Reduced Parametrization for Discrete-Time Multivariable Adaptive Control

Copyright @IFAC Adaptive Systems in Control and Signal Processing. Grenoble. France. 1992

REDUCED PARAMETRIZATION FOR DISCRETE-TIME MULTIVARIABLE ADAPTIVE CONTROL Y. Mutoh* and R. Ortega** ·Department of Mechanical Engineering. Sophia University. Kioichyo. Chiyodalcu. Tokyo. Japan ··Department of Electrical Engineering. McGill University. Montreal, Canada

ABSTRACT We are interested in the following problems of discrete-ti me multivariable adaptive control: First , give a procedure to incorporate plant prior information, (e.g, relative degrees of the transfer matrix entries, plant symmetries or bounds on its impulse response), to reduce the number of adjustable parameters and try to avoid ill-conditioning in the control law calculation. Second, provide a self-contained elementary derivation of the delay structure of multivariable systems and highlight its importance in adaptive control. Third, present a complete global convergence proof of an adaptive controller for minimum phase systems with nontrivial delay structure which is only generically known .

1

INTRODUCTION

The problem of adaptive control of multi-input multioutput (MIMO) linear time invariant systems has attracted the attention of researchers for several years. As pointed out in (Sastry and Bodson, 1989), in spite of all the researcher's efforts several open issues remain to be clarified and the field is sti ll at the stage where unverifiable assumptions are imposed to carry out the stabi li ty proofs. In this paper we are interested in adaptively controlling nondecouplable systems. In particular we address the problem of prior knowledge incorporation to determine a controller parametrization wi th a reduced number of parameters and such that. as explained below, we have a handl e on the issue of con trol calculation singularities. It is well known that a key piece of information required for direct model reference adaptive control of scalar systems is the pure delay. Its importance stems from the fact that, being an invariant of the system, it determines the set of matchable reference models. Thus. ill-conditioning in the control calculation is prone to happen when the delay is not exactly known., In the MlMO case the problem is considerably compli cated because the delay structure, and conseq uently, the set of matchable reference models, is defined now by a polynomial matrix known as the interactor (Goodwin and Sin. 1984). It is worth mentioning at this point that typically in the literature the derivation of the interactor concept requires some preliminaries on rational matrix algebra which sometimes obscure the issues of importance for adaptive control. One contribution of our paper is an elementary explanation (in terms of attainable re ferences and the plant Markov parameters) of the MIMO system delay structure whi ch underscores its importance for adaptive control. It i~ clear frolll the discus,~iol1 above that for succ:,essful MIMO adaptive contro , some prior knowledge a.l'>out the Interactor matrix; has to be assl,mH:d . A brief rcview Qf son}c o f the assumptIO ns made in the literature 0 lows. '1 he sllllples t case of kn own diagonal interador, which leads to a straightforward exte nsion of the scalar design, was studied in (](oivo, 1980, 130risson , 1979). The case of nondiagonal known interact or may be found in (Goodwin and Sin, 1984) . A key contribution due to (Johansson, 1987) is the proof that the coefficicnts of the interactor need not be known but may be also estilllated. In (Dugard, Goodwin a nd de Souza, 1984) it is claimed that. globa l convergence can be insured in this case provided the diagonal powers and maximum row powers of the interactor are known. A simil ar claim is made in (Ortega, et. aI., 1985) where this prior information was further reduced to the diagonal powers and the largest power of all the interactor entries. We would like to remark that both claims are apparently incorrect, see Remark 5.4 . Therefore, to the best of our kn owledge, there is no globally convergent scheme for plants with unknown non diagonal interaclor. In an important paper (Das, 1986) established an u[)per bound on the degrees of the interactor entries in terms 0 the degrees of the diagonal entries. This bound is improved and

239

shown to be tight in (Dion, et. aI., 1988) . That is, the authors prove that there are plants for which the upperbounds are satisfied with equality. This result is significant because it proves that with out further information we can not reduce the input-output delays, and consequently the number of adjustable parameters beyond the value provided by the diagonal term powers. Therefore, we conclude from here that to obtain reduced parametrizations and enhance performance we must assume another type of prior information besides the powers of the interactor diagonal term. Also, it is important to stress the fact that, to the best of our knowledge, except from (](ase and Tamura, 1990) and (Mutoh and lchikawa, 1991) there is no practically meaningful way of estimating these numbers in the literature. One of the objectives of our paper is to contribute, if modestly, to bridge this gap. In anothe r line of research initiated in (Singh and Narendra, 1984) the aim is to design a fixed precompensator to get a cascade connection with known diagonal interactor. This precompensator necessarily introduces extra delays that degrade performance and increase the number of adjustable parameters. Also, the diagonalization aUained with the proposed procedure is understood on ly in a generic sense. T hus, it may very well happen that the controll ed plant is arbitrarily close to the singular set leading to an ill-conditioned compensat ion law. See also (Tsiligiannis and Svoronos, 1986) where the diagonalization idea is pursued using left and right interactor matrices for discrete-time systems and (Tao and loannou, 1988) for the continuous-time case. It is clear that to enhance the practical value of the design we must avoid the inclusion of unnecessarily large delays in the loop and reduce, as much as possible, the nUlllber of adjustable parameters. Furthermore, we would like to dispose of sOllle guidelines to take care of the singul ar ity problelll mentioned above. T hese objectives can be aUained only if we abandon the objective of interactor diagonalization and we can extract from the avai labl e plant knowledge more detailed information on the interactor structure. The main contribution of this paper is to provide a procedure to estimate a su itabl e interactor structure from plant prior information, like for instance, thc rcl ative degrees of the transfer matrix entries, plant sym lllctries or bounds on its impulse response. The procedure relics on generic linear independen ce tests of the 1,Iant Markov paralneters and is particularly useful when the p ant has a n ondiagon~1 interactor. An aHcrnative proccdure to estimate the interactor structure uased on Wol ov icll 's algorithlll (Wolovich and Fall" 1976) was recently reportl'd in (I
2

SYSTEM DELAY STRUCTURE AND MATCHABLE MODELS We considcr a squarc MIMO strictly propcr plant desc ribed

by y(t) = 1'VI)U(t)

(2.1)

where y(I.),u(t) E il"', Z-1 is the delay opcrator and 1'( 7. ) E R"'xm(z) - the set of m x III real rational transfer matri ces . The transfcr matrix has a polynomial expansi on in terms o f the Markov paramclers 11, E RH':,'" as 1'(Z-I)

= L H,z-' .=1

(2 .2 )

We will further assume that the transfer matrix is full rank, that is IT(z)1 '" o. This assumpt ion ensures output function controlla bility (Goodwi n and Sin, 1984, p.132) . The proposit ion below establish es a key relations hip between the set of r~ferences attainab le with causal control laws and the plants Markov paramet ers. Propos ition 2.1 Conside r the plant (2.1). For a given positive integer N, define a nonsingu lar polynom ial matrix L(z) as N

L(z) =

L

L;z;

L;

E

Rmxm

AT

~ [R~, ··· , Rn [ }: 1i, [,11 ,=k

(3.3)

To solve this problem we need some structur al propertie s of the matrices fj that we proceed to establish below. B. Nullity Increas ing Proper ty First, notice that

(2.3)

i=O

(3.4)

The following statemen ts are equivalent. 1) There exists a causal control sequence u(t) such that (2.4) y(t) = L-I(z)r( t) for all times and all reference signals r(t). 2) The coefficients of L(z) satisfy LNH] = LNH2 + LN_IH I = ... = LNH N- I + ... 11. = LNHN

+ ... + LIHI :

Thus, it is clear that (3.5) dim[Ke r(f))] ~ dim[J(e r(f)+I)]' Vj ~ 1 Also, space. where dim[J( er(-)] denotes dimensio n of the null we see that if for some j (3 .6) dim[K el'(f))] = dim[J( er(f)+d ]

+ L2HI

full rank

= 0 (2.5a) (2.5b)

that is, if the shaded vectors are linearly independ ent, then

PH.OOF : From (2.1) we have

L( z )y(t) =

L( z)T(z-l) u(t) N 2 [LNlIl z N- 1 + (LNH2 + L N_ IlIdz - + ...

The following lemma will be instrume ntal for further developments. Lemma 3.1 Let 11 denote the Smith-M cMillan degree of T(z). Then, (3 .8) dill1[Kel·(l'n+d] = dim[Ker (I'n)] it is 1984) Sin, PROOF : In Lemma 5.2.2 of (Goodwi n and nco function output implies T(z) of larity nonsingu that shown trollability. This in its turn insures that (3.9) rank[H I , lI 2 , · · · , Hn] = m

00

+(LNIiN + .. . + LJld zo + L M;z-;)u(t)

(2.6)

i=1

where M, E R",xm. Equating (2.6) and (2.4) it is clear that law . On the (2.~a) is needed to insure causality of the control other hand , 11. full rank allows us to obtain u(t) for alll·(t). 0 Remark 2.2 Notice that the choice of Lo in (2.3) is arbitrary . Remark 2.3 In the scalar case the solutions of (2 .5) are arbitrary polynom ials of degree d, with d the relative degree of the plant. Remark 2.4 It is clear from (2.6) that for the polynomial matrices L( z) satisfyin g (2.5) we have (2.7) }0.!, L(z)T(z ) = 11. : full rank

On the other hand, we see that

o (3.10)

which is a property common ly used in the definition of the interaclo r matrix (Goodwi n and Sin, 1984) The main purpose of our paper is to present a procedur e to estimate the structur e of L(z) matrices satisfyin g (2.5), which hereafte r will be referred as interacto r matrices . To this end, we will establish in the following section a rank property of the systems truncate d Toeplitz matrices which will prove instrumen tal for our purpose. Even though this property can be ('st"lJlish ed frolll other consider ations involving the systems st.ruct.ure at infinit'y we prefer here to present first. an elementary derivatio n of the property and postpone to the end of the section the discussio n of its relations hip with the structur e at infinity.

3

SOM E PRO PERT IES OF THE SYST EMS TOE PLIT Z MAT RICES

It!: Therefor e, (3.9) insures that. (3.8) holds. 0 Frolll LrllllT;1a 3.1 [·I·lld ~he dis<;ussion above \V~, can establish the fo low11lg key nu lty 11Icreas11Ig property 01 1 J Fact 3.2 There exists an integer w E [1, n], such that dim[I(el ·(l'.)] < dim[Kel ·(r 2 )] < .. .

< dim[K el·(l'",)] = di III [f( er(f'",+d] = .. .

(3 . 11)

We are in position to present t.he lIIain res ult of this paragra.ph . Proposi tion 3.3 Conside r the nonsingu lar plant (2.1) . Then, for all solution s of (2.5), (i.e., all ini."'." I.or matri ces ofT( z) )

(3 .12)

N = w with w as in Fact 3.2. PROOF: First, let li S rewrite (2 .5) in the compact foml

o (3. D)

A. Relatio nship with Interac tor Matrix Let us define the systems j-th trun cated Toeplitz matrices as

1'] =

[0] E R"'x",

o

liT

o

j=1,2, ... (3 .1)

liT

0

With t.his notation and (2 ..5) we can define the set of interac tor matrices as the set of matrices L(z) whose coefficients solve the algebraic equation s (3.2)

subject to the restrictio n that

Now, we will consider the case of N > wand, without loss of generality, take N = w + I. In this case, (3.11) illlplies that (3.H) dim[K el·(f.,,)]= dim[J(el ·(l'N)]. shaded the that (3.4), in w = j This in its turn implies, taking columns are linearly independ ent. Therefore, LN solution of (3.13) should be identically 7.ero, which is a contradi clion. To complete the proof let liS consider the case N < w, and again, set N = w - I. Frolll (3. 11) we now have (3.1~) dim[J( el·(l' N)] < dim[K er(f"')]. there is This implies that [H~, ... , HTJ is not full rank , that are less than III linearly independ ent columns and 11. can not be nonsingu lar . 0 Remark 3.4 The significance of the nullity increasin g property (3.11) and Proposit ion 3.3 is that by checking the nullity of

240

the matrices I'J, i.e. checking linearly independence of columns of the Markov parameters, we can determine the degree of the interactor. Furthermore, as will be explained later, we can use (3 .13) to estimate an interactor whose structure is suitable for adaptive control purpose.

C. Structure at Infinity We will show in this section that a sharper version of the nullity increasing property can be directly derived from the systems structure at infinity (Kailath, 1980). To this end, we recall the following well known fact. Fact 3.5 Given the transfer matrix (2.2), there exist m x 111 bicausal transfer matrices U(z), V(z), N(z) and unique integers fl h fm such that

:s :s ... :s

U(z)T(z)V(z) = N(z) N(z) = diag[z-f>, z-h,'" , z-/m]

(3.16) (3.17)

One says that T(z) has m zeros at infinity of order {fl, h,"', fm}. 0 Itationalmatrices U(z), V(z) and N(z) can also be expanded using their own Markov matrices as follows.

Another important fact, relevant for our purpose, is that the structure at infinity, e.g. {fl, h,"" f",}, can be derived from knowledge of the relative degrees of the entries of T(z) (Kailath, 1980, p.45l). Specifically, if we let Ok be the minimum relative degree among those of k-th ordered minors of T(z) which is not 0, then,

01 Oj - OJ_1 (j = 2,,,, , m)

D. Example Consider the transfer matrix described by

£y

2

Since U(z) and V(z) are bicausal matrices, Uo and Vo are nonsingular. The Toeplitz matrix of N(z) of order j, say 6.), is given by

o

1::.)=

(3.19)

NJ-2

Fro[mVY'I~\ (2.2)

and

T(z) =

U~I

U~2

rankr j = rank6.j (Vj

u,1

= 6.)

~ z

1 Z

~ z

(3.29)

as follows 0

2:

1).

1

diag[l, 0, ... ,0] if fl < h diag[O, 1, " . , 0] if fl < h { diagl,I,O,""O [ ] i f fl= f 2< f 3

r~

(3.22)

with ".n obvious definition for the remaining NI,. Also, we have

N)=O,

jlfl,h,"',f",

(3.23)

N OIV, if we denote by

1 1

1 1 2

(3.21)

On the other hand, (3.17) implies that

=

1 1 3 0 1 0 0 2 0 0 3 0 1 0 0 0 2 0 0 0 3 0 0 0 0 0 0 0 0 0

0

1 1 1 1 1 2 1 1 ;} 0 0 0

1 2 3 1 0 0 2 0 0

(3.31)

0

0 1 1 1 0 1 1 2 0 1 1 ;} 0 0 1 0 0 0 2 0 3 0 ;} 0

t t t t t t t

0

1

1 1

t t

t

1 1 2 1 1 3

(;\.32)

(3.24) and in general rank

f\ + ak

0·25)

Noti«' t.hat if k = fn j otherwise

= 1,2,··· ,111

aIm = m

0

Note th,~t column vectors with symbol" I" Me regardpd as linearly imlependent vectors in 1\. From this and (3.25) wc have al = 2, a2 = 2, a3 = 3, a4 = 3, "', then it follows that

of zeros at infinity of order j

for.i = 1,2"", fm. Then, it is clear from (3.19), (3.21) and (3.22) that

and

L:U z

0

(3.20)

= number

t

First, the structure at infinity of T(z) is eva.luated. Sincc, '\1 = 1, 02 = 1 and 03 = 5, t.he orders of the zeros at infinity ,He given by {fl, h, h} = {I, 1,3}. From these we conclude t.hat. w = 3. To illustrate (3.25) let us construct r~ from the Markov matrices

This means that

t)

t

(3'11~' [it ~lows ~hat

V1

~~2

N[

(3.28)

Reml\rk 3.6 The derivation
Uo + Ulz- + U2 z- + ... , U) E llmxm Vo + VIZ- I + V2 Z- 2 +"', V) E n",xm (3.18) Nlz- I + N 2 z- 2 +"', NJ E nmxm I

U(z) V(z) N(z)

~~I

previously established in (Dugard and Dion, 1985)

(J.2G) (3.27)

Further, it is easy to see that w as defined in Fact 3.2 is the largest zero order at infinity, i.e. w = fm. A fact that was

241

which is consistent with (3.25)-(3.27).

4

Estimation Structure

of

the

Interactor

As pointed out in the int.roduction t.he main object.i\·c of this paper is to provide a systC'lllat.ic procedure to use plant. prior knowledge for thc estilllation of an interact.or structure without additional dcla.ys a.nd minilllllll1 IIIllllber of unknown par,unders. To at.tain (.he first objective we Illust a.bandon (.he interaclor diagonalizat.ion object.ive and we have to cslilllate the structure of the ofr-diagona.l tenns. Towards this end, wc willllse the nullity increasing propert.y of Fact 3.2 which will be tested by checkillg linear independence in the generic sellse of the Markov parallleters. Therefore, we will require t.h" gellcric kllowledge (i .e. , whether they are zero or nonzero) of the first w + 1 Markov parameters, where w itself will be deterlllined by the procedure. As shown above the nullity increasing property can also be checked if we know the relat.ivc degrees or the entries of the plant transfer Illatrix. As will bccollle clear laler,

the procedure of linear independence testing is more general s ince it allows us to incorporate other kind of prior knowledge to refine the interactor structure estimation. For the second objective - obtain a reduced parametrization for the interactor - we propose an output reordering procedure that yields a lower triangular interactor whose diagonal entries coincide with the zero structure at infinity of the system. An interactor which has this property will be called a regular interactor. Also , the corresponding rj will be called regular Toeplitz matrices. A. Determination of a Regular Toeplitz Matrix An algo rithm to carry-o ut the nullity inc reasing ch ecking and the o utput reorde ring (i .e., reord ering th e columns of r) 's ) n eed ed to o btain regular T oe plitz matrices is given in the app endix . As a res ult of the algorithm we o bta in ne w T oe plitz ma tri ces o f th e fo rlll

r, ~

'* ~ ~ ~o I ----.,--

-."..J

~

where a~)'s are some real numbers . Further, the regular interactor is of the form

(L(Z))),k (L( z ))j,j (L(Z))j ,k

zl;

:

io: :

"I) E Jl 12

(4.2)

Th c al gori(.hm in~ur cs th a t th e columns includin g th e s had ed p or ti o ns ("(I ~ "15 a nd "(7) a rc lin carl y ind cpc nde nt among th c ll1selves whil e t he re m a inin g ones a rc lin earl y d ep c nd c nt. Spcc ifi ca ll y, "18 a nd "19 are lincar ly d cpc nde nt t o "I7, while "16 dep e nd s o n "I,,"Is a nd "17' No ti ce th a t s in ce th e firs t th ree colull1ns a re a ll lin early indcpe ndc n t we h ave

dim(K er( r w+.)) = dim [i( er( l' w) )

(4.3)

a nd, accord in g t o P roposit io n 3.3, w e ffectively d e fin cs th c hi g h('s t powe r o f th c in t(' rador. Note t h at th e st [11ct ure a.t infinit y o ( (.hi s ~y~ t (' 111 is {fl , h, h } = {I , 2, 3 }. (w = h = 1)

B.

Dd( ~ rI1lil1a t io1l o f R cg ulnr 111tc rac tor S t ru c tur e With l'w+ 1 in rcg nl a r fo nll a s ui table solut.i o n o f (:J. 1.1) lIl ay be (o ulld (rOl ll t he fo ll owin g proposition. Prop() ~ iti()l1 4 .1 Co ns id er t he alge brai c cqn at io n (1. D) wi t h N = w (= f",). ASS IIIIIC l'w is in t. hc rcg lll a r fOI"lIl . Lcl (I'k)) h(, t he j-t h row o f L k , k = 1,2,'" ,w, j = 1,2 , ' " ,m. T he n , a ~Cllllt. i () n o ( (T 13) with A- filII ra nk a nd L(z) rcg ul ar is prov id cd I,y

:

~

j01 :

0 0 0 0 0 0 0 a~,l 0 1 0 0 1 0 0 0 0 0

a~,1

a~ ,2

1 a~, J

=0

0 0 0 0 0

(o\.G)

whc re th e a~ ·k 's a re so mc real numbe rs . The validity of th e solut.io n fo r th e firs t col1l111n is o bvi ous . Fo r th e second COIUIlIl1 , by the constr uct ion o f ['w, its fifth column d e pe nds o nly o n th e fo u r th o ne a nd we can se t to zcro t he rem a.ining ele111c nts o f the column. A sil11il a r reasonin g a pplies for th c solutio n t o th c third col1l111n . To q u alify as a n int e raclo r t hc solutio n give n in (,1.6) 1IIl1s t. satis fy th e full ra nk co ndition (3.3 ) . For conve nience we writ.e hc re (3 .13 ) fo r o ur exal11pl e

a.

= b", .. , "112],

:0:

:

I?y'!

ioi

io:

W(' d efin c fo r co nve ni e nce th e colullln vecto rs

l'w +I

:0:

~

0

~

ioi

io:

aa

let

k :::; m

where (3k = max[jk, 2). To illustrate ttle proof procedure we will, without loss of ge nerality, consider the case of the regular Toeplitz matrix (,1.1) . Applying th e proposition 4.1 t o this example we see that (3 .2) b ecom es

0-

OJ CT'-I. CT'~ Oi wh e re s h ad ed p ortio ns a re linearl y ind c pc ndc nt column vcctors in 1'), whil e th e rem aining vecto rs are lincarl y d c p cnd e nt. Mo re ~ p ec i ri ca ll y th e foll owing pr op c rtics h old . (1'. 1) In cac h column bl oc k o f r), linearl y ind e p end e nt co lull1n vec (.ors (s h aded porti on) a ppcar co ntinu o usly fr o m t he le ft. (!'.2) Fo r a ny j, a ny lin ea rl y d ep e nd e nt vector in l') can IJe c xp r('sscd as a lin ea.r co m bin atio n o f (s h ad cd) li ncarly ind cpe nd e nt vccto r o f l'). Thi s t wo p ro p e rti es ch a r ac t e ri 7.e wh a t we call a regul a r Toc pli t z m a trix . As will be s how n in t h c nc xt pa rag raph t h e soluti on o f (3.2 ) a nd (3.3 ) in this case y ield s a rcg ular inte rac(.o r. To fix tile i.de a.~ u ~ cQn ~ id e r a case w1lrr c w = III = Th e prop osc( a lgo nt Im, III Its final stc p, WI give a ne w r w+1 Il1 atr ix in regul a r for m as

(4.5)

+ 1 :::;

if j

0,

0 0 0 0 0 0 I 0 0

0 0 0 a~ , I

1 0 0 0 0

a~ , 1

a~ ·2

0 0 0

1 a~· I 0 0 0

(.J.7)

-;;r

0

0

whe re again th e colu111n s with sha.d cd po rti o ns a re lin e il.rly ind epe nd c nt , see (-1.1) . Becausc of thc la tt cr a li<.I the s tructur e o f the proposed soluti o n o f L,'s it is ob vio us th al t.he ri ght. hand side matrix is full rank , whic h ill its turn insures th,~t A is 11 0 11· sing ul a r. T he inte raclor lII a trix corr csp o ndillg to thc pro p osed soluti o n is

L(z ) = [

a~'~

3 1 a . Z3

z2

+ a 3 •1 Z2

~2 ~]

a~·2 z3

(4.8 )

z3

R c mark 4 .2 An illlpo rta.llt fcature o f a re gul a r intc ract.or is t.h "t it. is a. ro w pro pc r lII at. ri x, th a t is th c h igh cs t P O WN ill ea.c h row does IIUt. excecd t hc powc r ill it.s co rres po lldin!\ c1i,,!\· 0 11 il l tCI"II!. Al so, du e to t hc for m of LJ th c rc is 11 0 fir s t. po wer ill t.h c o ff-di "go llal t NJIIS. As a cO ll s('qll c nce o f t.hrsc prnprr t.i l's t hc 1I11111 bc r o f 11 1I ("<'r t" i 11 p" ra lll('\.(' rs i ~ rcdll ccd . If 11' (' ("O IlI PM(" fo r ill s t. a llf c, wi t h t.h c ini.e ra rt.o r s t.r uct urc o bt a ined b y o nly kn owin g th e d i"gCl II ;d "ncl Jn a xilll ".\ powe rs wc n o t.i ("(~ i.i. " t. w'e 1I",nagcd to redll cc t hc nlllllbc r o f unkn o wn p a ra111c t crs fro lll 7 t.o 'I. Notwi t hs t,"ldillg th e fad that it is no t c1 ciH wh cre thi s kin d o f p rio r infor lll ilt. io ll lIIi g ht. b c a v"il able fron!. Rem a rk 4 . 3 Sin cc th e first. s ulJIIli ssio JI o f th c prcsc nt p"pe r we becam c a.wa rc o f ( La fay, ct a i, Iga O) whe re thc id c a of Ollt put rco rdc rin g t o o bta in a rcgul a r intcraclo r is a lso expl o r('d .

C . All Example with Kllown Relative Degre e s

(J'I,+ .))

Considc r a 3-inpu ts 3-outputs strictly prop c r, nonsingulM pl a nt wh osc rcla.t.i ye d eg recs o f th e tran sfc r mat rix e ntries arc gi vc n by .

(LJ,l)

(L f,-.))

D[l i.J = (L 2 ))

[a~, l, a~·2 , . .. , a ~,)-l , 10,' . . , 0]

(L.)) (L o))

[0, .. · , 0) [0, ... ,0)

1 3 4] 2 4 5 [234

(-1.9)

Fro m this we c an ge nerically d etcrmine th e Markov pa.nullct ers as

(j = 1,2, ... , Ill)

(4 .4)

242

[0U 80 8], Ii2 = 0

110 = 0, 11 1 =

[:

..

0], H~ =

H4 = [ : : ***

8008], Ii3 = [:

:

. 0] 28

[ :

..

(4.10)

:]

***

r2

Following the procedure of appendix we notice that and are left unchanged and at the third step we get a matrix 0 0 0

0 0 0 0 0 0

..0

0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 U 0 0 0 0

0 0

U

0 0

0 0 0

0

.. .. 0 ....0 0 ..

..0 0 ....0 .. .. ....

Fr0111

r~

t t

~ 1Id

0

()

..0 ..0 ....0 ()

and

a' ,) E R

,~

(5. 2)

r( z) = diag[ z/;)

(5.3)

0 0 0

0 0 0 0 0 0

dcg( S)

0 0

0 0

++

t

t

0 0 0 0 0 0 0 0 0 0 0 0 0 U 0 1 0 0 0 a~ , 1 0 0 0 0 1 0 0 0 0 0

0 U 1

pr o p os iti o n 4.1 we gc t

=

dcg( R )

=

(G .G)

m a x[deg(A ), d eg (B ))

and th e co effic ient m a trix o f powcr ze ro o f S is A as d efi II cd in (2 .7) , thus inv ert.ibl e. In th e scq llel we will ass umc th a t th c sys tc m is l11inilllll1l1 phasc. T he co ntrol o oj ectivc is to in s ure

e( t) = y(t ) - )·(t ) -+ 0 as t

(G.7)

-+ 00

wi t h a ll inte rn a l s ig ll als b o unded , wh er(' )·( t ) is a vcdo r o f d esircd refe rences wh ose co mpo ne nt.s )·.( t ) a re ass ull1cd t o h p kn own a.l k as t. f , s t e ps in adv~n cc (sce I\ c mark G. I) . T o a l l .",. the co ntrol o bj ective , we propose t he ad a p t ive co nt.r oll er

0

()

(!J.G)

wh e re

-+

..0 00 00 00 00 0 .0 0.. ..00 ..00 000 000 00 0 U 0 0 U 0 0 .... .... ..0 ..0 0.. ..0 .. 0 0 0 0 0 0

+ R( Z- l) y(t )

L( z )y(t) = S (Z- I) U(t) 0 0 0

0

0 0

h + a") zl.-I, + aif"1 ) I,z +=... / , - f}

Wc canwrit. e (2. 1) in pr edi ctor fo rm as 0 0 U 0 0 0

It. is cl ear that t o get th e ind ep e nd ent col1l111n s at th e Icft1l10s t sid e o f the lllatrix bl oc k wc must exchange the second and third COI1l111n s. This co rrespo nds to exchan ging th e seco nd and third p ,, 1, ·,,1 s ign~l s . If we go two steps b eyo nd in the algorith111 we ,. th a t di m[!,'e l'(1'6)) = dim[J(c"{I's) ) thu s w = 5 and th e rcol< i<- l"1n g proce dure is fini s hed yiel din g a ne w r ~ 0 0 0

a") o

0 0

0 0 0

++

t

r3

M. ,){ z) = { 0

As shown in Theorem 5.2.4 of (Goodwin and Sin , 1984) , if we fact or the tran s fe r m a t rix as

..0 .0 ..0 ..0 0 0 0 0 0 0 0 ..0.. 0..0 ....0 ..00 ..00 U..0 00..

1\ =

where M{ z) is lower triangular with zeros in the main di~go nill 'and the remaining t erms of the form

0 0 0 0 0 0

0 0 0 0

0

0

0

0

0

0

0

0 0 0 0

(·1.1 2)

3 ,1

wh e re S, (Z- l) , i1.( Z- I ) a nd Ai,( Z-l) are p ol)'n 01 l1i al 111 at rices wi t h time va ry in g coe fii cie nts o f th e sa llle s t.ru ct ure ~s S (Z- I), R( Z-l) alld M( Z-l) res pcct ive ly. Rcmark 5.1 It is cl ear fW IIl (5.8) th a t th e e frectil'e illput Oll tput de lay, th a t is t he numb er o f s t e ps a.head t.h a t wc ne cd I he refe re nce cO lllpo nents to be kn own , is de te r mill ed I,.\' th e 111 " trix M{ z). T he b es t s it. ll ilti o n b eing of co urse t h e c ase wh (' n th e int.e ra ct.o r is di ago n a l. Rcmark 5.2 Th e proce dure d esc ribed in th e prc l'i o ll s sec l io n will Icad t o a b e tt er es t.illlati on o f t.h e s tru c t.ure of M( : ) a nd conseq ue ntly to a redu ct io n in the nUIIloe r of a dju s t. a hl e p a[ 'UlI ct.crs.

a! ,2 a4

Fro m (5.5)-(5.8) wc get t he crror cqll a t io n

0

l'( z )c(t ) = -,5,(Z-I)II(t ) - ll ,( z- l) y(t ) -Af,( z) J' (z )J·(t ) + M( z )l'( z )e(t )

(l~ , 1

0

(·1.1.3)

0 a~ · I 0 0 0 0 0

(5 .9)

whc re

[) , =n,-(·)

(05. 10)

To a na l)'ze (.5.9) it is co nve ni e llt to co ll side r

it

ro w hy row d e-

sc ri pt.io n as

whi ch prov id cs a sol1lt.i o n t o t.h e p ro bl cl1l . It is e asy to chcc k t h a.l we a re cx pl oiting t he pa~ t i c ul a r di s t.ribu t io n o f ~cros an d no nze ros to c xp ress th e gc nCrI C hn ca r depc ndc nce. 1 he co rrcspo ndin g in t.c r~ cto r s tru cturc is

el(t + Id dt+h) c", (I

+ I".)

-e,(tf",,(t)

-e (I)T "'At ) + J1f 2

2 •I

( z )edl

+ fd

(0, . 11 )

- e".(lf"'".(I) + JII""I( z)el(t + Id + JIf". ,( ,.. _I)(z) e",_l(t + I,..-d

+.

wh e re we h a l' c i 11 t. rod JI("('d S(J IIIC " IJI'io ll s d c filli t. io ll s for 0,(1) " lid

"'.(1), a lld JII •. )( z)is th e i-j c lllr y " f JII ( z). In t h(' ncx t scct i01l , w c will pr('se nt a g loba ll y co nvc rgc nt adapt ti n ' cont roll e r for n01 I-dcco u p lablc 11I1I1t.i va ri able syste l1ls usi ng t.h" l1J1.c ract.o r l1I a trlx whosc s tru c t. ur e IS ge nc rI call y cst l1u at.cd by t.h c a b ove m et ho d .

5

A Globally Convergent Adaptive Controller

Fr om t.he'
L( z ) = [I - M{ z ))r( z ), rE Rmx m[z)

(5 .1)

243

Wc ar(' ill pos iti o ll t.o pr('s en t. t he ," a in r(,Sl d t o f t hi s s('c l ill ll . Propos it.ioll 5.3 T he ("Tor (''III ;d iOIl (0, . 11 ) wi th t. he c,;fi,,1
B.(t ) = B,(t p~(t)

",.(t - f ,)

I.) + >"-2 - (-) p, t

" ,( t), 0 < >'. < 2

= 1 + L 1I",,(t - I.)W

(G. 12) (5 . 1.3)

for i = 2, 3, ... , m

2(t) _ {P?_l{ t ), if I. = 1.-1 p, ll1ax[p~_l(t) , ... , f1~_l(t

+ 1.-, - I.)L

15. H )

o t.h e r\'·ls('

6

with 11 . 11 the Euclidean norm, is golbally convergent. That is, e(l) ....... 0 as t ....... 00 with all signal bounded. PROOF: From the first equation of (5.11) and (5.12)-(5.14) wc conclude, invoking standard arguments, that (5.15)

thus

and BI(t.) is bounded. Now, for the second error consider the quadratic function

(5.1G) which "fler ~Ollle st.raightforward bOllnding yields

wherc wc have defined

(5.18) Nolice [1'0111 (5.2) that £(1) is either zero (if 11= /2) or clse it <"olll"ills terms of ~I(t) and its pa~t, i.c., el(t -J),"',el(t + I1 - 12)· Wc will a~Sllll l e that II i 12, thc othcr case being t.rivially h;lIldled. Frolll (S. I 7) wc get the bound

hold~ for all ('( > O. From .(5.19) (by adding up both alld (t ~Ilrficiellll'y larg(' , we GI ll est".IJlish tire existcllce of Cl > 0 ;lIld C 2 ~Ildl tlr;d.

which ~id('s)

e2

(t) 11 . Il P2(1) 2


-

11 I

W)

P2(t)

11 2

+ C2

(5.20)

11'11('1'(' 11·1\2 dellot.es I.hc (2 II Orlll. Thc prohlcm to prove co nvcrge'lIce o [ C~(t)/f'2(t) frolll (5.20) is that, even thollgh wc have (', I"I,li s h('d (.':i.I.':i), t.his does not illlply that ~(t)/p2(1. ) will go t.o ze ro hecallse of thc prcsence of past terms of el(t). This difficlllty lIIot.ivates o nr definition of P2(t) in (5.H). With this lIol'lllaliz"lioll sigual w(' ca ll prove that for SOllle C 3 > 0 (5.21)

Conclusions

The ll1ultivariaLle adaptive control for plants with unknown delay structure has been considered in this paper. First, the clementary relationship between the structure of the interactor matrix and the plant's Markov parameters was provided. Second, using this relation, a procedure was proposed to estimate the structure of interactor matrix for the unknown plant. The relativc dcgrees of the transfer matrix entries can be used as a prior information. The estimated interactor is a row proper polynomial matrix having a reduced number of paramctcrs. From this point of view, this so-called regular interactor has a suitable form for adaptive control. Finally, the paper gave the proof of global convcrgence of an adaptive controller for the multi variable unknown plant with nondiagonal interactor matrix. To the best of our knowlcd~e this is the first proof of global convergence for nOllllccouplal>le systcms whose intcractor is only generically known. Acknowledgements Thc authors would like to thank J.M.Dion for having brought to their attention thc pa.pcrs (Vcrghese and I
References Borison,U.(J979). Automatica, IS, 209-21S D
;lIld frolll (5.15) alld (.5.20) cOllclude that e2(t)/P2(1) ....... 0 as

APflfMlfX

/ ~ 00.

F()llowin~ t.h e sallle lill e o f reasolling wc can cstablis h also (h;d r ,(I)/I',(I) ~ 0 for i = :3"", m. Also, noting that p",(t) :::: I\,,-I(t) :::: ... :::: /'I(t), we havc

(0,.22) The proof is cOIIII'Iet.ed frolll t.he lin ('ar 1)()lIlIdedn('~s condit.ion n[ IIlillillllllll "h"se systellls

p~,,(t) ::; C4 sup lI e(r)1I

2

+ C,

Construcl lie malrix r. and add lie first K colUJlnS lolheselD. Search for the firsl "E(l. m-KI such lhal Ue culum K+" of r. is linearly independenl lo lhe elements of D.

(5.21)

O~T~ t

and Ihe key t.echllic"llclllllla (Goodwin and Sin, 198·1). Ilf'lIlark 5.4 '['0 (he be~t of our knowledgc this is t.he first proof of glol>al COII\'crgence o[ an adaptivc controll e r for an ~III\I 0 s)'S1 elll wi( h unknown nontiiagonal interador 111"( rix. It if worlh pointing out that the proof in (Dugarci, et.. al., I !)R·') is "pparell( I)' illcorrect since it is erroneo usly clai lllcd Ih;t!. (h(' regressors (1),(1) in that paper) are linearly houllded J,y (he corrc'pondillg prediction errors (c(t) for i = 1,2"", m). Onc way (0 relllove t.his naw is to dcfine a new normali zat.io n sigll,,1 a~ (-'I. D). A s illlilar mistake is lIIade in (Ortcga, el. aI., 1985).

244

YES

0 .. 0" .. · are sets of veclors of dimensioo 2m . 3m.'" respecli veiy.