Optimal Control Coupled Fields in the Process of Induction Heating

Copyright © IFAC Control Applications of Optimisation, Visegnid, Hungary. 2003

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OPTIMAL CONTROL COUPLED FIELDS IN THE PROCESS OF INDUCTION HEATING V. Dikusar l , Daria V, Filatova1, M. Grzywaczewski J , M. W6jtowid IRussian Academy ojSciences, Moscow, Russia 2Kielce University, ul. Krakowska 11, Kielce, 25-328, Poland J Radom

Technical University, ul. Malczewskiego 20A, Radom, 26-600, Poland

Abstract: In this paper is considered the problem of optimal control of process described by boundary problem for partial differential equation. As an example of optimal control of thermal and electromagnetic coupled field is shown inductive heating of aluminum slabs with minimization of used energy for obtaining the set end field of temperatures, taking into consideration phase constraints of maximum temperature and the greatest difference temperature at this moment (thermo stresses) and ended constrains of Tchebyshev type connected by temperature straggling. Copyright © 2003 IFAC Keywords: maximum principle, optimization problems, constraint problems, partial differential equation, minimax techniques.

I. INTRODUCTION

et al., 1988). With requirements posed in the problems of optimal control of the heating process application of such imprecise models may cause high safety coefficients, respectively, and consequently a decrease in the value of quality functional determined in such away. Determination of optimal control of such an object is possible only using numerical methods. The specific problem (a high cost of stopping process of plastic deformation due to an improperly heated slab, a long time of heating) creates an obligation to solve the problem with always guaranteed effect (purposelessness of stochastic approach).

Recently, due to the development of modem technologies on one hand and application of modem computer techniques for control of technological processes on the other, a number of problems of optimal control of coupled electromagnetic fields, temperature and thermal stresses has been formulated and solved by Wojtowicz, et al. (2002) , Grzywaczewski, et al. (1991 and 1989), Gorbatkov, et al. (1988), Siratori, et al. (1986). There is a frequent need in modelling to take into consideration the dependencies on temperature of such properties as specific electric and thermal conductivity, specific heat, Young modules, etc. of environments in which these fields occur.

This paper considers a sufficiently general formulation of the optimal control problem connected with induction heating of conductive bodies taking into consideration phase, terminal and control constraints. Both in the mathematical model of the process (heat model) and in the constraints the dependence of parameters of the environment on temperature is taken into consideration.

Mathematical modelling of the induction process of slab heating using models of constant coefficients (average, parameter values of a heated body) creates discrepancies between computer and physical experiments by 15-20% for aluminium, by 50% for titanium, by 100% and more for steel (Gorbatkow

213

After justification of the adopted optimal control algorithm attention was concentrated on irregular phase points theoretically predicted by Milutin (1992). Due to the application of limited variation using the local maximum principle, situations may occur in which optimal trajectory determined by the maximum principle can not be avoided in admissible areas. For optimal control with participation of such a situation one must get prepared a priori.

Tmax~,t)-TmiD~,t) ~ c 3 (Tme

(t»

(6)

V~,t)ea

3. OPTIMAL CONTROL ALGORITHM Stationary conditions are obtained by classic scheme Dubovicki - Milutin (Milutin, 1992 and Dubiwickij et al., 1980). Determination of the necessary conditions of the first order requires the following stages:

This paper presents numerical algorithms of optimal control based on the local maximum principle and conclusion related to it. The considered theoretical problem is illustrated by an example of the determination of optimal control of inductive heating of aluminum ingots. Irregular phase points occur taking into consideration constraints on the maximal temperature difference of a heated body taking into consideration the dependence of maximum admissible temperature difference from average temperature of the heated body.

10 Approximation of the set decrease value of functional (I): Mo={weWII(wo+w)
2. PROBLEM FORMULATION

WO

-

0. 0 ,0.; (i el), 0. The optimal control can be formulated as follows: Find control u = u(t) minimizing the functional l(u) = fu(t).~,t).d~.dt ~ min

intersection

~=(X"x2,x3)e9l3,

a

n

ue9l',

te9l',

(2)

v,!en,

(of admissible directions) with top at point wO were constructed from directions w e W the same as for

or

A(T)· On =q(,!,t,T) , V~,t)erxl, ~.lr below relations were considered as constraints

every w it is possible to find this neighborhood of

inequality

zero Q(w) c Wand number Co > 0 that for every

(3)

weQ(w),O 0 it is possible to find W(c) e W the same as

- control constraints: u l ~ u(t) ~ u 2 , i = l,k

were

To guarantee this property of cones approximations are realized in accordance with the following scheme: a) cones no (decrease of functional) also nj (i e l)

+ u(t)· ,~,t, T) V~,t)erxl, T~,to)=To~

0.

Fulfillment of the above condition trust of approximation requires additional openness of cones no, n j (i e/).

model (boundary problem) of the object was considered as equality constraint

where

nj (i el),

M;(ieI),N,Q (Q - a certain neighborhood of

3

yA(T)YT~,t)+

no,

zero).

= 0. x I, t = [to,t,], E 91 is domain with segments of smooth boundary r = in, the heat

c(T) or;,t) =

cones

equivalent to non empty subset intersection set Mo,

(I)

u where

local minimum, by convex cones thus that nonempty subset

- end process constraint

max IT(~, t l )

- T· (~)I ~ &1 (4) n where T· (~) - a priori given the results function of temperature

cw+W(c)eN (i.e. g(wo+cw+W(c»=O). For operator g if it is continuously differentiability acc. to Frechet at point WO it is possible to do it using Lusternik theorem (Dikusar et aI., 1989)

- phase constraints 2° Euler equations. In accordance with DubovickiMilutin theorem (Dikusar et al., 1989) intersection of systems cones (11) is empty only then if there exist linear functionals Wo (no> E V', w;(nJ e v' ;

°

I for maximum temperature maxT~,t)~c2

(5) u 2° for greatest difference temperature of heated body

i el,

Cl1{n) e V·

conditions:

214

which fulfil the

following

El:

(!, !:,t)eQxQx/~~, /)el(,

1) wo(Q o) ~ 0, w;(QJ ~ 0, i el, w(Q) ~ 0

P2:

~,!:,/)eQxQx/~(!:, /)ea ,

(i.e. support functionals for cones (11); 2) condition of non triviality is fulfilled

A - measurable set in 6, ~,P2 - projectors.

=

Ilw 1 + L !l w;Il+Ilwll > 0

b) boundary equation for t tl (tranversality conditions) 'I'~, il)=-(TO~,il)-T·(!)).dv, V!eQ (8)

o

;et

3) Euler equation is fulfilled wO+Lwi+w=O

r xI , V~,i)erxl

c) boundary conditions on

iet

A Or;r ='I' ·qrO

on

Deriving Euler equation it is enough to construct a general form of support functional for every approximated cone and next the sum of supports should be equated to zero in order that the support set could be nontrivial.

(9)

d) local maximum principle h(/) =

fH(ao ,IJI~,/),U(/),!,/) . d!~max

(10)

n where H = u(th6(!,I,TH'I'~,/)-ao)­

3° Analyze Euler equation. The equation is enough to transform in a simple form or to deduce proper conclusions from Euler equation.

2

-m l (I) · (-u(t) + u l ) - m2(I) · (U(/) - u )

Viel

(11)

Local maximum principle can be written in the form of necessary conditions of extreme

4. MAXIMUM PRINCIPLE

,

h" = 0

On the basis of the above method was formulated a local maximum principle for problems (1 )-(6). Let (T o ,!{) - optimal pair and A(TO):;e 0 on

°

Remark. If H is convex to u local maximum principle is equivalent to maximum principle

a,

H(ao,'I'~,/),'l.,9) ~ H(ao,'I'~,/), 'l.° ,!,/)

c(TO) > on 6 .Then there exists a nontrivial set of ao,

Lagrange multipliers mi~, i)

a

i=l,k

1

=> 91 measure

-

that ao

~O

d p,

number,

Vu e 91 r I (u l ~ u(/) ~ u 2 )

T*

wl =

&I} ,

In the case under consideration the function H ( 0. 0 ' \jI ,!!,~, t) is convex to control 'l. . It results from analysis of (16), that optimal control u(t) has the following form

'I'~, i) :

dE> - measure

u(t) = u

on 6, mi~' i) : II => 91 nontrivial function which fulfills the following equation 1

J(u(t) · ~(~, t, T) .(\jI(~, t) -ao)) · d~ > 0,

o

u(t) = u if

The above Lagrange multipliers set must fulfill the following conditions: a) conjugate equation

·d~

< 0,

f(u(t).;~" ,TH'I'~, /)-ao )) · d! = o. o

where arbitrary nonnegative measures on

n = QxQxI

;

The solution thus formulated of the problem of optimal control was obtained with accuracy to parameters. If phase constraints do not activate and have not intervals of singular control then optimal control has intervals with control u 2 and intervals

da+ is measure concentrated on

M~ ={~,!:,/)en I ro~,t)-ro(!:,/)=&3} dais measure concentrated on

Mn ={~,!:, /) e n ITo~,/) - ro (!:,t) =-&3} ,

with control u l . The considered problem was parameterized. The value of obtained parameters, i.e. length of each intervals of control is unknown temporarily.

then where del

,da2(A)

))

if

dE> is measure on 6, defined by the follows. Let

= a(Jr'(A))

0

or

Or (7) W(!:, i)-ao . Wr ©)·d! ·di = dp

de[

2

J(u(t) .~(~, t, T) .(\jI(~, t) -0.

o

(A(T O)V2'1'~, i) + c(TO) Or;r~, i) +

dal(A)

2

if

mi (!:,i)·~o~,i),i)=Oi=l,k .

'I'~, i)·

(13)

dv,

function , d p ~ 0 nontrivial Radon 6 concentrated on set on

= ~ e Q : IT~, il) -

Mr

'I'~, i),

(12)

= a(Jr'(A))

215

5. LIMIT POINTS METHOD Solution of full problem requires the following stage.

(18)

I. On the basis of preliminary analysis taking into account maximum principle we need, if it is possible to parametrize the problem (i.e. find class of optimal control with accuracy to parameters which we need to count); to foresee initial approximation of optimal control, to consider technical chances of realizing chosen control.

a o J;(;!,/, ,T)·d! -rn, (t,) + rn2 (I,)] = 0 n In accordance with theorem of Karateodori (Milutin, 1992), because (17) is convex hull (16) l:$r:$n+1 (19) There is a close connection between the number of points r and k of the quantifies of control parameters which constrain ended manifold (Milutin, 1992)

2. Choose a method of optimal determination: - direct use of local principle maximum; - use of the limited points methods.

X = {~:IT(~, t 1) - T\~)I~ E, V'~ En}, (20) where c must be considered as an unknown parameter.

control

Ended manifold (19) is obtained by the iterative method beginning from the simplest one - parameter control for k = I . For each k - parameter control the

3. Make analysis of the trajectory behavior after activity phase constraints (possibility of existence of irregular phase points)

least value (I) Cmin

into

(2)

(n) _

> Cmin > ... > C min - cinf where the process is continued until C = c, is achieved. Using k -parameter control one can not achieve a smaller radius of ended manifold (decrease the final

For the case if: irregular phase points and segments of specific control do not exist; control is linear operator of the equation of heat conduction (2); separate components ui:s u;(-r) :$ Uj2 are subject to

temperature scatter) than c~~. The condition of achieving c~~ is the number "r" of limit points

constraints; then on the basis of the local maximum principle optimal control has a known form of alter

u; ,ui

is determined taking

consideration that

4. Make analysis of the existence of intervals of singular control and also arrangement of these intervals.

nattily operating control phase constraints.

C(kJ mm

obtained. And thus if r

or control activates

=k , then

C

> c~~ ,

= k + I then C > c~~ and if r = k + 2 then C > c(k) mm = c°nf , where this case must be considered

- if r

I

The classical formulation of the problem is the determination of optimal control in the class of nparameter control under minimization (1) with assumed accuracy of heating (4) (constraint of temperature scatter). This constraint finds the following reflection in the conditions on tranversality (8): for all points ~, I,) ( :! En), which constraint is not active, i.e.

IT~,/)-T'(E>I
II'~,/)=O

as the case of degenerated parameter control k + I , where the value of the final parameter .1 k +1 ~ 0 if

6. COMPUTER REALIZA nON OF OPTIMAL CONTROL ALGORITHM The algorithm of optimal control was based on the local maximum principle (Rapoport, 1993), method of limit points and the penalty method. Consideration of the dependence of the environment parameters on temperature led to the appearance of irregular phase points (Milutin, 1992) connected with activation of constraints on maximum of temperature difference of heated body. (Fig. 1-3).

(14)

at points C!, 1,), ! E n in which this constraint is activated (limited points) for a conjugated function a measure appears r(:!,/)-T'(:!) = c, 1I'(9)::::>-~j ~o r(!,t)-T'<:!)=-c, 11'(9)::::> ~j ~ 0 (IS)

Next a relationship was obtained between the quantity "r" limited points and the number "k" of control parameters. For parameterized control u = u(.1".1 2 , ••• ,.1n ). (16) At the end of the process for maximum principle has the form

1 = I,

The algorithm consisted of two parts: - control until transition to the irregular phase point (control with maximal power) - classical optimal control after transition of the phase point.

the local

h(~ = JH(ao'II'~,t,),u~),:!,/,).d:!

(17)

The control object was simulated by the multigrid method. Problems determination heat source and technique simulation were described by W ojtowicz, el al. (2002), Grzywaczewski, el al. (1991 and 1989),

n

and

216

Gorbatkow, et al. (1988), Siratori, et af. (1986). The optimal control in real object was realized with the use of inventor equipped with a microprocessor controller.

differential equations with ended constraints of Chebyshev type. Taking into consideration the depended of the coefficients of the mathematical model on temperature, the existence of irregular phase points was observed. For this case a computer algorithm of optimal control was modified by using the penalty function. Fulfilled phase constraints in the situation of existence of irregular phase points required calculated a priori coordinates of these points.

T!C it

-.t-------------------~~~~==

-

REFERENCES Wojtowicz, M., M. Grzywaczewski, V. V. Dikusar and L. Pietrasik (2002). 3aoa'lu OnmUMaJlbHOZO

Fig. l. Maximal Tmax(t) and minimal Tmin(t) temperature of ingot heating and control u = u (t) , where I is stage of basis heating, 2 is

ynpa6J/eHW! 3J1eKmpO-mefV/06OlMU npol/eccaMu,

262 CTp. BbI'IHCnHTeJlbHhIH ueH-rp P AH, MocKBa. Grzywaczewski M. and E. Rapoport (1991). Optimal

stage of equalization of temperature (transition of phase point), 3 and 4 are sleep of optimal trajectory after constraints, 5 is transport to the deforming device

Control of Electromagnetic Field Problems Coupled to Temperature and Thermal Stress in Induction Heating. Proceedings of Conference of

the Computation of Electromagnetic Fields, Sorento. Gorbatkow, S. and M. Grzywaczewski (1988). K aHaJlWY umepo-anpOKCUMamU6HOZO Memooa OM mpeXMepHblX HeJlUHeUHblX 3aOa'l 3J1eKmpOnp0600Hocmu, I-hBeCTHSI AH CCCP:

,......

,......

3HepreTHKa H -rpaHcnopT, 2, CTp. 101-llO. Grzywaczewski, M., E. Rapoport and A. Stochniol (1989) . Optimal Control of Electromagnetic and Temperature Fields in Induction Heating, International Symposium Electromagnetic Fields in Electrical Engineering, ISEF'89, LOdZ

Fig.2 Impossibility of fulfilling constraints on maximum temperature difference of heating body for case of optimal control without taking into consideration a priori existence of irregular phase point

Siratori M., T. Mijosi and H. Macusita (1986). Bbl'lUCJlUmeJlbHCUI MexaHuKa pa3pyweHuu, 344 CTp. ~P,MocKBa

Dikusar,

V.V.

and

A.A.

Milutin

(1989).

Ka'leCm6eHHble U 'IUCJleHHble MemoObl 6 143. HaYKa, npuHl/une MaKcUMYMa, CTp.

MocKBa. Dubiwickij, A.la. and A.A. Milutin (1980). TeopHSI npHHUHna MaKcHMYMa. In: Memoobl meopuu 3KcmpeMaJlbHblX 3aOa'l 6 3KoHoMUKe, CTp. 6 51 . HaYKa, MocKBa. Rapoport E.Ya. (1993). OnmUMWal/W! UHOYKl/UOHHOZO Hazpe6a MemaJIJIa, CTp. 278. Me-rannyprIDI, MocKBa. Milutin, A.A. (1992). Chebyshev Approximation in the Problems of Parametric Optimization of Controllable Process. Avtomatika Telemehanika, 2, pp. 60 - 67.

Fig. 3. Technique of overcoming an irregular phase point (penalty method)

7. CONCLUSION The algorithm of induction heating process optimal control with phase and ended constraints was presented in this paper. Principle of maximum for calculated optimal control of systems with distributed parameters was proved. Very effective computer algorithm was built for multidimensional problem of optimal control described by partial

217