renders the influ­ence of the disturbance unobservable by that particular output.This fruitful domain of research was opened by the works [9]and [6], and then continued by various authors (see, e.g., [13],[10], [5]). The problem of noninteracting control with stabilityrequired some extra effort in view of the existence of certain

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obstructions that do not have a counterpart in the case of linearsystems, but eventually was solvedin [8], [17],and [2].

The notion of invariant manifold led to the introduction ofthe conceptof zero dynamics, the nonlinearanalogue of the dy­namicsassociated withthenumerator of a transferfunction. Thezero dynamics of a nonlinearsystemis a dynamical systemde­scribingall (forced) statetrajectories thatareconsistentwiththeconstraint that the output is identically zero. As in the case oflinear systems, it can be shown(underappropriate "regularity"hypotheses) that those trajectories are actuallyfree trajectoriesof a feedback-modified system,whosestate-space is a subman­ifold (a controlled invariant submanifold) of the original statespace. If all suchtrajectories converge as time tends to 00 to anequilibrium, the system has properties analogous to those of alinear minimum-phase system. This notion,developed in a fewpreliminary conference paperswhoseresultsare summarized in[3],hada significant impactin theanalysis of theproblemofout­put trackingand, aboveall, in the systematic designof adaptivefeedback laws for systemsaffected by parameteruncertainties,as shownin the monographs [12], [11].

REFERENCES

[1] G. BASILE AND G. MARRo, "Controlled and conditioned invariant sub­spaces in linear system theory," J. Optimization Th. & Appl. 3:306-315,1969.

[2] S. BA'ITILOTII, "A sufficient conditionfor noninteracting controlwith sta­bilityvia dynamicstate feedback," IEEETrans. Aut.Contr., AC-36:1033­1045, 1991.

[3] C.I.BYRNES AND A. ISIDORI, "Asymptotic stabilization of minimum-phasenonlinearsystems," IEEETrans. Aut.Contr., AC-36:1122-1137, 1991.

[4] R.F. CURTAIN AND H. J. ZWART, An Introduction to Infinite-DimensionalLinearSystemsTheory, Springer-Verlag (Berlin),1995.

[5] W.P. DAYAWANSA, D. CHENG, T.J. TARN, AND W.M. BOOTHBY, "Global

([, g)-invariance ofnonlinearsystems," SIAMJ. Contr. Optimiz.,26:1119­1132, 1988.

[6] R.M. HIRSCHORN, "(A, B)-invariant distributions and disturbancedecou­piing of nonlinearsystems," SIAMJ. Contr. Optimiz., 19:1-19. 1981.

[7] A. ISIDORI, Nonlinear Control Systems: AnIntroduction, 3rded., Springer­Verlag (Berlin),1995.

[8] A. ISIDORI AND J. W.GRIZZLE, "Fixedmodesandnonlinearnoninteractingcontrolwith stability," IEEETrans. Aut. Contr., AC-33:907-914, 1988.

[9] A. ISIDORI, A. J. KRENER, C. GORI GIORGI, AND S. MONACO, "Nonlineardecouplingvia feedback: A differential geometricapproach," IEEETrans.Aut. Contr., AC-26:331-345,1981.

[10] AJ. KRENER, "(Adf,g), (adf,g) and locally (adf,g)-invariant and control­labilitydistributions," SIAMJ. Contr. Optimiz., 23:523-549, 1985.

[11] M.KRSTIC, I.KANELLAKOPOULOS AND P.KOKOTOvIc,NonlinearAdaptiveControl Design, Wiley(NewYork), 1995.

[12] R.MARINO AND P.TOMEI, Nonlinear ControlDesign:Geometric, Adaptive,Robust,PrenticeHall (NewYork), 1995.

[13] H. NUMEUER AND AJ. VAN DER SCHAFf, "Controlledinvariance for non­linear systems," IEEETrans Aut. Contr., AC-27:904-914, 1982.

[14] H. NUMEIJER AND AJ. VAN DER SCHAFf, Nonlinear Dynamical ControlSystems, Springer-Verlag (NewYork), 1990.

[15] PJ. RAMADGE AND W.M. WONHAM, "On the supremalcontrollablesub­languageofa givenlanguage," SIAMJ. Contr. Optimiz., 25:637--659,1987.

[16] J'M, SCHUMACHER, "Compensator synthesis using (C, A, B)-pairs,"IEEETrans. Aut.Contr., AC-25:1133-1138, 1980.

[17] K.G. WAGNER, "Nonlinearnoninteraction with stabilityby dynamicstatefeedback," SIAMJ. Contr. Optimiz., 29:609--622,1991.

[18] lC. WILLEMS AND C. COMMAULT, "Disturbancedecouplingby measure­ment feedbackwith stabilityor pole placement," SIAMJ. Contr. Optimiz.,19:490-504, 1981.

[19] J.C. WILLEMS, "Almost invariant subspaces: An approach to high gainfeedback design-Part I: Almost controlled invariant subspaces,Part IT:Almost conditionally invariant subspaces," IEEE Trans. Aut. Contr.,AC-26:235-252, 1981,and 27:1071-1085, 1982.

[20] W.M. WONHAM, Linear Multivariable Control: A Geometric Approach,1st edition, Lecture Notes in Economics and Mathematical Systems,Vol. 101, Springer-Verlag (New York), 1974; 2nd edition, Applicationsin Mathematics series,Volume 10,Springer-Verlag (NewYork), 1979.

A.I. &J.C.W

322

DECOUPLING AND POLE ASSIGNMENT IN LINEARMULTIVARIABLE SYSTEMS: A GEOMETRIC APPROACH·

W. M. WONHAMt AND A. S. MORSE:

1. Introduction. The current interest in linear multivariable control hasled to several algebraic results with important applications to system synthesis.In particular, the problem of decoupling of individual system outputs by meansof state variable feedback was studied by Rekasius [1], Falb and Wolovich [2]and Gilbert [3]; the problem of realizing arbitrary pole locations in the closedloop system transfer matrix was investigated by Wonham [4] and Heymann [5].In the present article, new results are obtained along these lines. In § 3, the problemof neutralizing the effect of disturbances with respect to a specified group ofoutput variables is solved. In § 4, the concept of a controllability subspace isintroduced and its relation to pole assignability is investigated. This materialis preliminary to the formulation of a general problem of output decouplingin § 5. In §6 and § 7, necessary and sufficient conditions for decoupling are ob­tained in two specialcases; the results of§7complement and extend those obtainedpreviously in [1], [2J and [3]. In each case, the problem of pole assignment issolvedcompletely.

Our viewpoint is that such problems are usefully treated in a geometricframework in which both definitions and results become intuitively transparent.In this way, entanglement at the outset in a thicket of algebraic calculations isavoided. Of course, for applications, it is necessary to translate the geometriccriteria into matrix operations suitable for computation. This matter will beconsidered in a future article.

2. Notation. The control system of interest is specified by the differentialequation

(2.1) X(I) = Ax(.t) + Bu(t'

with x an n-vector, u an m-vector and A, B constant matrices of dimension, re­spectively, n x nand n x m. Here and below,all vectors and matrices have real­valued elements. Script letters denote linear subspaces: 8" is real n-space; 'I '1 isthe orthogonal complement of the subspace .'1"'; 0 denotes both the vector zeroand the zero subspace.

If K is a matrix, {K} or .K is the range of K, and. ·i' '(K) is the null space ofK. If K is of dimension J.l x v and 'I .' ' c till, we write K- 1 1 . for the subspace{z:zet8''',KzE 1"} C s:

The controllable subspace of the pair (A, B), written {Alat}, is defined as

{AI~} = &4 + A~ + ... + An-lYl.

• Received by the editors February 3~ 1969,and in revised form June 4. 1969.t Office of Control Theory and Application, NASA Electronics Research Center, Cambridge.

Massachusetts 02139. The work of this author was supported by the National Aeronautics and SpaceAdministration while he held an NRC postdoctoral resident research associateship.

: Office of Control Theory and Application, NASA Electronics Research Center. Cambridge..Massachusetts 02139.

Reprinted with permission from SIAM Journal on Control, W. M. Wonham and A. S. Morse,"Decoupling and Pole Assignment in Linear Multivariable Systems: A Geometric

Approach" Vol. 8, 1970, pp.1-18.

323

Thus, {AltI} is the largest subspace of 8" which the control u(·) in (2.1) caninfluence. Observe that {Alai} is an A-invariant subspace of In.

With (2.1), we consider the auxiliary equation

(2.2) y(t) = Hx(t),

where H is a constant q x n matrix. The vector y is the output.Equations (2.1) and (2.2) play no essentialrole but serveto guide the investiga­

tion.

3. Localization of disturbances. In place of (2.1), consider the perturbedsystem

(3.1) x(t) = Ax(t) + Bu(t) + De(t),

where D is a constant n x d matrix and c;( ·)is a disturbance input. If u(t) = Cx(t)+v(t) (where v( · ) is an external control input), then the output y( · ) will be un­affected by all possible c;( · ) if and only if {A + BCIP}} c ~K(H). This suggeststhe problem: given A, B, !J) c tI", .IV c tfn, under what conditions does thereexist an m x n matrix C such that {A + BCI9}} c ..AI? If C exists, the effect ofdisturbances is, in an algebraic sense, localized to ./V:

THEoREM 3.1. There exists C such that {A + BCI~} c ,iV· if and only if~ c 1/;wherel' is the maximalsubspace such that

(3.2)

Furthermore j/" is given by "I/' = j"{lI), where

(3.3) 1,'(0) = .,N'~ j"(i) = rv: 1) n A - l(aJ +,,"(i-1}),

i= 1,2,···,v,and v = dim .A":

Here and below, "maximal" ('''minimal'') mean l.u.b. (g.l.b.) with respect tothe usual partial ordering of subspaces by inclusion.

To prove the theorem we need two auxiliary facts.LEMMA 3.1. Let XiEIPI, UiEsm, i = 1, "', N, and write X = (Xl"'" XN),

U = (U1'···' UN)' Thereexistsanm x nmatrix Csuch that Cx, = Uhi = 1",·, N,if and only if %(X) c ..¥(U). C always exists if the Xi are linearly independent.

The simple proof is omitted.LEMMA 3.2. Let "f'"' c tin. Thereexists anm x n matrix Csuchthat (A + Be) 1, .

c 1''"' if andonly if Ar" c 91 + 'I':Proof. Necessity is clear. For sufficiency, let VI' ••• , VII be a basis of 1<

Then AVi == BUi + Wi for some Ui E 8 m and Wi E r; Choose C, by Lemma 3.1,such that CVi = -Uh i = 1, ... , u; then (A + BC)v; = Wi'

Proof of Theorem 3.1. For sufficiency, (3.2) implies l' C ..f' and A l' . c 91+"f': By Lemma 3.2, there exists C such that (A + BC)"f' c l' ~ Then

{A + BCI,q)} c {A + Belt '} = 1" c . ,~<

The maximal property of 'j' ' was not required.For necessity write {A + BC'~} = If: Then

(3.4) lI'~ c .'V: A }f" c d4 + 1/':

324

If '~is the class of all jf' c tin which satisfy (3.4), then clearly 0 E 1Yand 11'isclosed under addition. Hence, 1f'''contains a (unique) maximal member 'II: Then~ c 'II'" c 'I" and r: satisfies (3.2).

To prove the second statement of the theorem, observe that 1"'(0):::> r;and if ; '(i- t) ~ 1~ then ,,'(0 ::> 1/ n A - 1(~ + 1"') = '1/~ Thus, t"(i) ::':) '/.''' for alli; and since ;,'(i) c t ·(i-1). there is a least integer j such that ~.(;) = r '(j) if i ~ j.Since 'f '(j) :::> ;'- and 1,/"0) satisfies (3.4), tJ 'Cj) = 1- ~ Clearly, 0 ~ j ~ v: and iff} c 1-'weeven have 0 ~ j ~ v - dim ~.

Remark 1. Theorem 3.1 depends essentially on the fact that the class 11'determined by (3.4), or equivalently

'1Y= {lI": 1r c . J' n A - 1(~ + 11' ")} ~

has a maximal element 1': Furthermore, 1" is defined constructively by means of(3.3). This fact will be used without special comment in the following sections.

4. CoatroUabiUty subspaces. In regard to the system (2.1), suppose that asubspace 91 c I" is selected and that it is desired to modify the system in such away that ~, but no larger subspace, is completelycontrollable. This aim is to berealizedby feedback ofstate variablesand by formingsuitable linearcombinationsof control variables: that is, by setting u = ex + Kv, where K is an m x Ill'

matrix for some m' ~ m. Then (2.1) becomes

x = (A + BC)x + BKt'

and we require

(4.1) {A + BCI{BK}} = .-Jt,

Condition (4.1) can be expressed more neatly by noting that {BK:· c tI and thefollowing.

LEMMA 4.1. If rJ c !M and {AI~} = 91, then {AI~ n JI} = ~jt. Conoersely, {!'{AliM n ~} = fJI, thereexists a matrix K such that {AI{BK }} = iJt.

Proof. {AI!f} = fJI implies rI c ~, so !f c 91 n ~, and thus ..Jt = lAI~:­c {AlflI n ~}. Also, A9t c ~ implies A(aJ n at) c .~; by induction Ai(:jJ n .11)c ~,j = 1,2,··· ,and so {AI~ nat} c YI.

For the converse, let bi , i = 1, ... , m, be the ith column of B and let :"i~

j = I, · · . , m'} bea basis of 81 n 91. Then

m

rj = L k;jb;,i= 1

j = I ~ , .... m',

for suitable ki j , and we set K = [ki j ] . This completes the proof of the lemma.By Lemma 4.1, we can pose the synthesis problem as follows:Given A, Band rJI, find conditions for the existenceof C such that

(4.2) {A + BCI~ n9t} = JI.

If such a C exists, we call Yl a controllability subspace of the pair (A, B). Observethat at = 0 and fJl = {AI~} are controllability subspaces.

Controllability subspaces can be characterized as follows.

325

THEOREM 4.1. Let A, B, .~ c I" be fixed. iJI. is a controllability subspace ~r

(A, B) if andonly if(4.3) A.JI c ;~ + .Jt

and

(4.4) .~ = if,

where ~ is the minimal subspace such that

(4.5) ~ = .'if n (A~ + :M).

Furthermore, :i = .\jI(P),where p = dim :JI and

.~(O) = 0,(4.6)

;=1 .. 2.···,1l.

WriteC for the classof matrices C such that (A + BC~ c .'iI. To prove thetheorem we need two preliminary results.

LEMMA 4.2. Let~ c .rJl. For all C E C,

Jl n• + (A + BC).j = .:JI n (Abf + :JI).

Proof. Let C E C. Then (A + BC).j c .rN and A.J + .M = (A + BC~.J + .~.By the modular distributive rule for subspaces,

.:Jt n(A~ + ;M) = .if n [(A + Be).} + .A]

= (A + BC~~ + .~ n ;JI,

LEMMA 4.3. I)' C E C then

(4.7)i

L (A + BCP-l(~ n :it) = ;.jf(i).

j= 1

where the sequence [JI(l) is defined by (4.6).Proof. Equation (4.7) is true for i = I, If it is true for i = k - I .. then by

Lemma 4.2,Il

L (A + BCY- l(aI n ..JI) = ~ n .JI + (A + BC)af(Il- I)

j= 1

= .JI n (A9r(k-l) + .rM)

= .;J1(11.,.

Proofof Theorem 4.1. By Lemma 3.2, C is nonempty if and only if (4.3) istrue. Let

(4.8) .lJI = {A + BCltf nat},

Then Ce C. By Lemma 4.3,

"..jI = L (A + BCY- I (~ n .:Jr.) = .Jttll) = ;:JI(P».j= 1

326

Conversely, if .'1t = ~(n). then (4.8) is true for every C E C. It remains to show that(4.5) has the minimal solution g,(p). By induction on i in (4.6), it is seen thatJt(i) c ~. i = 1,2•. · · • for every solution .j of (4.5), and that the sequence .jf(it ismonotone nondecreasing. Hence, there is Ji ~ P such that .ljI(l) = ..Jt(Il) for ; ~ JL:in particular, .tjf(p) c if and ;jf(II) satisfies (4.5),

Remark 2. If:~ is a controllability subspace, then it was proved incidentallythat

~~ = {A + BCI.qd n ~jf~

for every C such that (A + BC).~ c ~jf. This fact will be used later without specialmention.

Consider now the problem of assigning the eigenvalues of the restriction ofA + Be to fJt. It will be shown that there is complete freedom of assignment andthat simultaneously the control v introduced earlier can be made a scalar: i.e., in(4.1) K can be made an m-vector(111' = 1). For this, recall [4] that a subspace .1' isA-cyclic if there exists x E fl' such that {AI {x l} = PI'; that is. if .Uj' contains agenerator x. Thus we can take m' = I ifand only if.~ can be made (A + BC)-cyclicand ~ n PA contains a generator.

THEOREM 4.2. Let (4.3) and (4,4) hold, and let ~1' ••• , ~/' be arbitrary realnumbers (p = dim 91). Then C canbechosensuchthat (4,2);s trueand.'1I is (A + Be)­cyclic with characteristic polynomial

(4.9)I'

; ..p - L ~iAi-l,

i=1

lf 0 ¢ be fJI n fJI is arbitrary, C call be chosen so that, ill addition.. b generales .11.Proof', By Lemma 4.3 and Theorem 4.1,C is nonempty and

(4.10) {A + BClbit n .jf} = .jf

for every C E C, Choose C1 E C arbitrarily and write A + Be 1 = A I' Letb1 = b E!JI n~ and let PI be the largest integer such that the vectors

bt,Atb l , ••• 'I A/ll-lb l

are independent. Put r 1 = b, and rj = A1rj - 1 + b1,j = 2,···, Pl' Then riE.1fand the ri are independent. If PI < p, choose b2 E~:Jt n .A such that r1" • • • , '."."

b2 are independent; such a b2 exists by (4.7). Let P2 be the greatest integer suchthat

are independent, and define

rp t + i = A1' PI + i - 1 + h2 " i = I....... P2'

Then r. , , .. , 'Pl are independent and in~. Continuing thus, we obtain eventuallyr l' • · · , rP independent and in .~, with the property

ri+l = Atri + bi ,

where hi E Jt n94. Now let C2 bechosen such that

Be2';'= hi'

327

i = I....... p - I.

; = l , ... " p,

where bp E;;t n JI is arbitrary. Since bi = BUi for suitable Ui' and the r, areindependent, Lemma 3.1 guarantees that C2 exists.The situation now is that

i=I~···.p-l ..and

By independence of the r i'

.: A 1 + Be2H r 1 }-;. = .11;

that is, YI is cyclic relative to A + B((~l + e2) with generator r, = b, E.* n ;14.It is wellknown [4] that now an n-vector c can be found such that A + B(C1 + ('2)

+b1c' (restricted to (1) has the characteristic polynomial (4•.9). Setting b, = Bgfor suitable gEl"', it follows that the matrix

C = (~I + C2 + gc'

has all the required properties.Remark 3. The result that any nonzero vector indI n ;~ can serveas generator

is an extension of the useful lemma in [5].Remark 4. If 9t = 8", (4.3) holds automatically and (4.4) amounts to

{AI&4I} = tin, i.e., complete controllability of (A, B). Then Theorem 4.2 yields theknown result [4] that controllability implies pole assignability. The constructionjust used furnishes a simpler proof of this fact than that in [4J.

It will be necessary later to compute the maximal controllability subspacecontained in a given subspace .C/. For this, let '1~ be the maximal subspace of .(/which is (A + BC)-invariant for some C (recallRemark 1 following Theorem 3.1 ):and let C("Y)be the class of C for which (A + BC)"f~' c -1':'

THEOREM 4.3. If C E C("Y), the subspace

(4.11) jj = {A + BCI31 n 17'}

is the maximal controllability subspace in .V}.Proof. By (4.2)and Lemma 4.1,~ is a controllability subspace. Furthermore,

by Lemma 4.3 with C(1,,4) in place of C. ~ is independent of C EC(T) and so isuniquely defined. Now suppose

;j = {A + BCI&it n di}, .lI c .Yi

•

Since fA is (A + Be)-invariant and 17" is maximal, there follows .iI c '1~. Letf" = ;j E9 f~ By the construction used in proving Lemma 3.2. a matrix C existssuch that

cs s c«. xE.i~

Then C E C(Jil, and

~ = {A + BCI~ n~}

c {A + BCI8I n 1~'}

= ~i;

that is.. ~ is maximal.

328

s. DecoupUng of output variables: Problem statement. Consider the outputequation (2.2), with

(S.l) H=

where Hi is of dimension qi x n.. i = I, ... 'l k, k ~ 2" til + .. , + 'It = 'I. Then(2.2) can be written

(5.2) i = 1, ... 'I k ..

where Yl is a q,-vector. The vectors Yi may be regarded as physically significantgroups of scalar output variables. It may therefore be desirable to control com­pletelyeach of the output vectorsYi individually, without affecting the behaviorof the remaining Yj, j =F i. This end is to be achieved by linear state-variable feed­back togetherwith the assignment of a suitablegroup of control inputs to each Yi'That is, in (2.1) we set

(5.3)k

U = ex + L «»;i= 1

For Vi to control Yi completely, we must have

(5.4)

where Pltj is the range of Hi' Since the ith control r, is to leave the outputs Yj'j :#: i, unaffected, werequire also

(5.5)

Recalling the equivalence of (4.1) and (4.2), we can express conditions (5.4)and (5.S) more neatly as follows. Write tI" = I and

(5.6) i = L··· 'I k .

Then our problem is: Given A, Band . ~ i , ·· · 'l • ti, find a matrix C and controlla­bility subspaces ~I' · · · , ..Jt" with the properties:

(5.7)

(5.8)

(5.9)

~~i = {A + BC'~ n .Jtd- ...Jti + . t·; = 8,

:Jti c: n .Jj~

i*i

;= L··· .. k ..

i = 1.. k ..

;= 1. ., k.

Here (5.8) and (5.9) are equivalent, respectively, to (5.4) and (5.5).The relations (5.7}-(5.9) provide a geometric formulation of the problem of

simultaneous decoupling and complete control of the output vectors ..vi" . · . "J'k'Thus stated, the problem definition is both natural and intuitively transparent.

Weobserve that the output matrices Hi play no role beyond specification ofthesubspaces. f'jOl Since the Hi need haveno special structure, the.• "; are similarly

329

unrestricted. Nevertheless, weshall rule out trivialities by tacitly assuming:(i) . i·i #: I, i = 1, · · · , k,

(ii] The subspaces 0 ~ °t are mutually independent. 1 In particular, the .• iaredistinct and

(5.10) 0 l i :F 0, ; = 1~ · . · , k.

(iii) The pair (A, B) is completely controllable, i.e., {AI~~} = if.For if (i) fails, then for some i, . t i = I'; that is, Hi = 0 and Yi == O. If (ii] fails,

then for some i,

, t· 't n L .r OJ =F 0j~i

or, by taking orthogonal complements,

.•'i+ n.i'j~8j~i

and (5.8) must fail. For (iii), if {AI~} = 81 #: 8 we can write I = 1'1 ~ 82 and(2.1)as

Xl = A1x1 + A3X 2 + Blu,

X2 = A2X 2 ,

where X;E"i, i = 1,2, and {Allbf!} = "1' The problem is unrealistic unless A2

isstable(i.e., the pair (A, B)isstabilizable [4]). Hence, wemay assumeX2(t) == 0 andtake as starting point

The problemcan then be reformulated with 81 in placeof I.We tum now to the determination of necessary and sufficient conditions for

the existence of a solution to (5.7)--(5.9) in two special, but interesting, cases.In the following sections, ~i denotes the maximal controllability subspace

such that

(5.11 ) :li en ..Jj~j*i

i = 1,,·· ~ k.

The ~i are constructed according to Theorem 4.3.

6. Deeoupllng when rank (H) = n. Our assumption is equivalent to

(6.1) "n .i: = o.i= I

That is, there is a one-to-onemapping of state variables into output variables.THEOREM 6.1. If (6.1) holds, then the problem (5.7)-(5.9) has a solution {r and

only if(6.2) ; = 1, ..... k .

1 Equivalently.. the row spaces or the Hiare mutually independent.

330

Proof. If the problem has a solution :JI" i = 1, · .. , k, then by maximalityof the ~i' i = 1, ... , k, there follows:A i c .ii' and (6.2) follows from (5.8).

Conversely, suppose (6.2) holds. The jii are mutually independent; for.. by(5.11) and (6.1),

s, n L~Il C [n .'j] n [L n "f\:] en. "j n"f i = O.Il~i j~i Il:l=i V~1l j~i

Let C, bechosen such that

~j;i = {A + BCi/:A n ~d" i = L,·,. k .

Since the ~jii are independent there exists" by Lemma 3.1" a matrix C such thatCr = C;r (reti i , i = 1. ..... k).. i.e,..

Then(A + BC)r = (A + Bei)r .. re.ji" i = I .. ···. k .

J; = -fA + BCI~ n JJi }· .. i = L " .. k :

and C. together with the ~,' satisfy (5.7)-(5.9).Remark 5. ByTheorem 4.2, the C, can be chosen so that A + BC;, restricted

to ~i' has any desired spectrum. Hence, the same is true for A + BC.Furthermore.there exists b, E &f n~i such that

~i = ·:A + BCI{b':·:'" i = L ., .. k .

7. DecOlipUng when rank (8) = k, Our assumption is equivalent to

(7.1) dim:M = k,

Here the situation has been simplified by narrowing the choice of generatingsubspaces fJI ns; The same assumption was made in [1], [2] and [3]" with theadditional restriction that the outputs Yi be scalars.

THEOREM 7.1. //(7.1) holds, then the problem (5.7>-(5.9) has a solution ~r andonly ~r

(7.2)

and

(7.3)

·ji +".i = 8,

k

;;f = L ~ ns;i= 1

; = L ., ... k .

Furthermore, if C, Jt l' · · . , ·:Jtk is any solution" then

(7.4) i = L··· .. k.

i = L···. k .

Proof", Part 1. Suppose C, ~ l' ... " .ilk is a solution. The necessity of (7,2)

follows, as in the proof of Theorem 6.1. To verify (7.3), write

~ n ..II; = :iI; E9 [ !11 n .</1; n j~j .lIi] •

331

The ~i are mutually independent; in fact,

HI, n L!/Ij C :Mi n L ;M n ~j c .oA; n (~ n .it;) n L·~.i = o.i_' i-' j e i

Recall that the:A i are (A + Be)-invariant. Then

.11; = ·~A + B('l:jfi~' + ·ji~

where

.Ai C{A + Bcl L ilti}C L .Jtj C L n .1"~ c. t;.!j *' i j ¢ ; j ¢ i Il *j

Therefore. by (5.8),

{A + BCI~':' + . t ; = A,

and since . Ji #: 4 there follows ~i "# 0, i = 1....... k. Therefore

Ie "

dim L ~i = L dim ~i = k:i=1 i= 1

so

(7.5)

"and

.~ = ~A I ~ ... Ea :M"

dim..M; = 1.. i = l .... · .. k .

Since~, c: fM n :iii C ~ n Jilt it follows that (7.3) is true.Proof. Part 2. To verify (7.4),it is enough to show that the subspaces ~M n .iI;

are independent. For then..

and so

(7.6)

dim (fjI nJii ) = 1.. ; = 1....... k ..

i = l.. ..... k.

Assuming (7.6) is true, let.ji = ~i G) ~i and choose Ci • by Lemma 3. t .. such that

(A + BCi).,ii C '~i" eir = C,., re .if;, ; = 1.. ... "k .

Then C, E C(aI,) n C(~,), so that

;it i = {A + BCil~ n ·A'd"= ·fA + BCil;~ n ~d'

= .A;

which proves (7.4).We proceed to show that the.-A n ~ii are independent. Write

J;r = Ls;j¢;

332

It is even true that

(7.7) :An.lin~jr=O, i=L···.k.

On the contrary, suppose (7.7) fails for, say, ; = 1. If dim (j n ~j1) = L then

(7.8) :J4n~l c.iT.

If dim (at n ~1) ~ 2, and

(7.9) ;A n·ji ¢. ~r, i = J, 0" • k ,

then

Jl = 1, . · · , k - 1, that is,

dim [.f ~ n ';;i] ~ 3;,= 1

and by induction

dim [.r ~ n !.Ii] ~ k + I,,=1

a contradiction. Thus (7.9) is false; combining this result with (7.8) there follows

(7.10) &f nd(l. c fl,:for somecxe(l,···, k). It will be shown below that there exists C2 such that

(7.11) (A + BCac)9l(l C ~2; (A + BC(l~: c ;~:.

Assuming (7.11) is true, we have

~(I = {A + BC,JBI n :i(l.} C {A + BC21~:} c ~: c .. i';.

and therefore (7.2) fails for i = ~. With this contradiction, (7.7) is established.It remains to verifythe existenceof C2 0 For this we need the following result.

LEMMA 7.1. Let 'IJ"~ 11" be arbitrary. There exists C such that

(A + Be)l-" c j ~

if and only if'(A + Be) 'II' c 'JI'

A1,'c9l+ 1~

A11'cal+ 'II~

A(1' n ~"') c :11 + 1" n ~:

Proof. Necessity is obvious. For sufficiency, write

l' + 'H" = tAO ~ (1 . n ';f") ffi 1i~

333

where 1-' c 1 ~ ,*~' c 11 ~ Bythe construction of Lemma 3.2, C can be chosen suchthat

(A + BC)( t . n 'II ') c t' n 'H ~

(A + BC){' c: 1 ~

(A + eci« c 'N~

This completes the proof of the lemma.Consider now s; ~:. Clearly A;j2 c J8 + s; A-l: c ~11 + .j:. By (7.3)

fA = ~ n:12 + aJ n~:, and so

A(:i:l n;1:) c (:M + .12 ) n (~ + ,~:)

= 1~ + (;M + ~2) n.j:

= 111 + (~ n ~j: + ·j2) n ~:

= .eJI + ~2 n Ji:.

Byapplying Lemma 7.1, the existence of Cf/. is finally established.Proof. Part 3. We now prove that (7.2) and (7,3) are sufficient conditions for

existence ofa solution. Let fi be the maximal subspace such that

(7,12) 1~' en. t j,j:Fi

i = 1.. ..... k.

It is enough to check that the f; are compatible, in the sense that there exists Csuch that

(A + Be')1;' c t~·.

We show first that the subspaces

1~'" = L 1~'j'¢i

i= L·,·~k.

are compatible. From (7.12) there follows

An c: at + '-:1= :JI n ~. + 'f~1

= ~i + t~,

(by (7.3))

;= 1.. ..• _k ..

where ~i = 1M n ·t~. By Lemma 3.2.. there exist B, with {Bd' = Jli't and ('i" suchthat

i = L··· .. k .

Choosing a basis {VI' ... , vll

}- for 1~' + .,. + t~', we define C such that

k

BCv,. = L ec«;;= I

334

\' = I .. ··, .. }l.

Then

(7.13)

(A + BC)"f7 = (A + ec, + Ji BiCi) ",-,1'

c (A + BiCi )"f71 + L B4ji*i

c 'f-r + L 1j'j¢i

i = 1..','. k .

This proves compatibility of the 1li. Now define

1'; = n 'f"j,

Clearly, 'I i :::> f;', i = I, · · · ~ k. By (7.13),

(7.14) (A + Be)'! i c i'i.

and, furthermore by the second condition of (7.12),

(7.15) r , c n L n ,t~ = n .I j.lj~i 2:1:j m¢'2 j*i

i = 1.. .•. 't k.

i = 1,,··· .. k..

By (7.14) and (7.15), the j'j satisfy the conditions imposed on the '1~' in (7.12).Since the "fi are maximal, there results 'I i c j~, and, therefore, 1; = '1~' ..i = 1,"., k.

Remark 6. If the conditions of Theorem 7.1 are satisfied, then

(7.16).t iii = .± {A + BC/£f n;ji} = {A + BCj.± £f niii},-I ,= 1 ,=1

= {A + BClaI} = {Alai} = 8.

We turn now to the problem of pole assignment. In contrast to the situationof§6, it is no longer possible, in general, to vary the spectrumof A + Be on each:Ai independently. The following example shows that certain eigenvalues ofA + Be may even be fixed for all admissible C.

Let tf = 8 3• k = 2 and

[I 0 0]

A = 1 I I ~

001B = [: ~l

[0]l.

Ii = ~ .

2 This identityand itsdual, Ln L = L, are readilyestablished by using the (modular)distributiverule for subspaces.

335

It is easily checked that (S.7H5.9) have the (unique) solution

and that C must have the form

C = [('1 0 OJ° ° "2

with arbitrary c1 ' C2' Then

det (A + Be - A./) = (1 + ('1 - A..HI - A.)( 1 + ('2 - A).

Observe that the eigenvalue A. = I, belonging to the eigenvector (0'1 1'10)' of A + BC",is fixed,

To discuss the present case in general, we introduce a suitable decompositionof 8. Assume that the problem of (5.7H5.9) has a solution C, .jl' ... , .11k .. andlet C denote the class of matrices C for which (A + BC~~i C ~j;, ; = L .... k.We know that the spaces :l; are the unique solutions: for simplicity of notation"write Jt; for JI/. Define

(7.17)k

tfo = n ·*r,i= 1

and let tli beany subspace such that

(7.18) .lJti=8i~(.'JtintfO)' ;= I .. ···,k.

In the following, J denotes the set of indices (1, ... , k), Jo the set (0, 1, ... , k·).In intersections and summations involving 91's, the index ranges over J: in thoseinvolving 1'5, the index ranges over J 0 •

LEMMA 7.2. The subspaces tli have the properties

(7.19) 4 o E9 8 1 ~ ••• ~8k = I ..

(7.20) (A + BC)tIi c li+ 4 o, ieJo.. CECa

Proof. Assertion (7.20) is obvious by the fact that the :iii are (A + Be)­invariant. For (7.19), observe first that

9l i n tI0 = !JIi n n .~1 n fJtr = ·:Jli n ;Jt;j:;';

and so, if i e J,

8 i n (80 + L tSJ

j ) c: s, n (8 0 + .<Jtj)j*Oj:Fi

(7.21) = A'i n~r

= Gj n :Jti n ,CJtr

= o.Now for arbitrary subspaces ,Yi, i = 1, 2, 3, if

,y! n ('~2 + ,Y3) = .V1n '</2 + ,Y~ n .</3,

336

(") Ii.Ion t!.+j~2'i =/o n / . + / o n J 28j

Ie

= 1 0 n L lSj.j=2

then

Applying this fact and using (7.21) we have

(Ie ) k

8 1 n 8 0 + i~2 8) = 0 = 8 1 n A'o + 8 1 n j~2 8'j.

and therefore

Repetition of this argument yields, after k - 2 steps"

(7.22) 10 n (11 + jt2 Ii) = 10 n It = O.

Equations (7.21)" (7.22) state that the 8 i 'l l E J0' are independent. Finally" by (7.16) ..Ie" k

L Ii = L (8 i + 8 0 ) ~ L ~.;ti = I.i=O i=1 i=1

Remark 7. If the 91; are independent, then 8 0 = 0 and G; = Jt;9;E J.For; E Jo let Pi be the projection on 8; along k¢; 8 j 't and now let C E C be

fixed.LEMMA 7.3. Let BI n '~i = {hi}' ieJ. Then

(7.23) tI; = {~(A + BC~{Pibd'}' i e J,

Proof By (7.18) and (7.19), ti = Pialj • By (7.18) and (7.20),n

P/Jti = Pi L (A + BCy-l{bi }j= 1

n

= L [Pi(A + Bc)]j-l{Pibi:'j= I

= {~(A + BC~{~bd'}'

LEMMA 7.4.

(7.24) .4 n 8 0 = O.

Proof. By (7.3)and (7.7),

~ n 4 0 = ( i Y4 n ::11;) n n.:Jt1i= 1 j= 1

= (iM n 91 1 + i ~ n rJli ) n .':1It n n,lt1;=2 j=2

= ( i ~ n .11;) n niN!i=2 j=2

337

This completes the proof of Lemma 7.4.Next let C = C1 bea fixed member ofC, let C2 E C, and write D = C 2 - C' l :

thus A + Be 2 = A + BCI + BD. Now bie;~i c ~';i + ~IO (jEJ): and (7.20)yields BDtli c Ii + 1 0 (ieJo): therefore

(7.25) i.j e J; i r ).Also, using (7.24)

(7.26)

Write

(7.27)k

BD = L h.jc/j~j= 1

where, as before, .~ hj } = ~!d n ..;tj' Then

(7.28) PiBDlj=Pibi,litrf'j=O, i#j, ie i, jeJo .

Wecan now compute the spectrum 1\ of A + 8('2' Define

(7.29) Ai = ~(A + B(' d, i E J 0,

By (7.26) and (7.29),

(7.30) Po(A + Be2) = Po(A + Be2)(1 - Po) + AoPo,

and by (7.28) and (7.29),k

PitA + Be 2 ) = Ai + P;BD L Pj

(7.31) j=O

= Ai + Pib;d;Pj , ; E J.

Suppose A. E A.. with corresponding (complex) eigenvector ~. A brief calculationfrom (7.30), (7.31) shows that either (i) for some i E J, Pi~ #- 0 and (Ai + Pibid~)Pi~

= APi~' or (ii) ~ = poe and Aoe = leo Conversely, if Ao~ = Ae for 0 -# ~ E &o - or(Ai + p;bida~ = A.~ for 0 '# ~ E IJj and some i E J, then AE 1\. Therefore

k

1\ = U Ai'i=O

where Ai" ; E Jo.. is the spectrum of the restriction of Pj(A + Be2) to & i- By (7.30)"Ao is independent of the choice of C2' i.e., is fixed uniquely by the requirementC E C. On the other hand, for; E J Lemma 7.3 states that 8:j is the controllabilityspace of the pair (Ai' ~bi). Hence.. any choice of Aican be realized by appropriatechoice of d, : indeed, for any W E 8 there exists d, such that

I {W'X for xeA'i'ld·x =

I 0 for x E L 8'j.j*i

These results are summarized in the following theorem.THEOREM 7.2. Let the conditions oj' Theorem 7.1 be satisfied. (r C E C., the

eigenvalues of A + Be can be partitioned into k + 1 disjoint sets

338

(7.32~

where

no =. dim (.n ]11)'J= 1

n, = dim (.ji) - dim (~i n J#r)~ i E J .

The set Ao and the integersni (i E J0) are fixed for all C E C. The sets Ai (i E J) canbe assigned freely (by suitable choice of C E C) subject only to the requirementthat any Aii with 1m Aij =1= 0 occur in Ai in a conjugate pair.

Remark 8. If basis vectors are chosen in the tSi , then the system differentialequation can be put in a simple "normal" form. Let

and~x ~ (A + BC2)x + Br .

Multiplying through by 1'; and using (7.30), (7.31), we obtain

Zi = (Ai + PibidaZi + PiBv.. i E J ,

Zo = Po(A + Be2)(Z 1 + ... + z,,) + Aozo + PoBt:.

Let K be an m x m (= k x k) matrix such that BK = [bi ..• bk] and put r = K \V~

W == (WI'···' WIe)'. SincehiES; Et> tRIo, we have

hi = Pb, + PObi == bi + b;o·Adopting nrdimensional representations of the z., etc... we see that (7.32) can bewritten as

(7.33)i, = (Ai + £'~)Zi + hiWi~ i E J ~

k

to = L AOj : j + Ao=o + Bolt\'.i'* 1

Equation (7.33) exhibits the system (2.1) as an array of k decoupled subsystems,each completely controllable by an independent scalar input "';'1 plus one addi­tional subsystem which is driven by the others and by w, Finally, since ·;li n &'0= .'Jti n atr c .f', it follows by (5.8) and (7.18) that s, +.~; = ef~ that is, H;&;= .If;.

Remark 9. The decoupled system is acceptable in practice only if the eigen­values in the fixed set Ao are all stable. It is possible to check for stability of Aoas follows. Recall that Jt i C tl i + tf 0 (i E J) and note from (7.20) that A(&;i + a0)

c Ii + tf0 + YI (i E J). Furthermore,

~ + 80 en. 1j, i E J.j=/;i

It follows by Theorem 4.3and the maximality of the .Jt; (='~i) that

:Jt; = {A + BClat n (I'i + 10 ) :,

for any C with the property (7.20). That is, (7.20) is both necessary and sufficientthat C e C. Thus, to compute Ao it is necessary only to compute the spectrum ofA + Be0 (restricted to tf 0) where Co is any matrix such that (A + Be0 )($10 C ~ () •

339

Concluding remark. This article represents a preliminary investigation ofthe generaldecoupling problem formulated in § 5.The results for the specialcasesof §6 and § 7 suggest the possibility of a complete and detailed geometric theoryof linear multivariable control, in which the concept of controllability subspacewould playa central role.Specific problemsfor future study includenot only thatof § 5 but also the problem of decoupling by adjunction of suitable dynamics(augmentation of the state space), and the problem of sensitivity. As formulated,decoupling represents a "hard" constraint, an all-or-nothingalgebraic property.Of course, for applications a quantitative approach via "soft" constraints mightalso prove rewarding.

It is clear that an adequate qualitative theory of large linear multivariablesystems iscurrently lacking;and equallyclear that, withcomputers,such a theorywouldfind wideapplication.

REFERENCES

[1] Z. V. REKASIUS. Decouplinx of multivariabte systemsby means of state tariable feedback. Proc.Third AllertonConference on Circuitand SystemTheory, Urbana, Illinois,1965, pp. 439-447.

[2] P. L. FALS AND W. A. WOLOVICH, Decoupling in the design and synthesis ofmultivariable controlsystems, IEEETrans. Automatic Control,AC-12(1967), pp. 651-659.

[3] E. G. GILBERT, The decoupling of multivariable systemsby statefeedback, this Journal. 7 (1969),pp.50-63.

[4] W. M. WONHAM, On poleassignment inmulti-input controllable linear systems, IEEE Trans. Auto­matic Control, AC-12 (1967), pp. 660-665.

[5] M. HEYMANN, Pole assignment in multi-input linear systems, lbid., AC-13 (1968). pp. 748-749.

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