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On Line Identification of MIMO Linear Discrete Stochastic Systems
Cop\rig-III © IF:\( : Id t' lllil i( ;lIiOIl alld ESlim 'lIioll I ~H'G. York . L' }\" 1 ~1~'G
S\ SI l' 1ll 1\lr;IIIl<.:Ic.:1
ON LINE IDENTIFICATION OF MIMO , LINEAR DISCRETE STOCHASTIC SYSTEMS Zhang Zhong-jun, Li Dong-feng and Yuan Tian-xin DI'lmr/mm/ of J::ll'rlriml J::lIg inl'lTillg (11111 C()mIJII/1'!' Sril'lIn', SllIIlIglllli J i(l() T ()lIg ['11 il 'I' n i/.\' , Slltinglllli, Thl' P I'OIJ{,"" /?"IJII/JIir o{Chill(l
Abstract. In this paper the structure identification and parameter estimation of MIMO linear discrete stochsstic systems are discussed. Using Luenberger's observable canonlcsl form wlth steady-state Kalman fl1ter1ng representation, an on-llne ldentlflcstion method is developed. The structural identlflcation of the systems ls determined by residual error method by developlng a recursive algorlthm. Parameter estimation ls made by recursive extended instrumental variable method which gives an asympto~c unbiased estlmate. A recuesive algorithm for the residual-square~ is thua obtained. Computer simulation results indicate that the proposed methods are sufficiently well to be used for adaptive control p~pose.
Keywords. systems.
Discrete systems, Identification, Linesr systems, Stochast1c
INTRODUCTION
possible structure identlfication on-line. (3) A recursive extended instrumental variable method is proposed for system parameter estimation.
MIMO linear discrete stochast1c system is one of important systems existing extensively in engineering, social and economic fields. To obta1n their mathemstical modela, i.e. to identify the systems, ia one of the major problems for control theorista.
INPUT-OUTPUT RELATIONSHIP OF MIMO LINEAR DISCRETE STOCHASTIC SYSTEM MIMO linear discrete stochast1c system is usually described by the following equations:
By the application of Kslman filter theory Astrom, Mehra, Panusks and many others solved the parameter estimation problem ysing the maximum likelihood method ( AstrOID, 1980 ) and the extended state variable method (Panuska, 1979) respectively. However, prior knowledge of system structure should be available in applying their methods. Using steady state Kalman filtering expression and the correlation function method, Tse and Weinert proposed another identification method (Tse and ~einert, 1975) which has the advantage to identify the system structure and estimate its psrameters at the same time. The shortcoming of this method is that we can only use their identification algorithm for off-line calculations. Suen and Lin (1978) proposed another off-line identification method uslng the residual error method for system structure identification and the least squares for parameter estimation of the deterministic part of systen In order to improve the above methods,this paper presents an on-line algorithm to identify structure and paramEter estimation of M~ ." .) ::'lnear discrete stochastic sy st e ms. The new features o f t h is paper are as follows: (1) Using the steady state Kalman filtering expression, the input- ou tput relationship with Luenber~er's observable ~anont cal form ~s d erive d . (2) Th e stru ~ ture identification is c arried out by usin~ the residual error method. A recursive algorithm of residualsquare-sum is al s O derived, which makes
X(~+I )=iiX ~ ~l+B u (A)+;J \ ~)} ;:\ ~)=CX i ~)+ .).; (,()-t-
v (,()
\ 2 -1)
where X(~)E Rn, u(~)E RP, Y(~)E RID, w(,()ER n and v\~)ERm, w(~) and v(~) are zero mean normal mutually independent white noise sequence with the following covarianC'ee:
Elw ( ,()wT(j)J=W~j Efv (~)VT(j)} = V ~
}
(2-2 )
j
Assup'",e t hat (A, C) is Ob servable and (A , E) is contrOllable, using stead y s tate Kalman filterin~ expression with the parameter matrices - expressed in Luen b erger's observable can c nical form to get the inn ov ation e4uation of the sy stem (2-1) as f o llows:
X(~+II~) = AX \ ~ /.,6.-1) + Bu (~H- F £. (-()} , y
(l ) =
CX( l
\ tt-I)
f- D:.Jl ttH
t.
(~)
1::>- 3 )
where i(AI~-I) is th e single-~te p optimal prediction, E. ,.,t ' = ;:, f!.. ' -CX \ ,(/1<.-I)-D u\ ,u i s t he innovation sequ ence, F is the S6in o f t~e steady-state Kalman fi~ter1n ~ . whi~e
,' _[ :, Ij/,;-l II A(.",-- --,- - --------j a~.I: a,c.;. .,.'. a.'i ,~') lIS:;
Zh~llIg ZhOllg~jllll.
I -P'Hi
Li DOlIg-tellg alld YU
(J;'d=(~+ " P"4-<-' ··' P<;.p] ; Y'i= l r;;), Yip.",. r.:;,pJ ;
a.< =( tl.."I ,"· ' et,:">,, ; ... ; 11;';',1, " e.:=[ 0 0 10" oJ ; J/M= I
' , a..ii,;li ; ''' ; Cl,.A.,I ,'
,t1.:..,l/.]
,?1Utx,! J-q
L= I, l ,",nt
i.th .I• .,td
Comparing equation (2-4) with the inputoutput relationship of deter~inistic observable canonical syste~ equation with non-dynamic part (Guidorzi,1975), it is found that equation (2-4) has only one additional term: _m.. J/. ( 2- 7) 11,(.,(+~) = ~ j: !i; ,~ 4(.,(+'1) 'C <
observability index of the syste~), y; _ (" mi ll J.{ ,;/; ~ ~ <
f
i
<1-1 mLllf~+I , Y,j i~j
Si~ilar to the ~ethod suggested by Shrinkhande, Milal and Ray (1 980), we can obtain the input-output relationship of above ayste~ as expressed by the following equation:» p ~ u. (A.tJIi.) = 4-1 ~£!. aN, V;(I,+£-/)-t F-~ ~il',,,. '4.(~,f}) ~< !=I I I .=/ , . .
+ 1.=--( ~£ Y;.1',pJ.jJ~fi) i:24 and Y.i.J ,J,
(L=I , t , .. ,1It)
, (2-4-)
I
satisfy the following
1° I~ . .. , . . . . . .. 0 0 0 - - - ---r. 1
I '
J;; - - - - 1>,.(') .'
.
....... . ....
bM(~) .. . . . . . . . . .
I'
·: ·: · b.l,id,.
(2-5)
0
- - - - - - - - = - 1-:. . . . . .. . .
I:
which is the random input to the system. It can be easily proved that 1£..:( ~ -rJ{) is a K-step autocorrelated coloured noise with zero mean. Although system (2-1) is not identifiable (Li,1983;Tse and Anton,1972) , but it can be identified through system (2-3) or (2-4) . STRUCTURAL IDENTIFICATION In atructural identification of a system, it requires to find out the observability indices of the varioue SUbsystems JI..: (i= 1, 2,' · ·,m). As equation (2-4) represents the relationship between the output with the inpute and other outputs ,eo it is the basic relationship that will be used for structural identification. We know from the definition of )/9 that Yij depends not only on Y.: but also on YJ • Thus' f we use equation (2-4) directly to identify the structure of the system, it would be the same case as confronted with Guidorzi, the value of J{: can not be determined separately with each of them to be determined successively by using previoue results (Guidorzi,1975l. In order to identify each observability index independently, we rewrite eq.(2-4) as follows j__ 1<·, 1 , f Ii: , • ".·(~tJi.)= i;i!. ~ tZ;I' f;'; ,l ~ (A."'JJ t7i. Sf .1=1 ,.. , ,,,, g~I/I'("t~--I) -t l::cz:. ""r J=O fOI
t
10
I:
with
b.,
r... r.u .. ... ')';,/100 .. 0
e/
0 . . " ,, 0
0
.. .. f.,1 " e.,o .. o 0
e, :1;0),' ,
a,
10 I .
I" '
- - '0
1;.,1/)
e",o"·
0
f.' (~+Y,,;) ....
=
(3- 1 )
={, <, " ', ?!!.-)
~ fi n:~£-e( Ii +JJ
l ".t q;O
wher
(2-6)
y.: =[ V~~.\tl)J
( 3 - 2)
; ~t =[ 1i.~h')]
1;.
t ( ~ + K)
U,(~+/), , U,( i~""')
1J.p ( ~+J) , '
U,(~ ~JI
X"cY.,:J = [ :
!{/~). ' Zl,(I.'Il; tK) ... tLpr}.")" 'lil~-Y.'tJQ Y,M,tk) .. -!tlJ,--!l't<)
,,,,{A,--J{,tJj
10
. ! .:(!;J '" r;.<~7'.;t/) .. · y", (I.)
I'
'N.iA·K·1)- "Y<. (II. :)I,,+K)" . i ..:(~+K-o"'J~(~-y'; H<)
1 :
"'-( ~ --".
I
Y,,: = X..:(J.(.:)I1.: (Y,,:) t~·
o
' J";''l.e~l i2..
Rt, i1. '-j,t!/i<-t..+J.--')'t 't,J~+Y..'J I
Now , eq.(3-1) is only related to observability indices ~ , which enable us to identify the order of each subsystem independently. Although some terms on the right hand side of eq.(3-1) may be linearly dependent (e.g. when ){j
-I ':"" I'
)/...;
( i
I·
r;v,0· .. .. ·0
,
(J
- - - - -10
r;.Yt,· ..
)='1= 0
0 10
.. .
6h
14H i
:\(illlo Line ar Disc re te Stoc ha stic S\·st CIll S
Let
(3 -3)
Z,,=X..:(t{)bl..:(.Y.:)
then eq.(3-2 ) can be ~i m plified to the following: form:
Yi. = 'Z;. +
~i
uir
T'
= IfK -+
T Ktl
PK~I = PI( -
(3 - 4)
According to the principle of the residual error method (Suen and Lin , 1978 ), the estimated value of the residual-square-sum is A T -t ) e.i.(?t.<.) =Y..: (I-X..:(ll;.)X.<. (rl. i ) y,,0-5) where X!(Il,} is the Moore-Penrose pseudoinverse of X.: (1tl) • Eq . (3- 5 ) can be further written a~
"': . [ZI(l-~(Il.)xtclU»Z~ + E.f?!CI-x,:en.i.)Xr(Il.lV71..:} e..(fI.,) = n. ... < v.. ; Efr{(I-X;. (lI,JX!C Il .i.I)'l..< J, n.~ ,;I,,; , (3-6)
¥(Kfl)
-:<.;/K))/CX:;+'PI(X/(tftJ)
Y(J<+/) AI<+/ I
fI
(+~fK,(K+I)
. . . ... (3-j)
Ifr.
where
€,;"(~.)= ~(n <)/IIY"i~;
e;(I/;.t-I)=
a,(II":tl)/II YNII~
(..:= /, Z. , ...• m) When Z.<. ien 't a linear combination of the column vectors of X.<. (fI.;') , ZICI-X,;.(/1.Jx1CI!i.»)z..: ie the residual-equare-sum given by the least squares. It is the error due to the discrepancy between the actual system and the mooel, i . e. an error resulted from the mode1,~tructure. The expression E[~ifr~~(n.JX!(lI.';)f"J is the error resulted from n,oi se. As ~4(~~YA:) is a )/,,; -step a utocorre1at~d coloured nOise, it had been proved one of the authors (Li,1983) that when ~..: is a stationary zero mean ~ -step autocorrelat,ed noise. the value of E[~.l(I-X«~;)Xl(A4»)fA} will be independent of the value of n~ for aufficiently large K. Therefore, calculating e..:,(n..:)for different ILL when K is suffi ciently large, we can determine the structural parameters }I... (.i..=I • .z., ·.. ,1fl) from eq. (3-6 ) by varying the value of e.<.(fl..A ) •
for dlfferent ?lA. (?Z": = 0 ,1,2 ,' ' ·,n;.,.,~) for sufflcient larE!e N=K. (3) From e;(Ili..+/)-~!.(~;')~E wh ere e. is a preaesigned value, which le determined by the prior estimation, calculate n!. Then v!.:=n1 . (4) Repeat the above procedure for different 1 (i= 1,2. · ·· ,m) until all the estimated values of J/,.: (i= 1,2,'" ,m) are obtained.
With the use of the recursive algorithm for calculating the Moore-Penrose pseudoinverse suggested by Sinha and Pille(1971) we can develop a recursive algorithm for calculating the residual-square-sum e.i.(It...) . Omitting the subscript i of the subsystem and replacing n~ with subscript K and letting
After all Y": (i= 1,2,··· ,m) are obtained, we proceed now to estimate the eystem parameters.
YK-t/=LYd 19tKtl)J ; XK-tl --LX~J [XT(K";J
i
When k<.k' (1. is the column number of the matrix XK): " T T..I. r ~ (I T\ eA 1(11 = eK -t 1f1(W<"'! XK+I'T'K - lfl( CKti YWJ) t VWf! -Xl(flc;';Hj
1.
Cl -&fIXJ..')K
-t If:' -t
cf>K
/(JKfl =
f'Xj(;{K+/
- ( P,;XK+' (!X" XK+o/-r G.KAJ<+I ( PK,(KfllJ/ ;(;(r( ~KXKf( &1(11 = &oK - &/(XI(+/ CB.I(XK+li'/tkl 8.KXJ(fo/
All lni tial values 4>. , VJ• • taken as zero and 610= 1 . When k '"
el(+l=
~ 11
TA.
IS VOL 2- H
= PK XKTI/(:f.KtI fk.X<:t, "f i)
with
IJ .
~1
+ ffl(ffJ(f-XJ+,Ci<.1i)
(i.=I , .t, · , :t!)
.t=1
( - CZAJ. ~ ..{ ::=
l
(4-1)
A.-I
= zii.tEJJ.:*...l + E,..:(~ 0
J
A.
(4 -<..)
.li. >;/..:
(.i~/,." .J J:t ~ y,..: (1 :::-1.2./,., ,(-') (4-3)
,
Apparently, e..:(.) is still a normal white noise sequence wlth zero mean . Eq . (4-1) can be replaced by the following vector equation:
Y/=xtcy,u,£)B..:.-t-
Et
(4-4)
where
(y..;(~+I) ... . .. Y~ (~+I<)r
U,("'+I) .. · 2(, ("'-u.· +~
.. Upctf.+,) .. ·!i.p ll, -J/it,)
u,~+K) " · IU~+i<-}I') · .. /¥(A,+i
I£/ UI.:fk-r, J
£,C",) · · " ' &('(.-Jl'+O· " [" (~) """U"-y':")J
6l.l =[a.AI'Y~I·" a,:././ .. ' a;"lIl,JI~~· · · ai.""/ ~.i.JI,.;,1
may be
r
-1U<11'XJ+, CI-Cr<+ /:t;f/ )cf>K ex",!
P. ,
: T
e..:.«(J
E, (~+I<-1} .. . £' (~~K-YJ ... [..,(~+K-<)" ·&.d+'l'..I1)
(3-7)
11 e K T If" CKt/ Xt
where
,(1+Jc-~·+Jo.!r-0 " },CN.~J(..".)·· M~K-i..+¥,.-/) .. ·!.(~+K.-r.)
g(M!) ;(7:+1
PI<+/ = PI( + IJ.I( AK+I (&.I(x~+lf( /- xlt, PKXKH)/xJ,+I&1<4c+'
2.
'" Y,:-1
r<. ~1r(~ -11<+J1.J .. . Yltf.-Y.i+j ··.,;1'/M-Xt y....Y. ··f.,rA-J!.+i) X~ (y, U,£)== . : : :
(j(KflJ-X;',I<)/XJ+/ f9.KXm
e..
1:i! a;'J,j !j(,(-Ji'~-')-ti=tF~;i)U;(I.-Jl.:+JJ tY.< >. ("'--YA: ~ .) + lii r;-. r..;,J. '?+~ r e,.: (~
y~(.,f)= i"II-/
yt =
= 61K :f.K+1/xf+,6!K XI<+I
4>K+I=
eq. (2-5) and the definition of ~i , eq. (2-4) can be rewritten to the following forms:
Fro~
.
where the pseudo-inverse of XMI is denoted by Xt:+1 which is partitioned into two parts : the submatrix e; snd its last column CH' , then we can obtain the following recursive algorithm ~r calcu~ating the residual-aquare-sum eK as follows :
- 'I(I(+OX.;+/
PARAMETER ESTIMATION
XTK-I"I-_r~ '<...)( ,rT :C I Kf/]
~KN = Y;~/ (I - XKt/ xt-l"/) YK~/
CMI
In carrying out the above on-line structurel identification to obtein all ~ , we may use different groups of observation data. For example, we may use the firet N groups of observation data to obtain )/, and then use the successive N groups to obtain Y,z. , etc.
1
' .. P.d';,1· ~~., f
Yi()!,-/)"
· ~"
.}':',(
r;,o'.;~J, "I .t;D,mJ
Using the recursive extended matrix method
I-IHH
Ihang
Zhong~illn .
Li Dong-feng and YlIan Tian-xin
the estimation value ~.:. of parameter eA.. (i=1,2,"',!:l) can be co!!!puted. But the obtained estimator ~ is unsatisfactory because it is fenerally asymptotically biased. In order to overcome this shortcoming, we propose another method named the recursive extended instrumental variable method for esti matlnE the system parameters. #(
Let Xi.
be subdi vided into blocks as
Xi( V.IU)%(1JI 4 \~.-J4"+>i/):I ··· :, 41I1~~-)li+l)i, . :I Y~{04-){ty.,,)\I
.. :y;(~r~+,)\u.~~t')\ ·
I
with every block as a K- d i!:lensional vector Fro!:l the properties of the recursive extended matrix !:lethod, we know that the reason for aSY!:lptotic bias of the estimate ls due to the fact that Ef is correlated to S01!le column vectors of X:CV.!I..E.) • This difficulty can be overcome by the following arrangement:
".1<1"1
..... 1(+/
..... K+1
YiOl;-f)." " l'iD,/ ...
Xl =
)
.... "tl
"J(tl
"'1<+1
""Ktl
AK'f'I
"'1(.,.,
..:. =la':/,~/ · ··a.L', I ··· a.ill'l.>t..· ·' 12"".1PiJI.:,,1 · ·A',rA),
i 1l,«~-J;:-r')\;'4(~~)\iuf(4-)i'+I~
I
("'I(tl
l}Ki-'
('iT
t~(~): . ! [ IKCA-ltt1)i.J .:,G", c!(():.J :E",K(~_){+I)l I
where
[
'" K+l
Y:.;(Y..:-<),'IJI. .. ·
'" K+J
Ii ....
JT
¥1(K-Y.:.~')···y/(K-Jl.:+I) · " Ylll.(I(-y'/+)l.(Jl!)·--Y.. (K-J1,+f)l.ld KtIY Itj(K-Y..'ii)··· U,(I(+/)-' U,(Jc-P;:+I) E,(K) '" £'(K->i+O ··. ["CI<) " . £"'(K-Jl..:+I) ]
a~tf:"[y,(K-?'.:+Jf./)!,(J(->i.~/-f)· · ·V,(K-J{'+I)'" ~~(K-;{+J{('''J) s..-f{K-I!.:+ Y..;v.-v-I)- ·· K;, .. (K->itl) .. . 1.. 1.1<) '" !{(K-X.·tl) " · 'fI/(K-y.i+y"n);· f",(k-j:t-I) "/(K+I)-;;., I(,(J{-Jl,::::)- " U,(K..) ' " U,(I(-Jl.'tJ) £, U() ··· t,(K-Jl~·tl) ··· £..,(K) .. -t",CK-JHJJ] A
1,.<'<.-)I.;+J0.) ""
f~(K-y":+)I,.;J.)
JIJ ~ ~
.
0<. .,,: ("-=1,.1..;" , m ) 1> y.: J. =1.<',"',;'-1 The factor >-'(I
.
A.
lA. (Kt-I)
The initial valuea are assigned as follows " ' . =0 Si
m.)
A.="Z ..
(
~ ==1. oZ.:' :.i.-I
(4-6)
J= I. ,z ,"· ';<
ij(!/,u.,£)
ls the jth row of Z{,
H~nce for ~~ Y.:
=o}
E{z2cU,U,f.) E..:./(} E { Z~T( y, u.£) xf (g, U, E.)} '\= 0
(4-7)
ThuB , under the ergodic stationary condition, we have o. I(T PAK.,oo / m. Z.. CY,u ,E.) £" K 0 }
/(1 =
(4-8 )
pJ im. z1'Cy, U,E.) x1 (r, u,f)/ K ~ 0 1< ....
K
As a consequence, we may choose Z~ (!,U,t) as an instrul:lental variable and calculate the esti!:lated para!:leter 'B.. K fro!:l the following equation:
e.. '"
/(=
[I(T
I(
P.:=iXI
.
0
where 0(, i a a very large number. For different values of i the estimator 6t (i= 1,2," ',m) of parameter B..:. can be easl1y computed and then the matrix A of system (2-3). As for the elements of 1!latricea B, D, and F, they can be then calculated from eq.(2-4) and (2-5). The following recursive formula is useful 1\1<+1 /'\ -'" AT 61. = B!'_(~K-£(K+~e(K+J)) to get the estimated value of variance matrix~ of the innovation aequence E(~) A.
COMPUTER SIMULATION
Let the system to be identifled (es shown in Fig. 5.1) is expressed by the following equationa: . 3 X(.t+I)%(-~2S ~, ~J xc~) t fo. ) -0>5 -1·3 0.8
y(~!)= [~
~ ~1
[3 x(A)
J
wftJ+ wcA..)
-tVI)U.(~J+ V(~)
where U~) is a blnary pseudo-random signal wi th ampl1 tude ! 1, W(R.) and V(.Q) are zero mean normal whlte noise sequence with
EfW(.\)Wq)J =[0.;' 0.:, ; o
0
)4; EfVC~)l(j1=[O:5•.:~
ltl4J
0.25
I(T Y. :.K Z..:.CY,U,t)
/1-1
Z~ Cr,U,f)x.. (V,U,f.)J
C,:=,-Z, " ' 7I'l)
(4-j)
which can be proved to be asymptotically unbiased. For practical calculation, £ ia usually replaced by its estil:lated value f as in ordinary extended l:latrix method. Adopting the recursive algorithm (Soderstrol:l, Ljung and GustarsBon,1978), we can derive the following recursive algorithm of the extended instru1!lental method:
WeAl V(~)
Fig. 5.1 block diagram of the system to be identified Obviously . the structure parameters ~ and ~ of this system are 1 and 2, respectively Let the noise to signal ratio be 1
Rnls (L.) =
r;r-/rJr'
=(5:.. utC.4.) /~ yf(tf.))2 ~~I 1,f=1
then fro!:l the given data, it is found
14H9
I\limo Linear Discrete Storhastir S\'SIl'IllS R1t.ls\l)= 30.24%;
R71 lS ( 2 )=38 . 85%
By computer sfmulat1on, we w1ll ~et the follow1n~ results: (1) The reB~lta of structure 1dent1f1cat1on are g1ven in Fie. 5.2.
a 7 ~t1/;).·
e{(n)
0
,
N=200 N=60 )1=2
~=f
0./
0.1 7l.
o
n ~tUt)
0
3
2
I
o.4~5T 0./~/6
o
4-
3
2
4-
et'tn) 0.711 1
0.0560 0. 0537
, CI1.
F1g. 5 .2
4 -2.
I
0
3
"~,,. e<. (n.) 0.5707 0.048J 0.0356
0.a5l6 0.0506 0.0#6
' ' ew ) 0.6z<8 0.<'339 0.061t
3
2
71-
0. ~3·1
0.01# o.ob?! 0.0661
the results of structure ident1ficat1on
(11) The results of parsmeter est1mst1on sre given 1n the follow1ng table w1th RElV denot1ng recurs1ve extended 1nstrumental varisble method and RELS denot1ng recurs1ve least squares method.
IJentifica.tiOIl Method
Nllmber of RIIAS
2000
400
2000
- o. r1f6~
-off!O
- M9 8&
-0. ?/f9
-0.1160 (0.0.2.18)
-Mf j j (0.0061)
- 0.,z03/
-0.2.53 0
-0./683
-O.;,IU,
-0.2.~it3
-0.~63
tt'Zo.I
-0.03f/1
0. 000
-0. 0 743 -o. 043~ -0. o~/6 (o.O.2'~)
~2../
0·9347
M67'(
0·1.273
O·f6?C
4 .063f
3. ~616
4.019-3
3.6333
3·7g'l-
3.1;.J..¥-b
0.74-£6
07077
Yio.I
0./4-91
'9;,.<.
0./!S7 0.11<.6
a./I .2. A
a.1( . 1 A
?';, . I
~o '" Yt'. 1 A
A
'fio.Z
a;~/ . 2.. - (.J.i 93
'"
A
9.z
REL5
( Mean 4AcI St/V1.dud DeVlid,-oll of) P4rIJ.meter £sti~ti01l. jor 4 RUAS 2000 400
400
"-
GI
R.E L V
R.ElV
o...z.I.I -O. Mb~ 0.t 08/
11:.2.2.'
~
0
o.IM (0.0301)
I
3·~f:'
4 .0173 (0.04,z0)
3.fm
3.43 00
3.681<' (0. 146/)
3 .7HT(00316)
3·7{'"
0.70 91
o. 6~7&
0.6J6& ((}.O~'5)
0-7109 (O.O]ll)
0 .7"<'0
P.~!bJ
a.ISV'
0.06~0
0.0#6 (0.05!!)
0.07T!, (O,{JI]
0.,u,1fi,
0.v 01
0.2614-
0.16i1(o.osn.)
o.-2.57f (0. 0 <.36)
O.<.ts o
0.0-274 . 0./ 71.2...
O.o.3FO
0.0013 (o.PbtJ)
0.0.431 (d.03b/l-)
0.°3 0 6'
- 1. 3.<.86
- 1. 0 111
-I..zrlfl (0. 0 31°)
-/3<.10 (0.01/- 00)
-0.6b13 0.t1 04
- 1. on7
-0.5261 -0.5'05"6 0.6tbL
0.6370
o.BIK
0.<.771
0.0.<.91
8 ·It-4- F8
7- 4674 71t l 3
1'2.0.,
K·45'°4 0.14 03
o./~63
0.0785
o.o6Jt
y<..•.z.
-O. /ZlO
-0./oS9
-0.041F
oo/q,b
ft,
0.7S'16
0.!643
0· 74M
afb4f
,A.
[2
-/ -0.2.)
0·9146 (0.o.2j9)
~.z..I "
Q 'f2.1 tu
(0. 01'7)
0.000/ (O.OIOS)
t,o.,
"" 87<.
(0.01 14-)
True VaJ,tJ.e
-o.Otf/
(O.o~/3)
0)
-0. 6fIJS (0.0<'#) -o.Molf (O.OlOb )
o. 7f<. 6( o.oq,oj)
0.1'/3/ (O.ollr)
0. J 10 7 (o..o,S'6)
0.<.?-2.6 (0. OSS3)
8. jllS¥
~.4s<, (0./lf6t)
(0,/0/0)
0./ 0-<'-3 (0.0484) 0.14<,0 (0.041r-7) -0.11;.5'7 (o..oS87) -0.ofI1 (o.o-4Y1) o.tl,fo ( 0.04//)
o./N.3 (o..oZTo)
4-
o.orf/l-I
- /· 3 -0.6.S" 0./
0·3 t.~
0,/-<-0<--0 . /39-j
o.!'/JlI
-MAN -0.41.z.8 -a.fI,041 -0.4150 - 0. ~
- a.tlPT
-0. fl, 081f -0.41<.3
-0.4/ 0
-0. 4 04-1 -0.4ISo - 0..4,,0 (0.04 3<.)
/.ofI;.3
/.f 0<.9
I./ftz
/.U' 02
0. 0836
0. 0 /54
/.1515
f.7Nl>
-0 .40/0 (o.olf!)
!.IMf, (0.<527) 1.1060 ( 0.0<'78) o.021S
7
1/<'/3
0.0/" /
In the above table .r2=t.II~-BJ2ind1cates the arcur8cy of para~eter estimation.
Zhan g-
149()
Z h()ng-~jlln ,
Li Dong-fe ng and Yu a n Tia n-xin
CONCLUSION This paper presents an on-line identification algorith~ for MIMO stochastic linear discrete sys t em based on the steady-state Kalman filter representation with Luenberger's observable canonical for~. Recursive algorithms are developed both for on-line structural identification, and on-line unbiased parameter estimation. Even under the condition that the noiae to Signal ratio is greater than 30% and the data length is relatively short (N 60), satisfactory results can still be obtained These prop osed metho d s can also be used without any modification for problems when system noise and observation noise are correlated. REFERENCES Astrom, K. J. (1980) , Automatica, Vol. 16, pp551-574. Panuska, V. (1979), Proceedings of the 18th IEEE Conf. on Decision & Control Including the Symposium on Adaptive Processes , Vol.2, pp927-932. Tse, E. and Weinert, H. L. (1975), IEEE Trans. Autom. Contr. Vol.AC-20, pp603-613. Suen, L. C. and Lin, R. (1978), IEEE Trans Autom. Contr. Vol.AC-23, pp 45R-464. Shrinkhsnde, V. L., Milal, D. P. and Ray, L. M. (1975), IEEE Trans. Autom. Contr. Vol.AC-25, pp 307-312. Guidozi , P. (1975), Automatica, Vol.ll, pp361-374. Li Dongfong, (1983), Ms Thesis, Shanghai Jiao Uni vers i ty. Tse, E. and Anton, J. J. (1972), IEEE Trsns. Autom. Contr. Vol.AC-17,pp637646 . Tse, E. and ~einert, H. L. (1973), IEEE Trans. Autom. Contr. Vol.AC-18,pp 687688. Sinha, N. K. and Pille, 'wo (1971), Proc. IEEE Vol.118 , No.8, ppl044-1046. Soderstrom, T . Ljung, L. and Gustarsson, I (1978), Automatica, Vol.14 , pp231-244.
S\ SI l' 1ll 1\lr;IIIl<.:Ic.:1
ON LINE IDENTIFICATION OF MIMO , LINEAR DISCRETE STOCHASTIC SYSTEMS Zhang Zhong-jun, Li Dong-feng and Yuan Tian-xin DI'lmr/mm/ of J::ll'rlriml J::lIg inl'lTillg (11111 C()mIJII/1'!' Sril'lIn', SllIIlIglllli J i(l() T ()lIg ['11 il 'I' n i/.\' , Slltinglllli, Thl' P I'OIJ{,"" /?"IJII/JIir o{Chill(l
Abstract. In this paper the structure identification and parameter estimation of MIMO linear discrete stochsstic systems are discussed. Using Luenberger's observable canonlcsl form wlth steady-state Kalman fl1ter1ng representation, an on-llne ldentlflcstion method is developed. The structural identlflcation of the systems ls determined by residual error method by developlng a recursive algorlthm. Parameter estimation ls made by recursive extended instrumental variable method which gives an asympto~c unbiased estlmate. A recuesive algorithm for the residual-square~ is thua obtained. Computer simulation results indicate that the proposed methods are sufficiently well to be used for adaptive control p~pose.
Keywords. systems.
Discrete systems, Identification, Linesr systems, Stochast1c
INTRODUCTION
possible structure identlfication on-line. (3) A recursive extended instrumental variable method is proposed for system parameter estimation.
MIMO linear discrete stochast1c system is one of important systems existing extensively in engineering, social and economic fields. To obta1n their mathemstical modela, i.e. to identify the systems, ia one of the major problems for control theorista.
INPUT-OUTPUT RELATIONSHIP OF MIMO LINEAR DISCRETE STOCHASTIC SYSTEM MIMO linear discrete stochast1c system is usually described by the following equations:
By the application of Kslman filter theory Astrom, Mehra, Panusks and many others solved the parameter estimation problem ysing the maximum likelihood method ( AstrOID, 1980 ) and the extended state variable method (Panuska, 1979) respectively. However, prior knowledge of system structure should be available in applying their methods. Using steady state Kalman filtering expression and the correlation function method, Tse and Weinert proposed another identification method (Tse and ~einert, 1975) which has the advantage to identify the system structure and estimate its psrameters at the same time. The shortcoming of this method is that we can only use their identification algorithm for off-line calculations. Suen and Lin (1978) proposed another off-line identification method uslng the residual error method for system structure identification and the least squares for parameter estimation of the deterministic part of systen In order to improve the above methods,this paper presents an on-line algorithm to identify structure and paramEter estimation of M~ ." .) ::'lnear discrete stochastic sy st e ms. The new features o f t h is paper are as follows: (1) Using the steady state Kalman filtering expression, the input- ou tput relationship with Luenber~er's observable ~anont cal form ~s d erive d . (2) Th e stru ~ ture identification is c arried out by usin~ the residual error method. A recursive algorithm of residualsquare-sum is al s O derived, which makes
X(~+I )=iiX ~ ~l+B u (A)+;J \ ~)} ;:\ ~)=CX i ~)+ .).; (,()-t-
v (,()
\ 2 -1)
where X(~)E Rn, u(~)E RP, Y(~)E RID, w(,()ER n and v\~)ERm, w(~) and v(~) are zero mean normal mutually independent white noise sequence with the following covarianC'ee:
Elw ( ,()wT(j)J=W~j Efv (~)VT(j)} = V ~
}
(2-2 )
j
Assup'",e t hat (A, C) is Ob servable and (A , E) is contrOllable, using stead y s tate Kalman filterin~ expression with the parameter matrices - expressed in Luen b erger's observable can c nical form to get the inn ov ation e4uation of the sy stem (2-1) as f o llows:
X(~+II~) = AX \ ~ /.,6.-1) + Bu (~H- F £. (-()} , y
(l ) =
CX( l
\ tt-I)
f- D:.Jl ttH
t.
(~)
1::>- 3 )
where i(AI~-I) is th e single-~te p optimal prediction, E. ,.,t ' = ;:, f!.. ' -CX \ ,(/1<.-I)-D u\ ,u i s t he innovation sequ ence, F is the S6in o f t~e steady-state Kalman fi~ter1n ~ . whi~e
,' _[ :, Ij/,;-l II A(.",-- --,- - --------j a~.I: a,c.;. .,.'. a.'i ,~') lIS:;
Zh~llIg ZhOllg~jllll.
I -P'Hi
Li DOlIg-tellg alld YU
(J;'d=(~+ " P"4-<-' ··' P<;.p] ; Y'i= l r;;), Yip.",. r.:;,pJ ;
a.< =( tl.."I ,"· ' et,:">,, ; ... ; 11;';',1, " e.:=[ 0 0 10" oJ ; J/M= I
' , a..ii,;li ; ''' ; Cl,.A.,I ,'
,t1.:..,l/.]
,?1Utx,! J-q
L= I, l ,",nt
i.th .I• .,td
Comparing equation (2-4) with the inputoutput relationship of deter~inistic observable canonical syste~ equation with non-dynamic part (Guidorzi,1975), it is found that equation (2-4) has only one additional term: _m.. J/. ( 2- 7) 11,(.,(+~) = ~ j: !i; ,~ 4(.,(+'1) 'C <
observability index of the syste~), y; _ (" mi ll J.{ ,;/; ~ ~ <
f
i
<1-1 mLllf~+I , Y,j i~j
Si~ilar to the ~ethod suggested by Shrinkhande, Milal and Ray (1 980), we can obtain the input-output relationship of above ayste~ as expressed by the following equation:» p ~ u. (A.tJIi.) = 4-1 ~£!. aN, V;(I,+£-/)-t F-~ ~il',,,. '4.(~,f}) ~< !=I I I .=/ , . .
+ 1.=--( ~£ Y;.1',pJ.jJ~fi) i:24 and Y.i.J ,J,
(L=I , t , .. ,1It)
, (2-4-)
I
satisfy the following
1° I~ . .. , . . . . . .. 0 0 0 - - - ---r. 1
I '
J;; - - - - 1>,.(') .'
.
....... . ....
bM(~) .. . . . . . . . . .
I'
·: ·: · b.l,id,.
(2-5)
0
- - - - - - - - = - 1-:. . . . . .. . .
I:
which is the random input to the system. It can be easily proved that 1£..:( ~ -rJ{) is a K-step autocorrelated coloured noise with zero mean. Although system (2-1) is not identifiable (Li,1983;Tse and Anton,1972) , but it can be identified through system (2-3) or (2-4) . STRUCTURAL IDENTIFICATION In atructural identification of a system, it requires to find out the observability indices of the varioue SUbsystems JI..: (i= 1, 2,' · ·,m). As equation (2-4) represents the relationship between the output with the inpute and other outputs ,eo it is the basic relationship that will be used for structural identification. We know from the definition of )/9 that Yij depends not only on Y.: but also on YJ • Thus' f we use equation (2-4) directly to identify the structure of the system, it would be the same case as confronted with Guidorzi, the value of J{: can not be determined separately with each of them to be determined successively by using previoue results (Guidorzi,1975l. In order to identify each observability index independently, we rewrite eq.(2-4) as follows j__ 1<·, 1 , f Ii: , • ".·(~tJi.)= i;i!. ~ tZ;I' f;'; ,l ~ (A."'JJ t7i. Sf .1=1 ,.. , ,,,, g~I/I'("t~--I) -t l::cz:. ""r J=O fOI
t
10
I:
with
b.,
r... r.u .. ... ')';,/100 .. 0
e/
0 . . " ,, 0
0
.. .. f.,1 " e.,o .. o 0
e, :1;0),' ,
a,
10 I .
I" '
- - '0
1;.,1/)
e",o"·
0
f.' (~+Y,,;) ....
=
(3- 1 )
={, <, " ', ?!!.-)
~ fi n:~£-e( Ii +JJ
l ".t q;O
wher
(2-6)
y.: =[ V~~.\tl)J
( 3 - 2)
; ~t =[ 1i.~h')]
1;.
t ( ~ + K)
U,(~+/), , U,( i~""')
1J.p ( ~+J) , '
U,(~ ~JI
X"cY.,:J = [ :
!{/~). ' Zl,(I.'Il; tK) ... tLpr}.")" 'lil~-Y.'tJQ Y,M,tk) .. -!tlJ,--!l't<)
,,,,{A,--J{,tJj
10
. ! .:(!;J '" r;.<~7'.;t/) .. · y", (I.)
I'
'N.iA·K·1)- "Y<. (II. :)I,,+K)" . i ..:(~+K-o"'J~(~-y'; H<)
1 :
"'-( ~ --".
I
Y,,: = X..:(J.(.:)I1.: (Y,,:) t~·
o
' J";''l.e~l i2..
Rt, i1. '-j,t!/i<-t..+J.--')'t 't,J~+Y..'J I
Now , eq.(3-1) is only related to observability indices ~ , which enable us to identify the order of each subsystem independently. Although some terms on the right hand side of eq.(3-1) may be linearly dependent (e.g. when ){j
-I ':"" I'
)/...;
( i
I·
r;v,0· .. .. ·0
,
(J
- - - - -10
r;.Yt,· ..
)='1= 0
0 10
.. .
6h
14H i
:\(illlo Line ar Disc re te Stoc ha stic S\·st CIll S
Let
(3 -3)
Z,,=X..:(t{)bl..:(.Y.:)
then eq.(3-2 ) can be ~i m plified to the following: form:
Yi. = 'Z;. +
~i
uir
T'
= IfK -+
T Ktl
PK~I = PI( -
(3 - 4)
According to the principle of the residual error method (Suen and Lin , 1978 ), the estimated value of the residual-square-sum is A T -t ) e.i.(?t.<.) =Y..: (I-X..:(ll;.)X.<. (rl. i ) y,,0-5) where X!(Il,} is the Moore-Penrose pseudoinverse of X.: (1tl) • Eq . (3- 5 ) can be further written a~
"': . [ZI(l-~(Il.)xtclU»Z~ + E.f?!CI-x,:en.i.)Xr(Il.lV71..:} e..(fI.,) = n. ... < v.. ; Efr{(I-X;. (lI,JX!C Il .i.I)'l..< J, n.~ ,;I,,; , (3-6)
¥(Kfl)
-:<.;/K))/CX:;+'PI(X/(tftJ)
Y(J<+/) AI<+/ I
fI
(+~fK,(K+I)
. . . ... (3-j)
Ifr.
where
€,;"(~.)= ~(n <)/IIY"i~;
e;(I/;.t-I)=
a,(II":tl)/II YNII~
(..:= /, Z. , ...• m) When Z.<. ien 't a linear combination of the column vectors of X.<. (fI.;') , ZICI-X,;.(/1.Jx1CI!i.»)z..: ie the residual-equare-sum given by the least squares. It is the error due to the discrepancy between the actual system and the mooel, i . e. an error resulted from the mode1,~tructure. The expression E[~ifr~~(n.JX!(lI.';)f"J is the error resulted from n,oi se. As ~4(~~YA:) is a )/,,; -step a utocorre1at~d coloured nOise, it had been proved one of the authors (Li,1983) that when ~..: is a stationary zero mean ~ -step autocorrelat,ed noise. the value of E[~.l(I-X«~;)Xl(A4»)fA} will be independent of the value of n~ for aufficiently large K. Therefore, calculating e..:,(n..:)for different ILL when K is suffi ciently large, we can determine the structural parameters }I... (.i..=I • .z., ·.. ,1fl) from eq. (3-6 ) by varying the value of e.<.(fl..A ) •
for dlfferent ?lA. (?Z": = 0 ,1,2 ,' ' ·,n;.,.,~) for sufflcient larE!e N=K. (3) From e;(Ili..+/)-~!.(~;')~E wh ere e. is a preaesigned value, which le determined by the prior estimation, calculate n!. Then v!.:=n1 . (4) Repeat the above procedure for different 1 (i= 1,2. · ·· ,m) until all the estimated values of J/,.: (i= 1,2,'" ,m) are obtained.
With the use of the recursive algorithm for calculating the Moore-Penrose pseudoinverse suggested by Sinha and Pille(1971) we can develop a recursive algorithm for calculating the residual-square-sum e.i.(It...) . Omitting the subscript i of the subsystem and replacing n~ with subscript K and letting
After all Y": (i= 1,2,··· ,m) are obtained, we proceed now to estimate the eystem parameters.
YK-t/=LYd 19tKtl)J ; XK-tl --LX~J [XT(K";J
i
When k<.k' (1. is the column number of the matrix XK): " T T..I. r ~ (I T\ eA 1(11 = eK -t 1f1(W<"'! XK+I'T'K - lfl( CKti YWJ) t VWf! -Xl(flc;';Hj
1.
Cl -&fIXJ..')K
-t If:' -t
cf>K
/(JKfl =
f'Xj(;{K+/
- ( P,;XK+' (!X" XK+o/-r G.KAJ<+I ( PK,(KfllJ/ ;(;(r( ~KXKf( &1(11 = &oK - &/(XI(+/ CB.I(XK+li'/tkl 8.KXJ(fo/
All lni tial values 4>. , VJ• • taken as zero and 610= 1 . When k '"
el(+l=
~ 11
TA.
IS VOL 2- H
= PK XKTI/(:f.KtI fk.X<:t, "f i)
with
IJ .
~1
+ ffl(ffJ(f-XJ+,Ci<.1i)
(i.=I , .t, · , :t!)
.t=1
( - CZAJ. ~ ..{ ::=
l
(4-1)
A.-I
= zii.tEJJ.:*...l + E,..:(~ 0
J
A.
(4 -<..)
.li. >;/..:
(.i~/,." .J J:t ~ y,..: (1 :::-1.2./,., ,(-') (4-3)
,
Apparently, e..:(.) is still a normal white noise sequence wlth zero mean . Eq . (4-1) can be replaced by the following vector equation:
Y/=xtcy,u,£)B..:.-t-
Et
(4-4)
where
(y..;(~+I) ... . .. Y~ (~+I<)r
U,("'+I) .. · 2(, ("'-u.· +~
.. Upctf.+,) .. ·!i.p ll, -J/it,)
u,~+K) " · IU~+i<-}I') · .. /¥(A,+i
I£/ UI.:fk-r, J
£,C",) · · " ' &('(.-Jl'+O· " [" (~) """U"-y':")J
6l.l =[a.AI'Y~I·" a,:././ .. ' a;"lIl,JI~~· · · ai.""/ ~.i.JI,.;,1
may be
r
-1U<11'XJ+, CI-Cr<+ /:t;f/ )cf>K ex",!
P. ,
: T
e..:.«(J
E, (~+I<-1} .. . £' (~~K-YJ ... [..,(~+K-<)" ·&.d+'l'..I1)
(3-7)
11 e K T If" CKt/ Xt
where
,(1+Jc-~·+Jo.!r-0 " },CN.~J(..".)·· M~K-i..+¥,.-/) .. ·!.(~+K.-r.)
g(M!) ;(7:+1
PI<+/ = PI( + IJ.I( AK+I (&.I(x~+lf( /- xlt, PKXKH)/xJ,+I&1<4c+'
2.
'" Y,:-1
r<. ~1r(~ -11<+J1.J .. . Yltf.-Y.i+j ··.,;1'/M-Xt y....Y. ··f.,rA-J!.+i) X~ (y, U,£)== . : : :
(j(KflJ-X;',I<)/XJ+/ f9.KXm
e..
1:i! a;'J,j !j(,(-Ji'~-')-ti=tF~;i)U;(I.-Jl.:+JJ tY.< >. ("'--YA: ~ .) + lii r;-. r..;,J. '?+~ r e,.: (~
y~(.,f)= i"II-/
yt =
= 61K :f.K+1/xf+,6!K XI<+I
4>K+I=
eq. (2-5) and the definition of ~i , eq. (2-4) can be rewritten to the following forms:
Fro~
.
where the pseudo-inverse of XMI is denoted by Xt:+1 which is partitioned into two parts : the submatrix e; snd its last column CH' , then we can obtain the following recursive algorithm ~r calcu~ating the residual-aquare-sum eK as follows :
- 'I(I(+OX.;+/
PARAMETER ESTIMATION
XTK-I"I-_r~ '<...)( ,rT :C I Kf/]
~KN = Y;~/ (I - XKt/ xt-l"/) YK~/
CMI
In carrying out the above on-line structurel identification to obtein all ~ , we may use different groups of observation data. For example, we may use the firet N groups of observation data to obtain )/, and then use the successive N groups to obtain Y,z. , etc.
1
' .. P.d';,1· ~~., f
Yi()!,-/)"
· ~"
.}':',(
r;,o'.;~J, "I .t;D,mJ
Using the recursive extended matrix method
I-IHH
Ihang
Zhong~illn .
Li Dong-feng and YlIan Tian-xin
the estimation value ~.:. of parameter eA.. (i=1,2,"',!:l) can be co!!!puted. But the obtained estimator ~ is unsatisfactory because it is fenerally asymptotically biased. In order to overcome this shortcoming, we propose another method named the recursive extended instrumental variable method for esti matlnE the system parameters. #(
Let Xi.
be subdi vided into blocks as
Xi( V.IU)%(1JI 4 \~.-J4"+>i/):I ··· :, 41I1~~-)li+l)i, . :I Y~{04-){ty.,,)\I
.. :y;(~r~+,)\u.~~t')\ ·
I
with every block as a K- d i!:lensional vector Fro!:l the properties of the recursive extended matrix !:lethod, we know that the reason for aSY!:lptotic bias of the estimate ls due to the fact that Ef is correlated to S01!le column vectors of X:CV.!I..E.) • This difficulty can be overcome by the following arrangement:
".1<1"1
..... 1(+/
..... K+1
YiOl;-f)." " l'iD,/ ...
Xl =
)
.... "tl
"J(tl
"'1<+1
""Ktl
AK'f'I
"'1(.,.,
..:. =la':/,~/ · ··a.L', I ··· a.ill'l.>t..· ·' 12"".1PiJI.:,,1 · ·A',rA),
i 1l,«~-J;:-r')\;'4(~~)\iuf(4-)i'+I~
I
("'I(tl
l}Ki-'
('iT
t~(~): . ! [ IKCA-ltt1)i.J .:,G", c!(():.J :E",K(~_){+I)l I
where
[
'" K+l
Y:.;(Y..:-<),'IJI. .. ·
'" K+J
Ii ....
JT
¥1(K-Y.:.~')···y/(K-Jl.:+I) · " Ylll.(I(-y'/+)l.(Jl!)·--Y.. (K-J1,+f)l.ld KtIY Itj(K-Y..'ii)··· U,(I(+/)-' U,(Jc-P;:+I) E,(K) '" £'(K->i+O ··. ["CI<) " . £"'(K-Jl..:+I) ]
a~tf:"[y,(K-?'.:+Jf./)!,(J(->i.~/-f)· · ·V,(K-J{'+I)'" ~~(K-;{+J{('''J) s..-f{K-I!.:+ Y..;v.-v-I)- ·· K;, .. (K->itl) .. . 1.. 1.1<) '" !{(K-X.·tl) " · 'fI/(K-y.i+y"n);· f",(k-j:t-I) "/(K+I)-;;., I(,(J{-Jl,::::)- " U,(K..) ' " U,(I(-Jl.'tJ) £, U() ··· t,(K-Jl~·tl) ··· £..,(K) .. -t",CK-JHJJ] A
1,.<'<.-)I.;+J0.) ""
f~(K-y":+)I,.;J.)
JIJ ~ ~
.
0<. .,,: ("-=1,.1..;" , m ) 1> y.: J. =1.<',"',;'-1 The factor >-'(I
.
A.
lA. (Kt-I)
The initial valuea are assigned as follows " ' . =0 Si
m.)
A.="Z ..
(
~ ==1. oZ.:' :.i.-I
(4-6)
J= I. ,z ,"· ';<
ij(!/,u.,£)
ls the jth row of Z{,
H~nce for ~~ Y.:
=o}
E{z2cU,U,f.) E..:./(} E { Z~T( y, u.£) xf (g, U, E.)} '\= 0
(4-7)
ThuB , under the ergodic stationary condition, we have o. I(T PAK.,oo / m. Z.. CY,u ,E.) £" K 0 }
/(1 =
(4-8 )
pJ im. z1'Cy, U,E.) x1 (r, u,f)/ K ~ 0 1< ....
K
As a consequence, we may choose Z~ (!,U,t) as an instrul:lental variable and calculate the esti!:lated para!:leter 'B.. K fro!:l the following equation:
e.. '"
/(=
[I(T
I(
P.:=iXI
.
0
where 0(, i a a very large number. For different values of i the estimator 6t (i= 1,2," ',m) of parameter B..:. can be easl1y computed and then the matrix A of system (2-3). As for the elements of 1!latricea B, D, and F, they can be then calculated from eq.(2-4) and (2-5). The following recursive formula is useful 1\1<+1 /'\ -'" AT 61. = B!'_(~K-£(K+~e(K+J)) to get the estimated value of variance matrix~ of the innovation aequence E(~) A.
COMPUTER SIMULATION
Let the system to be identifled (es shown in Fig. 5.1) is expressed by the following equationa: . 3 X(.t+I)%(-~2S ~, ~J xc~) t fo. ) -0>5 -1·3 0.8
y(~!)= [~
~ ~1
[3 x(A)
J
wftJ+ wcA..)
-tVI)U.(~J+ V(~)
where U~) is a blnary pseudo-random signal wi th ampl1 tude ! 1, W(R.) and V(.Q) are zero mean normal whlte noise sequence with
EfW(.\)Wq)J =[0.;' 0.:, ; o
0
)4; EfVC~)l(j1=[O:5•.:~
ltl4J
0.25
I(T Y. :.K Z..:.CY,U,t)
/1-1
Z~ Cr,U,f)x.. (V,U,f.)J
C,:=,-Z, " ' 7I'l)
(4-j)
which can be proved to be asymptotically unbiased. For practical calculation, £ ia usually replaced by its estil:lated value f as in ordinary extended l:latrix method. Adopting the recursive algorithm (Soderstrol:l, Ljung and GustarsBon,1978), we can derive the following recursive algorithm of the extended instru1!lental method:
WeAl V(~)
Fig. 5.1 block diagram of the system to be identified Obviously . the structure parameters ~ and ~ of this system are 1 and 2, respectively Let the noise to signal ratio be 1
Rnls (L.) =
r;r-/rJr'
=(5:.. utC.4.) /~ yf(tf.))2 ~~I 1,f=1
then fro!:l the given data, it is found
14H9
I\limo Linear Discrete Storhastir S\'SIl'IllS R1t.ls\l)= 30.24%;
R71 lS ( 2 )=38 . 85%
By computer sfmulat1on, we w1ll ~et the follow1n~ results: (1) The reB~lta of structure 1dent1f1cat1on are g1ven in Fie. 5.2.
a 7 ~t1/;).·
e{(n)
0
,
N=200 N=60 )1=2
~=f
0./
0.1 7l.
o
n ~tUt)
0
3
2
I
o.4~5T 0./~/6
o
4-
3
2
4-
et'tn) 0.711 1
0.0560 0. 0537
, CI1.
F1g. 5 .2
4 -2.
I
0
3
"~,,. e<. (n.) 0.5707 0.048J 0.0356
0.a5l6 0.0506 0.0#6
' ' ew ) 0.6z<8 0.<'339 0.061t
3
2
71-
0. ~3·1
0.01# o.ob?! 0.0661
the results of structure ident1ficat1on
(11) The results of parsmeter est1mst1on sre given 1n the follow1ng table w1th RElV denot1ng recurs1ve extended 1nstrumental varisble method and RELS denot1ng recurs1ve least squares method.
IJentifica.tiOIl Method
Nllmber of RIIAS
2000
400
2000
- o. r1f6~
-off!O
- M9 8&
-0. ?/f9
-0.1160 (0.0.2.18)
-Mf j j (0.0061)
- 0.,z03/
-0.2.53 0
-0./683
-O.;,IU,
-0.2.~it3
-0.~63
tt'Zo.I
-0.03f/1
0. 000
-0. 0 743 -o. 043~ -0. o~/6 (o.O.2'~)
~2../
0·9347
M67'(
0·1.273
O·f6?C
4 .063f
3. ~616
4.019-3
3.6333
3·7g'l-
3.1;.J..¥-b
0.74-£6
07077
Yio.I
0./4-91
'9;,.<.
0./!S7 0.11<.6
a./I .2. A
a.1( . 1 A
?';, . I
~o '" Yt'. 1 A
A
'fio.Z
a;~/ . 2.. - (.J.i 93
'"
A
9.z
REL5
( Mean 4AcI St/V1.dud DeVlid,-oll of) P4rIJ.meter £sti~ti01l. jor 4 RUAS 2000 400
400
"-
GI
R.E L V
R.ElV
o...z.I.I -O. Mb~ 0.t 08/
11:.2.2.'
~
0
o.IM (0.0301)
I
3·~f:'
4 .0173 (0.04,z0)
3.fm
3.43 00
3.681<' (0. 146/)
3 .7HT(00316)
3·7{'"
0.70 91
o. 6~7&
0.6J6& ((}.O~'5)
0-7109 (O.O]ll)
0 .7"<'0
P.~!bJ
a.ISV'
0.06~0
0.0#6 (0.05!!)
0.07T!, (O,{JI]
0.,u,1fi,
0.v 01
0.2614-
0.16i1(o.osn.)
o.-2.57f (0. 0 <.36)
O.<.ts o
0.0-274 . 0./ 71.2...
O.o.3FO
0.0013 (o.PbtJ)
0.0.431 (d.03b/l-)
0.°3 0 6'
- 1. 3.<.86
- 1. 0 111
-I..zrlfl (0. 0 31°)
-/3<.10 (0.01/- 00)
-0.6b13 0.t1 04
- 1. on7
-0.5261 -0.5'05"6 0.6tbL
0.6370
o.BIK
0.<.771
0.0.<.91
8 ·It-4- F8
7- 4674 71t l 3
1'2.0.,
K·45'°4 0.14 03
o./~63
0.0785
o.o6Jt
y<..•.z.
-O. /ZlO
-0./oS9
-0.041F
oo/q,b
ft,
0.7S'16
0.!643
0· 74M
afb4f
,A.
[2
-/ -0.2.)
0·9146 (0.o.2j9)
~.z..I "
Q 'f2.1 tu
(0. 01'7)
0.000/ (O.OIOS)
t,o.,
"" 87<.
(0.01 14-)
True VaJ,tJ.e
-o.Otf/
(O.o~/3)
0)
-0. 6fIJS (0.0<'#) -o.Molf (O.OlOb )
o. 7f<. 6( o.oq,oj)
0.1'/3/ (O.ollr)
0. J 10 7 (o..o,S'6)
0.<.?-2.6 (0. OSS3)
8. jllS¥
~.4s<, (0./lf6t)
(0,/0/0)
0./ 0-<'-3 (0.0484) 0.14<,0 (0.041r-7) -0.11;.5'7 (o..oS87) -0.ofI1 (o.o-4Y1) o.tl,fo ( 0.04//)
o./N.3 (o..oZTo)
4-
o.orf/l-I
- /· 3 -0.6.S" 0./
0·3 t.~
0,/-<-0<--0 . /39-j
o.!'/JlI
-MAN -0.41.z.8 -a.fI,041 -0.4150 - 0. ~
- a.tlPT
-0. fl, 081f -0.41<.3
-0.4/ 0
-0. 4 04-1 -0.4ISo - 0..4,,0 (0.04 3<.)
/.ofI;.3
/.f 0<.9
I./ftz
/.U' 02
0. 0836
0. 0 /54
/.1515
f.7Nl>
-0 .40/0 (o.olf!)
!.IMf, (0.<527) 1.1060 ( 0.0<'78) o.021S
7
1/<'/3
0.0/" /
In the above table .r2=t.II~-BJ2ind1cates the arcur8cy of para~eter estimation.
Zhan g-
149()
Z h()ng-~jlln ,
Li Dong-fe ng and Yu a n Tia n-xin
CONCLUSION This paper presents an on-line identification algorith~ for MIMO stochastic linear discrete sys t em based on the steady-state Kalman filter representation with Luenberger's observable canonical for~. Recursive algorithms are developed both for on-line structural identification, and on-line unbiased parameter estimation. Even under the condition that the noiae to Signal ratio is greater than 30% and the data length is relatively short (N 60), satisfactory results can still be obtained These prop osed metho d s can also be used without any modification for problems when system noise and observation noise are correlated. REFERENCES Astrom, K. J. (1980) , Automatica, Vol. 16, pp551-574. Panuska, V. (1979), Proceedings of the 18th IEEE Conf. on Decision & Control Including the Symposium on Adaptive Processes , Vol.2, pp927-932. Tse, E. and Weinert, H. L. (1975), IEEE Trans. Autom. Contr. Vol.AC-20, pp603-613. Suen, L. C. and Lin, R. (1978), IEEE Trans Autom. Contr. Vol.AC-23, pp 45R-464. Shrinkhsnde, V. L., Milal, D. P. and Ray, L. M. (1975), IEEE Trans. Autom. Contr. Vol.AC-25, pp 307-312. Guidozi , P. (1975), Automatica, Vol.ll, pp361-374. Li Dongfong, (1983), Ms Thesis, Shanghai Jiao Uni vers i ty. Tse, E. and Anton, J. J. (1972), IEEE Trsns. Autom. Contr. Vol.AC-17,pp637646 . Tse, E. and ~einert, H. L. (1973), IEEE Trans. Autom. Contr. Vol.AC-18,pp 687688. Sinha, N. K. and Pille, 'wo (1971), Proc. IEEE Vol.118 , No.8, ppl044-1046. Soderstrom, T . Ljung, L. and Gustarsson, I (1978), Automatica, Vol.14 , pp231-244.