A New Method to Establish Steady-state Model of Large-scale System

Copyrighl © IFAC Large Scale Syslems. Beijing. PRC. 1992

A NEW METHOD TO ESTABLISH STEADY-STATE MODEL OF LARGE-SCALE SYSTEM Q.X. Chen and

n.w. Wan

InsliluJe of Syslems Engineering, Xi'an Jiaolong Universily, Xi'an, 710049, PRC

Abstract: In ttlis paper. under the weak assumptions such as only the step Cllilll81:S of system set puints lJeill8 the excitln8 si8nal. unklluwn structures and the dynamic parameters. approxillate linear d~'lia.ic model and the least sqllares estimatiun metllu~. for the lineal' slow tiwe-val'yill8 lar:Je-scale S),stCIII. the steady-state 8,lilJ est imate of the systcl is forled frolll ttle cst imates of the paramder5 and a parallel identification alaol'ittlm is pill fOI'~ard. The cOllsistellc, of the estimate and the conver8ence of tile parallel itel'atiulI are ana]yzed. Based on this consistency theorem, a pl'aHmatic metllUt! to uet tht' stl'01l8 cUllsistent estillate uf the stf': ..1y-state model vI' the larue-scale syslt-III is presented. Sillul~tiun study has already proved this. Key Words:

Lal'fw-Scalp S,stem; Hierarchical Steady-State Ident ificat ion; Stocl'ilstlc System; On-Lille Ol'timiLatlOn Control; Data Parallel Pnressing, existillg optimizatiun algorithllJs since this del'ivat ives. in a broad seJlse. imply the information about the optilllizatiun dil'cctloll of the lIext step. As a cOllsequence. hu. to esl i ma le these derivatives effectively from noise corrupted sampled data will be one of several keys to ilpruve the performm~es of most optimizatiun alyorithms available nOli.

INTRODUCTION Fro~

70' s to nOli. more and more literatllre has pllblished abuut the online opl iliOiLalion of ti,e indllstrial process. mllre and IIIOl'e interests hilve arisen in this field because utility and I'a. lDatel'ia] custs have lll'o.n _ith steady steps, In the mll~t of act \la I cases. indust rial processes are :ilow t ille-val'yinll and lIonl inear .i th Lnknololl strllcllll'e and parameters III lIatul't'. then t he opt i 11111111 shou I d ue researched 011-1 i lie in orciel' to heel' thl.'lII opu'atinu in the opt imal state. On aCCoullt of this point. a lut of integrated estimation and optillizat .J' :l!llCoritilllJs have br(ju:Jht up Sll far. The majority of these alUlJrith;s. on the otlter hand. has severe applicability conditions and is very sens i t i ve to t he process vr measurCIIICJlt noise. The systelJ set poiJlts. in the most practical cases. are sl'l liP previously in accordance lIith sOle steady-state optimization pr03ram

It is re8al'dul tllat nil the steady-state lodel est imatlull al8ul'itlims liP to no. ran pei'haps be cldssifi~d into three sorts. The first Olle is a class of deterministic ISOI'E methods which is well knOlo'1l ill this field. First put fonal'll by Robert s (Roberts. 1979) • this kind of alsori tl.;us. wldch applies a simple pel'tllrbat ion tectlnique. estilales the real process derivatives by their finite difference approxillations. Unfortunately. these al80r i thllls are very sensi ti ve to the stochastic disturbance since this silPle perturbation technique will likely ~ile out an incorrect research direction for the n~xt step of G~ti.izatioll in the eler8ence uf noise. At this lIoment, divergence of these al9urithlls .i 11 happen inevitably, The second one is the so-called conelat iOIl signal estilation aluori thll (BalJlJeruer & Iserllanll. 1~78; Gal'cia & Murari. 1981). Such kind of ulguri tillS. ho_ever, rl"(Jllires J specifIC test signal to excite the systel pel'sistelltb. Certaillly accelel'al11l9 estimation procedure. thIS special test si8nal. "ill unavoiriably disturb the Ilol'lIal opel'ations of the real process ill some extent. evell m"I'e to influence the stationari ty of it. file last one will be the t ..o-stage S)'stl'l identification technique .hich is tlie direct extension of the estimation technique applied in the deterlinistic ISUPE metltods to the

alld hence. fur lite sake of avuidan2e tu disturb tlte norllal operation of the prucess. shuu Id be c!lanued as slIIa II tiles as possihle lJy no use uf any other additiollal excitinC) Si8Jlal in each Ol't illiLatiun l'eriod, III the vie. of est iJllat iOJl. the experillental c[lJldi t ions art: v"r~ \oeilk alii! the iJlformatioll ulll,'llllt obtaiJled is fairly poor. therefore tile est illJatioll pl'oblell under litis situation wi 11 he a hard nllt to crack. Tlte dcrivut ivcs of tlte (;,01 IDlized l'erfuJ"mi1J1cc ~itll respect tu the s,stell set poiJlts. ill otlter .ol'ds. the steati J'state gain "f the inddstrial process in tlte Jloise-I'rce sillwtion. have tu be kuo_Jl 01' he estimated effiCIent I,. This stave dOlJillates the effects of lIany

III

stochastic cases (Lin.Han.Roberts &Wan. 1989) . This estilation approach delands the on-line structure identification and structure recursion to the industrial process. as a result. sOle artificial noises or PRBS are sOletiles necessary to be exerted on the several cOlponents of the real process. naturally. the pl'oces~ has to be disturbed allay frol ils norlal operations. In general. these defects above said will !ilit the applicable range of these al90rithls.

~here H,.I

are the 18 tr ices elelents ones and zeros.

L1: for each subsyst el 0=1.2 .... .. .. M):

...-

tC. .

.

.-1

(i)C.(i - I -

'-I

4...)+ ••

I -

(i)

...... (N) -

.....,

1- 1.2· .. ··· ••"'.,

(3)

Urn

B..(N) -

D,.,

1- 1,2······.11•.•

(1)

Urn

C, .• (N) -

c..

1- 1.2·· . ... , ••.•

(5)

(6) .here E dE-Tlotes the expectat ion operat.or. L3 : for each subsystel (i=I.2 • .. . .. .• ~). there exist s som e ">0 . such that the stochast ic disturbance _. (I) has hounded 4+" absolute lIolents; __ The set points change in the lode of step function. for the i-t.h suhsystel (i=1.2 ........ H). we assule that the s~t point vector changes froll a•.'.A(al •. :.· ... a,•. ,)' toa.£.(a,., ..... a,.,)'. nalely the usual step ('h~n!ll' ~ of the s~strl set points in the optilli zation procedure. L4 : for each subsyst 1'1 (i =1.2 ........ H) • each component of the vector a, is not equal to zero. and each ratio a•.• /a•. , is not equlII with each ot.her. and not equal to I. j=2.3 •... .. .• qj For each subsystel (i=I.2 • . ... • .• M). the set. point vector changes step hy step. n8lely:

,

{"Jo ... i <: T. + (j -

c•. /( l - 4..... - 1)-"J.,t

~••• ) +

~•.• ,.

.

~A"•.".(i -

(I)

'-',

(8)

I - 4•.••• )

+

(~)

I - I ..... )

- ;,(l»)',•• (,,(i)

-

;,(l») (10)

IIt,ere fI... is the .eight inu latri x of the i tll sllhs~' st' ~ 1 at the IIOl ent Qf k. L5: for each suh~ystel (i=I.2 •.... . .• M). t.he lIatrix fI•., is a diagonal and positil'e definite. And that:

H •. /EK''''I

i-l.2.· .. ,M

+ m••• )

'-I

~c•.,.,c,(l -

'-I

11

'-1

4•••••

wl,el'!' 11....,-'" • aTld for, each subsys t (·I. the sa.pled data lIullhl'r IS N. The least squares Irthod is applied for the estilat ion of the parameters in the is lode1. This est ilat.ion I'roredure parallel in the H suosystels. TIle optilllization index is that:

H:

~H•..,,(i)

1)m•••

+ ()' _ l)m.0•

.

J ••• At±(t,(l)

.. (i) -

+ ...... '

~D•.,...,(l -

l) -

I-I.

The intrrconnection along subsystels are presented by the interconnection latrix i,j-l.2.···.M,

,;:P',

where m•. " Ill• •• and "'... are the ordcrs of the output. interconnected input and set points of the approxilate linear lodel for the i-th subsystell r especti vely; I ••••• and I •.••• are the tille delays of t.he interconnected input and set points of the lodel for !-he i-th subsystel respective!.\'; Th e approxilate linear lod e l used in the identification for the ith suhsystel is that:

where ••.• , .... and ••.• are the orders of the output. interconnect.ed input and set points of the ith subsystel respect he b' 4.. •• ,\lid 4•.• are t.he tile dr.ln.Ys of the intpl'rormected input and set points of the I , .. s uhsystel respectively; ,,(1) -.(1) c, (l) alld .,(l) are thE:' vrc tors (lf ou tl'ut. interc:onnect.ed input. set points and sll.chnstic disturhance of the ith subsystel respect ively; besides • .4••• (l) .,"

H£[H•. /]

' i ...... T

(7) T, £ "'
Consider the linear slow tile-varying large-scale systel with M subsystels interconnected with one another in the fori of difference equations as follows:

tD",U),,(i -

Urn

and the whole large-scale syste. is stable exponentinliy ~hen each II,.,(i) D,.,(l) and C,.,U) are rep laced wi t hA,." 11,., and c.., rp.spec ti I'e)y . L2: for each subsystem 0=1.2 ...... . . H):

FORMULATION OF TH! PROBLEM

l) -

the

In order to obtain the consistency of the identification alGorithl. follo~in9 fundalental assulptions on the systel. stochastic noise and set points should be satisfied:

In this paper . a large scale slow tilevaryin9 linear industrial process. whi ch is cOJPosed of lany subsystcls having interactive connection with one another. is considered. In view of the present situations of the practical application. the structure of the real process is assuled unknown and the identification for this structure is IInnecessary. Besides. allY other test input signals excppt the usual step changes of the s)'stew .;et pnillts in the opthization procedure are not introduced in this estillat ion period. Under these fundalental assulptions. an approxilate linear lodel class is applied and thEsi.ple weighted least-squares .ethod is ellp!oyed for each subsystell . Accordin9 t.o the estilat.ed paraleters. an estilate for steady-state gain of the real process in the noise-free situation is ('ollstructed. The calculation of the estilat(' consists , of two stages: Pllrllllrl estilation and flllrallel iterat ion along subsystels. And then. the consistency of the estilllte and the convergence of the parallel calculation lire proved in two theorels respect iI'(,! y . It i s also 1I0rth lentionin9 that the assulptions about the stationarity and the distribution of the noise are not required in the proof. Based on the consistency theorel. a pragntic piecewise estilation lethod which is able to give out the strong consistent estinte of this large scale systell is presented. Silulation study ,is ilplelent.ed to verify the efficiency of this lethod.

f. . . . U),.u f:t

with

(2)

(lJ)

112

~

Uudcr the assullptious Li-l5. if tht large-s"ale sysle. (I) aud (2) sat isfies the follOliing condit ions: L6 : for each subsyste. (1=1.2 •... . ..• M) . 1

11

IImN~B( •• U>}-... ...... .-1

(12)

there exist so.eN,~m""( .......... ·•• ...... ). when Ijl";; N, • such that

!!!!! k~B( •• (l).~(l- j)} -

(29)

11

.-.

(30)

(13)

.... ).n

where "", U} is the s. -th co.ponent of the vector '" U} 5\=1.2 ....... ,po then we can conclude that: • Cl: there exist so.e N ... • such tbat for is any N~N.... the .atrix invertible in the sense of' a.s., i=I.2 . .... .. . H; j=1.2 ....... ,PI C2: if Cl holds. then the aatrix

there exist so.e· ·N.i> maxC ••.• + ~..,.••.• +. ~............. + ~.... ) when lil";; N•• such that " !!!!! N1 .-, ~B("U)M~Uj)} -

-'.01

(H)

....)...

where ........) .n and ....)... are the .atrice whose ele.euts are all finite . L7: there exist sOlle positi\'e iutegerN.~ for allY N?N ••• the noise .akes the .a t rix _. is pos i t i ve de finite. Where i=I.2 ... .. ... M and

(dla,block::lo.,,- dlaablockJ••.,l/)

is also invertible in the sense of a.s .• where

~B{S,•.,.} l ~B{S" .•,} i ~B(;,}

....................+...................... ;. ...............

i

~BIS::,") ~B{S, .•,} 1 ~B(ii.}

-.A

(15)

+...............

-" ••. ).A

••••••• •• •• •••• ••••• J.. ••••• •••••••• ••••••• •

!i NB 1 (-Y) ; MI i

1 ("'") NB,. I)

--p.

I),:(i -

_.'.

where , •.•., denotes the j-th diagonal ele.ent of the .atrix /I•.•

Ill• •• )

8".,,£

::10 •• A 11

"

_.',

....,.

~ •• (l - ...... ).:(i - I) ...... ~ •• U -

fII•••

>i(i - ......>

I

).~(i

-

~•.•••

-

(/~ + .-, ~::Io.,,~) ...,

(38)

11•• 11 A (~D•.• ~)

.-,

(16)

" •. (l :E --p.

(31)

s:,.r,....J I &:.........J s.....,.....J

1

" •• (i ~

11

~ •• (l - I)iu -

I~............ : : : : ~ .J..· ~ ·:......................... ::::·: :·: · I ·: :·: : :. ]

•

~ •• (i

1)

o-Y,

-

(39)

1).~(i - ~••••• - ...... )

(17) 11

:E,•. (i _.p

....,.>.:U

ij ..( l - "-~, -

t_,.,

- ~•.•.• -

I)

"-~., -

I)

1)"'(1-

" •• (i ~

"'Y,

m••• ).r(i

- ~ .•.• -

m••

i;,,(1- "-...,- 1)0(1- ~.•_ -

... _)

0-"

8 ..... A

.> (I8)

-

~"'(1- "-.• , - ........ (1 - "-~., - ...,) t_",

where J•. II~ and lI. .• ~ al"e the least squares esti.ates of JI...I •• and D.~~ respectively.

(19)

ii. A (:EM~(l- ~•.•.• 11

_-p.

-

1) ........

~#o

L8:

(20)

Definition: the steady-state noise-free counterpart of the large-scale syste. is that:

(21)

(40)

:E.r(l - ~•.•.• - ...... »)Y ..,, "

fliagbloci ( I.) I G

O.A 1j;;;~iJ.;;;i(j;i··ri

1

..·J

(41)

(22)

(42)

where i=I . 2 •. . . . ..• M and that:

.

(23) G.A.

when

C,•. )). '".) E

i#j

n( ....,+~....,+I) "'.

(24)

j=1.2 •......• M

A.,A. (/~ + 0-, tA..IJ

(44)

(~B 0-, •.I)

(45)

11. A

(46) ,

.. A

(I • ....... 1

.Hrl ........ Ht. •

)~'~ .

in•. ,

or

Equation (40) defines the gain .atrix of the counterpart as that in (43). (41) and (42) are the steady-state output and input of the systell in the noise-free case. A is well-defined. as fro. the assu.ption Lt; we can derive that :

(20)

"'t .•

doc(dlaablOCkAI - dl •• hlock1l••

113

H»

0

(47)

(32)

" +1 .•.".(1 2,;.8.-••

.

*-',

",•.• ),~(1 -

})

:E.8.-'.+I.,.".(l - 1)':(1- (. .•. , -

" ~.8._.,+1.,.".(1 -

__ I',

.

~.8'-'.+I .,.".U -

I)

._F,

I.r,

.

I;.8.-r,+I.,.".(l - .... ,) .... (1 - (.~., -

~.8._"+1_.,,.(1-

•-r,

I)

1)0(1- (. .•. , -

I)

.

:E.8.-"+1.,.,,.(1 - .... ,) ... (1 - t •.• _ -

I)

I_F,

--~,8,_,.._.",(t . . . ,

'-',

t •.•. , - ""_)

- ..... >...(t -

~ .•, -

""_)

~.8._,,+1., .... (1 - l)?,(t - (.._ - .....)

""

(34)

E.I.-".+",,4.U- ..... >,fu - ~. ".j

-

... .,)

,,,P,

It

a_r,

I) .... (t -

(33)

I-~I

~P.-r.+I"'JU.(.I:

",•.,),~(l - ..... ,)

It

-

d••••• -

1) ..... (.1: -
1)

...... ~P>-r,+t.,.JU;(.I: *.1",

4. .•., -

d••• •.

1) ..... (.1: -

m .•. )

•....

S~•.•..••. I£ (35)

N

~P.-r.+l .• "u,(.I:

N

-

d•••. , -

"".,) ..... (.1: -

d•.•• , -

1)

1_T,

......

~P>-r.+I ••. JU.(.I: -

'-1',

It

~P>-T,+I"'JU,(.I:

._T,

d•.•.• -

"".;)ur(.I: -

d•.•.• -

l)e;(.I: -

d•• , _ m.;) . .

. ....

d•• , -

m. •,)

It

-

4. .•.• -

4.., .• -

1)c[(l: -

1)

......

~P.-,.,+I .•• JU,(.I: -

*.1',

..

(36)

N

~P'-T.+I"'JU,(.I: -

.-T,

It

d•••.• -

m••• )e;(.I: -

1)

d•• , •• -

......

It

~P'-T,+I"'JC.(.I:

-

d•. , •• -

1)c;(.I: -

d•.

<., -

~P.-T,+t ••. A(.I: -

*-",

d•.•.• -

"" .• )er(.I: -

d•• , •• -

1)(';(.1: -

d•••.• -

1110 •• )

It

1)

1-T,

......

~P.-T.+I ••• JC.(.I: *.1',

-

d••• ..

m,.)

.

(37)

It

~P'-T.+I"'JC,(.I:

1-',

It

-

d••••• -

m. .• )cf(.I: -

d•••.• -

SilDi lal'lY • • e call ddille .1.. as est ilate of the matrix A

1)

......

~P.-r.+I.'.JC,(.I: --",

The sufficient the is

(49)

is the least squares estimate of the matrix c•. ,., 1=1,2 ........ m•• ,1 i=I.2 . ....... M Froll the lemma we can see that the estilate.1.. is also well-defined ill the sense of a.s.

T £ dlaaCA;:~ll•. ,,). J1

Because the elements in H latrix are all ones or zeros. this parallel estilation and iterative al~orilh. will work quite efficiently on the co.putel' network for hierarchical oplilization and control.

Theo rcm I(ronsist e nc~): Under the assumptIons I.I-LB. conSl er the large-scale systel (1) and (2). then the estilate of the steady-state ~ain matrix is consistent. na.ely thal: 11

i~ 1 t 2 I

Since it is unable to gct tile stron9 cOllsistent estillate of the steady-slate lode I of the lar~e- sc ale systel under only one staHe of the step funclion to excite. a Pl'aHlatic approach 10 achieve the strons consistent estilate by Icans of multiple stages hf step function 10 excite the systcm will be used here .

11

2,;1"., .••. ,." - 2,;lo .•.. P,

II-J_I

(50)

•.•.

J-l ... I

M IS,=- 1 t 2, ..•

A ~ (A, ,) .i.j .

co

t

1.2.,,·.M 1,1,.,

(55)

where p . (T) denotes the spectruI radius of the latrix T.

CONSISTENCY AND CONVERGENCE ANALYSIS OF THE ALGORITHM

where

"

'·1

t,....

Urn

d•• , -

(53)

(~B)

l',." £ (~t,." ..)

m••• )cr(.I: -

tlte

.1.• A( dl.ablockjl, ... -dlaablockll, ... H) -le dlaab1ockl'.... ) .here

d•.•.• -

"

E w·'"

(51)

A.• ~ (.1".,.,). i.} - J.2,,,·.MIA".'.J E R""(52)

Now. let us assule that. for the i-th subsystel in the j-th sta9c. the slstel set puint I'ector in the opt ilization procedure wi II cltanuc froll r!.. to"l . i=I.2 •. " .... H; This step changes will be parallel ill sultsyslcms in each stage . To satisfy the steady-stbte open-loop identifiabiliti of the syslel. we shall desi8n the set poiuts of the large-scale systel to chanuc in step tanner by Q sta!jcs. hCI'e Q denotes the dilension of

lo .••. , and i ".' .••. , denote both \lIe s,-th vectors of the latril:es A.., alld A•.,.,

respectil'ely; Because tllC Icas t squares es t i la ti on procedure is parallel in each subsystel . ill orrler to use the ability of the cOIPuler nelliork fully. a Jacobl-t)pe iteratil'e forllula lay be parallel appl i~d here to get the estimate A" 114

m.;)

..

the lur~ e -s.:a le system sl'l I'lliltt H·ctor. All stll' values of the systel set points in each sta~e should be in the feasihle set of the large-scale s~· stel. Defilic the set puint vector as: tr A [-iT -iT a;,;T] T Yj=

The

U,. a,: ....... "

suhs~' ste.

j=l.~ ......... O

(56)

.

:

: " " ","" .•,01 . ~J -- L.J "i" L.J

: . ~J + "',."Q ~

N ••••• • )

J-I

)-1

2: ••.• - ••.• - •..• -3 . .... I-~•. I-O.

A ••• (1) -

- O. 15(1

r- O. HO - 0.04.-··.... )

-

L-

-

0.31 (I

O. 11 (I - 0. 02.- u .... )

-

r 0. 09(1 - 0. 04.-.... ' .. ) A••• (l) -

(57) D, .,(l)

L- 0.01(1 + O.054-u ....) r 0.31(1 + 0.02.-.... ' .. ) = L- 0.13(1 - 0.04.-......·) r O. 18(1 - 0.03.-.... ' .. )

~hel'e ("., .1' . ... .. , ~~ .•, .g

are the <:rrol'S hecause of ti.e lilitation of the sampled da t a. i =1• 2 •• .. . .. •H; s, =1• 2 • . . .... •PI ' Frol the theorel I. as the lIumber of the sampleu data N-oo ~",,,,, ....... ~, .•,.• - 0 • then ti,e strulIg cUlls istellt estilate of the steady-st ate gain latrix uf the large-scale s~' stem CIII be obtained.

D... (l) -

l- 0.09(1 -

O. M.-....' .. )

r O. 13(1 - 0. 03.-.... ' .. ) D,.,(l) ..

l- 0.08(1 + 0.03.-···..,.. ) + 0. 04.-...... ·) L- 5. l( 1 + 0. 03.-......·) r 2.6(1 + 0. 04.-·""') L- 2. 3(1 - O. 03.-....' .. ) r 1.6(1 + O. 03.-~ ..... ) l- 1. 3(1 + O. M.-....' .. )

C,.,(1) ..

C,.,(l) -

J

0.07(1 - 0. 04.-......·)

+ 0.03.-.. •.... ») O. 29 (I + O. 02.-...."·) J - 0. 07(1 + O. 04.- u ",,») 0.22(1 + O. O~.- .. •.... ) J - 0.06(1 + 0. 04.-.... ' .. ») O. 12 (I - O. 03.-...... ) J - O. 1(I

- 5.9(1 - 0. 03.-......·»)

r 9. 8(1

C,.,(l) ..

It is qui te easy to extellu the theorems above tu a cluss of nunlinear slow tillevarying large-scale systems such as Har.l1 ~ stein and bilinear Iwoliear systems.

+ O. O~.-....II.») + O. 02.- u .... ) J O. 13(1 + O. 03.-....' .. ») O. 16 (I + O. 03.-....,.. )J 0. 03(1 + 0. 03.-.... ' .. »)

r 0.350 + O. 03.-....' .. ) O. 13(1 - O. 054-....... )

l-

A, •• (l) -

1

"

~~ .•,.J' vl- I-I 2:;i~., .•,.J' vl .+ ~.•,., ;-1

factors are chosen as

the paraleter latrices:

these vectors should be linearly independent ~ith each other . Then after Q stages step chao~es. we can uet from the theurel 1 that:

"

~eighting

/l•. ,., - o. 999P._,.,., + 0. 001 • P,.,., - 0.011 /l•. ,.• - O. 97P._,., .• + 0. 03 • /l,.,., - 0.011

9.5(1

+ O. 054-....' .. ) J

- 2. Ht - 0. 04.-...."·») 3. 8(1 - O. 02.-~""·) -

J

+ 0.02.-"·"'») - 0. 03.-...... ·) J

1. HI I. 8(1

• v,(1) = .,(1) + ~D, ...,(l...., I)

The noi se 4-1 where .,0) is a stational'Y white noise process. with norlal distribution. zero expectation. stalldard error vector [0.918.0.969\. The paraleter matrices : D, .z(k) 1=1.2.3. are that: I

SIMULATION STUDY

Consider the slow time-varying large-scale system "ith t\W cOllnec t ed suh s ~' s t ems :

lillear illter-

r 0. 5(1

subsy stem 1: ••.• - ••.•- ••. 1-3, 4•. 1 -4. 01 -0,

the parameter latrices; rO. 38(1 + O. 054- uOI ') loo 15(1 - O. 054- u .... ) r- O. 12(1 - O. 054- u A,.,(l) -

l

D,.,(l) -

.... )

0. 08(1 - 0.054-·· .. ' .. )

r- O. 07(1 - O. A•. ,(l) -

O. 17 (I + O. 05.- u ",,») 0.35(1 + o. 054- uOIZ.)J 0.09(1 + O. 05.- u .",)

l- O. 03(1 + O. 054- u

r O. 31 (I + O.054- u

l- 0, 11 (I -

- 0.13(1 - O. 02(1

054-~"''')

D,.,O)

~

",)

C•• ,(l) -

L-

r

D, .,(l) -

D,.,O)" D•. ,(l) -

J. 5(1

tll • •• z:::m.,z=z.2, tn • • I *=

+O. 03.-......»)

8.5(1

+ 0.054-...... ) J

- 2.0(1 - 0.05.-.... ' ..

»

4.2(1 - 0. 05.-·· .. ' .. )

J

0. 05.-0. ..... )

- 2.7(1

+ O. 05.-~""·)

H-

+ O. Os.,-·· .... »)

2.0(1 - 0.050-.... ' .. )

•

+ O. 154- u .... ) L- O. 15(1 + O. r- O. 14(1 + 0. 13.-· ..•.. ) L 0.1(1 + O. 154-~"''') ra.2 (1 + O. loo 2(1 _ O. 154-· ....·) -

11.-· ..... )

11.-· .....)

~. 8~8966

=:0

~.

387559

- 3.

I 7. 873022

08749~ I -

...

2. 170883

-

1.980081

8. 75~539

The step changes of the systel set point vector are divided into 4 sta~es. In each stauc. the lIullber of tile sallpled data N is 250. Tt:e values of the systel set point vector ill each staue are those as follows:

14.-· ....

+ O. 154-u .... )

I

~lii~

- I. 8~8607

0.2(1

0.25(1

I •

A(1000)-['~~:~~'~';~;;~"~~:~~';;;;;~..I..~~.·~;,;;;~~! ~~:~~';:';;~':' l

+ O. 14.-....."») 0. 13(1 + O. 12.-· ..... ) J - 0. 15(1 + O. ·») - 0.3(1 + O. 13.-··..... ) J O. l( 1 + O. 154-·· ..... ) ) -

1, 4•.•. I =:d •.

The real value of the gain latrix at the tile k=IOOO is that:

J

'. v,(l) - .,(1) + ~D,.,(l).,(lI) Th e nOise ,_, • • here .~1) is a stationary "hite noise process • • ilh IIlJrul distrihuliun . zeru eXl'ectation. stalldal'd enor vector [0.819.0 .8171 ' The parameter matrices : D,.,(t). =1.2.3. are that : 0. 3(1

0.17(1

- 0. 25(1

The weighting facturs are chosell as : /l•.•. , - O. 93P._, .•. , + 0. 07 • P,.,., - 0.011 /l•. , .• - O. 97P... ,••.• + 0.03 , P, .•.• - 0. 011 TIle illtercollllectill9 matrix along all subsystems is that:

- 3.7(1 - 0. 054-.... ' .. »)

r J. 2( 1 + O. 054-·· .. ' .. ) C•. ,(t) -

+ O. O~.-u",,») + 0.054-0. ..... ) J

+ O. Ole-......·) J - 0.06(1 + O. O~.-"""·») O. H(I - O. O~.-·· "''') J

+ 0. 05.-......·)

l- 2. 0(1 -

D... (l) ..

+ o. 13.-......·) ) + O. H.-......·)J 0.15(1 + O. 13.-""''') ) ro. 35(1 + o. 14.-""''') Lo. 15(1 - O. 12.-...... ) - O. 35(1 + o. 11e-" ..... )J

r- 0.24(1 + 0 ! 50-....... ) 0. 15(1 + O. 12.-....... )

The apprpxilate dynamic lode 1 structul'e :

l...., 2.5(1 + 0. 054-0."''') r 2. 2(1

l

+ o. 13.-·· ..... ») + O. H.- u ",) J

J

O. 19(1

+ 0.054-...."·)

D,.,(1) -

0. 23(1

O. 3HI

- O. 09(1

l- 0. 05(1 + O. O~.-""''')

- 0.3(1

l- o. I (I + O. 154-..•..·)

- O. 12(1

O. 02.-·· ..... )

r 0. 17(1 - O. .. ) l- 0.08(1 - 0.054-"''''') r O. 18(1 - O. O~.- .. "''') r 9.7(1

C,.,(l) -

+ 0.05.- u ''')J + 0.05.-....... »)

0.06(1 - 0.05.-...... )

",)

O~.-u.,

D, .,(l) -

)

+ O. H,-""''')

D, .,(l) =

J

The approxilate dynalllic mod el structure:

115

v. - (0. 3,\. 3: 0. 1.1. 6)'1

v~ -

(0.6,1.1: O. ~,\. 3)',

oi-

vl-

0.1.0. 7: 1.0,0.6)'1

(0. 8.0.9:0. 7.0.9)'1

v: - v- Cl. 5.0. 5: I. ~,O. ~)',

After these 4 sta~es, the the gain m· trix uf the system is that :

..1 ... -

8.311920 - 2. 6839~9

estilate of large-scale

- •. 911H3 i 5.1610U 9.• 95185 i - 3.852165

AClNOWLEDGE~ENT

This research project is supported by the National Natural Science Fund of P. R. China., Authol's are thankful for tltis sllPl'ort.

- 3. 602~~6l 5.• 5575~

····························.········.····· .......i ................................................ . •. 211276 7. 53~980 - 1. 625107

[ - 1. 55HI7 5. 650638

!

- ~. 157071 i -

3. 050622

9. 057006

This di~ital silulaqoll has Pl'oved that , ullder the contlit16n of quite lo~ sigllalnoise ratio, the alsoritllll expressed in this p~p e r is efficient and reliable to identifYinu the steady-state sain latrix of tlie lineal' Ial'ue scale systel .hich is very sl owly tile-varying.

CONCLUSION In this paper, fur th e linear slow lii.th tille-varyin!J large-scale sysleloJ I"ny int ercollnected sulJsystells, the estilate of lhe steady-state gain of the large-st:ale Syst el is first forled and a parallel identificati0n algorithm in subs)·st els is then put forliard. The cOlIs i s tency of the estilate and the cOllver[jencc uf the parallel iteration are proted. Based on the consistency theorem, a pra~lI a tic apPI'oach to achie\'e the strong t onsisteut estilate of the steady-state l odel of the system is presented . Digital silulation has proved that this approach is reliable and efficient. Because this al~0rithl utilizes the dynalic infor.alion fully, the tile spent is lIuch l ess thall that the pnre steady-state idenl ification algorithm takes, which is presented by Hasiewicz, H.(Hasiewicz, H. ,1988; 1989). AII the t heo r els abov e are easily extended to a dass of nonl inear largescale syslel such as Halllerstein, bil ineal' cases . REFERENCES Balber~er,W.

Slid R. Iserlann (I!178).Adaplive on - line steady-state ol'tilization of slow dynalic processes. AlItomatica. ll, 223-230. Garcia,C.E . and H.Morari (1981). Optimal operation of inte9rated processing systels. Part 1: open-loop on-line optili z ing control. AIChE JOllrnal, 960-968,

n,

Hasiewize, H. (1988). Two-stage procedure for para.eler estilation in largesca I e i nt CrCOllllec ted lineal' zel'o-Ielunry syslels. Int. .1. Syst . Sci.. 19, 497-516. Hasiewize, H. (1989). Applicability of I east-squares to tile paraleter est ilalion of lal'ge-scale nO-Ielol'y linear cOIPosite systels . Int. J. Syst. ~, 2427-2449.

2£l.:,

lin,J., C. Han . P. D.Roberts, and B.~.~an (1989). A new approach to st ochast ic Clpt ilization control of stead)' st ate systels usin~ dynaaic infonation. Int. J. r. o ntr . .~, 2205-2235. Robel'ts,P . D. (1979). An algorithl for steady stale systel optilization and paraleter estilation. Int. J. Syst. Sei. ~, 719-734.

116