Identification of Float Surface Temperature in Floated Gyroscope

Copyright ©IFAC Srd Symposium Control of Distributed Parameter Systems Toulouse , France , 1982

IDENTIFICATION OF FLOAT SURFACE TEMPERATURE IN FLOATED GYROSCOPE Hu Shunju and Yu Wenhuan Department of Mathematics, Nanlwi University, Tianjin , China

Abstract. The temperature distribution in the fluid in a floated gyroscope is described by a Dirichlet-Neumann's mixed boundary value problem of 2nd order linear elliptiC equations. Usin5 the Lions' function space optimization method,we estimate the float surface temperature by means of measuring the temperature on the part of the gyroscope hull. It is proved that the above problem is identifiable when the uny~own float temperature (the parruneter) varies in some admissible set. KeYVlords. Identification; distributed parameter system; optimal control; gyroscope; temperature control; bOlu1dary value problem. It is well known that in a floated GYroscope the inner gimbal is the sealed float which houses the rotor. The float is suspended in the fluid. It is necessary to know the float surface temperature in order to control it preCisely. In the paper \Ve will give a method,in accordance l... i th which the float surface temperature can be estin;ated by means of ~easurinG the tewperature on the part of the 3yroscope hull. We first describe the above temperature field by a 2nd order elliptic partial differential equations and the float temperature (the boundary value) is regarded as a unknown parameter. Employing the Lions' function space optir;;ization metilOd[3J,I. . e can estimate the parameter optimally. It is proved that the above problem is identifiable when the unknown parameter varies in SOI:ie

admissible set. lt,finally,must be pOinted out that the above results can also be applied to many practical problems, such as the Colli-Franzone anJ others' lvork on heart diseases(11,but they have only obtained the approximate results under the sarJe conditions as ours.

PROBLEi1 FOm:ULA1'ION

Suppose the float surface is ~ and that the hull is r. and that r. and Tj are the two-dimensional smooth closed surface. The region surrounded by ro and r; is fi. (as Fig. 1 ). Suppose the temperature field defined in .n is 1(1), H.!l. Then T(x) satisfies

@ B

348

H. Shunju and Y. Wenhuan

(1 ) (2 ) (3)

where R'j is the thermal conducti vi ty coefficient, which satisfies

perature \f is unknown, but we can identify it by means of the n:easurement 3{)) on the part B of the hull. We evidently can use a reSidual square to evaluate the merits of the identification. That is, when the parameter V varies in some admissible set 'llu and we look for a parameter from ~ minimizing the residual functional

1 ). ~~ are continuous piece by piece on 1i and Rj (1.) := ~. (x ) , (4) 2). the uniforrully elliptic condition, ..J.:l J .) 1 1. e. L 1; ~ .:J"" ~ ~ii 1i lJo ~ t::1 r f·' (5) ,-A.t' ,-, J where ~ is some constant.

r

0

aT

~

a)' -

.3

L

~T ~,o Ix) ~ ~"S(",~)

i:J=I:J

:J

(6) '

where n is a outward unit vector normal to ro or f, • The equation (1) is the conservation law of heat applied to an infinite-smial element of volume inn~ The equation (2) means that the float sLtrface temperature must be equal to some value. The equation (3) is the Fourier's law of heat conduction applied to the hUll. In the paper we can suppose f = o. On the resolvability of the problem (1) we have the Lerr~ 1. Suppose the condition (5) is s atisfied, beSides, '\rE HCcr.). f€ L'(Jl.) and ~fH-i(!;). Then the problem (I) is wellposed in H'(.1l). • With regard to the definitions of H'Cry and HS(.J\), cf. [4 J. Proof. I'lhen the boundary value are homogeneous the lelLI:la 1 is proved in [ 6 J. Then employing the trace theorem of the Sobolev's space [4] we at once obtain the Lex:una 1.

•

The solution of the problem (I),obviously,depends on V and we denote 1=~(Y) =~(~j~). In general the float surface tem-

Definition. The parameter ~ of the system (I) is identifiable in ~din accordance with the functional (7) if the following conditions are satisfied: a). there exists a unique optimal parameter ~ minimizing the residual (7); b). the optimal parameter U depends continuously on the measurement 3f l'(8).

UNI:,tUENESS OF CAUCHY PROBLEM FOa ELLIPTIC EQUATiONS Before giving the main results we quote from [2] the Lex:una 2 Suppose S is a open set relative to all and that H:c.nU,s) is the closure of (~(J1US) in H'(.II.), and suppose (5) is valid. If the function ~fH'(Jl.) satisfies

(9)

then ~ either must be a positive constant or satisfy .s"r~ ~slAf f , where ~+:: M"J((~OJ.

.A

S

I

Now we forn:ulate the Theorem 1. Suppose the conditions of the Lemma 1 are satisfied and meS B>o, then the problem Xf..fL

( 1 0)

Float Surface Temperature in Floated Gyroscope

only has the trivial solution in H'(Jl.). Proof. Let T be a weak solution of (10) in 1.f(Jl), then =0 Suppose the test dV 11 • function space of T is E ,then we have

'if/

O=J~ATdl

=- - 5 ~T}- d.s J.n\B 3)1

-5-N

ldJ

3ecause

~EE

is

=

~J('II",)+ (1-"JJ(tr..)

+ tr(TI~)'

(11)

'tf S ~ E • a~bitrary,we

VAf(O,I).

'*

When Vj"'1Ii,obviously,we have c,'Ifi (I~. If there exist such ",.,.'11', that76((I";)*Ye({,'I4~ Let T==TlV,)-Tlv.) ,then \Ve have AT=o, *\~:: 0 and

have

T1a'-::o. AccordinG to the

Theorem 1 we have ( 1 2)

(13 )

I

~

~

2). J\d.n.\S~O,:. ECIi~(.A\8).The condition (12) implies (g),according to the Lemma 2 we have SjtP T ~ .s~rT"',~o. :3i:nilarly f"-',... SJtf l' ~o • :. T =-0.

iT 3). 3-=0 on a part 0 f ci).n.\B and ~.o on

the other part of 0).11\8, it can be translated into the case 2). I In a word, we have T=O all.

Therefore, VI-Vi

< A JC1r,) t-

\;/>.(:(0,0),

(13) is valid if and only if one of the ~

r.o.

::::T\r,,:'O, this is contrary to '\Ij*"i . In a word, when 1.J;4='\J;,we have 7!l(c,v,J*"l'o(CI v,) all. Therefore,from (15) we have :T (,w; + (.-A)\I..)

followinb tr~ee cases takes place: 1 ). i>~ =0, therefore ~I =0 T is the ~, jjA.\~ v, ~ • solution of the homogeneous Neumann's boundary value problem for the homogeneous equations,but rl a ::: o ,:. T:::o.

(15)

5af'Yl a (C , Vj)-Ye14v-.JfJs,

J ~f 1115 +TTIT,!) 6

~.n\S

]a ~'B(Cof-tC,,\V;

+ (,_.\)clv. 4-C~Cf]-S}LdS

-A(.A)

:0.

~

J(A'\rI~ (l-o~)v.):::::

349

[,->. ).J(tr.).

V;fV,.

I

Prom [51 we also quote the Lemma 3. Suppose ICy) is a differentiable strict convex functional of ~ and that 6U"cl is a bounded closed convex set. Then there exists a unique elemen t \(f'll",d minimizing rev), i. e. 11101.) =

j"f

1('11"),

'II"~CU...d

and ~ is the optimal element if and only if r'(I4.)(v-l.\.'~o,

\!\rf.'lJ. .. 4.

(16)

I IDilllTI?lABILITY OF THE PARAi':ETER

Theorem 2. Under the assumptions of the Theoren; 1 and 9l reS), the runc tional defined by (1) is a strictly convex one. Proof. By the theory of the partial differential e~uations the solution of the problen; (1) can be represented by T l v)

=Co ~ + Cl

1r t (t

er '

( 14 )

Theorem 3. Suppose the assumptions of the Theorem 1 and (4) are satisfied and that the admiSSible set is

where ~ is a constant. Then there exists a unique element IH·'U...,j I:lini:nizing the residual functional (1) and the optimal parameter I( is char:icterized by ( 10)

350

H. Shunju a nd Y. Wenhuan

where t-=t(l() is the co -st ate ,whi ch is uetermined by

Employing the generalized Green's formula,w e have

( 19 )

+ S 1'0 [T{V)-T(L{~[

r Proof.

-t

\Ie first pro ve t hut ](v-J is a

c ontinuous Frechet differentiable functional ofV. In fact,according to (14) and (1) we compute easily the Gateaux d ifferen tial of Jrv) wit h respect to V; J ' rv-)h

= l i .....

g

<.

1'B [Thrth) - Trv-)]' iI

](v-)

J IS[T(V)-I((.O]['YoTlolJ-aJds,

==

of t h e opt i u:al parar:1eter U on the mea-

~tH'(~)

depends continuous ly on t he measuremen t ~ f I.:(.fl_), Proof.

We,first,consider the se t

\vi t h respec t to V-

K.,g {T{'\r)

l = 5 ['B(Cort(,v-tC.~)- ~Jl_ J[l'a(q+

.[ 1'[3

3J

JB 'IB( C,(V~hJ-C,VJ

u..\ - ~

L

(Cor-r- c,v+C'rf-jJd5

= JB Ll'a{C,h,f Js

V-f'U"d

---v

(lihll

0

l ).

(21 )

ele~ent u~~ud mini~i~inc

(1) . Next , suppose h= V- L{ and t~lat '\r in (20) e'1uals l{ and then by (16) and ( 20 ) we have '"Ye. [T(v)-TI

lOJ

['Xi T( It}

~ Jd.5 9- 0

-

'IJ f

'l1 lid •

is

::l.

~P..cJ

( 22 )

Then using ( l J ) ,we reduce (22) . Jecaus e

hrv) -T(II} llr. -= '\1'-1-\ , = r,

J.

(24 )

eusi lJ that '.;ne map P, : V~T('\r) is a one - to - one continuous rr~ap from CUa.<4 ont o K. By t ~ e compactness principle of t h e Sobolev ' s space [ 4] 'UQ,tj is also 1 8. co mpact subset of H '(r.). Thus by the Tikhonov 's lerrma(1) ,in accordance with which every c ontinuous one-to-one map is a hO::leo mo.rphise .we have that t h e map P, is '.i i:orneomo r phism . Then , we consiJe r the set

,,,.Jl.

0.

I

c l c v:? d convex subse t of H'(r.J. Be -

T! (V)

•

lI)

0..

is defined by (11) and i t

sati s fies

At,IV)-T(IO]=O ,

of

IS

from a compact set to a Hausd orff set

J

B

'V

, where

T(v)

cause TlIi)::::C.t+c,v+c,'f , k i s a clos e d conve x Is u. bs r: t 0) r H'(lll. I t is proved

By the Lemma 3 t h ere exists 3. uni 1 ue

T(vj - T(\4 )

H'Ul.)}

e

B

j

f

5ol"~io"

](vH) - J(v-)- J((v)

(,'["-;- (1-
(18)

'tj V-E. 'U ....d. I Concerning the continuous depenJance 8

.J.

(20)

vie can pr ove t h at ;r '(v-lk is the Fr~chet differential of as well, because

ds

the Theo rem 3,the op timal p a ramet e r

T \\T I - ~] d.s •

• [ )'/l

t

,

SL
rl I(Vtfth)-:J(v ) }

-L

Q-HO

~

(\I'-l{)

TlI(J-SldJ

)'8

B

=:.13 T ('Ir),

T(-V-) ft<.

( 2 5)

1.

By [ 4 ] ',,,e have t hat the trace ;;:'3.p 1'( v-) ~

onto

N.

Pl :

con timi'Jus one froe k And by the Theorem 2 we also

Tel"") is

,1

have tha t the inverse map PI-I: Te(v-)I'7T(v)

Float Surface Temperature in Floated Gyroscope

is a single value map from N to K. As K is a compact subset of H'(Jl), by the Tikhonov's lemma we have th8.t the map ~ is also a homeomorphism. Besides, the residual functional (7) can be regarded as a distance in l' (B);

351

and T(a.c) is the solution of the problem

(26 ) To seek the optimal paraJ;Jeter UE- ~o..d minimizing (26) is obviously equivalent to seeking the project TIi(~) of ~ in the convex closed set N, which is a closure of N in t(O») Le. ~

IITs(w) -~ Ilt'IS)

i"S

Ta(o.I}~#

IlTIl('V")-~\I~ ~

(s).

Finally,we point out that for the nonstationary system

~.L~- t<,(l,iJ l)x aT J

aT f\ T It ,~}~ -~~ -

(27)

By the Theorem 3 there exists such a unique element TB''')ENCN that (27) is valid. Then by means of that the project map in the Banach space is continuous [51,we obtain that the proj ect map P!j: Tol") a conti!1uous map from e(S) to H ~ (13). In a word,by the continuity of the maps p·'and p-'we at once obtain that l I , the map ~,~'~:~~~ is a continuous one from t'IS) to Ht Ir.). I

3. . . . .

The equation (19) and (1) can be solved by the finite element method and the inequality (18) can be solved by the optimization methods.

1

=

IS

=:

oX',

fCJr,C),

T ro Xl G,T)

l',It'"'o

L..." ':j='

=- tTo

,

J

)

(X,tJf--'1.X(o,T)

El ~v

r,X'(C,T)

-

~

rp

I,

o",.n..

it is possible to de termine '\T u.ni rluely by means of the measurement ~ on B, here are omitted these results.

REFEREtICE3 Combining the Theorem 3 with the Theorem 4, we h'ive that the p'ir3llleter \J" of t he system (I) is identifiable when '\J varies in ~o..cl defined by (i 7 ). And the optinial parameter (the unk..'1own float su.rface temperature) L( j.s determined by the followinG ( 10)

1 •

Colli-Franzone,p.,E. Taccardi,and C. ViGanotti (1976). Un metodo per la ricostruzione di potenziali epicardici Jai potenziali di superficie , L.lI.A. Pavia. 2 • Gilbar.;,D.,and N.S. Trudineer (1977). Elliptic Differential Bquations oi' 2nJ order, Sprin..;cr-Verl3.!;,

Berlin, Ch~p.8. . !,ions,J.L. (1 ?68 ). Controle Jphcal de 3ys:ewes Gouvernes par des Equations aux Derivees Partielles, JaQthier-Villars,Paris,Chap.l. 4. Lions,J.L.,and E. iiagenes (1 97 2 ). Non-homogeneous Eound~ry Value Proble~s ~Dd Applications,Vol.I,Sprincier-'Terlag,Berlin,Ch'1p.l.

3. WIlere 11 is the solution of the probleI:l in

At=:o. \

P\r. -= n

Sl.

,

..£Ll =0 ' ) ay If,\!3 \-{f- L~ = 1'8 TI"'-~

352

5.

H. Shunju and Y. Wenhuan

rmrtin,Jr.R.H. (1976). Nonlinear Operators aDd Differential Equations in Banach Space,A Wiley- Interscience Pub.,N.Y., Chap.1. 6. Showulter,R.E. (1977). Hilbert Space Nethods for Partial Differential Equations,Pitman Pub. Limited,London,Chap.4. 7. Tikhonov,A.N.,and V.Y. Arsenir. (1977). Solutions of Ill-posed Problems,V.H. Winston and Sons, Washington D.C.,Chap.2.