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A New Approach to the Analysis of On-Line Control
Cop yrig h t © IFAC Ide ntificat ioll a nd Syst e m Pa ra meter Estima tio n I!JH5. York. U K. I "H:,
A NEW APPROACH TO THE ANALYSIS OF ON-LINE CONTROL Jiang Ji-chen Heilongjiang Institute of APIJliNI Mathflllatirs. Harbin , The People's Re/JUblir of China
Abstract. Tbia pap.r .qui vallllc. of control and convergeac. of control are d.fined. We first introduc. n .... notions of J-equivaleace and £-J equivalence wh.re J is a p.rformance index. Then ... e givan a def inition of a n.w converg.nce bas.d the above notions. According to th.s. equivalence it is ea.ier to find a control ...hich mak~. ~h ••rror bet .... en the performance ind.x and optimal R8rformanc. index d.crea.. aB tim. iaoreases. In addition. it i, conveniant to carry out theor.tical analy8is to 80•• of Itftevn sy.tems. Besid.s a •• riea of re.ults are giv.n the prop.rti •• of J-equival.nc •• On. of ...hich is that the control of ••If tuning r.gulator will be convsrgent to lIinillUB varianc. control in terms of J-equivalenc. only if clos.d loop output of the sy.tem is bounded uniformaly. the e.timation of paramet.rs converget in quadric m.an. Key...ords. Adaptiv. control; atochastic syst.m analysis ; conv.rg.nc. in equivalenc., self-tuning r.gulator. INTRODUCTION With the continou. d.velopment and broad wid. spread application of comput.r control, onlin. control i. b.co.ing mora and mora important. S.lf-tuning regulators are on. of such on-line control m.thods. Lots of work has been made on theoretical analysi. of one of important characteristics of self-tuning regulators __ approximation (Astrom,1973; Cains, 19801 LJung, 1980), though it i. "very difficult" (AstrOm, 1977). In the above mentioned pap.rs anal.i. vaB don. by taking a metric of control spac. aa an approach to describe a neighborhood of control value thus good characteristics that und.r o.rtain conditions on-line control gradually approach.s .om. opti.al control a. tim. go.s on vera given • In this pap.r .... vant to define a n.v "neighborhood" relation ba.ed on control ind.x, that i. equivalenc. to J and £ - equi valanc. to J • Using thi. d.finition, differeDoe b.t....en onl in. control and opti.al control indic. can be directly analys.dl if both indio. approach a. tim ••lapses, the r.liability of u.ing thi' online control ...il1 not be qu •• tioned further more , in 80me cAs.s, it can be si.pler and ea.i.r to analyse probl .. s vith this method .
I J (Ylt )) - J(1~'IIl ~ E vaera l It' 1. an output e»rr.1Ipondent · to the mo.ent. H.re J(-) is an ind.x function; if u,ltl,u,lt l ~ U{l f-J) tben Il,lt) IIIld lI,lt) are call.d £-J equivalent control.
J-equivalent and €.-J equivalent. control have Uta following prop.rti •• : 1· set l Y" I: J(yl.)crl t lt1)} ha. a one-t",-one correapoad.n" ...ith set U: (I) 2• •
y
Ut l])C
= U~lr)
Utylt- ]) and
for £"'>0, £.\o.we have
.
-
y
('\UtlE",-J') ""
3 J( . ) 11 a funotion ...i th a UIlique m1l:li.\I. (maximum) . If J( Y~tl) -m1n(8&x)JIY!'I) , th.. for &nl ~'tl ~
be
l'
Ut IJ I
.qual to
'1'1')
its corr••poadent output ~\t)
vill
•
,
y'
In fact, suppose that th.re is a LIItl~ UtI]) , iu output is ;y'(t), and that y\t)* Y'lt) , th.. Jiyltl):> « ) JIY·'t l)
thus w. get a contradi tio ••
,
4· Suppos.(s) is described by the folloving BOd.l y(t)
= G- ( 1It).
(, )
Ultl) -t E(t)
G( · ,·) is a continuos function. EQUIVALEI'ICE OF CONTROL AND ITS SIMPLE PROPERTIES Definition 11
In a .ystem(s), a subeet
t UtiJ) of
ad!lil8ibla control sat Uj, is call.d J-set of equivalence to J control of the output y( t) correspoadent to the t, i t for tfttl' U:H)v. have J(~) =1( y"lt)) ...h.re Y~t) 11 an output corre.pondent to u·tt) If OI,t~),~,It) E U;
disturbance ind.pendent of moment: wh.r. Then
to the .o_.t t, if for any
lltl t
U;<£-J)v. have
E
E'ltl
=
(1'
•
constants. L.t J(y(t»-Ey~(~) U~( ~l are clo •• d •• ts.
Proof: L.t u, E Ut' I E-I) and V. ~,. u• w. denote the outputs of lIk ail! ..... by ~\tl and y~~) reep.ctively, k-1,2,... y For any giv.n £ > 0 .... show u'*f V, ( E- I) By the continuity of G(' ,.) , vhen n i. large enough, v. have 1 ~·()tltl}~ ,lItt..,)J) -1f'()flt), ( U,~ for Vly q '70 ...here ~
Definition 21
.t adaissi bIe control •• t U..
Eqt)-O.
~~ is stocha.tio l/ltl , l.flt ) vith 2-th
v. have
U(t-I))) I <
~"
A
(10( , lilt-,» = I " . 'I.\t-') , .. . , Ul O)
IJ ly"'tt-I ) - J (~l~)H
=
=1 E~\>1l'i: )'( U': (1(t - I)) -t(l"!.. EG-(}y(-t" ( I~.' l.f{t-')
-0"'1
364
Jiang Ji-chen
I J I'1(t) - J(,,{.(t)) I ~ £ 1~'(f(t~(u~ U"-')) - ~·(yt",(ut..ult-'))I
Hence ~
EI?I='!, . . ,
I
1 J('i~(ti)
- J[7',I')1 ~
I J(y~(t» -JIY.lt') I+ IJ(Y.lt) -J(Yl')!I< [ t 'l,..
For 'Lv is arbitrarily choeen, we obtain 1 J lyit.C)1 - Jl'Ylt)) 1 '" t.Therefore U{IE-JI is a closed eet. By property we obtain U~lJI is closed.
Ii
1"
5' Suppose(s) is a linear system define. by the following: Ylt) = 0. , 1[+.-') """ + ()..","!tt-mj-t + b,Wf.- I) + .+ b"u. lt- n )+ elt) (2) where 'fIt) , Ult) , (- 00 4 t ". co ) are one-dimensional output and control respectiYely. e(t) is a stochastic disturbance and E e(t)- 0 , If in the meantime J(.) is a nonnegative convex function, then Urc<-J!. U(l J) are all conyex set. Here .,·l~) : Jl:YH-IJlJ.~",J(y(t», and 1l-tl is all the outputs system(2) at t. In fact, let LL.lt). Lj,lt) ~ U{,ll -J) denote their correspondent outputs by 'j,t+.>, "j,l~). By the definition, IJ(Y,(+.» -Jlrlt » I ~ ~,
Let
.\
~ lO' I ) ;
U!'(tl
or
o.,'Ylt-tl
t
b , u.~ (t-')
"fl"e'::
'I-
et.. YCt- ~J + + b"l.l ('\.-h.) + elt)
0.,:/(+.-1) .. ... ...
+
,,(Q,Ylt-I)'" Q,:ytt-l)+ ..
+ b, U.,I-( -I) t
....
+ b. U Ct -
( I-Aj(Cl''Ylt- l) ...
..
Q",:Y+
It.)
-+ e [*)) t
0,,)'1+.-11 ....
+o.... )'et -"')+
b, Udt -I) + .. +- b,(.LCt-n.)f- elt» Y'lt) ~X ,-,,,It) + 1I-),') ,},.ct) . t
that is
Note that "j~lt.>
I:: YCt).
so JC'j* lt») ~ Jl'Y'(+.»)
Hence
: (J lAY,lt) + ll -A! )"I")1
- Jly'Ct)!]
~
[>..J l}lt») ...
~
A I J ()" It-)) - ]l)" (f.») I +
*
A~ T (1-
(I-A)](),lt!) -
An
(1-
JCY-lt>!J A,) \ Je Y,ctl!
- J O'l+.» I
=&
We thus know that Url L-J) ia a convex aet. Again by property'" we know U~ lj) 11 alac a convex set. CONVERGaWE IN EQUIVALENCE TO J UlIing the concepts of J -equi valence and t -J equivalence, we CID define convergence in equivalence to J. Suppose there ill a dynamic aystem with stochastic disturbance , we consider two different control strategies C and E. Let their controls at time t be U.'lt) , U' lt-) respectively, and their correspondent outputs be "j'l". "j
1:.;00>
I).e(t.).
and say that control strategy E is a control strategy equivalent to J.
0
RellJ1'k: Generally speaking, we cannot tell which of convergence in equivalence to J and common convergences ia weaker. E.g. : in Fig. , is the relation between the output and the control ot the system at an instant to . Suppose that the outputs of uOft · u•. u. &re 'Yopt . 'Y" '1. respectively. Eaay to see that we atill have IU,lt.J- Uop' (i..o) 1 < IU,lt.) - Lloptlt.)I
though I Jl'}',lto)) - Jl)'opt 1"'» I" I J(y,c~J) -J(y'p'lt.J) I where
&
, we have
by
=
Lt'(O) - Uoptl~) 1I ~
wbere U,pt (i.) denote the optimal control under some definition of the eystem at an instant k; 11 ' 11 is a norm dependJmg on conditions. If we find suoh a control that the difference of index funCtioDl IJl~(i.» -J("}01't1/U I tends to decrease as time goea on, we will reach the destination and aatiyify the demand of the control. That is why we introduce difinitions 3.
= 1\'\""
Jl),t"»
1),lt)1 .
CONVERGENCE IN EQUIVALENCE TO J OF A SELF-TUNING REGULATOP
= A U,lt)
and the output of y\~'
IJey,lt))- J(Y'lt') I ~ + li- A) IL.lt) ,
One of the most important problems to use on-line control is ensure tbat the control approximates (converges) to acme optillal solution. With practical consideration , we need not seek for a control u'cl>.) such tbat
AstrOIl and Wittenmark (1973) diaculaed the following linear .tocha.tic lIystem with unknown parameters ' (3)
where
At 1)"
)~+
a.,r-' ... ··· ..
a(p =b,r" .. c, (J) ;: r" +-
b,t -~
Q..
... + b..
b, 1: 0
C. ]'-' + '.' +- C ..
e(t), too O. ~ I. t 2 -.. are a sequence of Gau8B random variables 11(0 ,er), and independent of '/Ict), \:l.It!. In Eq. (3) , q is a rer.ard operator: q(y(t»-y(t+1). Polynomials B(z), C(z) bave all their zeroes inside the unit circle. In AlltrOm (1973) an arithmetic to the self-tuning reiUlator of syst .. (3) - - a method of on-line control havin, be.n widely used for recent years, WaS obtained by combinin, lesst square estimation and minimum yariance strategy. Under a.~mption that parameters are convergent and that the output of closed loop sYlltem has 2th order ergodic, etc, they proyed that the abov ...entioned regulator conyerges to a minimum variance controller. We ebtain the following result. by analysing ~e self-tuning regulator with convergence in equYilance to J. Theorem I Consider .ingle-input and s~ggle-output linear system (3) with unknown time-invariant parameter'l i f (i) 2th lIIement of the output y(t) (too 0, 'it. f 2... -) of cloaed loop system exists, and ia uniformly bounded; (ii) the estimation Ba) of parameters converges to their true yalue S" in quadric mean; the for the control U(R) of the mini ,num variRJlce control of system (3) ,t the instant k ,we have Uo(A.) ~1.I~) where lI\~)is the minimum yariance control of \3) • First we prove a lelllla For conYeniece, we giye some notatioDs fA. ~ (y(t). y(t-l) .. . j !t-nv+ l ) tlR'" A
( LL .lW.-I). · .. UC4,-I\..TIJ) u.~ ~ .,:t'( t~", t ~~. U~~). B(~)) = ~ .:(ct;,(U.O:),e(~)
(tL.U;~) ~
365
Analysis of On-line Control
tAt· ;t'(f..... l<4. u~~j. Bot) ~ ~ .:z:'(~1. Cl).. D;~). 8~)
;tcffo-1.
((.t .
$.(k)U.
T
O;~J. § (R.»)~ (-~,((,;Y(fu- . , -;;{",IR)9'rk.-
p.(~ifu U(£.-I) ....
T
.' .
tr
t-1-(A.J~(Vii(R.-,O)
;t.(f~-I.(u.Oi~).8?t)~ (-01.,9((.. >- .. · -~m"j(/I.,-rn--l) + p. ().. + ~,~, u.( k - () + ... ;. f.f,l V- (6. -J..) ) Leama I Suppose there ex1ats an admissible control subset U" c. U,,- . ltI-. u~ ~U". k-1 ,2, ..... ve denote the output set ~f \tby 'I' . k-1,2, ... if for any 1).. D;:: ~U~ and t '" ~ ~ are unifonaly held
r
..,'" , lu · " 16.L.Jl~ U;"').8('-»)
""(1'1,~.U.·Ut~).") ( -'-0 ll ... IA--+0 .... [,L
-
then
_
I Et.W,... (li.•. O;~).O(~)- E..:t'(Y~.(I.t,,-,D~~)1 A-", 0 Proofl note that u... ,u.~fU'''c.U .. and .:t('f~.l· . D;:).6(,,) ia a convex function, and
ia its unique minimua.
B7 t'undmental Leam. ot optimal control (Aatroa, 1970) ve have
E..t.'<~~.
= 11\ ..... E.t'(f,.... (<(·U;.::).B(k'j = 1n~""Et.t~.<.(u. U:~)B(I.') ~( ~~,. 0:~. ~(,,) t U.. U' rl><. u;; }(~»)W·
)
we change (3') into ~ It
= - o(,'~<-t) -
1'4. t I)
' .. -
01", ;f , t"""t 1 ) +
+[3.(lI.l't)+.· . + (JJ.U l -t-,L)) +/:. ( t+i+l)
(4) The coefficients 01,( and (3.: of the .oove equality vere obtained by estimating (1, ... b .. , disturbance t (t) is k-order moving average sum driven by noise e(t). Hence t.lt -rA-tl) ~s independe~t of ~ C1"".(u.U;:;o'>. EUt+p-+I)=o and ge.. (t+-~-tl)= ( j In Astrom and Wittenmark (1973), the paraneters 8·~ (- "" .. ' -01•. ra.. N~) of model (4) vu eetimated recursively vith least equare and then combined with minimum variance control, an arithmetio Qt self-tuning regulators wa~ obtained. In order of to prove in the above arithlletic
U.. ~u: J vhere u.;, is the minimum variance control at instant k, we first prove that E u.;. . E <4" are uniformly bounded since a certain instant. In fact, by (3) \Ht-Iu A(~)/(3(~) 'Y lt ) -+ A.C(~)/B«(P e(t)
=
U,(t):::
~·AC'P/I3«P y(t) -r Aq.~C(~)/13(~) €lt) + (C, [~) + C.(~)/B(~»)e(t)
= ( AI(P ~ A.(PlB(~))'y(t) siailarly
E ;t'(y,~, (14. D~:),II")
-=
=~
~.t' ~
1.,.
(U.
D;.~,. 8")
E.t·(f",.w. D~:) . 8~)-= In<-'" Et'
m1.n.
1).'1".'0;; • If')t U"
,,'fo..0:::1. O~)' u. m.l.n..
J1Cx.) ~ ~
tl{.:Ko )
x't,)tX
+ l'I'\l)..."X (tl(X) - )10(.)
>:(1,
.l, ' ~ ~
vbere fi are functions defined on
X. x(fi) ia a
value of x dependent:.. on f i • i=, . z . For V u. , D~= ~ U*, y... q~ are uniformly held
I E.t'(Y,(~.(u. O~~), 8(~) hence mAl)'.
(i).... AI [P = 0.: ~ '+
G(p = c:"
By e18llentary inequili ty : x'~,)tx
where A.O). C'(P are polynomials with degreu less than that of B{z) and irreducible with B(a).
- E.t''
t,
CL)
-k
(4t.,)
_I
J . .. ... + a. + C' J 1..•. )+ ".. + C(41., 0.-
By theory of z-transform, decomposing B(z), we obtain S(J)::(l-);)(J-r.) .. ·( }-Y.",) , IY; I
=-
I E.trf",t~ .D;::).9(I.,J) - El'
lI.(f.,., fm, 914i, ()') ~U~ Therefore
"w.
E;t'c1t-<, lu,D~~J. 8(l) ~ ~ C((Y."IU.U;:J./J'I)+
v.t~... U:::.8('J)~U"
+
L(llt'.U':,O'HU·
[E.L·(f... cl).,U~J.8·(,,)) -E.1\r:(U,lt',IJ~)) U, ft-<. u:~ ,8('),0' ) ~ u~ m<>-'X
V E > o.
3
N.
tor (,. > rJ
•
~ E..t,
Siailarly ve can prove
3N. tor ft,'" E.t( fl",(I). , U~~), 8')
_
\fE.>o.
or
l!'0
lI.\r., . u:~./I~)t:
!
uqtdr.~./J('»)tU·
U"
lE .t'( ~~. (14 . O~~). 01\)
-
~
~ Et(i'l"lU.D~).e(kJ+t
E.t'(fA~' (it •. D:~),e(6.J)I~ 0
Theorem proof : The model of system (3) is the folloving : )'(~) T Q.1~tt -I ) + .. . +- CL. )"'--IL)
+ ... +b .. lA..lt-&.-"->t \
+-
= b, «CH"+I
(el-t.,+ C, elt-I) +
c~el'--IL)J
UBing identity
~'C(P = Ai~) F(~) + ~rp
(Flp=J"+j,)t' .... ·-+f·'
)t
...
Similarly
00
,>..~·C(P/B(~)
-\" };,
~
_' -I
)
q~, S,Y,
>F
Without losl of generality, we assume the degresl "/",: -h. ~ -Iv hence U,'tl= (A,(~) + i. (i~.rjrj -I)Jy(t) +
ot polynomial. +
(C, l~'" ,=0 ~ (r ~:~j"r~~~j-'J) e lt, 1.:"
:.: ct')'(ttk.)" C·'e. lt t le)
i- , " t-
o:,·''jl~). c."'elt)'t
+ j::o f.. f' {~;rlyl.. -j-, ) "s4Ylelt-j- ' j) ."SI
+
= c,q)
366
Jiang Ji-rh e ll
c ~ m().'Y. {I C"~'I) I"'" _ V/-t ) ~ (n.-IJ(S+})yi ( I~l~- ) - ')I+ lelt-j-!) I
Vt
....
-
.
.
lE .:t'( for all t
Q:I -
~ ;~,(ll...t.,C)(\~lttt >l + le..ltH)1)
(.)0,
+ fo y/tl
~ Vt,
! I.I. l '-) I
Since
l v-· (ont ))
<.
0-
Tt
V- ,
m. .. '" ~
1
IJ
EV: _
E u.~t. ) ~ t
,
~
2
V- lm )
~
t
>z.1
-V'·"" t
<. +
'
.)-1
~I
+
""
+ In.- ')(5t~)-;-=r-
hence ..SU:f E tt(t) ~ C.. ~ In the following, we assulle Assulle
V:
I
J
" :0
!
)
C~
a.1~\tt">I-t-CIe.l-ttk)1 +
to
i. (Q"C)C\'9'(tt~)1 +\e.(t+.i..)\) + J;.Yjlt )
is the output correspondent to II1nill118 variance control u.'lt) at instant t. Vt
I
Then
= v't.) - ~ Ca
0
...
j¥."'.'
l.I.t
u.. u* :
EII8(tJII'
A <+ 00
<.
~ Ll'(+.) , :;
,j",
(~-r,)
~
1-, .fo.~ -'- q-1~)
_
IT.I
91!~
ll)
It}
= --"-'- ... ~t- - .. +--.1L..+ (1- Tt)"
( J-Y,)'"
I(~"P
Uo. , tu..' {, C: J
ll-T1 )
9i':~.
( Ill '
+ --, -t- ~+~+ -- . +~
(3 -1~ )· ·
'('t31
q-r~ )
For ever), pod tive integer
f3.(6Jh(~)ii. ("-,{)J -l-",,9l~'- '
. - olrnY(6.-m-')t
+poLl. -- - +F'Pt'jj.(6.-,t.»)J' , E { loI.,-~,(~)\ \~(A.)h' - -+ \~"'-~'" (~)\l9(R.-m.-1)\+ 1~(6.) -~.I\J.L.,\ t - - - 'r\~(')~lP-J
(Q.S)
'
A,()! / B(J)
(r
t
t;", u'lt)
We can decollpose rational function
Ef t t1 , ( 1.1., Ut;), 8(~» - ~(ri."lU , U;;I), e~) y = E- { (-Q'(~J'!i(6.J - - - - ,Q~ (~)'9(~-m-l) +p,(~)u.+ T - - - -+
SUf
IJ~)=q-r,)
then i).'lt) , u. lt ) e Ul', For an)'
iA Lt)
(11) ~([i.q~,~)\+ I(.«-t,)\»)/m) ~ too
n(
Taking into account of
we can simplarl)' prove
V" ~ {u..:
i -e
Rellark : If B(z) haS multiple root.,
V)-(t)
~~'l-t) H~(t.)
we have:
u\t) _
J
Ul\:)
00_
~ V(~"
u.'(t) \
(..
E.-
!;
f.-+ I),
<2:
In fact, by (ii) and that e(t) i, independent of '/I(t , , lilt) and E elt, ~ 0 , E e'lt ) '" 0-' , we know the estimated par8lleters converge (a.s ) (Ljung, 1970). And b)' (iii) we know they converge in quadric lIean. Hence by (i) and Theorem we get
.. _
"j',t)
otm>
I
,
;(
Corollar), 2: For system (3), if (i) t:'I{'(+') <. A
h
CN>A
-t ",i l
- J l:J°(H~tI)
.. ; ~ , u'CH
~
l)..(t)
(CN ::: COrt5t_
00
for the 88,ae 7N,
t
Corollar)' 1: If the output of system (3) is a stationary proceaa and the e.tillated par8lleters converge a in quadric lIean, then we have
(11i)
eN
l9l t t P- t')
U,l-H k.t')
Ai+ ll\.-11LS.~)~T}Ai\. 1
1 2 A·{CCLtC>-+t.
~
{(CtH).h,
{N"N.1, when
3max
or
B)' Fatou lemma EV/ ~ SI/.f {EV~~)J • For ever), t and 11, note that EI!(~H·A , f.Y(· )<. A ( A is a constant ) and use Minkowski in equation for sever~ tilDes we have:... _ _ (E.lV:;')] ~ = rE {(Qtc) (~ qYltH)I-+ Ieltt.\.)\>J -t
+ l:"' -Vjlt)}1.!. ]1
? N, , but l).lt) Eo If' _ there exists an N, when
ut:: ),CI") I < I 1iI-t
Set _
l1.l'lt),
.." 'c::-"Yt-" (LI(t), - -UI'I.·' /" - U- t·,) , G)\ I E..L ,-,J,S(t») -E 11(1"r:-.." , (.«-1.),
V';' ': V.. - f;. . v) l~} Eridentl)',
f•., ,(Q(t), o~:: ) ,6(tJ) - E;t'( ~t."
~
O-<-rd = + ("t:-t-k+I)!
By that we can eaail)' know that the following hold. uniforlD1), for any lJ. E: U" :
IE.t·(r,~ , (u.O~:,'). g((._)) - Et'(fl"(U. , U;~) , 8*;1~O The index function of lIinimum variance control is Jl'~ltl) = E)'\t) , In order to show ill"l t~\-\) -+~. Lilt) + --- -+r,pL Ult -l)'-E(_ol,'Y lb - - -ol",9(t-m-,) tp.lA'lt) + ... tMLLlI...t.-.t;l j
lE(-
=I E[C1.;, l~I'), O~~ ), 0-) - £.i'cft-"lU'lt),D~~), i9
I ~ \ I:1'C~" ( U.l~), D~::) , (;l~) - Et '- f.. ,CUlt), O~::l , §l-t»I t
...
~ lE :z"(Tt-' , (u.(t.),O~~), 8(t.')- E..iCT" ,(U<'{), O:_~),
*)
e-) \
Sinoe U.lt ) t- U~ U.·lt)t U~appl)'ing the above lemma, for an)' 00, 3 fI, , we have:
"t.- :
.
I ~
1:- ~ 1-1.,;.
l1~T;rt '" ~~'r\I-tH'r't -- -+ r! r' J!tC-1)! IU) + '" )
we evolve every item of ,iI) as above in t" a infiM te seri .. sillilar to the case without multiple roots, we can obtain operator solution of difference equation of L4~ ) we know that E ii'('),EU.'~") are bounded uniforal), b)' the salle analysis as before.
-M.l.llu..(k-,(.)I]"
.::: (Cm+ 1) + Cm+- J.) Cm • .L-')/ 2'1 -C..' ·E 1 8(~J -8' 1I~ 0
""r~
q- 1",)
-I
NUMERICAL EXAMPLE The following exallple chosen froll Astrom (1967) is the situation when .elf-tuning regulators is applied to quantitative production of paper. ~(to)::: \,2i; 'Y(t-') -0·49s ~l'-') t Z-3 0 11LIt-l)+ 2 -02S I/.,t· 4)+Vtt) 'V,-\:)
=
O.J8 2 ( 1-1.4 38f'1-o-ssf')
e(")
(!-I - ~8Jr +o'4-95f') (!1-f) 1\
Setting /1. '" ,-5' we get the following graph Fig.2 (paraMeter value is 70g/m 2 ) of outputs of minimu. variance control and self-tuning regulator when the)' are in operation. In the graph bold line denote the lIinimuM varitnce and thin line denote the self-tuning regulator. It is eas)' to know the avera~e bias of the output. of the two abavelIentioned controls aotuall)' tend. to decrease ~aG uall)' a. tille inc'r ealle CONCLUSIONS In this paper oonvergence of on-line control is discune.d. elluiyalence to J. equivalence to £'-J
Analysis of On-line Contro l
ana
convergence in equivalence to J of control are given from a practical backgroud and vithout 10e8 of rigor their simple properties are discussed. Through discussion about self-tuning regulators, ve obtain oonvergence in equivalence to J of selftuning regulators under such coaparatively veaker conditione as uniformly bounded outputs of cloaed loop system and convergence in quadric mean of the estimated parameters. REFERENCES Astrom, K. J., B. Wittenmark (1973). On self-tuning regulators. Automatic, ~, 185-199. Astrom, K. J. (1970). Introduction to Stochastic Control Theory. Academic PreD', New York. PI'. 259-262. "tram, K. J. (1967). Computer control ot a paper machine. IBM. J . Res. Development, 11, 381-405. Aetrom, K. J., U. Borisson, L. LJung and B. Wittemmark (1977). Theory and applications fo selftuning regulators_ Automatica, ll, 457-476. Caines, P. E., D. Dorer (1980). Adaptive control of systems subject to a class of random parameter variatimee and disturbance. In M. Hazevinkel, J. C. Willeme (Ed.). Stochastic SYstems. D. Reidel Pubishing Company, London, pp. 421-433. LJung, L. (1970). Consistency of L.S. identifioation method. IEEE Trans. Autom. Control, 21, 5-6. Ljung, L. (1980). The ODE approach to analysis of adaptive control systems. Joint Automatic Control Conference. San Francisco, Augu.t, 1980. Huth, E. J. (1977). Application ot the z-transfora method. Transform Method vith Applications to Engineering and Operations Research. PrenticsHall , Inc.PP.90-94.
0 Ti "lC
Fig. 2
~o
l
'"
"'in]
lA'",, ) ,
t.hi_ line
Cl et )
tIli.e 11ne
,
90
%7
A NEW APPROACH TO THE ANALYSIS OF ON-LINE CONTROL Jiang Ji-chen Heilongjiang Institute of APIJliNI Mathflllatirs. Harbin , The People's Re/JUblir of China
Abstract. Tbia pap.r .qui vallllc. of control and convergeac. of control are d.fined. We first introduc. n .... notions of J-equivaleace and £-J equivalence wh.re J is a p.rformance index. Then ... e givan a def inition of a n.w converg.nce bas.d the above notions. According to th.s. equivalence it is ea.ier to find a control ...hich mak~. ~h ••rror bet .... en the performance ind.x and optimal R8rformanc. index d.crea.. aB tim. iaoreases. In addition. it i, conveniant to carry out theor.tical analy8is to 80•• of Itftevn sy.tems. Besid.s a •• riea of re.ults are giv.n the prop.rti •• of J-equival.nc •• On. of ...hich is that the control of ••If tuning r.gulator will be convsrgent to lIinillUB varianc. control in terms of J-equivalenc. only if clos.d loop output of the sy.tem is bounded uniformaly. the e.timation of paramet.rs converget in quadric m.an. Key...ords. Adaptiv. control; atochastic syst.m analysis ; conv.rg.nc. in equivalenc., self-tuning r.gulator. INTRODUCTION With the continou. d.velopment and broad wid. spread application of comput.r control, onlin. control i. b.co.ing mora and mora important. S.lf-tuning regulators are on. of such on-line control m.thods. Lots of work has been made on theoretical analysi. of one of important characteristics of self-tuning regulators __ approximation (Astrom,1973; Cains, 19801 LJung, 1980), though it i. "very difficult" (AstrOm, 1977). In the above mentioned pap.rs anal.i. vaB don. by taking a metric of control spac. aa an approach to describe a neighborhood of control value thus good characteristics that und.r o.rtain conditions on-line control gradually approach.s .om. opti.al control a. tim. go.s on vera given • In this pap.r .... vant to define a n.v "neighborhood" relation ba.ed on control ind.x, that i. equivalenc. to J and £ - equi valanc. to J • Using thi. d.finition, differeDoe b.t....en onl in. control and opti.al control indic. can be directly analys.dl if both indio. approach a. tim ••lapses, the r.liability of u.ing thi' online control ...il1 not be qu •• tioned further more , in 80me cAs.s, it can be si.pler and ea.i.r to analyse probl .. s vith this method .
I J (Ylt )) - J(1~'IIl ~ E vaera l It' 1. an output e»rr.1Ipondent · to the mo.ent. H.re J(-) is an ind.x function; if u,ltl,u,lt l ~ U{l f-J) tben Il,lt) IIIld lI,lt) are call.d £-J equivalent control.
J-equivalent and €.-J equivalent. control have Uta following prop.rti •• : 1· set l Y" I: J(yl.)crl t lt1)} ha. a one-t",-one correapoad.n" ...ith set U: (I) 2• •
y
Ut l])C
= U~lr)
Utylt- ]) and
for £"'>0, £.\o.we have
.
-
y
('\UtlE",-J') ""
3 J( . ) 11 a funotion ...i th a UIlique m1l:li.\I. (maximum) . If J( Y~tl) -m1n(8&x)JIY!'I) , th.. for &nl ~'tl ~
be
l'
Ut IJ I
.qual to
'1'1')
its corr••poadent output ~\t)
vill
•
,
y'
In fact, suppose that th.re is a LIItl~ UtI]) , iu output is ;y'(t), and that y\t)* Y'lt) , th.. Jiyltl):> « ) JIY·'t l)
thus w. get a contradi tio ••
,
4· Suppos.(s) is described by the folloving BOd.l y(t)
= G- ( 1It).
(, )
Ultl) -t E(t)
G( · ,·) is a continuos function. EQUIVALEI'ICE OF CONTROL AND ITS SIMPLE PROPERTIES Definition 11
In a .ystem(s), a subeet
t UtiJ) of
ad!lil8ibla control sat Uj, is call.d J-set of equivalence to J control of the output y( t) correspoadent to the t, i t for tfttl' U:H)v. have J(~) =1( y"lt)) ...h.re Y~t) 11 an output corre.pondent to u·tt) If OI,t~),~,It) E U;
disturbance ind.pendent of moment: wh.r. Then
to the .o_.t t, if for any
lltl t
U;<£-J)v. have
E
E'ltl
=
(1'
•
constants. L.t J(y(t»-Ey~(~) U~( ~l are clo •• d •• ts.
Proof: L.t u, E Ut' I E-I) and V. ~,. u• w. denote the outputs of lIk ail! ..... by ~\tl and y~~) reep.ctively, k-1,2,... y For any giv.n £ > 0 .... show u'*f V, ( E- I) By the continuity of G(' ,.) , vhen n i. large enough, v. have 1 ~·()tltl}~ ,lItt..,)J) -1f'()flt), ( U,~ for Vly q '70 ...here ~
Definition 21
.t adaissi bIe control •• t U..
Eqt)-O.
~~ is stocha.tio l/ltl , l.flt ) vith 2-th
v. have
U(t-I))) I <
~"
A
(10( , lilt-,» = I " . 'I.\t-') , .. . , Ul O)
IJ ly"'tt-I ) - J (~l~)H
=
=1 E~\>1l'i: )'( U': (1(t - I)) -t(l"!.. EG-(}y(-t" ( I~.' l.f{t-')
-0"'1
364
Jiang Ji-chen
I J I'1(t) - J(,,{.(t)) I ~ £ 1~'(f(t~(u~ U"-')) - ~·(yt",(ut..ult-'))I
Hence ~
EI?I='!, . . ,
I
1 J('i~(ti)
- J[7',I')1 ~
I J(y~(t» -JIY.lt') I+ IJ(Y.lt) -J(Yl')!I< [ t 'l,..
For 'Lv is arbitrarily choeen, we obtain 1 J lyit.C)1 - Jl'Ylt)) 1 '" t.Therefore U{IE-JI is a closed eet. By property we obtain U~lJI is closed.
Ii
1"
5' Suppose(s) is a linear system define. by the following: Ylt) = 0. , 1[+.-') """ + ()..","!tt-mj-t + b,Wf.- I) + .+ b"u. lt- n )+ elt) (2) where 'fIt) , Ult) , (- 00 4 t ". co ) are one-dimensional output and control respectiYely. e(t) is a stochastic disturbance and E e(t)- 0 , If in the meantime J(.) is a nonnegative convex function, then Urc<-J!. U(l J) are all conyex set. Here .,·l~) : Jl:YH-IJlJ.~",J(y(t», and 1l-tl is all the outputs system(2) at t. In fact, let LL.lt). Lj,lt) ~ U{,ll -J) denote their correspondent outputs by 'j,t+.>, "j,l~). By the definition, IJ(Y,(+.» -Jlrlt » I ~ ~,
Let
.\
~ lO' I ) ;
U!'(tl
or
o.,'Ylt-tl
t
b , u.~ (t-')
"fl"e'::
'I-
et.. YCt- ~J + + b"l.l ('\.-h.) + elt)
0.,:/(+.-1) .. ... ...
+
,,(Q,Ylt-I)'" Q,:ytt-l)+ ..
+ b, U.,I-( -I) t
....
+ b. U Ct -
( I-Aj(Cl''Ylt- l) ...
..
Q",:Y
It.)
-+ e [*)) t
0,,)'1+.-11 ....
+o.... )'et -"')+
b, Udt -I) + .. +- b,(.LCt-n.)f- elt» Y'lt) ~X ,-,,,It) + 1I-),') ,},.ct) . t
that is
Note that "j~lt.>
I:: YCt).
so JC'j* lt») ~ Jl'Y'(+.»)
Hence
: (J lAY,lt) + ll -A! )"I")1
- Jly'Ct)!]
~
[>..J l}lt») ...
~
A I J ()" It-)) - ]l)" (f.») I +
*
A~ T (1-
(I-A)](),lt!) -
An
(1-
JCY-lt>!J A,) \ Je Y,ctl!
- J O'l+.» I
=&
We thus know that Url L-J) ia a convex aet. Again by property'" we know U~ lj) 11 alac a convex set. CONVERGaWE IN EQUIVALENCE TO J UlIing the concepts of J -equi valence and t -J equivalence, we CID define convergence in equivalence to J. Suppose there ill a dynamic aystem with stochastic disturbance , we consider two different control strategies C and E. Let their controls at time t be U.'lt) , U' lt-) respectively, and their correspondent outputs be "j'l". "j
1:.;00>
I).e(t.).
and say that control strategy E is a control strategy equivalent to J.
0
RellJ1'k: Generally speaking, we cannot tell which of convergence in equivalence to J and common convergences ia weaker. E.g. : in Fig. , is the relation between the output and the control ot the system at an instant to . Suppose that the outputs of uOft · u•. u. &re 'Yopt . 'Y" '1. respectively. Eaay to see that we atill have IU,lt.J- Uop' (i..o) 1 < IU,lt.) - Lloptlt.)I
though I Jl'}',lto)) - Jl)'opt 1"'» I" I J(y,c~J) -J(y'p'lt.J) I where
&
, we have
by
=
Lt'(O) - Uoptl~) 1I ~
wbere U,pt (i.) denote the optimal control under some definition of the eystem at an instant k; 11 ' 11 is a norm dependJmg on conditions. If we find suoh a control that the difference of index funCtioDl IJl~(i.» -J("}01't1/U I tends to decrease as time goea on, we will reach the destination and aatiyify the demand of the control. That is why we introduce difinitions 3.
= 1\'\""
Jl),t"»
1),lt)1 .
CONVERGENCE IN EQUIVALENCE TO J OF A SELF-TUNING REGULATOP
= A U,lt)
and the output of y\~'
IJey,lt))- J(Y'lt') I ~ + li- A) IL.lt) ,
One of the most important problems to use on-line control is ensure tbat the control approximates (converges) to acme optillal solution. With practical consideration , we need not seek for a control u'cl>.) such tbat
AstrOIl and Wittenmark (1973) diaculaed the following linear .tocha.tic lIystem with unknown parameters ' (3)
where
At 1)"
)~+
a.,r-' ... ··· ..
a(p =b,r" .. c, (J) ;: r" +-
b,t -~
Q..
... + b..
b, 1: 0
C. ]'-' + '.' +- C ..
e(t), too O. ~ I. t 2 -.. are a sequence of Gau8B random variables 11(0 ,er), and independent of '/Ict), \:l.It!. In Eq. (3) , q is a rer.ard operator: q(y(t»-y(t+1). Polynomials B(z), C(z) bave all their zeroes inside the unit circle. In AlltrOm (1973) an arithmetic to the self-tuning reiUlator of syst .. (3) - - a method of on-line control havin, be.n widely used for recent years, WaS obtained by combinin, lesst square estimation and minimum yariance strategy. Under a.~mption that parameters are convergent and that the output of closed loop sYlltem has 2th order ergodic, etc, they proyed that the abov ...entioned regulator conyerges to a minimum variance controller. We ebtain the following result. by analysing ~e self-tuning regulator with convergence in equYilance to J. Theorem I Consider .ingle-input and s~ggle-output linear system (3) with unknown time-invariant parameter'l i f (i) 2th lIIement of the output y(t) (too 0, 'it. f 2... -) of cloaed loop system exists, and ia uniformly bounded; (ii) the estimation Ba) of parameters converges to their true yalue S" in quadric mean; the for the control U(R) of the mini ,num variRJlce control of system (3) ,t the instant k ,we have Uo(A.) ~1.I~) where lI\~)is the minimum yariance control of \3) • First we prove a lelllla For conYeniece, we giye some notatioDs fA. ~ (y(t). y(t-l) .. . j !t-nv+ l ) tlR'" A
( LL .lW.-I). · .. UC4,-I\..TIJ) u.~ ~ .,:t'( t~", t ~~. U~~). B(~)) = ~ .:(ct;,(U.O:),e(~)
(tL.U;~) ~
365
Analysis of On-line Control
tAt· ;t'(f..... l<4. u~~j. Bot) ~ ~ .:z:'(~1. Cl).. D;~). 8~)
;tcffo-1.
((.t .
$.(k)U.
T
O;~J. § (R.»)~ (-~,((,;Y(fu- . , -;;{",IR)9'rk.-
p.(~ifu U(£.-I) ....
T
.' .
tr
t-1-(A.J~(Vii(R.-,O)
;t.(f~-I.(u.Oi~).8?t)~ (-01.,9((.. >- .. · -~m"j(/I.,-rn--l) + p. ().. + ~,~, u.( k - () + ... ;. f.f,l V- (6. -J..) ) Leama I Suppose there ex1ats an admissible control subset U" c. U,,- . ltI-. u~ ~U". k-1 ,2, ..... ve denote the output set ~f \tby 'I' . k-1,2, ... if for any 1).. D;:: ~U~ and t '" ~ ~ are unifonaly held
r
..,'" , lu · " 16.L.Jl~ U;"').8('-»)
""(1'1,~.U.·Ut~).") ( -'-0 ll ... IA--+0 .... [,L
-
then
_
I Et.W,... (li.•. O;~).O(~)- E..:t'(Y~.(I.t,,-,D~~)1 A-", 0 Proofl note that u... ,u.~fU'''c.U .. and .:t('f~.l· . D;:).6(,,) ia a convex function, and
ia its unique minimua.
B7 t'undmental Leam. ot optimal control (Aatroa, 1970) ve have
E..t.'<~~.
= 11\ ..... E.t'(f,.... (<(·U;.::).B(k'j = 1n~""Et.t~.<.(u. U:~)B(I.') ~( ~~,. 0:~. ~(,,) t U.. U' rl><. u;; }(~»)W·
)
we change (3') into ~ It
= - o(,'~<-t) -
1'4. t I)
' .. -
01", ;f , t"""t 1 ) +
+[3.(lI.l't)+.· . + (JJ.U l -t-,L)) +/:. ( t+i+l)
(4) The coefficients 01,( and (3.: of the .oove equality vere obtained by estimating (1, ... b .. , disturbance t (t) is k-order moving average sum driven by noise e(t). Hence t.lt -rA-tl) ~s independe~t of ~ C1"".(u.U;:;o'>. EUt+p-+I)=o and ge.. (t+-~-tl)= ( j In Astrom and Wittenmark (1973), the paraneters 8·~ (- "" .. ' -01•. ra.. N~) of model (4) vu eetimated recursively vith least equare and then combined with minimum variance control, an arithmetio Qt self-tuning regulators wa~ obtained. In order of to prove in the above arithlletic
U.. ~u: J vhere u.;, is the minimum variance control at instant k, we first prove that E u.;. . E <4" are uniformly bounded since a certain instant. In fact, by (3) \Ht-Iu A(~)/(3(~) 'Y lt ) -+ A.C(~)/B«(P e(t)
=
U,(t):::
~·AC'P/I3«P y(t) -r Aq.~C(~)/13(~) €lt) + (C, [~) + C.(~)/B(~»)e(t)
= ( AI(P ~ A.(PlB(~))'y(t) siailarly
E ;t'(y,~, (14. D~:),II")
-=
=~
~.t' ~
1.,.
(U.
D;.~,. 8")
E.t·(f",.w. D~:) . 8~)-= In<-'" Et'
m1.n.
1).'1".'0;; • If')t U"
,,'fo..0:::1. O~)' u. m.l.n..
J1Cx.) ~ ~
tl{.:Ko )
x't,)tX
+ l'I'\l)..."X (tl(X) - )10(.)
>:(1,
.l, ' ~ ~
vbere fi are functions defined on
X. x(fi) ia a
value of x dependent:.. on f i • i=, . z . For V u. , D~= ~ U*, y... q~ are uniformly held
I E.t'(Y,(~.(u. O~~), 8(~) hence mAl)'.
(i).... AI [P = 0.: ~ '+
G(p = c:"
By e18llentary inequili ty : x'~,)tx
where A.O). C'(P are polynomials with degreu less than that of B{z) and irreducible with B(a).
- E.t''
t,
CL)
-k
(4t.,)
_I
J . .. ... + a. + C' J 1..•. )+ ".. + C(41., 0.-
By theory of z-transform, decomposing B(z), we obtain S(J)::(l-);)(J-r.) .. ·( }-Y.",) , IY; I
=-
I E.trf",t~ .D;::).9(I.,J) - El'
lI.(f.,., fm, 914i, ()') ~U~ Therefore
"w.
E;t'c1t-<, lu,D~~J. 8(l) ~ ~ C((Y."IU.U;:J./J'I)+
v.t~... U:::.8('J)~U"
+
L(llt'.U':,O'HU·
[E.L·(f... cl).,U~J.8·(,,)) -E.1\r:(U,lt',IJ~)) U, ft-<. u:~ ,8('),0' ) ~ u~ m<>-'X
V E > o.
3
N.
tor (,. > rJ
•
~ E..t,
Siailarly ve can prove
3N. tor ft,'" E.t( fl",(I). , U~~), 8')
_
\fE.>o.
or
l!'0
lI.\r., . u:~./I~)t:
!
uqtdr.~./J('»)tU·
U"
lE .t'( ~~. (14 . O~~). 01\)
-
~
~ Et(i'l"lU.D~).e(kJ+t
E.t'(fA~' (it •. D:~),e(6.J)I~ 0
Theorem proof : The model of system (3) is the folloving : )'(~) T Q.1~tt -I ) + .. . +- CL. )"'--IL)
+ ... +b .. lA..lt-&.-"->t \
+-
= b, «CH"+I
(el-t.,+ C, elt-I) +
c~el'--IL)J
UBing identity
~'C(P = Ai~) F(~) + ~rp
(Flp=J"+j,)t' .... ·-+f·'
)t
...
Similarly
00
,>..~·C(P/B(~)
-\" };,
~
_' -I
)
q~, S,Y,
>F
Without losl of generality, we assume the degresl "/",: -h. ~ -Iv hence U,'tl= (A,(~) + i. (i~.rjrj -I)Jy(t) +
ot polynomial. +
(C, l~'" ,=0 ~ (r ~:~j"r~~~j-'J) e lt, 1.:"
:.: ct')'(ttk.)" C·'e. lt t le)
i- , " t-
o:,·''jl~). c."'elt)'t
+ j::o f.. f' {~;rlyl.. -j-, ) "s4Ylelt-j- ' j) ."SI
+
= c,q)
366
Jiang Ji-rh e ll
c ~ m().'Y. {I C"~'I) I"'" _ V/-t ) ~ (n.-IJ(S+})yi ( I~l~- ) - ')I+ lelt-j-!) I
Vt
....
-
.
.
lE .:t'( for all t
Q:I -
~ ;~,(ll...t.,C)(\~lttt >l + le..ltH)1)
(.)0,
+ fo y/tl
~ Vt,
! I.I. l '-) I
Since
l v-· (ont ))
<.
0-
Tt
V- ,
m. .. '" ~
1
IJ
EV: _
E u.~t. ) ~ t
,
~
2
V- lm )
~
t
>z.1
-V'·"" t
<. +
'
.)-1
~I
+
""
+ In.- ')(5t~)-;-=r-
hence ..SU:f E tt(t) ~ C.. ~ In the following, we assulle Assulle
V:
I
J
" :0
!
)
C~
a.1~\tt">I-t-CIe.l-ttk)1 +
to
i. (Q"C)C\'9'(tt~)1 +\e.(t+.i..)\) + J;.Yjlt )
is the output correspondent to II1nill118 variance control u.'lt) at instant t. Vt
I
Then
= v't.) - ~ Ca
0
...
j¥."'.'
l.I.t
u.. u* :
EII8(tJII'
A <+ 00
<.
~ Ll'(+.) , :;
,j",
(~-r,)
~
1-, .fo.~ -'- q-1~)
_
IT.I
91!~
ll)
It}
= --"-'- ... ~t- - .. +--.1L..+ (1- Tt)"
( J-Y,)'"
I(~"P
Uo. , tu..' {, C: J
ll-T1 )
9i':~.
( Ill '
+ --, -t- ~+~+ -- . +~
(3 -1~ )· ·
'('t31
q-r~ )
For ever), pod tive integer
f3.(6Jh(~)ii. ("-,{)J -l-",,9l~'- '
. - olrnY(6.-m-')t
+poLl. -- - +F'Pt'jj.(6.-,t.»)J' , E { loI.,-~,(~)\ \~(A.)h' - -+ \~"'-~'" (~)\l9(R.-m.-1)\+ 1~(6.) -~.I\J.L.,\ t - - - 'r\~(')~lP-J
(Q.S)
'
A,()! / B(J)
(r
t
t;", u'lt)
We can decollpose rational function
Ef t t1 , ( 1.1., Ut;), 8(~» - ~(ri."lU , U;;I), e~) y = E- { (-Q'(~J'!i(6.J - - - - ,Q~ (~)'9(~-m-l) +p,(~)u.+ T - - - -+
SUf
IJ~)=q-r,)
then i).'lt) , u. lt ) e Ul', For an)'
iA Lt)
(11) ~([i.q~,~)\+ I(.«-t,)\»)/m) ~ too
n(
Taking into account of
we can simplarl)' prove
V" ~ {u..:
i -e
Rellark : If B(z) haS multiple root.,
V)-(t)
~~'l-t) H~(t.)
we have:
u\t) _
J
Ul\:)
00_
~ V(~"
u.'(t) \
(..
E.-
!;
f.-+ I),
<2:
In fact, by (ii) and that e(t) i, independent of '/I(t , , lilt) and E elt, ~ 0 , E e'lt ) '" 0-' , we know the estimated par8lleters converge (a.s ) (Ljung, 1970). And b)' (iii) we know they converge in quadric lIean. Hence by (i) and Theorem we get
.. _
"j',t)
otm>
I
,
;(
Corollar), 2: For system (3), if (i) t:'I{'(+') <. A
h
CN>A
-t ",i l
- J l:J°(H~tI)
.. ; ~ , u'CH
~
l)..(t)
(CN ::: COrt5t_
00
for the 88,ae 7N,
t
Corollar)' 1: If the output of system (3) is a stationary proceaa and the e.tillated par8lleters converge a in quadric lIean, then we have
(11i)
eN
l9l t t P- t')
U,l-H k.t')
Ai+ ll\.-11LS.~)~T}Ai\. 1
1 2 A·{CCLtC>-+t.
~
{(CtH).h,
{N"N.1, when
3max
or
B)' Fatou lemma EV/ ~ SI/.f {EV~~)J • For ever), t and 11, note that EI!(~H·A , f.Y(· )<. A ( A is a constant ) and use Minkowski in equation for sever~ tilDes we have:... _ _ (E.lV:;')] ~ = rE {(Qtc) (~ qYltH)I-+ Ieltt.\.)\>J -t
+ l:"' -Vjlt)}1.!. ]1
? N, , but l).lt) Eo If' _ there exists an N, when
ut:: ),CI") I < I 1iI-t
Set _
l1.l'lt),
.." 'c::-"Yt-" (LI(t), - -UI'I.·' /" - U- t·,) , G)\ I E..L ,-,J,S(t») -E 11(1"r:-.." , (.«-1.),
V';' ': V.. - f;. . v) l~} Eridentl)',
f•., ,(Q(t), o~:: ) ,6(tJ) - E;t'( ~t."
~
O-<-rd = + ("t:-t-k+I)!
By that we can eaail)' know that the following hold. uniforlD1), for any lJ. E: U" :
IE.t·(r,~ , (u.O~:,'). g((._)) - Et'(fl"(U. , U;~) , 8*;1~O The index function of lIinimum variance control is Jl'~ltl) = E)'\t) , In order to show ill"l t~
lE(-
=I E[C1.;, l~I'), O~~ ), 0-) - £.i'cft-"lU'lt),D~~), i9
I ~ \ I:1'C~" ( U.l~), D~::) , (;l~) - Et '- f.. ,CUlt), O~::l , §l-t»I t
...
~ lE :z"(Tt-' , (u.(t.),O~~), 8(t.')- E..iCT" ,(U<'{), O:_~),
*)
e-) \
Sinoe U.lt ) t- U~ U.·lt)t U~appl)'ing the above lemma, for an)' 00, 3 fI, , we have:
"t.- :
.
I ~
1:- ~ 1-1.,;.
l1~T;rt '" ~~'r\I-tH'r't -- -+ r! r' J!tC-1)! IU) + '" )
we evolve every item of ,iI) as above in t" a infiM te seri .. sillilar to the case without multiple roots, we can obtain operator solution of difference equation of L4~ ) we know that E ii'('),EU.'~") are bounded uniforal), b)' the salle analysis as before.
-M.l.llu..(k-,(.)I]"
.::: (Cm+ 1) + Cm+- J.) Cm • .L-')/ 2'1 -C..' ·E 1 8(~J -8' 1I~ 0
""r~
q- 1",)
-I
NUMERICAL EXAMPLE The following exallple chosen froll Astrom (1967) is the situation when .elf-tuning regulators is applied to quantitative production of paper. ~(to)::: \,2i; 'Y(t-') -0·49s ~l'-') t Z-3 0 11LIt-l)+ 2 -02S I/.,t· 4)+Vtt) 'V,-\:)
=
O.J8 2 ( 1-1.4 38f'1-o-ssf')
e(")
(!-I - ~8Jr +o'4-95f') (!1-f) 1\
Setting /1. '" ,-5' we get the following graph Fig.2 (paraMeter value is 70g/m 2 ) of outputs of minimu. variance control and self-tuning regulator when the)' are in operation. In the graph bold line denote the lIinimuM varitnce and thin line denote the self-tuning regulator. It is eas)' to know the avera~e bias of the output. of the two abavelIentioned controls aotuall)' tend. to decrease ~aG uall)' a. tille inc'r ealle CONCLUSIONS In this paper oonvergence of on-line control is discune.d. elluiyalence to J. equivalence to £'-J
Analysis of On-line Contro l
ana
convergence in equivalence to J of control are given from a practical backgroud and vithout 10e8 of rigor their simple properties are discussed. Through discussion about self-tuning regulators, ve obtain oonvergence in equivalence to J of selftuning regulators under such coaparatively veaker conditione as uniformly bounded outputs of cloaed loop system and convergence in quadric mean of the estimated parameters. REFERENCES Astrom, K. J., B. Wittenmark (1973). On self-tuning regulators. Automatic, ~, 185-199. Astrom, K. J. (1970). Introduction to Stochastic Control Theory. Academic PreD', New York. PI'. 259-262. "tram, K. J. (1967). Computer control ot a paper machine. IBM. J . Res. Development, 11, 381-405. Aetrom, K. J., U. Borisson, L. LJung and B. Wittemmark (1977). Theory and applications fo selftuning regulators_ Automatica, ll, 457-476. Caines, P. E., D. Dorer (1980). Adaptive control of systems subject to a class of random parameter variatimee and disturbance. In M. Hazevinkel, J. C. Willeme (Ed.). Stochastic SYstems. D. Reidel Pubishing Company, London, pp. 421-433. LJung, L. (1970). Consistency of L.S. identifioation method. IEEE Trans. Autom. Control, 21, 5-6. Ljung, L. (1980). The ODE approach to analysis of adaptive control systems. Joint Automatic Control Conference. San Francisco, Augu.t, 1980. Huth, E. J. (1977). Application ot the z-transfora method. Transform Method vith Applications to Engineering and Operations Research. PrenticsHall , Inc.PP.90-94.
0 Ti "lC
Fig. 2
~o
l
'"
"'in]
lA'",, ) ,
t.hi_ line
Cl et )
tIli.e 11ne
,
90
%7