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JOINIAUTOMATICCONTROL CONFERENCE

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— ~~~~~~THE ROLE OF THE INTERACTOR IN DECOUPLING

1 . L

PETER L. FALS and WILLIAM A • ~ ~VXCH* I ~ ..•~~-.._ _ _ _

Lefsohetz Center for Dynamical Systems, Division of ~~Applied Mathematics Brown University, Providence, R.X. 02912

~~~

Abstract relating the interactor to decoupling via

Given any proper rational transfer state feedback are developed. The

matrix, T(s), a special lower left tn- question of triangular decoupling witharbitrary pole assignment is examined in

angular polynomial matrix, ~~~~~~

called section 3. Some comments and extensionsthe interactor has been defined and are considered in section 4.shown to be (together with the rank ofT(s) ) a complete invariant under dy— 2. The Interactor and State Feedbacknamic compensation. In this paper , theinteractor is used to develop results ~~

Let S be the set of all properrational transfer matrices T(s) of full

decoupling and pole placement via feed— rank pback. For example, it is shown that T

such that the first rows,

triangular decoupling with arbitrary pole T (s), are independent. If T(s) is an

assignment is always possible using statefeedback and that decoupling with element of S, then the interactor is

arbitrary pole assignment is always defined by means of the following lemmas

possible using dynamic compensation. ((1]):

Lemma 2.1: Let T(s) be an m x m ele-ment of S. ~~1~en there is a uniquenonsingu].ar matrix

~~~~ of the form

f f

1. Introduction ~~~~ 1~.r(5)t

~i~~

(8 ~~,. . . ,s ~) (2.2)

Given any proper rational transfer wherematrix T{s), a special lower left tri-angular matrX~c, ~~~~~~

called the r 1 0 ...interactor has been defined and shown tobe (together with the rank of T(s)) a HT(s) 1h21

(s) :~ (2.3)

complete invariant under dynamic compensa-tion ((1)). In this paper, the interactor Lhml (5) 1

~’m2~~~

... iJis used to develop results on decouplingand pole placement. In particular, it is and h~~ (s) is divisible by s (or is

shown that a square system can be decoupled zero) such thatvia linear state feedback if and on).y ifthe interactor is diagonal, that tri-angular decoupling with arbitrary pole u r n

~T(s)T(s) (2.4)

assignment is always possible using statefeedback, that decoupling with arbitrarypole placement is always possible using with nonsingular.dynamic compensation (state feedback andinput dynamics) ((2]), and that certain of Lemma 2.5: Let T(s) be a p X rn ele—

these properties are generics. ment of S Ith p c m~~~~hen there T~ a

In section .2, the interactor is de- un~i~ue nonsi~~~Tar lower left triangular

fined for reference and some basic results p )C p matrix F T (s) of the form (13)such that

*Thi$ work was supported by the Air Force Office of D D CScientific Research under Grant APOSR 77-3182.

~ r~~r~sn~ rip

iiki _

*PR 1$ 19T81456

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-S., ., ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~. .•

ii’

~lix

~T(s)T(s) K.~. (2.6) 0

—

0Lemma 2.7: Let T(s) baa p x m ale— a —1ment of S ~IEh p> m and let Tm tIY $ 1 ... 0be the nonsingular matrix consisting of 0 1 0the first m rows of T(s). Then there S(s) : : (2.10)is a unique nonsingular lower left tn —angular p x p matrix F;~~(s) of the 0 s 0form—

r~rn 0~1 ~~

.

I I (2.8)

~~Y1(5) i2(s~J and {~1, ,am} being the column

where y (s),y~~(s) are relatively left degrees of P ((4]). The feedpack pair1 (F,G) can be chosen to obtain anyprime and ~~~~

is a nonsi~gular lower arbitrary column proper P~ G~’~ having

left triangular (p—rn) x (p—rn) matrix in the same column degrees as P(s). It fol—Rermite normal form with monic diagonal -lentries such that lows that T (s) P(s)P (s) and itiC —l

u r n F; (s)T(s) = inverse, P~ ,0(s)P (s), are both proper

T £ and hence, that linear state variablefeedback can be represented by the dy-!L~~ a constant matrix of rank in namic compensator T (a) (i.e., by

whose final P - m rows are zero. postmultiplication of T(s) by ~~~~~~The interactor is of critical This leads to:

importance in questions relating to de-coupling as will be shown in the sequel. Proposition 2.11: The interactor

Let T(s) be an element of ~~ is an invariant under linear state van-Then it is well-known that T(s) can be able feedback.written in the form R(g)P 1(a) whereR (s),P (a) are relatively right prime Proof: By definition, lix F;~~(s)T(s) —

polyncmia~. matrices and P(s) is column s-~~proper. Under linear state variable with K~ nonsingular. However,feedback of the form u — Fx + Gv , G non- lix P(s)P~~ (s) — C and so it followssingular, the open loop transfer matrix F,GT(.) is transformed into the cloud loop that lix F; (s)R(s)P 1 (s)transfer matrix TF,G(s) given by T F,G

TF,G S =R (5)(P($)—F($)r’G=Rc$)ç~~(5)(2.9) ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ — is

nonsingular. The proposition follows froawhere F(s) — PS (s) with the uniqueness of the interactor.

Th.oram 2.12 (Cf. (4)) A system charac-terized by a nonsinqular, proper, rational .3in x in transfer matrix T(s) can be dc-coupled via linear state variable feedbackif aM only if the interactor isdiagonal. —

1457

.1

- . .g~~~~~

- - -~ ,‘~

i~ ,1~ _ , ,_

~~~~~~~~~~~~~~~~~~~~~

Proof: If T(s) — R(s)P~~ (s) can be polynomial of degree f~ + P ji thendecoupled using state feedback, then D(s)R(s) is a column proper polynomialthere is a feedback pair (F,G) such matrix with deg d~ = the i—tb colum.that R(s)P

~~G(s) — D(s) with D(s) a degree of P(s). It follows that there

diagonal transfer matrix. Since a feedback pain (F,G) such that

is invariant under state feedback, ~~~~~~ = D(s)R(s) and hence, that the

lix F;,~,(s)D(s) — K a diagonal non- closed loop transfer matrix—

= R(s)R 1(s)D~~ (s) = Rd(s)D (s). Thus,singular matrix and so, ~~~~

is mdiagonal.

Conversely, if F;~~(s) is ~ (f ~ +~~~) poles can be assigned

arbitrarily using this technique. Ifdiagonal, then F;T(s)R(s) can be deg(det K(s)) q and deg(det Rd(s))written in the form P ,.(s) for an m

appropriate feedback pair (F,G) (where p~ = p, then deg (det i ( s ) ) q — p

T(s) — R(s)P~~ (s)) since R(s) is m

nonsingular (i.e. det R(s) ~ 0). It and ~((f~+~~) = n — q + p = n — (q.-p) .1

(s) The q - P poles which cannot be assign-follows that — R(5)P;~G ed correspond to system zeros (those of

— F;;1(~) is the diagonal matrix with det’R(s)) which must be “cancelled” via—f state feedback.

diagonal entries a ~~. In other words , It should also be noted that ifT~,~~(s) is the transfer matrix of an T(s) is a p x m transfer matrix in Sintegrator decoupled system. with p < m, then the above results can

be used by simply adjoining additionalSuppose that T(s) is a non- rows to R(s) in an appropriate fashion.singular , proper, rational m x mtransfer matrix add that

~~~~ is Theorem 2.14: Let T(s) be a proper

diagonal so that T(s) can be decoupled rational p x m transfer matrix withusing state feedback. How many poles can p < m and suppose that T(s) is of fullbe arbitrarily assigned while simultane- ra~k p. Let D(s) be any properously d.coupling the system? To answer Pi~Tonal ~~~ p diagonal tran~~er matrixthis question, let T(s) — R(s)P~~ (s) such that is proper. Thenwith R(s),P(s) relatively right prime there is an element T (s) of S suchand F(s) column proper. Then a(s) C —

can be written in the form that T( s)T~~

(s) — D(s).

R(s ) — Rd (s)R( s) (2.13) Proof: An immediate consequence ofTheorem 4.5 of (1) .

where ad (s) is a diagonal matrix with This theorem (cf. (2)) essentiallydiagonal entriis nj(s) such that r (a) states that decoupling with “arbitrary”is the greatest common divisor of the ith pole placement is always possible using

a combination of state feedback and inputrow of K(s ) • Let Pj — deg ri (s) and dynamics (i.e. so-called dynamiclet f~ be as in th. definition of compensation).

If D (s) is a diagonal matrix

with diagonal entries d~ (s) where each

d~ (s) is an arbitrary (Hurwitz)

1458 —

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I

3. Triangular Decoupling Using lix A(s) (F;T(s)R(s)U(s)1~~ 0 (3.4)

In this section, it will be shown andthat triangular decoupling witharbitrary pole assignment is alwayspossible using state feedback. El 0 0 ... o

Theorem 3.1: A system characterized by I~2l~’} 1 0 ... 0

a nonsingularq proper, rational m x ~~

E(s) . . . (3.5)

transfer matrix T(s) can always betriangularly decoupled with all closed Ie~1

(s) e~(s) . ... 1

loop poles arbitrarily assigned usinglinear state variable feedback.

p f —l It follows from (3.3) that D( s) — A(s)

K(s) ,P(s) relativ;ly right ;rime + E(s)F;T(s)R(s)U(s) or, equivalently,

lower left triangular and P(s) columnproper. Let U(s) be a unimodularpolynomial matrix which reduces

D(s) = A( s) + F;T(5)l~~5Ufl5)

~T ( s) R ( s) to row proper , lower left ( 3.6)

triangular Hermite normal form i.e., + (E(s)~ IJF;T(s)R(s)U (s)

( a(S) 0 ~ ... Let P(s) — A(s) +

~~~~~~~~~~~~~ Then

I P(s) is triangular and det F(s) =

q21(s) q22(s) 0 ... 0

F;.p(5)R(5)U(5) — I : : det D (s) — fld~ (s) in view of (3.6). By

4 (~) ... %,~(5~j virtu, of (3.4),

(3.2) ~~~~~ ~

(s) 1F;T(s) k(s)U(s)I’ — I (3.7)

where qj j(s) is a nonzero monic poly—nosnial and deg qj j(s) — k~ . Note that which implies

~ k~ — n — deg (det F ( s ) ) — ~~~ ~~~~~~~~~~~~~~~~ l (~ ) )~~1. — I (3.8)

deg(det F; (s)R(s)).- T In other words, P(.)U ‘(a) is colui nNow let D(s) be a diagonal matrix prop.r and of the same column degrees as

with diagonal entries d~ (s) wher• •ach ~~~~~~~~ (a fortiori as P(s)). Thus,

d~ (s) is an arbitrary (Hurwitz) poly— there ii a_ f eedback pair (F ,G) such that

nomial of degree k~. If 0(5) is right ~P’,G~’~

— • A (s)U 3 (s)

divided by IF ;T(s)R (s)U (s)J ~~~, then + F;f (s)R (s) and dat ~~~~~

— ai~~~d~ (s)

- ~- D(s) (F;.~(s)R(s)U(5) i~ — A(s) IF;,~(s)R(s)U(s ) 1~~

~~~. Thus , th. theorem

+ (3.3) i. coem.nts and Hxtenaion~

wt ere Th. results obtained her. are

1459

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indicative of the role the interactor (5] W. Wolovich and P. Falb, On the-

can play in decoupling problems using structure of multivaniable systems,state feedback. Extensions to the case SIAM 3. on Control , 7 , 1969.of output feedback can be readily S

—

developed for , under linear output feed— (6] P. Palb, On generic properties ofback of the form u = —Hy + Gv, G systems defined by transfer matricesnonsingular, the open loop transfer to appear.matrix T(s) is transformed into theclosed loop transfer matrix TH,G(s)given by

T (B) = R(s)(P(s) — HR ( s) ]~~~GH,G —l (4.1)

= R (s)PH ,G (s) .

This leads, for example, to an immediatetranslation of Theorem 2.12 for outputfeedback.

Since the pole placement resultsare constructive, they lead to specificprocedures for implementing the requisitecompensators. However , the questions ofminimality and stability of the compensa-tors remain to be treated.

Finally, there is the question of - -

F. to what extent the results obtained hereare “generic”. The answer to thisquestion rests on showing that the inter-actor

~~~~ is, in an appropriate sense,

a continuous function of the transfermatrix T(s). This is examined in [6).

References(1] W. Wolovich and P. Falb, Invaniants

4 and canonical forms under dynamiccompensation, SIAM 3. on Control , 141976.

(2) W. Wonham and A.S. Morse , Decouplingand pole assignment in linear multi—variable systems: a geometricapproach, SIAM 3. on control, 8, 1970.

(3] ii. Wolovich, “Linear MultivariableSystems”, Springer—verlag , New York,1974.

(4) P. Falb and W. Wolovich, Decouplingin the design and synthesis of multi-variable control systems , IEEE Trans.Aut. Control, AC—12, 1967.

1460

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S L C UR I1V C.I. A ICATION OF THIS PAGE (When P u n MIe, V l)

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D ITLE ( i d Subtlti.) S. TYPE F REPORT S PrR r~r) COV€R EO

711E ROLE OF THEJ,N TERACTO R. IN DECOU PLIN ~~~~ ~~~~~~~-.

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8.— Peter L .fFalb I /~~

AF~~SR.~ 7_3l82y~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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Proceedings of the j 977 Joint Automatic Control Conference , San Francisco , Jun 22—24,FA27—1140 p1456-1460

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Given any proper rational transfer ’m rix, T(s), a special lowerleft triinangülar polynomial matrix , ~ Cs), called the interactorhas been defined and shown to be (to ther with the rank of T(s))a complete invariant under dynamic compensation. In this paper,the interactor is used to c*Jelop results on decoupling and poleplacement via feedback. For example, it is shown that triangulardecoupling with arbitrary pole assignment is always possible usingstate feedback and that decoupling with arbitrary pole assignment~,>DD ~~~~~~~ 1473 EDITION OF I NOV 83 9 OSSOL ITE

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20. ABS TRACT (Continued)

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