Sensitivity of Minimum-Time Problem for a Class of Nonlinear Distributed Parameter Systems

Copyright © IFAC 3rd Symposium Control of Distributed Parameter Systems Toulouse . France . 1982

SENSITIVITY OF MINIMUM-TIME PROBLEM FOR A CLASS OF NONLINEAR DISTRIBUTED PARAMETER SYSTEMS D. P. Petrovacki and D. M. Marganovic Faculty of Technz'cal Scz'enc es, 21000 NOlli Sad ,

Yugoslat~'a

Abstract. The analysis of sensitivity of the solution on thermophysical parameters of materials as well as on the parameter characterizing nonlinearity of problem has been performed in this paper by the application of the sensitivity method on the minimum-time problem of nonlinear heat conduction process. Apart from sensitivity functions defined for the original optimal control problem (of the system with distributed parameters), sensitivity functions for corresponding transformed optimality problem for systems with lumped parameters have been deflned ln this paper as well. After the establishment of relations between these functions, the problem of their calculation is reduced to the calculation of sensitivity functions for systems with lumped parameters. Solution is obtained by the application of a numerical procedure based on Warner'S algorithm. Keywords. Distributed parameter systems; optimal control; sensitivity; approxlmate method; nonlinear systems; heat systems; maximum principle. I NTRODUCTI ON Many real physical systems are by nature distributed and their mathematical descriptions are in the form of partial differential equations or integral equations. Typical representitatives of such systems, Distributed Parameter Systems (DPS), are many industrial processes in which heat conduction process in a given materia 1 is incl uded. In the various areas of engineering application the optimal control of DPS is often needed . For example, in the metallurgical industries it is required to control the temperature distribution in a metal by its heating in the furnace. An important situation then arises: to control heating in such a way to obtain a specified temperature distribution in metal in the shortest time.

skii (1979), Ray (1978) and Robinson (1971). Linear cases have been treated in most of the papers where heat transfer problems are considered. In these linear situations thermophysical coefficients, which are temperature dependant, are often replaced by the corresponding constants. But this assumption is not always justified. Namely, temperature dependence of physi ca l properties, such as thermal conductivity of the material ,must be taken into account in many heat conduction problems in industrial applications.

It is a well known fact that the methods for solving the optimal control problem for DPS are incorporated with complex mathematical problems. Because of this fact, it is necessary to introduce some kind of approximation at the beginning or at a later stage of the solving process (Ray and Lainiotis (1978), chapter 2). One of the major advantages of early approximation is that the knowledge of the process and engineering experience are of much help in the approximation procedure . Also, early approximation avoids the solution procedure of sets of partial differential equations or integral equations. Simplification of the mathematical problems, by use of early approximation, leads to algorithms which are less complex than those for which distributed nature of the process is retained. Detailed surveys of the papers which deal with the control of DPS are given by Butkov569

Sensitivity techniques have been successfully applied in the design of optimal Lumped Parameter Systems (LPS) in which a vector of constant plant parameters varies from its nominal value, i . e . mathematical model of such LPS have some errors or uncertainties (Cruz (1972), Frank (1973) , Sage (1968)). However, the sensitivity of optimal DPS with respect to model uncertainties has been rarely treated (C hin and Shih (1972), Pedersen and Nardizzi (1972)). Calculation of sensitivity functions is the most difficult part of the sensitivity analysis problem. This problem is made much more difficult in case of optimal DPS due to the fact that sensitivity equations are given with the set of distributed equati on s as in Pedersen and Nardi zzi (1972), whose solutions yield the sensitivity matrices.In order to make the solving of these problems easier, (Chin and Shih ( 1972) ) , modal analysis have been used for a class of linear DPS.

D. P. Petrovatki and D. M.

570

In this paper we shall consider the influence of variation of constant system parameters for a class of nonl inear optimal DPS. In particular, variations of thermophysical parameters of material, as well as of the parameter characterizing nonlinearity of the problem, on the minimum time problem of nonlinear heat conduction are analized by means of the sensitivity theory. As in all nonlinear system theories, where the trea tmen t of non 1 i ne ar sys tems req ui res more formal and approximate methods (Ray and Lainictis, introduction, (1978)), approximate techniques will be also ap~lied in this paper Some results of the minimum tlme problem obtained in Petrovacki (1980) will be used; firstly, the procedure of transformation of DPS into corresponding LPS, and secondly, necessary conditions for optimality.

We shall analyse sensitivity of the nonlinear minimum time problem for DPS which is defined as follows: DPS under consideration is one dimensional heat conduction, defined by nonlinear partial differential equation of the form 5 "OT = 3... (k:(!) 1I I -I- g(t) (1) o IT 0.)( o)(i

= Ko

(1+ol.J) ; O~.x.~{,

o,;.i,;,tf

and the following boundary and initial conditions T(o,-I) = 1.1,(+) , T({,f)=U2 fi) , o"f.;ff (2) T(X, 0)==

Ta

o,;,.x.~1.

(2), which is in direct connection with the variation of medium temperature in the furnace. In such a description, eqn. (2), our attention is therefore narrowly concentrated on the slab exclusively which is regarded as a whole system. All influences of broader environment may be taken into account through the determination of the dependence of slab boundary temperature on the envi ronment. In this case, control of temperature distribution may be carried out through the boundary by means of an external variable and not by boundary temperature. Here, it is desired to determine the control inputs, whi ch may be ei ther g(f l or Uj r'f ) (F ~!Lj or both, such tha t they transfer the sys tem from the initial state, eqn. (2), to the given fi na 1 state

(3)

where: state function T(.x.,f) is the temperature in a solid;x is the space coordinate (o~x«,/) k(T) is the thermal conductivity wich depends on temperature; f is the time (o
o~x~L

UX,ff)=!1

(4)

in a minimum time. Thus, the cost function is i{

fo df

J=

STATEMENT OF THE PROBLE'1

kiT)

Mar g anovi ~

(5)

where the final time if is unknown. By the performance index, eqn. (5), one of the control problems, which often arises for the heat transfer systems, is defined, and it is particularly important in the period when these processes start. Also, an important practical problem is how to control the heating of meta 1 in furnaces to obtain a desired temperature distribution in metal in the minimum time. The optimal control problem for DPS described above has been approximately solved in Petrovacki (1980). The method used is based on approximation of DPS by the corresponding Lumped Parameter System (LPS). Transformation has been achieved through the application of the techniques for approximate solving of nonlinear partial differential equations (Galerkin method, Gauss principle and the integral method (Finlayson (1972), Soodman (1964)) .

In that article, the trial solution of eqn. (1) is assumed in the following form T (x, f) == Ut (I) -I{ilz (f ; - LLI (f)) + Cl (t) r:P (.x)

f.'

q; (x) =

X)2

X

( 1- T - (1- T )

(6 )

Selecting initi al and final val ues for a /f ) a-ioj=0 a /ij )=o (7) as the boundary and initial and final conditions, eqns. (2), (3) and (4), are exactly satisfied. Here, some general remar~s about the form of the trial function must be made. Firstly, a qualitative analysis of the solution of the optimal problem under consideration for the linear and nonl inear case has been the goal. This means that a high accuracy in the final results is not the main requirement. Also, this analysis gives the error which is made when the real nonlinear optimal control problem of heat conduction is considered as linear one (,,(=0) and that is a very important

571

Sensitivity of Minimum-Time Problem

information before one decides to take any linearization. Secondly, when the space functions are chosen so as to correspond to the real temperature profile, then the number of time variable quantities can be smaller.Thirdly, application of the integral method, for example,(the method which has been applied successfully to a broad class of engineering problems see for example Goodman (1964)) limits the number of time variable quantity in tra i 1 func ti on to on 1y one. These rema rks imply that the trial function, eqn (6),should give an adequate qualitative description for the treated problem. At the same time, we expect that the corresponding error will be of the same order as the errors in the approximate solutions for other engineering problems. In the relation (6) a ff ) is an unknown function of time which has been found by the use of approximation methods mentioned earlier. The ordinary differential equation for determin i n g Q (t) a re • 2 a (-t ) = Kt uf (f ) -f K2 Ul ffJ + K3 g ff } of K* Cl ff) f 1:'5 a (t) + KG a (f) U f f l-) f 1(7 Cl (iJ Uz f t) f <8 u, ft) f /('3 Llz(f) + K I () u,(t ) uz fl ) 1 Kl1 U7(fJ + Kf2 u: rtJ

whe re and

cA.= d a / df ,

0

LI,;

d

'!

Ut / a '! ,

(8) 0

u2 =

d ;. "2

,

j '" 1, 2,

'

!

12

Since the Pontryagin maximum principle has been used to obtain necessary condi ti ons for optimality, new state variables Q2ff) and ct3 (1) have been introduced (to eliminate UI (t ) and uz (t) from eqn. (8)) by relation a3(t ) =fl2(f)

) {a1

and new control variables v:, ( t ) and defi ned by

(i) =a (t)} ( g) ~ (t) are

a3 (f) =V2 (f )

( 10)

Then, eqn. (8 ), in respect to eqns, (9) and (10), becOOles cif rt ) = f, ( a 1, Cl2 , a 3 , 111) ~ ) (J) (11) The hamiltonian of the problem is If = 1 + f 1 f: f f2 V; 1- A 2-2 (12) where Pi (i= ~ 2, ~are the Lagrange multipliers which must satisfy -'

i=1,2, 3

( 13)

The control functionsZl;(t),z.;6')and gft) are of the bang-bang type. Combining (2),(3),(5) (7) and (9) boundary conditions are QI (O}=O

1

ad t;.'=o,

For the original minimum time problem for DPS, eqns. (1)-(5), the state sensitivity functions 8; rf,x,qn)and the control sensi tivity functions-Ujiff; Cj.n) are defined as follows:fJ,(f )=()T(X, t ) . i=~2J 3; /="2

a 3 fo) =0 a l ,'tf )= T1 , a 3 ftf ) = 7i

-I

, ,x, q/)

J q,. -;;UJt) /

0,ff,q. )= ~

In such a way, the original minimum time problem for DPS, eqns (1)-(5) has been transformed into the corresponding minimum time problem for the LPS, eqns. (8), (7) and (5).

A=- o H/ Jai

SENSITIVITY ANALYSIS

1.1

cfl

are constant parameters with the values depending on the used approximate method.

QzfiJ=ZJ; rf ) ,

This sensitivity analysis should be used for a more complete consideration of a previous optimal control problem for DPS, i.e. it should give useful information for the design of this class of nonlinear optimal problem for DPS.

.

q

I

ag(f)l

_n

; G (t,ljn)" ~

0) 15

t7

Cf., Cjn '" ]:n where
K; = K; ( 0( , 1:0 ) So , L)

a2 It )= ti, ff ) ,

The objective of this paper is to determine sensitivity of the minimum time problem of DPS defined by eqns. (1)-(5), for small variations of constant plant parameters appearing in eqn. (1), i.e. 0/. , So and Ko. The ana lys is of the in fl uences of a 11 these variables, is of il1lllediate practical interest, particularly the influence of the coefficient cl. . Its values are rather uncertain, because it is obtained as a mathematical approximation of experimental data.

V. . (I

(I

t7

) '"

o;)VicCf;lt ) I~n

-

e

i= f, 2, 3 j = 1, 2, 3 ;

(16)

K= 1,2 Relations among these functions can be simply established. Partial differentation of eqns. (6) and (9) in respect to any of system parameters C/; and combinning of so obtained results with (15), (16), results in ~ e,(t,.x. in ) = A2, (f, Cjn ) rH) [A3/ ,'f,:t. )-A2i If,in ) 1 cp (X) Afl ft, q,, ) (17) k/

J]on

Uti f~in ) =A2i

, Uz /(f,f:,, )=A 3 I fl,'jn )

Hence, by relations (17), the problem of determination of the original sensitivity function for optimal DPS, eqn. (15), is practically transformed into determination of the sensitivity function (16).

( 11f)

o2 / 0}= To,

, H(~ } = 0

So, the original optimal control problem has been reduced t() Nonlinear Two Point Boundary Value Problem (NTPBVP), eqns. (10), (11),(13), and (14). With known solution of that NTPBVP optimal control Iif (f ) and Ul !f) and the corresponding optimal temperature distribution T(.X; t) can be simply calculated using eqns. (9) and (6), respectively. For more details about obtaining necessary conditions for optimality see Petrovacki (1980).

SENSITIVITY EQUATIONS To simplify further consideration of sensitivity of the minimum time problem,without loss of generality, we shall assume that there is no heat generated within the solid ( g ff ) =0 ) and that the temperature on the boundary .x= ! is given in advance as li2 it ) = To -f ; ' T, -Ta ) f / tf (18) Thus, combining (9), (10) and (18) control functi on 112 has the form 1/2 = ( r, -To ) / tf ( 19)

D. P. Petrovacki and D. M. Mar ganovic

572

Now, the control input is only through the boundary x=o i.e. V; = 1/ Since the final time of the process tf' is not specified for further easier solution of NTPBVP we shall introduce new independent variable ?, ( o"r.. f)with its known final value, putting (20)

and introducing the new state variable by means of z ( 7) = If

z (7 )

(21)

Combining (18)-(21) with (9)-(14) we have datfT} / dT

= Kl V"(7) + Kz

dA2i / d'T= V

d Z , / dT = 0

( T; - Ta) 1- Z ( r) W1

do2. ( r }/dT = 7nT ) d z tr) / dr = 0

swi tching instants. Now, taking partial derivative of eqns.(22), (24) and (27) wi th respect to et, ' wi thi n the interval 7 k _1 ",7., Z"'k and respecting eqns (9), (10), (16), (18)-(21) the sensitivity equations with boundary conditions are obtained as follows d A1i / d'T= k, Vr I(/r 'lI f Kz:' (Tf-7;) -I- Z i J.f; + z L~/

(22)

d fii / dr= - ~ (P, Z , +- P"Z)-fI Z {2<01 /

= - 'p1 ( T )

z

tr) W3

d pz ( 7 )/drr = - f -/V'tj Wt where 2 ,

d /il / dT= - ~ (f, Zi

Kit 0 , + (Ks + K7 7O ) a 1 + K7f71 -To )'t 0 ,

W; =

+ K6

a1 + 2 K,! O2 + K10

(7:' -lO) 't+K8

r. + KI2 7,;2 +K'0

-

Yf't) =[O, 02. z p, Pt. pz]7

( 1) = 0

,

02. (t)=

71

A'i

(25) (26)

(27)

Pz (1) =0

But it can be seen that eqns. (22)-(24) define the system with the switching function. Thi s fact is the reason why the procedure for obtaining the sensitivity functions is differen t from the procedure for continous sys tems. Namely, at the switching instants sensitivity functions are di scontinuous so that the jump conditions have to be taken into account (Frank (1978)). Following Frank (D78), (page 179) the regui red jump conditions of the sensitivity ve c tor '[i=oi /'J qi in the swi tching instants ?k (k =~2J) are T

r? h; 1 Y- -t ch;

'!tic

L ?!; J _tk

t k ") hk] 1,- + ddh~f' 0: -.f) [:1 Ik J -

=

+

(0)

=A21 (0) = /;;,

+

A2i (1)=1;,

( 0)=

0

(30)

(1)=O

where I I ~/= K:/ a 1z + 2 K441 Ali -+ (K5{ + K7i T;,)af 1" (K5 r-1::7 To) At! + T [K;, (Tf-7;)a , + K7 (Tf -To ) A1t1+ k:io1a2t-K6AtiO:z + K6 a1 A2i -t k/" + 2 Kfl a2 Ati + ( K;,' +- K:ai To) Q2 + ( K8 + /(10 ?;)A21

a;

FT [K~i ( 71-1;,)01. + K10 (Tf -loJA 2 1

NOW, sensitivity equations from which sensitivity functions (16) can be calculated, may be obtained, respecting (9), (10), (16), (18)-(21), by the partial differentation eqns. (22), (24) and (27) wi th respect to Cl,

+

01

I

dei / dz-= -!?" Wt - If tt:1/

All (1)=

wi th boundary condi tions (obtained by the use of (14), (18)-(21) a 1 (o} =o J O2 (0) =10 , jJz ( O) =O

a,

?'iZ)-/F{ K;,

and: (24)

or in the vector notation ; = f (:! , 1T, 7) where 'i -is a col umn vector of the form

(.

V= -Z, s,qn{hI(1+/7j

~

and

/

To +/(71 If-7;)r

+ K6 Ati + K8i -+ /(10i ~ I / + K10i (1,-7;,)'1:+21(11(°2 T 2 K11 1\2i J

(23)

%:2 Kit 0 1 +K6 0'1. +K7 (71- To)T+K;; + K7 70

~.3 = KG

1I

a , 0 Z +K11 at + ( K8-1-Kl07; ) 0:z+

+ Kl0(71-1o)702.f KfZtT1 -To/'Z"2 + [K9 ( 71 - 10)+2,('12 7,; /71-10 J] '[" +Kg

/

+2 Kit Ati + /(Si + J(7i + K:i a2 -t /(6 A2i )

dPt (T )/d'T= -.!J"Tj Z / 'I } W:z dp~ (7 )/dz-

(29)

+ Kt2I l (r., - Ta )2 't 2 + [K'9 ( (T., - To ) - 2 k:21 ~ (Tr T;)jT-t K;i To + K~ T/

oz

Z ,-= 7J'li p... = ~P1. 1, -

tJC/-,.

e2l -=

10; == ?JKj / ~q.i

djJ2

d'l,'

,i=1,~3

df>z

1;, == 'drt,

; )= 4;2' J 12

Hence, sensitivity analysis for the original minimum time problem for DPS is finaly reduced to the solution of the system equations (29)-(30), couppled with system equations (22)-(24) and (27). Therefore, we have to solve the NTPBVP. When its solution is known, the original sensitivity functions e, (~J()qn) Ufi ( t, 'in ) and Uti U, Cf-n) can be simplycalculated using the equations (17).

T

?'rk

(28)

where 'rift: ,lk, hk and /kdesignated with the mark , ;:) or the mark ,'- ) define their values on the right or on the left hand side at

N~ERICAL

RESULTS

For solving NTPBVP, eqns. (29)-(30) and (22)(27), the Warner's algorithm (a kind of a shooting algorithm) is used because of its properties (for more details see, for example (Marganovic et al. (1979)). This algorithm is successfully used for solving a si-

~ensitivity

of Minimum-Time Problem

mi 1 arc 1ass of problems in Petrovack i (1980) and in Petrovacki and Marganovif (1981). For concrete numerical values of physical parameters, which correspond to a kind of steel s oi.=§fO-

5 0 "'910.5 kca ljocm J

Ta:

kcal/ msoc

/:;,=0.0111081 ;

1=0.1

IfI

(31)

71 =soo"c

100 "C

So the vector of nominal parameters is In=[510-

5

o.01f1081 ,

,

K8= - K5 ,

1(9

K{1=o<:. / sot'

K6=-701 / 5,,(2

= KS

J

J

K7=-3..'/~/

K10 = 80<. / 5 0 / K12=-901 / So/.2 )

2

(32)

For the material properties listed above and by the use of the numerical procedure, previously mentioned, the NTPBVP (22)-(24),(27) and (29)-(30) is solved with these coefficients. Once its solution has been established it can be noted, that the sol ution of the transformed minimum time problem, eqns (22)(24), as well as the corresponding sensitivity function defined with (29)-(30) are really known. Hence, by means of the relations (17) the original optimal sensitivity functions &-, and Uji are easily calculated. The jLlTIP condi tions in A1i and A2i , according to (28), are + _ K.1 ( A21 -A 2 i) A+-A-,·= 11 ' A + - A~ = 2 £ s ign {K1 f; .,. P; l-

2,

I

'(K,

F;;""

is marked by.c,.T ( 4T=e ·LlotJwhile the mark 4 Tit denotes the same value bei ng obtained through the direct calcu~ation of the original optimal control problem for DPS. On the basis of the resul ts computed, temperature deviations caused by the change of parameters ko and So is manifested in the si mil a r way as is the case for the coefficient of. • CONCLUSION

910.5J

Analysis of the influences of various approximate methods(Galerkin method, Gauss principle, Inteqral method)given in Petrovacki (1980) has shown that the influences on the solution of this class of problems are relatively small. In this paper the coefficients Kj which figured in eqns. (8), (22) and (23) were calculated using the Galerkin method and they are as follows (Petrovacki (1980)) 2 K1 =3.5 , 1(2= '1.5 , 1(3=-5/50 / , 1::,,=0
573

~;) /(Kf /r""lr)

Sensitivity functions A'i ( '(,g,,)) A 2 if'r,'l,,) and ZJr; gfl ) for vari ati on of of, ko, So a re shown in Figure 1, Figure 2 and Figure 3, respectively. In these pictures, according to the relation (17) ( U i = A 2 ;) ,values ' of A2i are at the same time the original optimal control sensitivity fun ctions U1i i 'T,ct,, ) The infuence of the swi tching function 1rrT) which figures in eqns (22)-(24), (29)-(30); on the sensitivity function is obvious in these pictures. In Fig. 4, Fig. 5 and Fig. 6 the optimal temperature sensitivity functions {fi ( '"C"",.x. / q.,, ) at five points of solid .x/I=o,O.2, 0.J,5",0"7, o'%for the variation of nonlinearity coefficient eX: , thermal conductivity .to and So which is the product of density and specific heat are presented, respectively. In figure 7, temperature deviations due to the change in the coefficient 0<:: for +5 % in relation to the nominal value at fivE' pOints of solid are given. The deviation calculated by means of the state sensitivity function (f,

The results obtained in this paper have given the opportunity of getting essential information about optimal control problem for DPS under consideration, resulting from the system sensitivity analysis (identification of the infl uence of the parameter variations on the solution;providing of rapid recomputation of the optimal problem for small variation in system parameters; etc.). It has been also stated (in this case) that the solution of the minimum time problem couppled with sensitivity equations, is only to a minor degree more difficult, than the solution of the minimum time problem alone.

It has been pointed out by the previous statement that the sensitivity function could be efficiently used for obtaining missing initial conditions when solving NTPBVP with shooting algorithms as a kind of the continuation procedure see, for example Hall and Watting (1976). Although the sensitivity of a class of nonlinear optimal DPS problem has been treated in this paper, we are of the opinion that the use of this method is not limited to this class only. We feel that the application of the results of the paper (wi th certain modifications) could be expected when solving those optimal control heating process problems, when a rapid change from the initial temperature distribution to the final desired one is demanded. REFERENCES Butkovskii ,A.G, (1979), Control of Distributed Systems (Survey), Avtom. i Telemkh. No 11, 16-65. Chin,K.C, and Shih Y.P., (1972), Low sensitivity optimal control of a class of linear distributed systems, Ing . J.Control, Vol.16, No.2, 325-336. Cruz,~ (1972), Feedback Systems, McGrawHill, New York. Frank,P.'1, (1978), Introduction to System Sensitivity Theory, Academlc Press, New York. Finlayson,B.A, (1972), The ~ethod of Weighted Residuals and Varlatlonal Prlnclples, Academlc Press, New York. Good~R.T, (1964), Application of Integral ~ethods to Transient Nonlinear Heat Transfer, Advances in Heat Transfer, Vol. 1, AcademlC Press, New York.

574

D. P. Petrovacki and D. M. Mar ganovic

Hall,G. and Watt,J, (1976), "1odern Numerical Methods for Ordinary Differentlal Equatlons, Clarendon Press, Oxford. Marganovlc,D., Petrovacki,D. and Baclic,B .• (1979), Use of Warner's Algorithm for Solving Some Heat Transfer Problems, International Conference on Numerical Methods ln Ihennal Problems ,Swansea . Pedersen,K.C. and Nardizzi,L.R . , (1972), Optimally Sensitive Control for Distributed Parameter Systems, Int.J.Control, Vol.16, No.4, 723-735. Petrovacki,D.P, (1980), The '1inimum TiJre Problem for a Class of Non-Linear Distributed Parameter Systems, Int.J.Control, Vol. 32 , No.1, 51-62 .

Petrovacki ,D.P. and Marganovi c ,0."1, (1981), The minimum time problem for Nonlinear Heat Conduction, IFAC 8th Triennial World Congress (Prepnnts). SesSlon 8, Kyoto. Ray,W . H, (1978), Some Recent Applications of Distributed Parameter Systems Theory A Survey, Automatica, Vol .14, 281-287. Ray,W . H. and Lalnlotls,D.G, editors, (1978), Distributed Parameter Systems, Marcel Dekker, New York. Robinson,A.C, (1971), A Survey of Optimal Control of Distributed - Parameter Systems, Automatica, Vol.9, 371-388. Sage,A.P, (1968), Opbmum Systems Control, Prentice Hall, Englwood Cllffs, New Yersey.

!

Al Az Az

10 7

10 5

I

j !Od

10

5

Fig. 1. Sensitivity h!~:tions At , A2 and Z for variation of a .

Fig . 2. Sensitivity functions At , A2 and Z for variation of k . o

. 1rf-·

Fig. 3. Sensitivity functions At , A2 and Z for variation of So.

575

Sensitivity o f Minimum-Time Prob lem

~s

!Er

la '

c.

. ".

,. .

"

~

0

0

:

10 ~c; ~

10 3 i

c;

10 "

"

1. 3

0 0

~ 00

" Z

t~

0

0

10

..

j

0 0,20

:J

0

.:.

~

0

0

0

0 :J

T 0,6

0.7

0,9

0,6

, c

"

0

." .

0

11

i - 10 5 :

0

•

0

+

Cl 0

•

.

"

c

0.20 0,45 0,70 0. 95

"0

0,2

c-

0

0.3

-

-Q,4

" 0

~ ---~ 0.5

0,6

0,7

T 0.11

0,8

I

"

A

-10 I

j ;.

2 '

-10 -

0

~

0

'

..

""

-10 3 -

•

. 0

60

•

$*

0 Cl

~

0

, Cl

o ,, '

0

-.

0

1

-10 0

0

10

.

":,~-I ~-----10

" " 0

xII

Symbol

0,1

"

"

1;

0

iI

~

0,5

0,4

0 0

10 0 e

0,95

-10 3

-la " -

.>

"

S:J

0

0

0,45 0,70

.: ~" i

i ago

I la ' J 8

"

x/l

Symbol

0"" I·

2

"

"C

,t

"

-la "

+

!

0

io

10

0

0

. . ". 0

0

_10 " "

Fig . 4. Optima 1 tempera ture sensitivity func ti on e for vari ation of a .

Fi g. 5 . Optima 1 tempe ra ture sens i ti vi ty function e for variation of k o'

D. P . Petrova~ki and D. M. Marganovi~

576

~&

-* *

10" ' .

10'3.

c:

~

" -

.

=

c

~ .

-Symbol

x/l

"

0

Q

10' · .

~

c

0.20

"

0.45

<.

0.70

c

0 .95

la' ; ,

o ~. I 0.1

r 0,2

0.4

0 .3

0.5

0.6

0.7

O.S

0.9

-10 ; j

-10 "o

_10-3 ~

" II C

=

. - 1" 0 !

o

g

= 0

o o

c

0 .\

Fig . 6. Optimal temp erature sens itivi ty function e for variati on of So' liT", liT

Symbol

x /l 0.05 0.1

1 -

0.25 0.5 0.75

o r"'_ _:::- . - - : - - - -

-1

-2 -

Fig. 7. Temperature deviation for 6a (6T) and directly (.".T* ) .

5 :, calculated by means of sensitivity function