Status of Noninteracting Control - IEEE Xplore

568

IEEE TRAh'SACnONS O N AlJTOM.4TIC

CONTROL, VOL. AC-16,NO. 6,

DECEMBER

1971

Status of Noninteracting Control

Abstract-The current status of decoupling theory for linear constant multivariable systems is described. The subject is treated in vector space terms and appropriate background concepts including invariant and controllability subspaces are discussed. Suggestions are given for translating vector space operations into matrix operations suitable forcomputation. The controllability subspace is used to formulate the restricted (static compensation) decoupling problem. Although the most general version of this problem is unsolved, there are known solutions for three special cases. A complete solution to the extended (dynamic compensation) decoupling problem is known. If a linear constantmultivariable system can be decoupled at all, by any means whatever, then it can always be decoupled using linear dynamic compensation. The internal structure of a decoupled system isdescribed in simple matrix terms. Using this representation, it ispossible to characterize the system pole distributions which may be achieved while preserving a decoupled structure. A procedure is outlined for synthesizing a dynamic compensatorof low order which will decouple a system.The procedure actually provides minimal order decoupling compensators for systems in which the number of open-loop inputs equal the number of outputs to be controlled.

I. INTRODUCTION

D

URING the period 1963-1970, significant advances have been made in the theory of decoupling or noninteracting control. I n 1963, Morgan [ l ] proposed a decoupling synthesis ut.iliaing a(staticallycompensated) state feedback control which result.ed in the first. study of the decoupling problem using the state space approach. The central existence question posed byMorgan u-as eventually resolved by Falb and Wolovich [ a ] in 1967. I n 1969, the present authors [3] formulated and solved a more general state feedback decoupling problem (u-ith dynamic compensation) and significantlyextended the class of systems t o which decoupling theory isapplicable. As a consequence of these and other results, decoupling theory is now at a high level of development. The objective of t,his paper is to describe this theory andt.0 identify t.he important cont.ributions upon which it is based. Roughly speaking, a multi-input, multi-out,put, d p a m i cal system is decoupled if each syst,em output can be independently controlled by a corresponding systeminput. Such a structure is desirable in a number of applications, including the control of turboprop engines [4], boilers [ 5 ] , [6], coupled core nuclear reactors [7], and VTOL aircraft, Manuscript received July 19, 1971. Paper recommended by work of A. S. Morse was supported in part by the Office of Naval Research Grant NO0014 67-A-0097-0020. The work of W . $1. Wonham was support,ed in part by the National Research Council of Canada under Grant A-7399. A. S. Morse is wit.h theDepartment of Engineering and Applied Science, Yale University, New Haven, Conn. W. M. Wonham is with the Departmentof Electrical Engineering, University of Toronto, Toronto, Ont.., Canada.

R. W. Brockett, Associate Guest Editor. The

[SI. No doubt, there are nmny other syst,ems to u-lich decoupling theory may be usefully applied. Our vien- is that, linear decoupling theory can be most easily understood if explained in a vector space setting. Conscquently, in this article we adopt the geometric approach of [9]. Suggest.ions are given in the Appendx for translating vector space operations into mat.rix operations suitable for comput'ation. Fundament-al geometric concepts including invariant and controlla.bility subspaces are discussed in Section 111. I n Section IV the controllabilitysubspace is used t o formulate t.hr restrickd (st.at.ic compensation) decoupling problem. Although the most general version of this problem isunsolved, there are known solut,ions for three special cases including the problem formulated by Morgan. These results are discussed in geomet.ric terms and the relationships betu-een t,hem and the corresponding Falb-Wolovich results are explained. A complete solution to the extended(dynamic compensation) decoupling problem (Sect-ion V) is known. Solvabilityconditions for this problem coincide with solvabilityconditionsfora corresponding open-loop decoupling problem; the significance of t.his resultis explained. I n Section VI the class of feedback matrices which decouple a fixed systemischaracterized. A simple matrix represent,ation is developed which exhibits the structureof a decoupled system in a useful n-a\-. Using this representation, it is a simple nlatter to describe the syst.cm pole distribut.ions which ma,y be achieved while preserving a decoupled structure. I n Sect.ion VI1 we describe a. systematic procedure for synt.hesizing a dynamic compensat.or of low order which will decouple a system. This procedure actually provides a minimalorder decoupling compensator for systems in which the number of open-loop input.s equal the number of outputs to be controlled. The relationship bet'ween pole assignment and compensator order is discussed. The possibility of using the geometric concepts of Section I11 t o formula,te and solve a variety of input-output structure problems is explored in Sect.ion VIII. Tn-o part,icular problems, triangula.r decoupling andpartial decoupling, are discussed. I n Section IX we provide a. brief historical a.ccount, of the contributions leading to the results present,ed in this paper.

11.

PRELIBIIXARIES

Notation and Background Algebra. BehJ-, capitalitalic lett-ers, 8 , B , c, ' . ., denote matrices; script letters, a,a, X, . . ., den0t.e linear vector spaces. The same 1ett.er is used to denote both a matrix

569

MORSE AND WOhqAM: STATUS CONTROL O F NONINTER.4CTING

and its map; a primedenotes t-ranspose. All mat,rices, maps, vector spaces, and polynomials are defined over the (field of) real numbers. Elements of a vector space are denoted by E , y , z, . * . ; the zero spa,ce is mitten as 0; the space spanned by a vector z is denoted by 3. The dimension of a space 'u is denoted by d ( W ) ; r ( M ) denotes rank ( M ) . The empt,y setis written as 4. A set of complex numbers is called symmetric if all nonreal element,s in the set. occur in conjugate pairs. We reca.11 certain basic definit,ions from linear algebra [IO]. Two spacesare called independent if their intersection is zero. The sum of two spa.ces, denoted by is the space spanned by the unionof their respective bases. The sum is called a direct sum, denot,ed by 0, if the two spaces are independent. If W c X, the orthogon.al corn.plentent of w , denoted by W , I is t.he set of all vectors in X which are orthogona.1 to all vectors in747. The notationM : U +-W means Jlr is a map fromU to W. If S c U, the image of S under M , denoted by AIS, is given by M S = ( v :v = M s , s E S} ; if S = U, M S is sometimes denoted by 3n. If 3 C '0, the symbol i P 1 3 { u:Mu E 3} denotes the inaeme image of 3 under N ; if 3 = 0 , X ( M ) = M - ' 3 denotes the null space of M. X is a basis m.atrix for Sn if MU = 3n and r ( M ) = d(m).If & = El @ E2, the projection from € onto €1 along & is the n1a.p P :€ + & satisfying Pel = el,.Pez = o for all ei E gi,i E !1, 2 ) . If A : % + X, @ c X, andd(T) = n, { A \ @ =) @ A@ .. An-1@. If 2, C X and AW c W, then 10 is ca.lled an A-invariant subspace of X. If T Iis a basis mat.& for V, the nmp corresponding to the matrixA which satisfies AV = V A is called the restriction of A to w ; this map issometimes denoted by A '0. The subspace z'is called cyclic if '0 = { A b} for some v E W ; v is called a generator of W. The spectrum of A is its set of (complex) eigenvalues list,ed according to multiplicity. If k is a fixed posit.ive integer, k { 1, 2, . . , k ] and ko { O , 1 , 2 , . . k}. A set of spaces ai, i E k, is denoted by { ( K i ) k . For this set

+,

+

+

+

In the follo\$hgdevelopment, full sta,te feedback isemployed. Of course, in practice act,ual measurement of the state is almost never possible, and so an observer [ l l 1, [12] might be required to estimate system statme. It. well is known, however, that for the system ( l ) , (2) any linear input-output structure realized via state feedback control may also be realized via "observed" state feedback control. Thus, assuming t,hat y and u may be measured a.nd t,hat (C, A ) is an observablepair,t.heproblem of decoupling with state feedback is the sa.me as the problemof decoupling with an observerplus output feedback. 111. GEOMETRIC CONCEPTS Both the formulationandsolution to the decoupling problem can be exceedingly(a.nd unnecessarily) complex if t,he wrong mathematical language is used. By using the language of linear vect,or spaces, it! has been possible to avoid much of this complexit,y and at the same timeto lay bare the essential features of the structure of decoupled systems. The vectorspace or geometric approach is based on the concepts of invariantandcontrollabilitysubspaces m-hich are discussed below.

Invariant an.d Controllability Subqnaces By starting wit.h the int.erna1system description (1) and employing a control law of the form u(t) =

a ,

2(t) = ( A

+

a* = n a,*.

Aw c w

A3

system

The cont.ro1 syst,em of interest is described by the differential equation .

(1)

a.nd the output relation

(5)

+ @.

(6)

The necessity of (6) is obvious. On the other ha,nd, if (6) holds for some 'u, one may compute an F E F ( W ) using the construction of Lemma 2 (see Appendix). For decoupling, we will be concerned withsets of inva,riant subspaces. A set, of invariant subspaces { TIi] Q is called compatible if n iEQ F ( W J # 4. If { w i } k is compatible, it is not difficult to show that

(2)

Here, u E %. is an m-vect.or, x E X is a.nn.-vector, and y E y is an r-vector; X, X, den0t.e the input, state, and output spaces, respectively, for the system. For simplicity, it is assumed that ( A , B ) is a completely controllable pair; i.e., { A CB) = X.

I

+ BF)U c W)

{F:(A

is nonempty. It is quite easyto show that F(W) # 4 if and only if

and

y ( t ) = C.(t).

(4)

+

F('0)

+ Bu(t)

+ B F ) x ( t ) + BGtl(t).

For much the samereasons that A-invariant subspaces (c.g., eigenspaces) are useful in describing the structure of A [13], ( A BF)-invariant subspaces are useful in discussing the effect of feedback ( F ) on t.he structure of the system (4).This brings us to our first concept. A subspa.ce '0 c X is called an invariant subspace of ( A , B ) if it is ( A BF)-invariant for some F; i.e., if the class

i #i

3(t) = As(t)

(3)

with v a.n ?%vector, there resuks the system

I

I

+ Gv(t)

Fx(t)

Au* c u* + ca and

nigF(ui) c

r l i F , (wi

-I-

(7)

w*).If

n F(WJ = i€Qn F ( W ~+ FU*)z 4 ;

(8)

ic,R

{ V i )Q is called stmzgly compatz3b. When this is so a.n F

E

570

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, DECEblBEE

1971

n ick F ( W I ) may be computed using the construction of of the folloning interesting fact:If & c @ and F(@) # 4, Lemma 3 (see Appendix). then Of particular importa.nce for the decoupling problem are (A+BF)&+@fI&=(A&+@)fI& inva.riant subspaces of ( A , B ) which are m a x i m a l relative to a fixed subspace of X. Write W(%) for the class of invari- .for all I" E F ( @ ) [9]. If F(&) # 9,this result may be apa,nt.subspaces of ( A ,B ) which are contained in aKxed sub- plied to the sequence of subspaces defined by a0 0 a.nd space X c X ; i.e., m ( ~=) { W : W c % n A - ~ ( ' u @)). i-1 It is clear that this class is nonempty [e.g., 0 E m(%)] a.nd ai= (A n (R j=O that it is closed under vector space summation. I t follows = (A BF)ai, @ fl a, i E n (13) that the class mustcontain a. uniquemaximal(largest) element Waf. This space may be computed directly from the where F E F(@j. The result is that given data. Theorem 1: W i t h % c X $zed, let ' u M be the maximal inai = Sf, i E n (14) variant suhspace of ( A , Bj contained in %. Then W" = vFJ where where 1.1 t d ( X ) and so 0, Sf (AS+' n a, i E n. (15) w0 = X, v i = X n A - 1 ( D ~ - ~ a), i E F. (9) Thus See [9] for a proof. S, = { A BFlB fl (RJ (16) For the specid case in which d(%) = n - 1, i.e.,

+

+

+

+

+

+

+

% = X(z'),

z # 0

(10)

the computat.ion of is particularly simple. For this case let d be the smallest nonnegative integer suchthat z'AdB # 0.

Kote that such an integer always exists, for if z'Ai-IB = 0, i E n,then z'( A @) = Z'X = 0; thus z = 0, which contradicts (10). Corollarry 1: If (10) holds, W M i s given by

I

whenever F E F(@). Clearly (R is st controllability subspa.ce if and only if F ( @ ) # 9 and & = s,. Even when @ is not a controllabilit,y subspace, (16) holds for all F E F( a).Under these circumstancesit is not difficult to show that S, is the largest controllabilit~7 subspace contained in @ [9]. This observation can beused to prove the following theorem. Theorem 2: W i t h X and W M defined as in Theorem 1, let @" beth.e mazimal controllability subspace of ( A , B ) contained in 31. Then

Proof: Woting (9) and t.aking orthogonal complements,

for all F E F(WJ3. For a complete proof, see [9]. Note that Theorems 1 and 2 provide an effective procewOL= 3, wiL = B A ' [ v ~ n- ~~ ~( ~ 9 1 i ,E p. (11) dure for computing (RM given %. First W* is computed via By the definition of d, ( A ' )i-19 c %(B')for j E d and (9); t.hen an F E F ( W 9 is determined using the construc(A')d9 n X ( B ' ) = 0. ,It follon-s by a simple computation t,ion of Lemma 2 ; relakion (17) provides the final result. that viL = (A')%for j E d . In part,icular 'ud' = Alternatively, one may compute @" by solving (15) for &i, (A')% andby ( l l ) , = 3 A ' [ x ( B ' )fl c i c d o S, n-ith (R replaced by I)";then = S,. Thisprocedure (A')%] = 3 -4'[x g ( d - l ) o (A')%] = Wd'. By induction, has the advantage of not requiring t,he computationof F. wjL = f o r j 2 d. Clearly, d 5 n - I = p, so w , = ~ Cont,rollability subspa.ces have the folloning important Wd', which completes the proof. property. Th.eorem 3: Let @ be a $xed controllability subspace and This corolla.ry, which provides t,he linkb e b e e n certain k t a(X) be aj.zed monic polynomialwith, degree ( a ) = d( a). matrix and geometric results, r i l l be used in the next secTh.ere exist8 an F E F ( R ) such that the characteristic polytion. nomial of ( A BF)I & i s &). If0 # b E @ fI (R is a&The second important geometric concept is the c.ontro1- trary, F can be chosen so th.at, in addition, b generates @. lability subspace. A subspace (R c X is called a controllaFor a proof of this t,heorem, see [9]. A procedure for bility subspace of ( A , B ) if for some F const.ructing F for the case = X may be found elsewhere & = { A BFIB ll @}. (12) in this issue. Controllability subspaces relateto linear systems as folKotice t.hat@ is ( A BF)-invariant; thus everycontrolla- lons. The setof states reachable from the zero state for the bility subspace of ( A , B )is an invaria.nt subspace of ( A , system (4)is given by { A BFI BS] and is a controllaBj. To charact,erize controllabi1it.y subspaces, use is made bility subspace. Conversely, if for fixed F and @ sa.tisfying (12), S is defined so t.hat BG = B (& 03 n @), @ will be the setof st,ates of the system (4) which are reachable from the origin.

+

xie xiEi0

+

+

+

+

+

+

571

MORSE AND WONHAbf: STATUS CONTROL OF NONIKTERtlCTING

There is also an alternative open-loop interpretation for controllability subspaces which is useful in st.udying certain problems. Let % denot,e the class of m-vector va.lued functions that is definedand continuous onT 5 [0, 11. Let. 4 : T X U + X denote the function

+(t, u) =

L'

eA(t-r)Bu(r) dr.

Let X c X be fixed a.nd define @tLvto be maximal cont.ro1IabiliOy subspace cont,ained in X . The following theorem provides the alternative interpretation for Th.eorem 4:Let Z denote th.e set of states x E X such that, for some u E U, @aTf.

dt,

U)

EX,

t E

T;

+(1, U )

= I.

Then

z

=

aJf.

This result is proved in [3]. Remark: We see that @t'w can be interpret,ed as t,he of set states of the system (1) which may be reached from t,he zero st,ate along a trajectory t.hat does not, lea,ve X ; in fact, it. is not difficult t>oshow t,hat,+(t, u)E (il" ( t E T) along all such trajectories. Clearly the choice of fixed endpoint point, t = 1 is not import.ant for t.hese statements t,o be true.

Invertfiility The quest.ion of whether a system may be decoupled is closely related t,o t.he question of system invertibi1it.y. Below, Theorem 4 is used t.o establish a simple condition for determining when the system (1)-(2) is invert.ible. L,et Y be t.he set of r-vect,or valued functionsdefined and continuous on T.Let, 6: U += Y be the function defined by e(u) =

c J' eA(t-T)Bu(T) d7,

t E T.

The system (1)-(2) is Zeft invwtz3le if e ( . ) is a one-to-one funct.ion;i.e., if O(u) = 0 implies u = 0. Clearly the system is left invertible if and only if the tmnsfer nmtrix C(X1 A)-'B is left invertible over thefield of rational functions in X. There is, however, an alternative way t.o check for t.his pr0pert.y. Write ' u M for themaximalinvariant (controllability) subspace of ( A , B ) contained in X ( C ) . Theorem 5: Th.e system (1)-(2) i s left invertible if and only if %(B) = 0 and f l @ = 0. Proof: Suppose (1)-(2) is left invertible. If w E X ( B ) a.nd u(t) =w ( t E T),then e(u)= 0; thus u = 0, sow = 0, proving X ( B ) = 0. If x E 'u" fl a,by Theorem 2 x E a".Theorem 4 providesa E U suchthat C+(t, a) 0 (t E T) and .x = +(l, a). Thus e($ = 0, implying = 0; therefore x = +(1, 0) = 0, proving V I f n 63 = 0. Suppose now that x ( B ) = 0 .and TI @ = 0. If e(u) = 0, +(t, u) E X ( C ) (t E T).B>- the remark belox- Theorem 4, +(t, u)E @tAv( t E T).But, since 'u" n @ = 0, by Theorem 2, @tAW = 0 or +(t, u ) = 0, t E T.Clearly Bu = 0. Since X ( B ) = 0, there follows u = 0, which completes the proof.

Historical hTotes To study the problem of disturbancelocalization [9], concept of an [14], Wonham and Morse introduced the invariant subspace of ( A , B ) and subsequently discovered the algorithm of Theorem 1. At approximat,ely t.he sa.me time (1968) Basile and Marro independently developedthe same concept. and discovered an almost identical algorit,hm [ l j ] . Earlier efforts relat,ing disturbance localization (also called signal or parameter invaria.nce [16]-[HI) and controllability were made by Wa.ng [19] and Levy and Sivan [20]. The closely related output^ zeroing"problem was studiedbyBrockettandIIesarovi6 [21], Weiss [22], Silverman [23], and Sivan [24]. Sivan in fact showed that Corollary 1 is true for t,he special case in which B is a vector. The concept, of acont.rollabilitysubspace [9] evolved from an att,emptt,o extend pole assignment results [25] to subspaces of thestatespace.Controlla.bility subspaces seem t,o be of fundamenta,l import,ance-to date they have beenused to effectively (andnat,urally)treatt,he combined problem of decoupling and pole assignment, (Section IV-Section VII), to invest,igate t.he synthesis question for various input-oubput struct,ures (Section YIII), t.0 obtain condit.ions for syst.em invert,ibility (Theorem5), t o identify a complet,e set of feedback inx7ariant.s for t,he pair ( A , B ) [26], and to clarify and extend certain results of observer theory [12],[26]. The applicat,ion of the controllability subspace to the question of syst.em invertibility is new herealthough a slightly more general version of Theorem 5 has also been independently esta.blished by Silverman and Payne [27]. For further discussion of t,he invertibilityquestion, see Brockett [28], Silverman [29], Sain a.nd Massey [30], and Dorato [31].

ITr. RESTRICTED DECOUPLING PROBLEM Formdation Suppose t,hat the out.put vectory for t,he system(1)-(2) consists of k subvectors yfw

=

C,z(t),

iEk

(18)

where yi is an rrvector; C and the C f are rela.t,edby C = [Cl', C2', ., C,']'. To avoid trivialities, it is assunled that C, # 0, i E k . Consider the (restricted) feedback law

-

which, when applied to (1), results in k(t) = ( A

+ BF)x(t) + B i€k G,vf(t).

(20)

Roughly speaking, the objective of decoupling is to determine F and the Gi so that, for each i E &, input ut can control output, yi without influencing t,he remaining outputs yj, j # i. To express t.his objective formally, let (i E k ) be the set of stat,es u-hich v i ma.y control. Thus & is the controllability subspace given by

572

IEEE TRANS-"IONS

=

+ BFIBg,],

{A

i E k.

(21)

Since y f is to be cont.rolled by vi for i E k, it. is required that2

ON AUTOMATIC CONTROL, DECEMBER

1971

therefore, tohave a simplecondition which can be used to determine when (28) is true. If { ( R f A w ] his compatible [i.e., (28) is true],

A(&")*

c a + (@")*.

To insure t,hat vi cannot influence yj for j # i, we must make certain that the stat.es vi can control do not affect yj for j # i. Thus

On the other hand, one may show that because of the ma.ximality of the a,"'and the special structure of the spaces conshiningthe [see (26)], tha,t if (29) holds, { Rf-''I k is strongly compatible (see 191). In other words,

manner. Given A , B , { X , ) h, Jind conditions for the existence of controllability subspaces ( R i (i E k ) of ( A , B ) such that

Conditions (27) and (29) are therefore suficient to insure the existence of a solution to RDP. Unfortunately, however,condition (29) is not necessary; if (27) holds and (29) fails, there still may be a solution t o R D P consisting of controllabilitysubspaces which are smaller than the &if". The following example illustrates this point. Let ei where i E 5 be the ith unitvector in five-dimensional space. Define

at

+Xt = X ',

iE k

(25)

n xi,

i E k.

(26)

ai c

j,#i

3ER

A set of controllability subspaces { &] 6 satisfying (24)(26) is called a solution to RDP. Condition (24) is necessary to insure that the set { k is compatible; i.e., to insure that the af may all be constructed nith the same F. Thus if (24) holds for a set of controllability subspaces { ( R f ] k , there is an F E ich F(@J such that (Rf}

n

at=

{A

+ BFIa fl a(], i E k.

Dificulties For k e d problem data, let (iE k) denote the mkximal controllability subspace such tha,t (26) is true. These subspaces are uniquely determined by A , B, { X t ) h and each may be computed via Theorems 1 a,nd 2. P.ropositiun 1: A necessary condition for the existenceof a solution to R D P i sthat

+ xf

i E k.

= %>

[ezl'

+ +

By a simple comput,at,ion &"' = el e2 e5 and &, = -%' ea e4 e5. Akhough (27) holds for these spaces, i t may = e5 and that (29) be checked that (a")* &-+r fl

+ +

I n addition, G f (i E k) may be computed so that Bst = fl at,thus insuring that (21) is satisfied. It iseasy t.0 verify that conditions (25) and (26) are equivalent to (22) and (23), respectively. Thus for vt to control y f , it is apparent t.hat 6 i f must be lorge enough so that (25) is true. To insure that v f cannot influence y j for j # i, af must be slnall enough so that (26) is true.

ai"

c1=

(27)

Proof: If { af]k is a solution to RDP, then satisfies (26). By the maximality of B f N , c RiA"and (27)folIon-s from (25). cRf

fails. Thus { @i"]z is not compatible. However, if one defines @ = @I'~ and @ = e3 e ( a ) where e ( a ) = e4 - (re5 and CY is a scalar, a solut.ion is at. hand. For fixed a # 1, it is easy to verify t,hat. is a cont,rollability subspace. Since a* = @I I7 6iz = 0, then Lemma. 3 provides an F E n is2 F((R,) # 4. I n addition, {a,)satisfies (25) a,nd (26) and is therefore a solution to t,he problem. Although for t.his example it is possible to find a set, of nonmaximal contro1labilit.y subspaces which can serve as a solution, in generalno systema.t,ic procedure isknown for finding solutions to R D P (if t,hey exist) when t.he will notr do; t,hat. is, n-hen { ( r t i M ) k satisfies the conditions of Proposition 1but is not compa.t,ible.It is for this rea.son t h t RDP, in its most general form, remui.ns unsolved. All is not lost, however. When certain additiona.1 const,raintsare added to the problem, compatibilit-ydifficulties evaporateandthe maximal controllability subspaces { 6 i f " f k provide a solution to RDP, if there is a solution at all. Forthese special cases, to be discussed below, t.he solvability question is completely answered.

+

6if"

It is clear that whenever (27) and

n F(R*"3 f 4

iEk

(28)

both hold, a solution t o R D P exists; e.g., under these circumstances { afATf} 6 can serve as a soIut.ion. It is useful, Note that this concept of output controllability coincides with that of [32] provided the Ci are of full rank.

Solution when R a n k (C) = n This assumption means that there isa one-to-one mapping of state variables into output variables, and thus this case is somelvhat special. R.ank (C) = n is equivalent to the assumption i€k

n X , = 0.

573

NORSE AND W-OKH.LM:ST.4TUS OF NONINTERACTING COXTROL

+ Xi

= E,

i E k.

(31)

a n at”.

=

Th.enrm1. 6: If (50) holds, a solu.tion to RDP exists if a,nd

only if

(35)

ick

If (36) is true, { 6 i l . c ” f ] k i s the only solution to RDP.

SufTiciency of (35) may beproved as inTheorem 7. Secessit,y of (35) a.nd t.he uniqueness of t.he solution are I j (31) h.olrls, ( at-$€} k i s a strongly compatible solution. Thisresult is proved in [9]. The proof of stficiency simpleconsequences of the hypothesis (34). For proofs, (necessity is establishedby Proposit,ion 1) amounts t.0 see [9]. shon-ing that (a”‘*)= 0 because ((R”) * C n ick X t = 0. This in turn implies t,hat (29) is t,rue and that { (Rf’If} k is Morgan’s Problem The original state feedback decoupling problem posed st.rongly compat,ible. by Morgan in1963 [I] is the special case of R D P for which Solution when R a n k (G) = m rank ( B ) = k (36) Here G = [G1, G2? . . ., Gk]. A41thoughthis assumpt,ion enables one to answer the solvability question for R.DP a.nd (it is thereforea. useful assumption), theredoes not appear rank (C,) = 1, iEk (37) to be any pract,ical reason for so constraining G. Rank (G) arethe simplifyingassumptions.Thus for the class of = n z means = nz or = %.Thus if { at)k is any fixed syst.ems being considered, the y r may be regardedas scalar solution t o R D P for which this assumption holds, output,s. Since (36) holds, Theorem 8 applies, thus providing a complete solutiont o t,he problem. However, becauseof the The assumption is therefore equivalent t.he to requirement special constraint on the C,, it is possible to developa solvability condition which, froma conlputational pointof 03 = n ai. ifk view, is particularly easy to use. For i E k, define d f as the least nonnegative integer such Theorem. 7: A sohtion to RDP satisfying (32)exists i f and only if tha,t @(tJEi

d(s)

s

a= i€k

a n a,”.

CfAd% # 0.

(33)

If (33) i s t.rue, { Rf-“]k i s a strongly compatible solution. Proof: Xecessity of (33) follows from (32) andthe maxinlality of the at”. For sufficiency, it d l fist be shown t,hat { at”f) k is a strongly compatibleset.. From (33), a = a fl afH a fl for i E k. There follows

(38)

It has already been noted in Section I11 that the d, exist. Define

+

Theorem 9: Morgan’s problem i s solvable if and only if rank (&I)

=

k.

(39)

This useful result, was discovered in 1967 by Falb and Wolovich [2]. It is possible to re1at.e condit,ion (39) to the corresponding geomet,ric condition (35). For simplicity t,his will be done only for the case k = 2; result.s for arbitrary k follow n4t.h minor modification. Let TI, 2 ) 2 be t.he maximal inyariant subspaces of ( A , B ) contained inX(C2),X(Cl), respectively. By Corollary 1,the satkfy ‘ufl=

where S f = e,’ and

Solution when Rank ( B ) = k

This condition is equivalent to the assumption that. the number of independent open-loop system inputs exa.ct,ly equal the numberof output vectors to be controlled. Rank Write X ( B ) = k also means d(@) = k.

081,

i E2

(40)

+ A’ej‘ + . . . (A’)d-lej‘, j # i; i, j E 2,

21 E

.z*

(A’)T2’,

(A‘)W.

X ( B ’ ) . It follows from the definition of

iE2

S,cX,

(34)

Theorem 8: If (34) holds, RDP has a solution if and only

if

Sf

(41)

d, that (42)

and

at n X

=

0,

i~

2.

(43)

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IEEE TRANSACTIONS ON A U T O W A ~ CCONTROL, DECEMBER

Using theserelations, it is possible to show the correspondence between the matrix and geometric results. Proposition 2: If k = 2, the following conditions for the solvability of Morgan’s problem are equivalent: rank (QB) = 2

(44)

n ( ~+p n ap.

=

+

+ +

+

+ + = ( X + 21) n (X + a2)

+

which is therefore equivalentto (45). To complete the proof, it is enoughto show that (44)and (46) areequivalent. If (44)istrue, r(B’) [&, 821) = 2. Clearly X, 31, 52 are independent and(46) is true. Conversely, if (46) holds, 31 n X = al fl (X &). From this and (43) there follon7 9 fl (X &) = 0 and a2 fl X =O. Thus d(B1 82 X ) = d(&) d(&) d(%) = n o r ’6.1 & X = X. Clearly @’ = B’(81 &), and since r((?B) = r(B‘e’) = r(B’[&, 321) = 2, (44)is true.

+ +

+ +

+ +

= - (QB)-@A [2]; i.e., F E F(aIM) fl

decouples the system see why this is so, note that (?A

u

= U @ %. Extensions

terms of matrices. Wit.h

the corresponding mat>rixrepresent,ations of the extensions are

A

=

[oA

B

00]

c, =

=

[oB

[C,, 01,

O0]

E

=

[; ;]

i E k.

If P:X + 3 is the project.ion of if on X along 5, then asa matrix

+

For Morgan’s problem t,here is no loss of genera.lit,y in a.ssuming t.hat,the C iand B are full-rank mat,rices. Under these circumstances, 6 B is a square 2 X 2 matrix. If (44) is true, (?B is nonsingular and the matrix

F

and

@

B , E ,Ci, i E k, respect,ively, are t,hen defined in the most obvious way. It is helpful to visualize t-hese extensions in

(46)

=

+ +

2 =X

A:x+i?,B:ii+%,E:%+iT, C t : x + ( y i , i E k , o f A ,

(45)

Proof: As a simple consequenceof Theorem 2, ‘Ut fl l f@, i E 2. Thus (45) maybe rewitten as @ = @ f l V1 @ n ‘Us. Takingcomplementsandapplying (40), there follows X = (X Wll) fl (X VZL) = ( X SI 31) fl ( X SZ 32); thus by (42) =

( 2 ) , (47), write

1971

where I is t,he ?I. X n identity. To decouple the outputs y f ( t ) = Cf3(t),i E k, consider the feedback control

G(t)

=

F3(t)

+ iEk Gioi(t).

mith t.his control applied to theextended system, t,hereresu1t.s

%(t) = ( A

+ (B+ E ) F ) % ( t )+

(B + E)Givr(t).

To

Theobjective of decouplinghere is the sameasfor RDP; the only real difference bet.xeen rest,ricted decoupling and t-he present, situation is the para.meterA, which (A’ F‘B‘)zj = 0, i E 2. may be freely adjusted. Thus, withX i= X(CJ (as before) But (A’ F’B’)’UtL = (A’ F’B‘) (Si 8i)= A‘St c and in a manner analogous to RDP, one may formulate ‘UfL;thus,t,aking complements, ( A BF)’Ui c V i . the extended decoupling problem (EDP). Clearly F E F ( q ) fl F(v2) c F ( B l M ) fl F ( G - ~ ) as Given A , B, {Xt]h, .find conditions for the existence of a claimed. space X (i.e., 6)and controllability subspaces Wt,i E k, of Using eithermat,rix or geometric argumenh, itis possible ( A , B E ) .such that to show that an appropriate G for decoupling is G ((%?)-I. For an explanation of this in matrix terms, see [a]. ~ ( a z, )0 (48)

+

+

+

+6BF

=

0 or

+ +

+

n

i€h

V. EXTENDED DECOUPLING PROBLEN

aa+%%+%=iT, i E

k

(49)

x+

It has already been noted in SectmionIV that RDP is an

gt c n zj, i E k. (50) iEh, unsolved problem. The situa.t.ionimproves, however, if t.he j #% class of adnlissible control lawsis broadened t.0 include dyN0t.e that in this formulation use is made of t,he fact that namic or integrator feedback. Under these circumstances, 9. it is possible to state andcompletely solve the more general %(CJ = xi Remark: It isknown [3] that. if GI is a cont,rollability extended decoupling problem. E ) , t.hen PW is a cs of ( A , B ) . subspace (cs) of ( A , B Fornzulation Conversely, if W c 3 and PW is a cs of ( A , B ) , then Fi is 8).Thus, for example, (A, B E ) is a A convenient way tate thefollowing important conof t'he factthat for anyfixed set of cs of ( A ,B ) , compa.tible clusion: If the system ( l ) ,(2) can be decoupled at all, by a n y or not, it is alxays possible to const.ruct a corresponding 7nean.s whatever, then it can always be clecoupled in th.e sense set of extended cs, { Gli)&, which is ~ompatible.~ With con+ of EDP. patibility of such extensions assured, t,he only rema.ining solva,bility requirement for EDP is t,hat. the spaces being VI. STRUCTURE OF DECOUPLED SYSTEMS extended be sufficient,lylarge; i.e., (51) must hold. Using the conceptsdeveloped so far, itis possible to disThe effect. of extension is easily illust>ra.t,edRrit.h t.he ex- cuss the problem of simultaneously decoupling a system ample of Section IV. Regarding ei as the it,h unit vector inand assigning its poles. This problem is approachcd below six-dimensionalspace (i.e., d(X) = e), ext.ensions W is by first describing t,he class of feedbackmatriceswhich el ez e5 and Gt2 _= e3 e4 t? (wit.h i? =e5 es) of 6 i l M preserve the decoupled struct,ure. Thcn by representing the and 6i2"', respectively,areobviouslyindependentand (decoupled) syst,em in simplenmtrix termsexhibiting therefore compatible. Thus, for this example, a1,G2 pro- syst.em struct.ure, t>hepole assignment question is easily vide a solution with ii. = 1. resolved. Under certain circumstances t,here are other conditions In thefollowing, E , { Gi} & is a fixed solut,ion to EDPj a.nd which may be used t.0 check for the solva.bility of EDP. { a i ) k is assumed to be st,rongly compatible. 1Vrit.e F = Consider, for exa.mple, the extended version of Morgan's F(6&) and B = B E (for RDP, B = B ) . For simproblem; i.e., EDP with constraints r(B) = k and r.(Ci) = plicity it. is assumed t,hat 1, i E k. For simplicit,y assume that B and t,he Cli,i E k, = 3. (32) are full-rankmatrices. It is easy t.0 show that thisproblem iB is solvable if and onIy if t,he (square) transfer matrix T(X) C(M - A)-lB is invertible over t.he field of rational This relation holds for solut,ions t.o RDP when either r ( B ) functions [33]. Since invertibility and left invertibility are = k or r(G) = 772. It also holds for solut,ions t.0 E D P when equivalent properties for square matrices, the conditions r(B) = k , provided such solut.ions are computed via the of Theorem 5 are also solvabilit,y conditions for this prob- const>ructions suggest,ed in t.his art.icle. Forthe more lem. Although t.hese conditions are easy tocheck, t,hey are general case in which (52) is not t.rue, see [3]. of litkle help in const,ructing a solution when oneexists. Feedback Class

+

+ +

+ +

+

nick

+

ai

To characterize F , several definitions are needed. Write Open-Loop Decoupling l o W* and let, Gi and Gibe chosen so t,hat Since the class of systems for which EDP is solvable is larger than the corresponding class for RDP, it, might be Rf = & i @ W i r lgo, i E k (53) conjectured that an even larger class of (linear) systems 8 9 , = B n (zi zo), i E k,. (54) could be decoupledusing contxol laws of more genera.1 structure. This conjechre is false. To understand why, it. It is easy to prove t.hat, = &lo G j (see [9]). is helpful to consider the open-loopdecouplingproblem Let Pi(i E R,) be the projection on 8i along (ODP). Ej. Given A , B, Ci, i E k, and arbitrary vectors y i E e , i E j #i E k o k, Jind controls u i E U,i E k, su.ch that4 Define H as H = { H :H = GiHiPi; all H , of appropriCk(1, ut>= yi, iEk ate size). For simplicit,y, and wit.hout loss of generality, C&(T, ut) = 0 , i, j E k,j f i. assume that %(B) = 0. The following propositmionprovides a useful characterizat,ion of F . Th,eorem 11 :O D P i s solvable for arbitrary y i E C ii f and Proposition 3: Let Fo E F . Then F1 E F if and only if only if - Fo E H . (55) aiM %(CJ = X, i E k. The proof of this t.heorem, which depends on the interpre- The proof of this proposition is straightforward. For arguments leading to similar results, see [9, pp. 13-17]. tat.ion of controllability subspaces provided by Theorem 4, is stra.ightforward a.nd may be found in [3]. Matrix Stmctupe For fixed Fo E F , let, i i= A B"Fo.To develop a cona For examde. using t h e construction of Lemma 4 wit,h @ t i = (i E k) and & 2 0, a s e t of independent (i.e., GL* = 0 ) and therefore

x

+

c ciao

+

W

f

compatible extensionsmay be obtained; for this solution, the number of requiredintegrators, A, is n*(@to) as specified in Lemma 4. 4 +, T, U are as defined in Section 111.

+

6 811 statemenh in this section also apply t o RDP; simply delete overbars the t,hroughout .

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I E E E TRANSACTIONS O N AUTOXITIC CONTROL, DECEMBER

venient. matrix representation for a decouplcd system, use is made of the following lemma. Lemma 1 :

Pix

PiAPi,

=

i E k

ci = CiP*, Pi&

=

(5G) (57)

i E k

j # i,j E ko, i E k

0,

{ P i x (P&)

=

i

Gj,

E k.

(”) (59)

+

Proof: For i E k , j E ko,j # i, P,xGj c Pi(&, Go) = Pix Pi = PiAP, and (56) is trur. Since a*) = 0 for i, j E k , i # j , there follows Go) = 0; t,hus ci = Pi = c,Pi and (57) is true. Relations (5s) follow from (54). Finally, for i E k , E i = Pix = Pi&& = P i ( A (&BSi) = PijLT1PiBS,] = { P$I Pi&,} and (59) is true. =

cj~~

0; t.hus P i x

cj(Wj+ cj(Ej+

cicjEdo

At this point, one could immediately draw a number of conclusions regarding the st.ructure of a, decouplrd system. However, since in practice it. is proba.bly most useful to have a simple matrix representation for a decouplrd system, n-e develop such a representation non- and draw conclusions later. For i E KO, factor Pi into the product of two full-rank matrices, Pi = LiMi, and define A i XiALj, Bi 5 ilf,BG,, c,Li.For i E k , define Aoi = X d L , and Boi= M d G , . Lemma 1’:

ei

Mix = A,Mi, ci = e,n/r,, =

iE k

(56’)

Ek

(57’)

j # i, j E ko, i E k

(5s’) (59’)

i

0,

1971

out.puts andwhich is driven by all the inputs. This t,hen is the matrix struct,ureof a decoupled syst.em.

Pole Assignme??$ To discuss the pole assignment, quest.ion, notethat, since each (Ai, B i ) , i E k , is a completely controllable pair, by Theorem 3 t.hc Hi,i E k , can be chosen so that. each A i BiAiwill haw some preassigned symmrtricspectrum. Since x0 is isolated from all out.puts, it cannot) affect. the input-output belmvior of t.he system; clearly the input,output. tra.nsfer matrix is determined solely by the system (61). Thus all the poles of a decouplcd system may be freely assigned with suitable H E H . When the eigenvalues of A” B”H arc considered, t.he situation is somrn-hat diffcrcnt; certain eigenvalues of A B H m a - be fixcd for all H E H. Since the spectrumof d BH equals the disjoint union of the spectra of the A i BiAi, i E ko, it is clear that any fixed eigenvalues of A BH must be fixed eigenvalues of A. Bogo. XOK t,he spectrum of R&o may be split into tn-o subsets, t.he first consisting of t h r eigenvalues of t.he restriction of A , R&o to {A,/@o} , and the second consisting of what is left,. The first of these two setsis obviously assignnblc u-ith suitable A since { A o j @o] is a controllable space. I t is easy to shon- that the second set is fixed for all €? [3]. Obviously this second set., call it A , is the set of cigenvalues of A” B H n-hich is fixed for all H E H. Thus the eigenvalues of A BH split into k 3 disjoint sets consisting of the spectrum of A i Bigj, i E k , t,he spect,rum of ( A o B g 0 )restricted to { A01 @o}, and A. Each set., except. A, may be freely assigned with H E H , and A is fixed for all H E H.

+

+

+ + + +

+

+

+

+

+

+

+

+

VII. A I m I m L EXTENSION For fixed A , B , { x i ]A , suppose EDP is solvable and conThese statements follow directly from Lemma 1; proofs sider the class of solutions t.0 t.he problem. Associated wit,h are t,herefore omitt.ed. each solut.ion is the integer fi = d(k)d i c h corresponds t o Consider now t,he decoupled system described by the number of integrators needed t.o implement t.hat, pari ( t ) = (A BH)Z(t) B Gic(t) ticular solution. For pra.ctica1 reasons, it. would be useful iEk (60) to find a solution in t.his class for n-hich A is as small as yi(t) = C+?(t), i E k possible. The problem of finding such a (minimal) solution apparently includes the solvability question for the correwhere H = &R~ GiHiPi E H. By introducing the nensponding RDP and is therefore unsolved. However, if the variables x i = M + T , i E ko (or equivalently, the coordia.dditiona1 constra.int, is imposed t.hat the a j we seek be nate transformation df where 1 1 ’ = [X,’,. . . , d l k ’ ] ) ,one cstrnsions of corresponding aiSf, i.e., may describe the syst,em (60) by PW, = aiM, i E k (63) &(t) = (Ai BjBi)Xi(t) BiVf(t) i E k (61) then itis possible to find a solution for which A is minimal. Yi(t) = For the specia.1 case where d(@) = X- (i.e., t,he number of open-loop sca1a.r inputs tot,he syst.em (1) equa.1the number of output vect.ors to bc controlled), every solut,ion t.o EDP ( ~ o ~&o)xo(t) C BoiL’i(t) (62) (minimal or not,) satisfies (63) and thus this case is solved ish completely. where Hi = fiiLt, i E ko. This represent,ation exhibit.s the Suppose t.hat a. minimal solution t.o the problem satisfysystem (GO) as k-independent completely controllable sys- ing (63) is found. Let-4be the set of fixed eigenvalues assotems (xi, i E k ) , ea.ch cont.rolling a single out.put yi, plus ciated with this solution. One can show that is uniquely one addit,ional system (xo),which is isolated from all t.he determined by thegiven system data and therequirement.

i E k.

( A , B i ) - i scompletely controllable,

+

+ e&)

+

1

+

+

+

i

+

MORSE AND 'S'OMHAM:

577

STATXX CONTROL O F NONINTERACTING

that the solution be minimal relative t.0 (63). Now it may happen that l i contains a subset, of bad (e.g., unst,able) eigenvalues svhich rendcrs t.his minimal solution unacceptable in practice. It. is possible t.0 remedy t.his situation by finding a second solution t.o E D P (ut,ilizing larger 6 ) x-it,h fixed eigensdues srhich are not. bad. In fa.ct.,wit'h sufficiently la.rge solutions may be const.ructedwit,h no fixed eigenvalues at all. To proceed, split A int.0 t,wo disjoint. symmetric sets Au (good eigenvalues) and A, (bad eigenva.lues). A solution to E D P is called good if itJscorresponding fixed set. of eigenvalues is disjoint srit.11Ab. Relat.ive to const.raint, (63), it. is possible t.0 find a good solut.ion to EDPfor svhich A is minimal. One can show that t,he set of fixed eigenvalues for such a solution is, in fact, a subset of A,. Below the procedure for finding a minimal solution of this t.ype is briefly described. For a more det,ailed account of this procedure, including associated proofs a.nd an exa.mple, see [3]. f i t

For the given data A , B, C, i E k, it,-isassumed that the aiwhave been comput.ed and E D P is solvable. Define 6 (6) t.0 be the maximal invariant (controllability) subspace of ( A , B ) contained in (@")*. For fixed F E F(6), let .(X) = characteristic polynomial of (-4 BFI and B(X) = charact,eristic polynomial of ( A BFI Find y(X) so that a(X) = fl(X)y(X). The set A is given by A = { X : y(h) = 01. Partition t>he set A into two disjoint symmetricsubset,s li, and &. This is equivalent to svrit,ing y(X) = y,(X)yb(X) where y oand yohave obvious interpreta,tionsandare coprime. Define W = 6 (?,(A BF))-l&. With aiE &i-wJ i E k, and

+

+

n

a,= n

i€k j # i j€k

c)

+

IR;'~ n 'O

use the construcOion of Lemma 4 t.0 compute a solut,ion { wit.h int.eger A = n.*(&). This solution still be minimalrelative to (63) and the requirement that, its fixed e i g e n d u e s not be bad. If A, = A, then y u = y and the corresponding solution is minimal relat,ive only t.0 (63) [3]. On the ot,her hand, if A, = 4, t,hen y b = y and the corresponding solut.ion is minimal relative to (63) and t.he constraint, t,ha.t there be no fixed eigenvalues.

ai]&

VIII. SYNTHESIS OF INPUT-OUTPUTSTRUCTURES So far, theonly type of input,-output, structure discussed has been the (diagonally) decoupled configuration. It is possible t.0 use the geometric concept,s of Sect,ion 111 to formulate and solve ot,her problems of system structure.' A general approach t.0 such problems is discussed below. If % (respectively k ) = 0, t,hen a(X) [respectivelyp(X)] = 1. This was suggested in [34]. For a different approach t.o such structure problems, see [35]. 7

Starting with the fixed data. A , B, 31,. where i E k , let kiand ki,i E k, be fixed subsets of k. These subset,sspecify a synthesis problem as follows: for t,he syst,em described by (18) and (20), find F and Gi, i E k , so t.hat f o r i E k , v i can control y,., j E ki,wit.hout influencing y p , q E &. As with decoupling, these requirements can be translat,ed into a formal algebraic problem. Skipping det,ails, it. is clear that the problem is t,o find a set of compatible controllability subspaces of ( A , B ) , { for which the following conditions hold :

at+ n

i

X,.= X ,

~

k

(64)

j€ki

aic

fl X,., i E k.

(65)

j&

An extended version of this problem may also be formulated along the same lines as EDP. Almost. everything host-n about t.he decoupling problem generalizes t,o the present problem. Let { I R i M } k be the set of maximal controllabi1it.y subspaces of ( A , B ) satisfying (65). For either the restrict.ed or extended version of t,he above problem to be solvable, it is clearly necessary that

If (66) holds, the ext.ended problem is solvable; if in addit,ion { I R t - " ; i ) ~is compatible, the rest,ricted problem is solvable. All stat,ements regarding pole assignment, for t>hedecoupling problem apply here without change. Only t.heresu1t.s pertaining to compatibi1it.y of { @;Ifli)k and to minimality do not, generalize to the present situation. Several specific synt,hesisproblems,fitting int.0 this general fra.mesvork have been discussed in the literature. Below, two of these are briefly discussed. fiian.gular Decoupling Problem In words, the system (18), (20) is triangularly decoupled if F and t,he G, ha.ve been chosen so t,hat v i can control y t , i E k , without influencing yi f o r j > i. The problem derives its name from the upper t.riangular structure of its associated t.ransfer f u n d o n matrix. The triangular decoupling problem (TDP) may be put. in t,he contextof the general -struct.ure - problem formulated above by defining k , = { i ) ,kh = 4, k f = ( i 1, . . , k} for i E k - 1. It is shown in [34] that condit,ion (66) is both necessary and sufficient for T D P t o be solvable; i.e., the set, ( k is necessarily compatible. Furthermore, t,here are no fixed eigenva.lues associated wit,h solutions to TDP. Apparently thereis not,hingto be gainedby considering the extended version Of the problem' For a discussion of T D P see [34], [36].

+

partial ~

~

~ problem ~ ~ ~

l

i

.

,

~

~

Stmt.ingwith RDP, suppose thatfor a single fixed integer E k, we dispense wit,h the requirement that vq must control yn. With all other requirements of R D P remaining the same, clearly pairs t,he (vi, y t ) , i E k, i # q,

578

IEEE TRANSACTIONS ON AUTOMATIC

must be decoupled in the usualsense. When a system isso configured, i t is called partially deeoupbd. More precisely, the partial decoupling problem (PDP) is specified by ki= { i) where i E k and i # q, kg = 4, a,nd Ei = { 1, 2, . . ., i - 1, i 1, . . . , k), where i E k. Wit>hout anyadditionalconstraints,the solvabi1it.y quest,ion for P D P is as difficult to answer as it is for RDP. However, if the additionalconstraint r(G) = m is imposed, a complete solution t.o the problem may be found. This constraint means that, (32) must hold for any solution to PDP. Clearly the same basic argument used to prove Theorem 7 may be applied here. It follows that wit,h r(C) = m , P D P is solvable if and only if the i E k, satisfy (32) and (64). For a more det,ailed t.reat,mentof PDP from a different. point. of view, see Sat.0 and L0prest.i [37].

+

Ix. HISTORICAL BACKGROUND There is considerable literature on the problem of noninteraction,with much of it. of hist,orical interest. only. Below we briefly survey this literature anddiscuss some of the ma.in contributions leading to the present theory. For more detailed accounts of the classical cont.ribut,ions prior t.0 1963, see Ravanagh [3S], Meerov [39], Slivinsky et al. [7], Tsien [40], Peschon [41], a.nd IIesarovib [42]. One of the earliest, (1934) known invest,igat.ions of noni n t e r a h o n n-as made in the Soviet, Union by Voznesenskii (see [39]), mho was concerned \\-it.h t.he control of power station turbines.In thiscountry, credit goes to Bolcsenbom a.nd Hood [4] for t,heir early (1949) eff0rt.s to develop a decoupling procedurefor t.he cont.ro1of gas turbineengines. Bythe mid-1950’s t.ransfer matrixmethods were n-ell established and Freeman [43], Kavanagh [44], and others were applying t,hem tothe problem of nonint.eraction. Kavanagh’s approach was to specify completely the desired closed-loop transfer matrix at. the outset and then to attempt tofind the corresponding compensating controller. As one might expect, this type of a.pproach led to funds.mental realizability problems (i.e., compensating transfer matrices which are not regular at, infinity) as well as st,abilityproblemsdue t.0 hidden pole-zero cancellations. Freeman [43] and othersconsidered these matters, but. the basic problems Rere never really resolved. Ot.her att.enlpts to design norlinteracting controllers using frequency domain methods (e.g., see [45]) also met 1vit.h 1imit.edsuccess. I n general, the status of decoupling t,heory in 1963 was not very satisfactory.I n spit,eof t.he large numberof papers devoted to the problem, there were fen: concrete resu1t.s. Without.making unnecessarily restrictiveassumptions, precise conditions were not known which would stat.e when asystem could be decoupled withaphysically realizable controller. The problem of potential instability due t.o hidden pole-zero cancellations was not really understood. Most. of the proposed synthesis techniques involved complicatedtransfermatrixma.nipulations which could not be easily implement,ed on a digital computer. It was also in t,he early 1960’s when the computational

CONTROL, DECEMBER

1971

(and conceptual) advantages of t.he state space formulat.ion were being discovered. I n 1962, Gilbert [46] successfully explained pole-zero cancellations using state space concepts (i.e., controllability and observability). Gilbert’s work plus t,he lackof signifimnt progress with the classical transfer matrix approach apparent,lyled Morgan t.0 formulate in 1963 the decoupling problem a.s a st,at,e feedback problem [l1. Norgan’s formulation and subsequent investigation were significant. because they st,imulat.ed research act,ivit,y in a new and fruitful direction. I n 1965, Rekasius [33] invest.igat.ed Morgan’s problem and obt.ained some useful resuhs extending earlier work. I n 1967, Falb and Wolovich [a] proved Theorem 9 and thus complet,ely solved Morgan’s problem. Theirresult was an import,ant, contribut,ion, being the first complete solution t.0 a decoupling problem for a significant, class of linear systems. I n addit,ion to establishing this result., Falb and Wolovich were able to characterize t,he class of decoupling feedback mat,rices (F) and also to identify the number of closed-loop eigenvalues which could be freely assigned. Unfortunat,ely thedefinition of decoupling used by Falb and Wolovich was difficult t,o understland, their charact.erization of F was cumbersome, and they did not, a.dequately describe t.he struct,ure of a decoupled system. ThesefactorsapparentlymotivatedGilbert (1968) t,o further investiga.te Morgan’s problem [47]. Gilbert was able to develop a simple characterization for F similar t.0 the cha.racterizat,ion provided by Proposition 3.He ident,ified t,he system eigenvalues fixed for all F in the class F and also described t.he det.ailed structure of a decoupled system using a mat.rix representation similar t.0 (61)-(62). I n 1969, Wonham a.nd Morse [9] generalized Morgan’s problem by formulat,ing in simple geomet,ric terms t,he restricteddecouplingproblem of Section IV. Use of the geometricapproach (or more specifically, the controllability subspace) result.ed in a t,ransparent, problem formulat,ion and a simple description of system struct,ure. Solvability conditions a.nd associat,ed pole assignment, results were obt,ained for the cases r(C) = ‘H. and r ( B ) = 12. Using different approaches, Sato andLoprest,i [37] and Silverman and Payne [27] solved R D P for t,he case r ( G ) = m. It hamsbeen known for a long time that system invertibility is related to the problem of noninteraction [4S]. I n 1965, Rekasius [33] showed t,hatsysteminvertibility is necessary and sufficient. for the extended version of Morgan’s problem to be solvable. Gilbert [47] provided an example of an invert.ible system which could be decoupled in an extended sense only. Although invertibility conditions such as Theorem 5 can be used to answer the solvabilityquestionfor the extended version of Morgan’s problem, withoutfurther interpret,at.iont,hey are of limitedhelpinconstructing decoupling controls and they provide little insight into the struct.ure of decoupled systems.hIoreover,invertibility does not, appea.r to be related t.0 the solvability question for t.he more general E D P when vect,or outputs are considered.

579

MORSE AND WONHAM: STATUS O F NONINTERACTING CONTFlOL

I n 1969, Morse and Wonham [3] formulated and solved efforts have been made t,o develop a decoupling theory for the extendeddecouplingproblem (Sect,ion V), described non1inea.r systems [57], time-varyingsystems [58], [59], the struct.ure of decoupled systems and answered related and disCributed systems [60], t,he considerable difficulties pole assignment. questions (Section VI), and devclopcd a, of these generalizationsof the problem haveso far inhibited systematic procedurefor synthesizing low-order decoupling subst,antial progress. compensators (Section VII).Thesecompensators were shown to be of minimal order for plants having a number APPENDIX of open-loop inputs equal to the numberof open-loop out.puts t o be controlled. Algebraic Comtructiun~ I n another study of EDP, Howze and Pearson [49] also Below the spaces X, 'U as well as the maps A :X X, showed that additional freedom in assigning eigenva.lues B : U X are fixed. could be achieved by usingdynamic compensation of suffiLemma 2: Let 'c' c X and suppose Aw c 'u a;there ciently high order. Basile and Marro [36], using the geoan F : X + U such that ( A BF)v c W. exists met,ric approach,andSilverman (501, Wang [51], and Construction: Write X = X1 0'c: for some X1 and let Q Silverman and Payne [27], using the matrix approach, all be the project,ion of X on XI along W. Let V be a basis have recently investigated various aspects of EDP. Morse matrix for W. Since QAW C QB, QAT' = Q S Z has a solu[52] and Silverman and Payne [27] have extended most of tion 2. Since V is of full rank, 2' = - V'F' has a so1ut)ion t.he results described in this art,icle to 1inea.r syst.ems with F'. Thus more general outputrela.tions of the form y i ( t ) = C,z(t) D*u(t)where i E k. 0 = QAV - QBZ +-

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= QAV

X. CONCLUSION The decoupling procedures discussedin this papera.pply to linear,const.ant, finite-dimensiona.1 cont.inuous systems. It is obvious that with minor modification these procedures will apply equally well t,o analogous sampled-data systems and even to discrete linear systems defined over finit,e fields (i.e., 1inea.r sequential machines). For all such systems, the theory provides effective procedures for determining when a system may be decoupled, and t,hen, how thesystemcan besimultaneouslydecoupled a.nd stabilized in an efficient way. I n spite of the generalit,y of t,he present theory, t,here remain a numberof important problems for future investigation. Probably t.he most. challenging of these is the solvabilit,y questionfor t.he general case of RDP; closely related istheproblem of finding minimal solut,ions to EDP. Keeded here, assumingthe geomet.ric approach isused, are syst.ematicprocedures for generat,ing nonmaximalcontrol1stbilit.y subspaces and, in addition, a simple procedure for checking when a. set, of subspaces is compatible. Another importa.nt endea.vor would be to develop efficient and a.ccurate computatrue.The dimensional relat.ion (69) is also true since d(%) = d(m) = q - T ( 8 ) = n*(ao).

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Remark: This a,pproachto extension is a generalization of the procedure given in [49]. Matrix Computations Below we very briefly outline one possible way of translating basic vector space operationsinto matrix operations suitable for computation. The translations are the obvious ones. More efficient. computational procedures, perhaps utilizing the generalized inverse, will no doubt be developed in the future. I n addit>ionto an accurate 1inea.r equation solver, three other basic computer procedures are required: reduction, 8

If 21

t

u,d(U/W) 3 d(%)

- d(W).

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581

bSORSE AND WONHAM: ST.4”ESCONTROL OF NONINTERACTING

parameter invariancein d p a m i c a l systems,” IEEE Trans. Automat. Contr. (Corresp.), vol. AG11, pp. 614-615, July 1966. [18] J. Preminger and J. Rootenberg, “Some considerat.ions relating t o control systems emploJ.ing the invariance principle,” I E E E Trans. Automnut. Contr., vol. AC-9, pp. 209-215, July 1964. [19] P. ,K. C: Kang, “Invariance, Fcont.rollability, and unobservablllty In dynamical syst.ems, I E E E T.rans. Automat. Conir. (Corresp.), vol. XC-lo, pp. 366-367, July 1965. [20] S.-Levy and R.. Swan, “On the stability of a zero-output system.” IEEE Trans. Autonzat. Coonfr. (Corresv.), vol. AG11. DV. 3i-C-1316, Apr. 1966. [21] R.&V. Brockett and hi. D. Mesarovib, “The reproducibility of multivariable spst.ems,” in Proc. 1964 Joint Autonmtz“ Control Conf., pp. 481686. [22] L. Weiss, “On a question related to the cont.rol of linear systerns.” IEEE Trans. Auiomat. Contr. (Short Papers), - ., vol. AC-9, pp. f76-177, Apr. 1964. [23] L. M. Silverman, “Propert.ies and application of inverse systems,’’ I E E E Trans. Automat. Contr. (Short Papers), vol. AC-13, pp. 436-437, Aug. 1968. [24] R.Sivan, “On zeroing t.he out.put. and maintaining it. zero,” IEEE Tram. Automat. Conlr. (Short. Papers), vol. AG10, pp. 193-194, -4pr. 1965. [25] ‘Ar. M . Wonham, “On pole assignment in multi-input. controllablelinear svstem.” I E E E Trans. Auiomat. Contr.. vol. AC-12, pp. 666-665, Dec. 1967. [26] ‘Ar. &I.Wonham and -4. S. Morse, “Feedback invariants of linear multivariable systems,” present,edat. the 2nd IFAC Symp. Multivariable ControlSystems,Dikseldorf,Germany, Oct,. 1971; also t o be published in Azctornatim., Jan. 1972. [ E ] L. M . Silverman and H. J. Payne, “Input-output structure of linear systems with application t o t.he decoupling problem,” SIAM J . Con.tr., t.o be published. [28] R. W. Brockett, “Poles, zeros, and feedback: Statespace int.erpretation,” I E E E T.ra.ns. Azdomat. Contr., vol. AG10, pp. 129135, Xpr. 1965. [29] L. M. Silverman, “Inversion of multivariable linear syatenq” I E E E Trans. Automat. Contr., vol. AC-14, pp. 270-276, June 1969. [30] h.1. L.. Sainand J. L. Massey, “Invertibility of linear t.imeinvariant dynamical systems,” I E E E Trans. Automat. Contr., V O ~ .AC-14, pp. 141-149, Apr. 1969. [31] P. Dorato, “On the inverse of linear dynamical systems,” I E E E Trans. Syst. Sci. Cybernetics, vol. SSG5, pp. 4348, Jan. 1969. [32] E. Kreindler and P. E. Sarachik, “On the concepts of cont.rollabilityand observability of linearsyst,ems,” I E E E Trans. Automnut. Cqntr., vol. AC-9, pp. 129-136, Apr. 1964. [33] Z. V. Rekasms, “Decoupling of multivariable syst.ems by meam of state variable feedback,” in Proc. 3rd Annu. Allmion Conf., 1965, pp. 439-448. [34] A . S. Morse and W. M. Wonham, “Triangular decoupling of linear multivariable syst,ems,” I E E E Trans. Automa.t. Contr. (Short. Papers), vol. AC-15, pp. 447449, Aug. 1970. [35] S.XI. Sat0 and P. 6 . Loprest.i, “Kew results in multivariable decoupling theory,” Aufonmfica,vol. 7, July 1971. [36]G. B a d e and G. hiarro, st.ate spaceapproach to non-interacting cont,rols,” Ricerch.e di Aulornaiica, vol. 1, 1970, pp. 68-77, Sept,. 1970. [37] S.$1. Sat0 and P. P . Lopresti, “On partial decoupling in multivariable control svstems.” in 1.970 Proc. Joint Automatic Control Conf., pp. 81i1819. ’ 1381 R. J. Kavanaeh. “The multi-variable vroblem.” Proor. Conir. ~. Eng., vol. 3, pi.’93-129, 1966. [39] &I.V. Xeerov, N u l t ~ u n u b l eControl Systems. Trans].from Russian by Israel Programfor Scientific Translations, 1968, ch. 6.Availablefrom U.S. Dep. Commerce, Clearinghouse for Fed. Sci. and Tech. Inform., Springfield, Va. 22151. [40] H: S. Tsien, Engineering Cyberndics. KewYork:hIcGrawH111. 1954. ch. . --~~ . - i [41] J. Peschon, Disciplines and Techniques of System Control. New York: Blaisdell, 196.5, ch. 3. [42] M . D. Mesaro&b, The Control o j Multimriuble Systems. S e w York: Wiley, 1960. 1431 H. Freeman. “Stabi1it.v and vhvsical realizabi1it.vconsiderations in the synthesis ofm;lt.ipoie control syst.ems,” A I E E Trans. (Appl. Ind.),vol. 77, pp. 1-5, Mar. 1958. [ 4 4 ] R. J. Kavanagh, “Multivariable control system synthesis,” A I E E Trans. (Appl. Ind.), vol. 77, pp. 42.3429, Nov. 1958. [45] K. Chen, R . A. Mathias, and D. W. Sauter, “Design of noninteracting control systemsusing Bode diagrams,“ A I E E Trans. (Appl. Ind.),vol. 80, pp. 336-346, Jan. 1962. [46] E. G. Gilbert, “Controllability and observability in multivariable controlsystems,” S I A X J . Contp., vol. 2, pp. 128-1.51, Feb. 1963. [47] --, “The decoupling of mult.ivaria.ble syst,ems by st,ate feedback,” S I A MJ . Coontr., vol. 7, pp. 50-63, Feb. 1969.

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[48] R.. J. Kavanagh, “Nonint,eracting cont.rols in linear multivariable systems,’’ A I E E Trans. (Appl. Ind.),vol. 76, pp. 95-100, May 1957. [49] J. W . Howze and J. B. Pea.rson, “llecoupling and arbit.rarypole placement. inlinear syst.ems using output, feedback,” I E E E Trans. Auionmt. Contr. (Short Papers), vol. AC-15, pp. 660-663, Dec. 1970. [do] L. hl. Silverman, “Ilecoupling wit.h state feedback and precomvol. pensat,ion,” IEEE Trans. Auiorrmt. Contr. (Corresp.), AC-1.5, pp. 487-489, Bug. 1970. [.?I] S. H. Waug, “Design of precompensator for decoupling problem,” Electronics Res. Lab., Univ. of Calif., Berkeley, Memo. ERL-h.1275, Apr. 1970. [.32] A . S. Morse, “Output controllahi1it.y and system synthesis,” SIAM J . Contr., vol. 9, pp. 143-lp8, AIay 1971. [.33] X. W . Gillies, “On the classficatlon of matrix generalized invewes,” S I A d l Rm., vol. 12, pp. 573-576, Oct. 1970. [.%I E. C. Gilbert, a.nd J. H. Pivnichny, “A computer program for the synt.heeis of decoupled multivariablefeedbacksystems,” IEEE Trans. Automat. Conlr., vol. AC-14, pp. 652-659, Dec. 1969. [5.3] T. S. Englar, “A program for interactive comput.ation in linear syst.ems theory,” NASA, Cont,ractor’s Rep. PiAS 12-583, May 1970. [d6] E. Yore, “0pt.imal decoupling cont.ro1,” in Proc. 1568 J o h t Aufornatic Control Conf., pp. 327-336. [57] S. Nazar and Z. V. Rekasius, “l>ecoupling of a class of nonlinear systems,” I E E E Trans. At~tomat.Contr. (Short Papers), vol. AC-16, pp. 2.57-260, June 1971. [58] W . A . Port,er, “Decoupling of and inverses for time-varying linear spstenq’’ IEEE Trans. Auion2at. Contr. (Short. Papers), vol. AG14, pp. 3i8-380> Bug. 1969, [.is] 2. Inai, “Studies on the noninteract,~ngcont.rol of multivariable systems,” Ph.D. diwert,ation, Kyoto Univ., Kyoto, Japan, 1970. [60] W . A. Porter, “Example of decoupling in a distribut.ive system,” IEEE Trans. Azdo?nat. Contr.(Corresp.), vol. AC-14, pp. 416417, Aug. 1969.

A. Stephen Morse“ (S’62-W67) was born in hIt. Vernon, N.Y., on June 18, 1939. He received the B.S.E.E. degree from Cornell Univet.sit.y, Ithaca, N.Y., in 1962, t,he h.1.S. degree from the Cniversit.?; of Arizona, Tllmon, in 1964, and the Ph.D. degree from Purdue Universit.y, Lafa.yet,te, Ind., in 1965, all in electrical en ine ring From 196, o 19,O he was associated with the Office of Control Theory and Application, NASA E1ect.ronic.s Research Center, Cambridge, hlass. Since July 1970 he has been on t.he faculty of t.he Depart.ment,of Engineering and Applied Science, Yale Universit.y, New Haven, Conn. His main interest i in system theory and he has. done rsearch in networksynthesis,optimal control, and multivariable syst.ems design. Dr. Morse is a member of Sigma Xi and Eta KappaNn. ,-’

W. M. Wonham (h1’64) was born in Montreal, P. Q., Canada, onNovember 1, 1934. He received the B.S. degree in engineering physics from McGill University,Montreal, and the Ph.D. degree in control theory from t.he University of Cambridge, Cambridge, England, in 1961. H e ha? been a.ssociat.ed wit.h t.he Control and 1nformat.ion Systems Laboratory at, Purdue Universit.y, t.he Research 1nstit.ut.e for Advanced Studies, the Division of Applied lIathemat,ics atBrown University, and the NASA Office of Control Theory and Application. Currently he is wit.h the Depart.n~ent of Electrical Engineering, University of Toronto, Toront.0, Ont.., Canada. He is act.ive in control t,heory research. Dr. Wonham is a member of the Societv for Industrial and -%TIplied Mathematics and is Associate Editor of the S Z A X Journal o n Coontrol.