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Dynamic optimization of a steel-making process in electric arc furnace
Automatica, Vol. 6, pp. 767-778. Pergamon Press, 1970. Printed in Great Britain.
Dynamic Optimization of a Steel-Making Process in Electric arc Furnace* L'optimalisation dynamique d'un proe&t6 d'61aboration d'aeier dans un four 61eetrique ~t arc Dynamisehe Optimierung der Stahlherstellung in einem Liehtbogenofen ~HHaMnqecI(a.q OIITHMH3RI_IHII npouecca ablpa6OTrH CTaJIH B 3Ylek'TpnqeciCOfi /lyroBofi
ne~m
A. G O S I E W S K H " and A. W I E R Z B I C K I t
The Maximum Principle is not only a pure theoretical concept but it can be also fruitfully used as a powerful tool to the design of optimal control systems involving real industrial processes. Summary--The paper presents the theoretical basis, technical design, and preliminary experimental results of the dynamic optimal control of a steel-making process in an electric arc furnace. As a performance measure of the process, the unit production cost is assumed to be a combination of the cost of electric energy and the cost of time. A system of ordinary differential equations is taken as the mathematical model of the process. The theoretical problems of process optimization and of control system synthesis is solved by means of the Maximum Principle. It is shown that, with some assumptions, optimum control can be accomplished by means of a peak-holding controller. The technical design of both open- and dosedloop optimal control systems, now being implemented, is based mainly on analogue techniques with respect to both control optimality and a temperature program. The experimental results of the optimal control achieved on the basis of prepared algorithms and presented in the paper confirm theoretical estimations. The possibility of considering an expanded problem nvolving optimization of a number of are furnaces with respect to joint constraints is also discussed at the end of paper.
Defined Symbol by equation
c
(22)
(15)
Co C*
(3) (16)
Cp
(1)
Cpm fl f2
A F
NOTATION Defined Symbol by equation
c*
F, H i i t
Purpose dimensionless cost o f 1 hr o f furnace usage
to io
* Received 16 June 1969; revised 30 March 1970. The original version of this paper was presented at the 4th IFAC Congress which was held in Warsaw, Poland, during June 1969. It was recommended for publication in revised form by associate editor A. Spang. The problem of optimization of steel-making process in an arc furnace which is presented in this paper has been investigated in a collaboration between the Chair of Automatic Control of the Technical University of Warsaw and the Division of Electrothermics of the Electrical Engineering Institute, Warsaw-Miedzylesie, as well as the Stalowa Wola Foundry by which the project has been supported. t Technical University of Warsaw, Chair of Automatic Control, Warsaw, Poland.
is Ai I
! /m
t,
(20) (2) (2) (2) (13) (7) (7) (20) (37) (25) (25) (36) (37) Fig. 7 (12) (20) (19)
J k
767
(21)
Purpose
dimensionless cost c, related to the constraint m a cost o f 1 hr o f furnace usage effective cost Co, related to the constraint Md instantaneous cost o f a supplied power cost Ce at the maximal heating power rate o f the charge temperature rate of the walls and b o t t o m temperature rate o f the r o o f temperature m o m e n t a r y performance index static performance index hamiltonian dimensionless arc current mean value o f i optimal arc current initial value o f the current t constant measurement error o f i amplitude o f r a n d o m disturbances searching step length arc current optimal arc current maximal arc current optimal are current, constant in time tap n u m b e r gain coefficient
768
A. GOSIEWSKIand A. WIERZBICKI Defined
Symbol K m rh md ms
M
gd Ph
Ph
p', pP Ph Phm Pi
Pt Pts
q q~
O Qeq Sd Ss
t ta T H~, I1p 0 ~, 13#
W X
Xd X1 X2 X3
xo xo
a
by
Defined Purpose
Symbol
by
Purpose
equation
equation
(6)
(15)
penalty function coefficient dimensionless mean cost of power (28) cost m corresponding to the dynamic optimal solution (23) admissible value of the cost m (33) cost m corresponding to the static solution (4) mean cost of power (4) admissible value of the cost M (20) dimensionless heating power (38) mean value ofph heating power of the furnace, number ~, fl (1) heating power (20) maximal heating power (37) probability of an arc current value (2) thermal losses (18) mean value o f P t (22) dimensionless performance index (27) minimal value of q (32) index q value corresponding to the static solution (3) performance index (6) performance index with the penalty component (30) sensitivity measure corresponding to the dynamic solution (35) sensitivity measure corresponding to the static solution time-variable (3) duration of the melting stage (21) charge time constant casting the furnace, number ~, fl loading the furnace, number ~, fl (2) thermal capacity of the charge (21) state variable: dimensionless temperature of the charge or bath (26) optimal temperature (24) final value of the temperature x (2) state variable: the charge or bath temperature (2) state variable: the walls and bottom temperature (2) state variable: the roof temperature (5) additional state variable: time t (5) additional state variable: the cost of energy (40), (41) output signal of a measurement device furnace number (40) forcing coefficient of a measurement device (23)
®
(40)
2
(41)
¢o
(25)
,g
(21)
Td
Ca
(29)
"~ds
(34)
0, ~,o, ~o
~, C,
(7) (7)
correction in optimizing feedback structure time constant of a measurement device relative error of the measurement of the state coefficient penalty coefficient, the coordination variable dimensionless time-variable final value of z, the melting stage duration final time rd corresponding to the optimal solution final time rd corresponding to the static solution costate, i = 1, 2, 3 costates adjoined to the additional state variables x °, x ° costate adjoined to the timevariable t costate Ot for the furnace number ~,/3 1. I N T R O D U C T I O N
THE GOAL of the steel-making process in an electric arc furnace is to obtain a high quality steel by means of melting selected steel-scrap, purging the melted steel, the "bath", of impurities and adding some ingredients to improve the quality. The main source of energy in the process is the heat supplied by a three-phase electric arc which burns between three graphite electrodes and the scrap, or bath. The process in an arc furnace is a typical charge-process; each charge, beginning with loading the furnace and ending with casting of finished steel, lasts about 6 hr for a furnace of an average load which may be about 30 tons. The average power consumption amounts about 0"6 MWh/ton, the average arc power--about 3MW. Each charge can be divided into three main stages: (1) the melting stage which lasts for about 0.5 of the whole charge duration time and demands about 0.7 of the whole charge energy consumption ; after melting, the bath is overheated to a given temperature, the "overheating temperature", the first slag is cast and oxidizing components are brought into the bath; (2) the oxidizing stage in which the impurities--such as coal, phosphorus and sulphur--are burned out; after oxidizing, the next slag is cast; (3) the refining stage in which the oxygen is reduced and the improving ingredients--such as manganese and chromium--are put in the bath;
Dynamic optimization of a steel-making process in electric arc furnace after refining, the molten steel is overheated and the casting of finished steel follows. 2. STATEMENT OF THE PROBLEM
The notion of "optimization" in an arc furnace described here, is that which results in the minimal cost of finished steel (zl/ton)* taking into consideration the cost of electric energy (zl/kWh) as well as the constant costs of time (zl/hr) during which a furnace is used. The problem is of a great economic importance: the energy cost for a 30 ton furnace amounts to 10 million zl per year and the constant cost of 1 hr of furnace usage can be estimated as 6000 zl. The optimization of a charge is usually related only to the melting stage [1]-[3], which determines the total energy consumption and influences strongly the total charge duration time. The oxidizing and refining stages are not included in the optimization because their duration and energy consumption, the temperature program, depends on chemical processes in the bath and are strictly determined by the process technology. The optimization of loading and casting of an isolated furnace is trivial: these functions must be performed as intensively as possible. Consider the simplified design of an arc furnace with only one phase of the arc circuit shown in Fig. I. Each electrode is supplied through a power cable
769
assume for the first approximation--apart from rapidly varying disturbances--that I = Io. Therefore the main controls of the process are: j
the tap number of the secondary transformer winding;
1 the arc current, or more correctly, the reference value Io of the arc controllers. It is easy to show [2] that the secondary voltage of the transformer should be kept as large as possible during the melting stage. Since the tap number is selected to do this, the only control during the melting stage is the current L Theoretically, the process can be divided into two sub-processes: the power conversion subprocess and the thermal one as indicated in Fig. 2.
LI
*l =fl (xl,xz, x3;I) Cp(I) P,(I)
i(~ =f,~(Xl, x2, x3) :t~ =f3 (xl, xz, x3)
F~G. 2. Power conveirsion subprocess and thermal subprocess.
The power subprocess is practically inertialess compared to the thermal one, and it can be desE[ I--"1
T
]llt
FIG. 1. Design of an arc furnace. C from the furnace transformer T which has several taps of number j = 1, 2 . . . . . on its primary winding; the taps are set by a tap-setting device that determines the nominal secondary voltage. The arc current 1is controlled by the arc controller, A C. This includes the electrode servomecharOsms, M, which lift or lower the electrodes according to the difference between I and the reference value Io. Because the arc controller acts rather quickly compared to the duration of the charge, we can * zl. is the abbreviation of Polish currency (zl.oty).
cribed for the optimization purposes by the two basic static relationships: Cp = Cp(l) ; Ph =Ph(I)
(1)
where Cp is the momentary cost of a power supplied (zl/hr) taking into account the active power as well as the reactive power and Ph is the heating power of the arc. Those characteristics can be determined either analytically--by means of an equivalent diagram of the electric circuit--or experimentally. Taking into consideration rapidly varying disturb-
770
A. GOSII/WSKIand A. WIERZBICK1
ances caused by arc burning those relationships should be treated as regressions. For approximate considerations, which are presented in section 3, the relationship Ce(1) can be assumed to be linear, the relationship Ph(I)--to be parabolic. An example of real characteristics (1), determined experimentally, is shown in Fig. 3. 5 4
o
furnace geometry and physical properties. It turns out, however, that the decisive part of the thermal power P1 depends mainly on the temperature x~ and not on the temperatures x2 and x 3. For that reason for approximate considerations, discussed in section 3, the model of the thermal subprocess can be reduced to the first of the equations (2) which can be further approximated by a linear equation of a simple inertial dement. For a given load, the performance of the melting stage can be expressed by the cost index Q=
¢0
)
l
I
I
[
f
td
[C O+
Ce(1)]dt
(3)
0
1
where td is the melting stage duration which is not given but determined by the overheating temperature xl(t~)=x~. The constant Co denotes the constant cost of 1 hr of furnace usage (zl/hr), which results from wages, redemption and so on. The mean power transmitted by the furnace transformer is constrained. Because the transmitted power is approximately proportional to the instantaneous cost, the constraint can be expressed as
6--
2
I0
20
30
L
40
50
60
tO
kA
Fro. 3. Cost of electric power (CP) and heating power
(Ph) versus arc current (I). Generally, a model of the thermal subprocess should be formulated as a very complicated partial differential equation with respect to the temperature x in the furnace space. That equation can be approximated, however, by a system of ordinary differential equations taking into consideration three mean temperatures as the process state variables: the charge, or bath, temperature x~, the walls and bottom temperature x2, and the roof temperature x3. The model of heating subprocess then assumes the normal state equation format
Pt(x" xz,
f '" Ce(I)dt 0
t~
~
(4)
where Md is a given admissible mean value. Now the optimization problem can be formulated: determine such a function of time I=I(t), tel0, td] which minimizes the performance functional (3) with the differential equations (2), the global constraint (4), and with the final condition xl(td)=x~. The solution of the problem can be based on the Maximum Principle [4]-[6].
3. THE METHOD OF SOLUTION Because of the existence of the integral constraint (4) let additional state variables, x ° and x °, be introduced and included in equations of the form
~ = f x(xl, x2, x33, I)
- W1, :[P'(I)-(xy
M=
x3)]
(2)
:Cz=f z(xt,
X2, X3)
(5)
:c3=f a(xl, x2, xa) where: charge,
W(xl)
Pt(xl, x2, xa)
is the thermal capacity of the
is the thermal power transmitted from the charge to the walls, bottom, and roof by conduction, radiation, and convection, the losses. The forms of the functions f t , fa, fa are rather complicated and depend on the kind of the heat exchange between walls, bottom and roof, on the
~g=c,(i) Thus, the initial problem can be formulated as a problem with the constrained ratio of the variables xO(td)/x°(td) at the terminal time td. It is useful, however, to reformulate the problem further by introducing a new cost index with a "penalty for violating the constraint" instead of the index (3). Thus,
Qe~= Q+K(M-Ma) 2 " I ( M - M # )
(6)
Dynamic optimization of a steel-making process in electric are furnace where K is a sufficiently large constant; a violation of the admissible mean value Md is associated, therefore, with a rapid increase of the functional (6). It can be shown [6] that with K ~ o o the reformulated problem, without the constraint, is strictly equivalent to the initial problem. The hamiltonian of the problem has the form
771
and then the necessary condition (12) can be rewritten in the form F(1)--- max F(l) for all t~[0, td].
(14)
I
The function ~/(Xl, x2, x3, t ) = ~ [ d / 2 ( t ) f 2 ( x l , x2,
X3)
H = - (C o - ~,o) _ (1 - ¢°)C1,(I)
+~k3(t)fa(xt, x2, x3)]
3
+ ~ dAf,(xl, x2, x3, I)+d/,
(7)
i=l
where the costates ~kare determined by the equations OH.
¢°=°;
(8)
The transversality conditions for g,o and ~b° have the form, of. Ref. [6]
Ox o
= K M 2 ( M - Ma)I(M- Ma) t> 0
ta
~b~= -~Q___2q ax o
(9)
= - - ~ ( M - M~)I(M- Ma) <<.0
t~
It follows from the form of the hamiltonian (7) and the conditions (9) that occurrence of the global constraint (4) leads to suppressing the constant cost of furnace usage and to increasing the cost of electric power; evidently Co>
,° .
(10)
Note that if this inequality does not hold then the furnace transformer is incorrectly designed. Because the process dynamics are determined mainly by the temperature xt it can be assumed
l¢,lfll,>l¢,2f2+a/3f31
wherefa>0, ~1>0.
(11)
As the inequalities (10) and (11) hold, the principle of optimizing feedback [6] can be applied and the necessary condition of optimality reformulated
maxH(l)=H(1)=O for all re[0, td]
(12)
I
where [ = l(t) is the optimal control. We introduce, therefore, the following instantaneous performance index
F=Ph(1)-Pt(xl, x2, x3)+r/(xt, x2, x3, t) (13) + c (t)
(15)
is a correction which represents the influence of temperatures x2, x3 on the optimal solution of the problem, whereas the constant Co*= C o - e ° . . 1-
(16)
--o
is the normalized constant cost of the time of furnace usage considering the constraint (4); the equality holds if the constraint does not influence the optimal solution of the problem. It should be noted that if the influence of the temperature x2, x3 on the process dynamics is negligible then we can assume t/=0. If the influence of temperature x2, x3 is small, then the correction rl is small compared to (Ph-Pt), and it can be determined approximately. Thus determination of the dynamic optimal control :(t) can be reduced to the peak-holding control of the instantaneous performance index only on the basis of the measurements of the powels Ph, Pt and the cost Cp and of an initial knowledge of the normalized cost C* without necessity of an exact knowledge of the functionsfi,f2,f3 and of the correction t/. This is the essential property of the optimizing feedback. In the previous papers mentioned earlier on the control of arc furnaces, the optimization problem has been treated only as a static, or steady-state, problem and the process dynamics (2) as well as the global constraint (4) have not been taken into consideration. When using this approach it can be assumed that the arc current I is constant in time. This current is then determined in such a way that it minimizes the performance function
Q~= Q[Co+ Cp(I)]
(17)
which is a static analogue to the performance functional (3). On the basis of the thermal balance
td[Ph(l)--Pls]=XatW whence td-
x~W (18) Ph(l)-Pts
where Pu is the mean, constant thermal power lost from the charge. The condition of the static optimality can be rewritten, therefore, in the form
F~- x~ W _Ph(l) - Pl, c
(I)
772
A. GOSIEWSKIand A. WIERZBICKI 4. SIMPLIFIED ANALYTIC SOLUTION AND APPROXIMATE SENSITIVITY ANALYSIS
and
Fs(]s) = max F~(I)
(19)
!
where F~ is the static performance index, and f,, the constant, static optimal control. The conditions (13), (14) and (19) appear to be formally similar, but they are essentially different. The difference between them is illustrated in Fig. 4. The static optimal heating power Ph(l~)
By choosing a sufficiently simplified model, the problem can be solved analytically and qualitative conclusions about the main features and the sensitivity of the solution can be obtained. According to the reasons given in section 2, the equations of the electric circuit can be assumed to be, in the dimensionless form, ph=l-(1-i)
2
where
Cp
Ph. ph=~,
I lm
Ph~
(20)
Ph denotes heating power, i, arc current, Phm, the
I
Ph
)
(a)
Ce=Cl,mi
i=--;
-C,
maximal heating power, I,,, the arc current which results in the maximal heating power, Cem, the cost of a power supplied at the maximal heating power. Assume, furthermore, a simplified equation for the dynamics
dx --=£=ph(i)-x
(21)
dv where
Cp
f . "t=--,
Xt X~----
T
/
/
J !
(b) -C(
_t
P~-,7 Fic. 4. Graphical illustration of the static (a) and dynamic (b) optimization of the melting stage in an arc furnace. can be determined graphically (Fig. 4a) by drawing a tangent to the plot Cp(Ph) from the point (Pis, -Co). Similarly, we can determine the dynamic optimal heating power Phil(t)] (Fig. 4b) if we assume Pt>>q and draw a tangent from the point [Pt(xl, x2, x3)-C~]. The thermal power Pl(xl, x2, x3) increases during the melting time; so does the dynamical optimal heating power Pn[f(t)] as indicated clearly in Fig. 4b. This property of the optimal solution has a simple physical interpretation: at the beginning of the melting period, the power of thermal losses P~ is small, the charge is cold, which implies a small heating power Ph and a high efficiency of the thermal subprocess; during the melting period, the power Pt increases and, in turn, the heat power Ph must be increased in order to shorten the period of large thermal losses, or to "compensate" for them.
-- Xl
kP~m xlm
x and xl should be treated as increments whereas the differentiation is understood to be with respect to the time-variable z; T denotes the charge time constant, k, the gain coefficient, xl,,, the maximal increment of the charge temperature which can be achieved by permanently applying the heat power Ph. Now, the performance index can be written as
q=
i)dz
(22)
where Q q = C-~mT,
Co c = Cpm .
The constant c is the dimensionless constant cost. The constraint of the mean power supplied assumes the form m = 1 I~d~ <~ma
(23)
TdJO
where
m d= ~Md . Cem Under the given end-point conditions x(0) = 0 ; x(%)=xn; ~ a - free
(24)
Dynamic optimization of a steel-making process in electric are furnace the problem of the minimization of the functional (22) under the differential conditions (21) and the integral constraint (23) can be solved analytically; the solution can be based on the Maximum Principle. Then the optimal control f = fix) is of the form
773
assumed that i = c o n s t ; then accordingly to equations (21), (17) and (18), x=[1-(1-i)2](1-e
-')
(31)
and
q,=(c+i)(za~)
(32)
m,=i
(33)
(25)
~ = 1 - ~ o e-~ where
and,
40-
1
l+c*+x/2c*+(c*) 2
c,_c-O° 1-¢°"
1-(1-02
As indicated above, c* can be interpreted as the dimensionless constant cost reduced according to the influence of the constraint of mean power supplied. The optimal temperature as a function of time 2 = 2(z), takes the form £ = (1 - ~o2e-')(1 - e - ')
/41= 1 - ~-2(1 - e - "9 ,isd
The equation (dqs(i))/(di) = 0 is transcendental; the simplest way to obtain the solution is to assume some values of i and to determine the values Zds(i) and q~(i) according to equations (32), (34). It is of use to introduce the ratio.
(26)
and the value of the performance functional 4, the constraint functional rh and the duration of the process cd can be expressed as
O=(l+c)Ta-~o(1-e TM)
(27)
s~(c, i)- qs(c' i)- O(c, C).lO0% O(c, c)
(35)
which shows how much worse are the results of static optimization than those of dynamic optimization. The results of that comparison and the sensitivity characteristics for a chosen value of c are presented in Fig. 5. It is worth mentioning that although the
(28) 20
.~ , , f
(34)
~d*=lnl_(l_i)2_xa"
l+c*+x/2c*+(c*)2+x a ]
18
a. . . . ~ - - - - ~ :;,. (29) ((1 - xd)[1 + c* + x/2c* + (c*)2]J
_ \
\ @,
16 14 12
It can be assumed c=c* if the inequality ma holds. If the inequality does not hold, the transcendental equation n~(c*) = m a should be solved and the necessary value c* < c determined. However, it is simpler to assume some values of c*~< c and determine the values ~a(c*), rh(c*) and q(c, c*) according to equations (27), (28), (29). Then, not only can the solution be obtained, but also the sensitivity characteristics of the solution with respect to the integral constraint and to the initial current t o = l - G 0 can be obtained also. It should be stressed that errors in determinat;on of that initial current can be caused by an incorrect estimation of such parameters as Cem, Ph or Ira. Therefore it is useful to present the results obtained as the values of the sensitivity measure [6]
to
I0 8 6 4 2 0
0 I
02
0'3
0.4
0"5
0-6
(30)
In order to compare the results of dynamic and traditional, static optimization, it should be
0.8
09
I0
zo 18 I6 14 12
o4
IO 8 6 4
:.,;7// I
2
sa(c, c*)- O(c, c*)- q(c, c)100%. O(c, c)
0.7
0
0"1
02
0'5
04
0'5
0"6
0.7
0"8
0.9
1-0
i, io
FIG. 5. Sensitivitycharacteristics of the optimal control, with respect to the chosen initial current value i or io, to the constant c* and to the integral constraint ma, for various c and xa.
774
A. GOSlEWSKIand A. WIF,RZBICKI
relative profits of the dynamical optimization are rather small, the sensitivity of the dynamic optimization with respect to the constraint of mean power supplied and to the choice of initial current is much lower than the sensitivity of the static optimization. It follows from this that the stronger the influence of the average power constraint, the more profitable is the application of dynamic optimization. In a similar way, [6], the sensitivity of the optimal solution with respect to the constant errors i, in the determination of optimal control t can be examined by assuming that
20 18
c=0"2 xd=0,6
16 14 12
S
I0 8 6
(o)
-02
!
:
-0"1
0
OI
0"2
i, 20
18
i = t + i,,= 1 + i,-~oe-"
(36)
16 14
or with respect to the amplitude i, of rapidly varying random disturbances ip of the arc current by assuming
=i+ip;
i=~;
ip=
t
+ i°,
8 6
P,(+ i,,)=
-
P,(0)=½
i.,
P , ( - i.)=
~h=ph(O--~(i.)2= 1--(1-- [)2__~(i.)2.
(37)
max F( i)= FO) .
-03
-o2
-ol
0
ol
o.z
0.3
i0 FxO.6. Sensitivitycharacteristicsof the optimalcontrol, with resl~et to the constant error ia (a), and to the amplitude & of random disturbances(b).
(38)
Some results of this sensitivity analysis are presented in Fig. 6. It should be emphasized that during the melting stage the disturbances of the are current are strong and the losses associated with them can amount to several percent of the optimal performance--as indicated in Fig. 6 either by dynamic or by static optimization. It is very important, therefore, to have a high performance of the arc controllers which counteract those disturbances. The sensitivity analysis presented above is related to the open-loop optimal control system. An application of the optimizing feedback corresponds, according to equation (14), to searching the optimal current f by the peak-holding control with the temporary index
F -p(i)-x c*+ i
S
2
(b) 0,
-
4
where [ is the mean value of the current, ip is the random component and P~ denotes probability. It is assumed here that the temperature x depends solely on the mean value of the heating power Ph which is in turn determined by changes of the mean current f= f(r). Then, according to equation (20),
i
12 S I0
(39)
Thus, the optimizing feedback results in full insensitivity of the system with respect to parameter variations in Ph,,, Ce,,, I,., k and T because the values p(i), x and i are determined by measurements. On the other hand, the sensitivity with respect to the setting the constant c*, the estimation of the mean power eonstraint's influence, and to the rapidly varying disturbances of the arc current, which cannot be suppressed by the rather slowly acting optimizing feedback, does not change. In order to estimate the influence of the measurement accuracy on the optimizing feedback performance, further sensitivity analysis has been carried out by means of digital modelling the cases when the state x, corresponding to the power of thermal losses Pt, is measured indirectly through an output signal y which is characterized by the time constant ® of the measurement inert[as and compensated by the forcing coefficient 6
Op + y=60:~ + x.
(40)
In determining the optimal control, the measured values of the signal y are put in place of the state x in the index F. The state x may be measured also with a relative error 2 y = ( l +2)x.
(41)
Another kind of the performance loss in the optimizing feedback system may be caused by
Dynamic optimization of a steel-making process in electric arc furnace finite searching steps of length Ai in the real peakholding controller of the index F. The sensitivity analysis results for these three cases are presented in Fig. 7a-c. It follows from them that the optimizing feedback system has a very low sensitivity to the measurement inaccuracies and a reasonable sensitivity to the searching step length since in a real system it can be assumed 2 < 0.1 and ® < 0.1 which results in s<0.5 ~o.
8:0
(a) 0 2
014
0,6
08
I0
8
S
0 I~
775
The results of those computations differed only slightly from the results presented above, rather quantitatively but not qualitatively. In order to verify experimentally the results achieved, the charges of 30 ton furnace in the Stalowa Wola Foundry were run during 6 months according to the optimal, or more correctly, the sub-optimal, program of arc current changes which were determined in advance. The program was conducted by a human operator who set the reference values of arc controllers. Then it was proved that the energy consumption for the melting stage decreased by about 5 per cent or by about 3 per cent with regard to the whole charge and 1 ton of steel, and the melting duration was shortened by about 11 per cent, or by about 5 per cent with regard to the whole charge and 1 ton of steel. The improvement of the cost index was about 8 per cent higher than that obtained in the theoretical research. This can be explained by the fact that the furnace had not previously been operated according to the static optimal conditions, as assumed in the theoretical studies.
O.I ,
(b) -- 0 'I0
_0.05
0
0.05
OIO
6. TECHNICAL IMPLEMENTATION
>,
For a technical implementation of the optimal control system for the process in an arc furnace the specialized control equipment has been designed and constructed. It consists of:
14l 12 l0 $
8
(i) modified arc current controllers and a remote controlled tap-setting device;
4 2
(c/
0
0.04
0100
0 112
O' 16
Ai
Fie. 7. Sensitivity characteristics of the optimizing feedback system, with respect to the measurements inertia O (a), to measurements error 2 (b), and to the step length Ai of the peak-holding controller (c). 5. ANALYSIS OF MORE ACCURATE MODELS A N D EXPERIMENTAL VERIFICATION OF RESULTS ACHIEVED
By applying more accurate models of the relationships (1) and the dynamics (2), several other computations have been carried out; the purposes for doing this were to obtain: (i) a verification of the modelling accuracy by computing the melting duration as well as the energy consumption when the usual are currents are applied and by comparing the results achieved to experimental data; (ii) a more accurate determination of the optimal current [(t) on a basis of the maximum principle and the numerical algorithms related to it [6]; (iii) a more accurate sensitivity analysis of various structures of the optimal control.
(ii) devices for automatic measuring and transducing the temperatures and electric power; Ctii) a device for generating the open-loop optimal control; (iv) an optimizer consisting of a computing device for the index F and the peak-holding controller; (v) a programming controller of the bath temperature that operates during the oxidizing and refining stages. A simplified block diagram of that control equipment is presented in Fig. 8. Application of a digital computer for the optimal control of the process in are furnace has also been considered. However, it has turned out that an application of a digital computer exclusively for that purpose is unprofitable. 7. POSSIBILITIES FOR F U R T H E R RESEARCH A N D APPLICATION
The application of a digital computer instead of the specialized control equipment described above may be profitable in an extended problem where the
776
A. GOSIEWSKIand A. WIERZBICKI
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t FIG. 8. Block diagram of the control equipment of an arc furnace. Symbols: Pno, xlo, lo, reference values of the heating power Pn, the temperature xl and the arc current I; A/o, the step of Io.
optimal control would be determined for the whole complex of a number of furnaces in a foundry. This problem is directly associated with programming the optimal operation programs for such complex. In principle, optimal programs for each furnace could be determined independently. However, common constraints occur here which follow from: (i) admissible power supplied for the complex of furnaces; (ii) constrained transmittance of the loading devices for that complex; (iii) constrained transmittance of the casting devices for that complex. In order to consider those common constraints there is a possiblity of applying multilevel optimization methods. On a basis of these methods, the numerical algorithm for determining the optimal program of the operation of furnace's complex has been proposed [6]. This algorithm is of an iterative nature: in the first computation, the first "iteration", the optimal programs for each furnace are computed independently; afterwards, "the penalties" for violating the common dependent constraints are determined, the computations are repeated and so on. An example of the results obtained from computations using such algorithm are presented in Fig. 9 where two furnaces are
involved. In the first iteration (i=0) the optimal electric powers, loadings and castings for two furnaces, denoted c~ and /y, are determined; the variables, ~kt, denote the co-states adjoined to the time which, after a sign change, can be interpreted as the cost of usage time of a furnace. After completing the first iteration and finding the violation of the common constraints, the penalty coefficients rt are determined. These coefficients permit repeated computations which influence the nature of "the time cost" variation, ~t, and they result in a speeding up of the operations for the ~-furnace, but a slowing down of the operations for/Y-furnace. In the subsequent iterations, the operation programs for both furnaces separate appropriately and violations of the common constraints are reduced to minimum. 8. CONCLUSIONS
It was shown that the problem of dynamic optimization of the melting stage of steel-making process in an electric arc furnace can be formulated and solved by means of the Maximum Principle. Due to the global constraint which has the form of the ratio of two integral functionals, a generalized version of the Maximum Principle was applied to the stated task. It turned out that the Maximum Principle in the case considered can be transformed into a so called optimizing feedback principle which
Dynamic optimization of a steel-making process in electric arc furnace izO
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permits synthesis of the optimal control system without a priori knowledge of the mathematical model of the process when it is of the first order. When the model is of higher order the same optimizing feedback can also be applied, but for proper operation, some additional correction is necessary. In a general case, the optimal control can be determined only by numerical computations. An analytic solution of the problem can be obtained for the simplified linear model of the process, and this solution makes it possible to analyse the sensitivity of the optimal control with respect to various errors in determining process parameters, to the global constraint and, what is particularly important, to the results of dynamic optimization with static solutions. The conclusions from the theoretical analysis have been applied to the effective control of the process in an electric arc furnace. The results of this application have confirmed the theoretical analysis and economic purposefulness of the research. Further investigation related to the optimal programming of a complex of arc furnaces shows that there is a possibility of applying multilevel optimization algorithms in order to solve this important problem.
REFERENCES [1] J. E. YEFRIMOVITCH" Automatic control of steelmaking are furnace (in Russian). Metallurgizdat, Moscow (1958). [2] L. McGE~ and A. METCALF: Controlling arc-furnace power. J. Control, 91-96 (1962). [3] A. N. SOKOLOV: Reasonable operation conditions of steel-making arc furnaces (in Russian). Metallurgizdat, Moscow (1960). [4] A. GOSIEWSKI and A. WmRZmCKi et aL: Automatic optimization of steel-making process in arc furnace (in Polish). Proc. 4th National Control Conference, Vol. 7, 73-80, Krak6w, 1967. [5] L. S. PONTRYAOIN, V. BOLTYANSKIJ, R. GAMKRELIDZE and E. MISHCHENKO: The Mathematical Theory of Optimal Process. Interscience, New York (1962). [6] A. WIERZBICKI: Maximum principle and synthesis of optimal controllers. Pt. III: Sensitivity and structure of optimal controllers. Pt. IV: Example of industrial application (in Polish). Archiwum Automatyki i Telemechaniki 14, 75-101, 187-213 (1969). R6sum6--L'article pr6sente la base th6orique, la r6alisation technique, et les r6sultats exp6rimentaux pr61iminaircs de la commande dynamique optimale d'un proccssus d'61aboration d'acier dans un four 61ectrique ~ arc. A titre de mesure des performances du proc6d6, le cofit unitaire de production est suppos6 ~tre constitu6 par une combinaison du cofit de l'energi¢ 61ectrique et de celui du temps. Un syst~me d'6quations differentielles ordinaires est pris comme mod~le math6matique du proc6d6. Le probl~me th6oriques d'optimalisation de proc6d6s et de synth~s¢ de syst~mes de commande sent r6solus ~t l'aide du Pfincipe du Maximum. II est montr6 qu'avec certaines hypoth/~ses la commande optimale peut 6tre r6alis6e au moyen d'un r6gulateur oxtromal.
778
A. GOSIEWSKI a n d A. WIERZBICKI
Le probl6me d'optimalisation du procddd d'61aboration d'acier clans un four /t arc pr6sent6 dans cet article a 6t6 6tudi6 en collaboration entre la Chaire d'Automatique de I'universit6 de Varsovie et la Division d'Electrothermie de l'Institut Electrotechnique de Varsovie-Miedzylesie, de m6me que la Fonderie de Stalowa Wola qui a financ6 la r6alisation. La r6alisation technique des syst6mes de commande optimale aussi bien en chaine ouverte qu'en boucle ferm6e, actuellement en voie d'applieation, est principalement bas6e sur des techniques analogiques en ce qui conceme aussi bien l'optimalit6 de la commande que le programme de temperatures. Les r~ultats de la commande optimale rbadis6e sur la base d'algorithmes pr6par6s et pr~sent6s darts l'article eonfirment les estimations th6oriques. La possibilit6 d'envisager un probl6me 61argie se rapportant/t l'optimalisation d'un groupe de fours/t arc par rapport /t des contraintes communes est 6galement diseut6e Ala fin de l'article.
Hinblick auf eine Optimalwertregelung und -temperaturprogramm aufgebaut. Die experimentellen Ergebnisse der Regelung auf der Basis vorbereitete und in der Arbeit optimalen mitgeteilter Algorithmen best~itigen die theoretischen Sch~.tzungen. Die M6glichkeit der Betrachtung der Problems einer Optimierung einer Reihe yon LichtbogenSfen unter Beriicksichtigung gemeinsamer Beschr~inkungen wird am Ende der Arbeit diskutiert. Pe31oMe--CTaTbg IlpC,R~l'aB.rl~leT TeOpCTHtI~Kyto OCHOBy, TeXHHqecKoe ocymecTn.qeHHe H HpeRBapMTeJH~H~e 3gcuepHMeHTaJlbHble pe3yBbTaTI,I ,RHHaMH'~OFO OnTHMaJIbHOFO yHpaBHeHHg npou~-,C~M BHpa6oTKH ffraHH B 3HegTpHq~KO~ ~yroBoil nCqH. B KaqecTBC Mep~ pa6oTH ~po~ecca, e~HHHtIHa~I CTOHMOCTb IIpo~yKl~H n p e ~ n o ~ a r a e T c a COCTOaI/~ei~ H3 KOM6HHaIIHH C°I'OHNIOCTH3nezTpO~HeprHH H CTOHMCTH BpeMeHH. CHCTCMa O 6 b l K H O B C ~ ~IH~pCHIIHabHblX ypaBHOHI~ B3HTa KaK MaTeMaTHqCCKag MOReJIb npoIlecca.
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Zm~mmenfs__~mng--Behandelt werden die theoretische Basis, der technische Entwuff und die vorl~iutigen experimentellen Resultate einer dynamischen optimalen Regelung der Stahlherstellung in einem Lichtbogenofen. Als MaB f'tir die LeistungsfS.higk¢it der Prozesse wird annommen, dab sich die Kosten ffir die Produktionseinheit aus denen ffir die der Elektroenergie und dem Zeitaufwand zusammensetzen. Als mathematisches Modell des Prozesses wird ein System gew~hniicher Diiferentialgleichungen angesetzt. Die theoretischen Probleme tier ProzeBoptimiemng und der Synthese des Regelungssystems werden mittels der Maximum-Prinzips gelfst. Gezeigt wird unter Benutzung einiger Annahmen, dal~ die optimale Reg¢lung mittels eines Optimalwertreglers durchgeffihrt werden kann. Das hier behandelte Problem in einem Lichtbogenofen wurde in Zusammenarbeit zwischen dem Lehrstuhl ffir Regelungstechnik der Technischen Universitiit Warschau, der Abteilung fiir Elektrow~irme des Elektro-Ingenieur-Instituts Warschau-Mi¢~lzylesie und der Stalowa-Wola-GieBerei, durch die das Projekt unterstiitzt wurde, untersucht. Der nun ausgef'fihrte technische Entwurf sowohl yon offenen als auch yon geschlossenen optimalen Kontrollsystemen ist hauptsiichlich auf der Analogtechnik im
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Dynamic Optimization of a Steel-Making Process in Electric arc Furnace* L'optimalisation dynamique d'un proe&t6 d'61aboration d'aeier dans un four 61eetrique ~t arc Dynamisehe Optimierung der Stahlherstellung in einem Liehtbogenofen ~HHaMnqecI(a.q OIITHMH3RI_IHII npouecca ablpa6OTrH CTaJIH B 3Ylek'TpnqeciCOfi /lyroBofi
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A. G O S I E W S K H " and A. W I E R Z B I C K I t
The Maximum Principle is not only a pure theoretical concept but it can be also fruitfully used as a powerful tool to the design of optimal control systems involving real industrial processes. Summary--The paper presents the theoretical basis, technical design, and preliminary experimental results of the dynamic optimal control of a steel-making process in an electric arc furnace. As a performance measure of the process, the unit production cost is assumed to be a combination of the cost of electric energy and the cost of time. A system of ordinary differential equations is taken as the mathematical model of the process. The theoretical problems of process optimization and of control system synthesis is solved by means of the Maximum Principle. It is shown that, with some assumptions, optimum control can be accomplished by means of a peak-holding controller. The technical design of both open- and dosedloop optimal control systems, now being implemented, is based mainly on analogue techniques with respect to both control optimality and a temperature program. The experimental results of the optimal control achieved on the basis of prepared algorithms and presented in the paper confirm theoretical estimations. The possibility of considering an expanded problem nvolving optimization of a number of are furnaces with respect to joint constraints is also discussed at the end of paper.
Defined Symbol by equation
c
(22)
(15)
Co C*
(3) (16)
Cp
(1)
Cpm fl f2
A F
NOTATION Defined Symbol by equation
c*
F, H i i t
Purpose dimensionless cost o f 1 hr o f furnace usage
to io
* Received 16 June 1969; revised 30 March 1970. The original version of this paper was presented at the 4th IFAC Congress which was held in Warsaw, Poland, during June 1969. It was recommended for publication in revised form by associate editor A. Spang. The problem of optimization of steel-making process in an arc furnace which is presented in this paper has been investigated in a collaboration between the Chair of Automatic Control of the Technical University of Warsaw and the Division of Electrothermics of the Electrical Engineering Institute, Warsaw-Miedzylesie, as well as the Stalowa Wola Foundry by which the project has been supported. t Technical University of Warsaw, Chair of Automatic Control, Warsaw, Poland.
is Ai I
! /m
t,
(20) (2) (2) (2) (13) (7) (7) (20) (37) (25) (25) (36) (37) Fig. 7 (12) (20) (19)
J k
767
(21)
Purpose
dimensionless cost c, related to the constraint m a cost o f 1 hr o f furnace usage effective cost Co, related to the constraint Md instantaneous cost o f a supplied power cost Ce at the maximal heating power rate o f the charge temperature rate of the walls and b o t t o m temperature rate o f the r o o f temperature m o m e n t a r y performance index static performance index hamiltonian dimensionless arc current mean value o f i optimal arc current initial value o f the current t constant measurement error o f i amplitude o f r a n d o m disturbances searching step length arc current optimal arc current maximal arc current optimal are current, constant in time tap n u m b e r gain coefficient
768
A. GOSIEWSKIand A. WIERZBICKI Defined
Symbol K m rh md ms
M
gd Ph
Ph
p', pP Ph Phm Pi
Pt Pts
q q~
O Qeq Sd Ss
t ta T H~, I1p 0 ~, 13#
W X
Xd X1 X2 X3
xo xo
a
by
Defined Purpose
Symbol
by
Purpose
equation
equation
(6)
(15)
penalty function coefficient dimensionless mean cost of power (28) cost m corresponding to the dynamic optimal solution (23) admissible value of the cost m (33) cost m corresponding to the static solution (4) mean cost of power (4) admissible value of the cost M (20) dimensionless heating power (38) mean value ofph heating power of the furnace, number ~, fl (1) heating power (20) maximal heating power (37) probability of an arc current value (2) thermal losses (18) mean value o f P t (22) dimensionless performance index (27) minimal value of q (32) index q value corresponding to the static solution (3) performance index (6) performance index with the penalty component (30) sensitivity measure corresponding to the dynamic solution (35) sensitivity measure corresponding to the static solution time-variable (3) duration of the melting stage (21) charge time constant casting the furnace, number ~, fl loading the furnace, number ~, fl (2) thermal capacity of the charge (21) state variable: dimensionless temperature of the charge or bath (26) optimal temperature (24) final value of the temperature x (2) state variable: the charge or bath temperature (2) state variable: the walls and bottom temperature (2) state variable: the roof temperature (5) additional state variable: time t (5) additional state variable: the cost of energy (40), (41) output signal of a measurement device furnace number (40) forcing coefficient of a measurement device (23)
®
(40)
2
(41)
¢o
(25)
,g
(21)
Td
Ca
(29)
"~ds
(34)
0, ~,o, ~o
~, C,
(7) (7)
correction in optimizing feedback structure time constant of a measurement device relative error of the measurement of the state coefficient penalty coefficient, the coordination variable dimensionless time-variable final value of z, the melting stage duration final time rd corresponding to the optimal solution final time rd corresponding to the static solution costate, i = 1, 2, 3 costates adjoined to the additional state variables x °, x ° costate adjoined to the timevariable t costate Ot for the furnace number ~,/3 1. I N T R O D U C T I O N
THE GOAL of the steel-making process in an electric arc furnace is to obtain a high quality steel by means of melting selected steel-scrap, purging the melted steel, the "bath", of impurities and adding some ingredients to improve the quality. The main source of energy in the process is the heat supplied by a three-phase electric arc which burns between three graphite electrodes and the scrap, or bath. The process in an arc furnace is a typical charge-process; each charge, beginning with loading the furnace and ending with casting of finished steel, lasts about 6 hr for a furnace of an average load which may be about 30 tons. The average power consumption amounts about 0"6 MWh/ton, the average arc power--about 3MW. Each charge can be divided into three main stages: (1) the melting stage which lasts for about 0.5 of the whole charge duration time and demands about 0.7 of the whole charge energy consumption ; after melting, the bath is overheated to a given temperature, the "overheating temperature", the first slag is cast and oxidizing components are brought into the bath; (2) the oxidizing stage in which the impurities--such as coal, phosphorus and sulphur--are burned out; after oxidizing, the next slag is cast; (3) the refining stage in which the oxygen is reduced and the improving ingredients--such as manganese and chromium--are put in the bath;
Dynamic optimization of a steel-making process in electric arc furnace after refining, the molten steel is overheated and the casting of finished steel follows. 2. STATEMENT OF THE PROBLEM
The notion of "optimization" in an arc furnace described here, is that which results in the minimal cost of finished steel (zl/ton)* taking into consideration the cost of electric energy (zl/kWh) as well as the constant costs of time (zl/hr) during which a furnace is used. The problem is of a great economic importance: the energy cost for a 30 ton furnace amounts to 10 million zl per year and the constant cost of 1 hr of furnace usage can be estimated as 6000 zl. The optimization of a charge is usually related only to the melting stage [1]-[3], which determines the total energy consumption and influences strongly the total charge duration time. The oxidizing and refining stages are not included in the optimization because their duration and energy consumption, the temperature program, depends on chemical processes in the bath and are strictly determined by the process technology. The optimization of loading and casting of an isolated furnace is trivial: these functions must be performed as intensively as possible. Consider the simplified design of an arc furnace with only one phase of the arc circuit shown in Fig. I. Each electrode is supplied through a power cable
769
assume for the first approximation--apart from rapidly varying disturbances--that I = Io. Therefore the main controls of the process are: j
the tap number of the secondary transformer winding;
1 the arc current, or more correctly, the reference value Io of the arc controllers. It is easy to show [2] that the secondary voltage of the transformer should be kept as large as possible during the melting stage. Since the tap number is selected to do this, the only control during the melting stage is the current L Theoretically, the process can be divided into two sub-processes: the power conversion subprocess and the thermal one as indicated in Fig. 2.
LI
*l =fl (xl,xz, x3;I) Cp(I) P,(I)
i(~ =f,~(Xl, x2, x3) :t~ =f3 (xl, xz, x3)
F~G. 2. Power conveirsion subprocess and thermal subprocess.
The power subprocess is practically inertialess compared to the thermal one, and it can be desE[ I--"1
T
]llt
FIG. 1. Design of an arc furnace. C from the furnace transformer T which has several taps of number j = 1, 2 . . . . . on its primary winding; the taps are set by a tap-setting device that determines the nominal secondary voltage. The arc current 1is controlled by the arc controller, A C. This includes the electrode servomecharOsms, M, which lift or lower the electrodes according to the difference between I and the reference value Io. Because the arc controller acts rather quickly compared to the duration of the charge, we can * zl. is the abbreviation of Polish currency (zl.oty).
cribed for the optimization purposes by the two basic static relationships: Cp = Cp(l) ; Ph =Ph(I)
(1)
where Cp is the momentary cost of a power supplied (zl/hr) taking into account the active power as well as the reactive power and Ph is the heating power of the arc. Those characteristics can be determined either analytically--by means of an equivalent diagram of the electric circuit--or experimentally. Taking into consideration rapidly varying disturb-
770
A. GOSII/WSKIand A. WIERZBICK1
ances caused by arc burning those relationships should be treated as regressions. For approximate considerations, which are presented in section 3, the relationship Ce(1) can be assumed to be linear, the relationship Ph(I)--to be parabolic. An example of real characteristics (1), determined experimentally, is shown in Fig. 3. 5 4
o
furnace geometry and physical properties. It turns out, however, that the decisive part of the thermal power P1 depends mainly on the temperature x~ and not on the temperatures x2 and x 3. For that reason for approximate considerations, discussed in section 3, the model of the thermal subprocess can be reduced to the first of the equations (2) which can be further approximated by a linear equation of a simple inertial dement. For a given load, the performance of the melting stage can be expressed by the cost index Q=
¢0
)
l
I
I
[
f
td
[C O+
Ce(1)]dt
(3)
0
1
where td is the melting stage duration which is not given but determined by the overheating temperature xl(t~)=x~. The constant Co denotes the constant cost of 1 hr of furnace usage (zl/hr), which results from wages, redemption and so on. The mean power transmitted by the furnace transformer is constrained. Because the transmitted power is approximately proportional to the instantaneous cost, the constraint can be expressed as
6--
2
I0
20
30
L
40
50
60
tO
kA
Fro. 3. Cost of electric power (CP) and heating power
(Ph) versus arc current (I). Generally, a model of the thermal subprocess should be formulated as a very complicated partial differential equation with respect to the temperature x in the furnace space. That equation can be approximated, however, by a system of ordinary differential equations taking into consideration three mean temperatures as the process state variables: the charge, or bath, temperature x~, the walls and bottom temperature x2, and the roof temperature x3. The model of heating subprocess then assumes the normal state equation format
Pt(x" xz,
f '" Ce(I)dt 0
t~
~
(4)
where Md is a given admissible mean value. Now the optimization problem can be formulated: determine such a function of time I=I(t), tel0, td] which minimizes the performance functional (3) with the differential equations (2), the global constraint (4), and with the final condition xl(td)=x~. The solution of the problem can be based on the Maximum Principle [4]-[6].
3. THE METHOD OF SOLUTION Because of the existence of the integral constraint (4) let additional state variables, x ° and x °, be introduced and included in equations of the form
~ = f x(xl, x2, x33, I)
- W1, :[P'(I)-(xy
M=
x3)]
(2)
:Cz=f z(xt,
X2, X3)
(5)
:c3=f a(xl, x2, xa) where: charge,
W(xl)
Pt(xl, x2, xa)
is the thermal capacity of the
is the thermal power transmitted from the charge to the walls, bottom, and roof by conduction, radiation, and convection, the losses. The forms of the functions f t , fa, fa are rather complicated and depend on the kind of the heat exchange between walls, bottom and roof, on the
~g=c,(i) Thus, the initial problem can be formulated as a problem with the constrained ratio of the variables xO(td)/x°(td) at the terminal time td. It is useful, however, to reformulate the problem further by introducing a new cost index with a "penalty for violating the constraint" instead of the index (3). Thus,
Qe~= Q+K(M-Ma) 2 " I ( M - M # )
(6)
Dynamic optimization of a steel-making process in electric are furnace where K is a sufficiently large constant; a violation of the admissible mean value Md is associated, therefore, with a rapid increase of the functional (6). It can be shown [6] that with K ~ o o the reformulated problem, without the constraint, is strictly equivalent to the initial problem. The hamiltonian of the problem has the form
771
and then the necessary condition (12) can be rewritten in the form F(1)--- max F(l) for all t~[0, td].
(14)
I
The function ~/(Xl, x2, x3, t ) = ~ [ d / 2 ( t ) f 2 ( x l , x2,
X3)
H = - (C o - ~,o) _ (1 - ¢°)C1,(I)
+~k3(t)fa(xt, x2, x3)]
3
+ ~ dAf,(xl, x2, x3, I)+d/,
(7)
i=l
where the costates ~kare determined by the equations OH.
¢°=°;
(8)
The transversality conditions for g,o and ~b° have the form, of. Ref. [6]
Ox o
= K M 2 ( M - Ma)I(M- Ma) t> 0
ta
~b~= -~Q___2q ax o
(9)
= - - ~ ( M - M~)I(M- Ma) <<.0
t~
It follows from the form of the hamiltonian (7) and the conditions (9) that occurrence of the global constraint (4) leads to suppressing the constant cost of furnace usage and to increasing the cost of electric power; evidently Co>
,° .
(10)
Note that if this inequality does not hold then the furnace transformer is incorrectly designed. Because the process dynamics are determined mainly by the temperature xt it can be assumed
l¢,lfll,>l¢,2f2+a/3f31
wherefa>0, ~1>0.
(11)
As the inequalities (10) and (11) hold, the principle of optimizing feedback [6] can be applied and the necessary condition of optimality reformulated
maxH(l)=H(1)=O for all re[0, td]
(12)
I
where [ = l(t) is the optimal control. We introduce, therefore, the following instantaneous performance index
F=Ph(1)-Pt(xl, x2, x3)+r/(xt, x2, x3, t) (13) + c (t)
(15)
is a correction which represents the influence of temperatures x2, x3 on the optimal solution of the problem, whereas the constant Co*= C o - e ° . . 1-
(16)
--o
is the normalized constant cost of the time of furnace usage considering the constraint (4); the equality holds if the constraint does not influence the optimal solution of the problem. It should be noted that if the influence of the temperature x2, x3 on the process dynamics is negligible then we can assume t/=0. If the influence of temperature x2, x3 is small, then the correction rl is small compared to (Ph-Pt), and it can be determined approximately. Thus determination of the dynamic optimal control :(t) can be reduced to the peak-holding control of the instantaneous performance index only on the basis of the measurements of the powels Ph, Pt and the cost Cp and of an initial knowledge of the normalized cost C* without necessity of an exact knowledge of the functionsfi,f2,f3 and of the correction t/. This is the essential property of the optimizing feedback. In the previous papers mentioned earlier on the control of arc furnaces, the optimization problem has been treated only as a static, or steady-state, problem and the process dynamics (2) as well as the global constraint (4) have not been taken into consideration. When using this approach it can be assumed that the arc current I is constant in time. This current is then determined in such a way that it minimizes the performance function
Q~= Q[Co+ Cp(I)]
(17)
which is a static analogue to the performance functional (3). On the basis of the thermal balance
td[Ph(l)--Pls]=XatW whence td-
x~W (18) Ph(l)-Pts
where Pu is the mean, constant thermal power lost from the charge. The condition of the static optimality can be rewritten, therefore, in the form
F~- x~ W _Ph(l) - Pl, c
(I)
772
A. GOSIEWSKIand A. WIERZBICKI 4. SIMPLIFIED ANALYTIC SOLUTION AND APPROXIMATE SENSITIVITY ANALYSIS
and
Fs(]s) = max F~(I)
(19)
!
where F~ is the static performance index, and f,, the constant, static optimal control. The conditions (13), (14) and (19) appear to be formally similar, but they are essentially different. The difference between them is illustrated in Fig. 4. The static optimal heating power Ph(l~)
By choosing a sufficiently simplified model, the problem can be solved analytically and qualitative conclusions about the main features and the sensitivity of the solution can be obtained. According to the reasons given in section 2, the equations of the electric circuit can be assumed to be, in the dimensionless form, ph=l-(1-i)
2
where
Cp
Ph. ph=~,
I lm
Ph~
(20)
Ph denotes heating power, i, arc current, Phm, the
I
Ph
)
(a)
Ce=Cl,mi
i=--;
-C,
maximal heating power, I,,, the arc current which results in the maximal heating power, Cem, the cost of a power supplied at the maximal heating power. Assume, furthermore, a simplified equation for the dynamics
dx --=£=ph(i)-x
(21)
dv where
Cp
f . "t=--,
Xt X~----
T
/
/
J !
(b) -C(
_t
P~-,7 Fic. 4. Graphical illustration of the static (a) and dynamic (b) optimization of the melting stage in an arc furnace. can be determined graphically (Fig. 4a) by drawing a tangent to the plot Cp(Ph) from the point (Pis, -Co). Similarly, we can determine the dynamic optimal heating power Phil(t)] (Fig. 4b) if we assume Pt>>q and draw a tangent from the point [Pt(xl, x2, x3)-C~]. The thermal power Pl(xl, x2, x3) increases during the melting time; so does the dynamical optimal heating power Pn[f(t)] as indicated clearly in Fig. 4b. This property of the optimal solution has a simple physical interpretation: at the beginning of the melting period, the power of thermal losses P~ is small, the charge is cold, which implies a small heating power Ph and a high efficiency of the thermal subprocess; during the melting period, the power Pt increases and, in turn, the heat power Ph must be increased in order to shorten the period of large thermal losses, or to "compensate" for them.
-- Xl
kP~m xlm
x and xl should be treated as increments whereas the differentiation is understood to be with respect to the time-variable z; T denotes the charge time constant, k, the gain coefficient, xl,,, the maximal increment of the charge temperature which can be achieved by permanently applying the heat power Ph. Now, the performance index can be written as
q=
i)dz
(22)
where Q q = C-~mT,
Co c = Cpm .
The constant c is the dimensionless constant cost. The constraint of the mean power supplied assumes the form m = 1 I~d~ <~ma
(23)
TdJO
where
m d= ~Md . Cem Under the given end-point conditions x(0) = 0 ; x(%)=xn; ~ a - free
(24)
Dynamic optimization of a steel-making process in electric are furnace the problem of the minimization of the functional (22) under the differential conditions (21) and the integral constraint (23) can be solved analytically; the solution can be based on the Maximum Principle. Then the optimal control f = fix) is of the form
773
assumed that i = c o n s t ; then accordingly to equations (21), (17) and (18), x=[1-(1-i)2](1-e
-')
(31)
and
q,=(c+i)(za~)
(32)
m,=i
(33)
(25)
~ = 1 - ~ o e-~ where
and,
40-
1
l+c*+x/2c*+(c*) 2
c,_c-O° 1-¢°"
1-(1-02
As indicated above, c* can be interpreted as the dimensionless constant cost reduced according to the influence of the constraint of mean power supplied. The optimal temperature as a function of time 2 = 2(z), takes the form £ = (1 - ~o2e-')(1 - e - ')
/41= 1 - ~-2(1 - e - "9 ,isd
The equation (dqs(i))/(di) = 0 is transcendental; the simplest way to obtain the solution is to assume some values of i and to determine the values Zds(i) and q~(i) according to equations (32), (34). It is of use to introduce the ratio.
(26)
and the value of the performance functional 4, the constraint functional rh and the duration of the process cd can be expressed as
O=(l+c)Ta-~o(1-e TM)
(27)
s~(c, i)- qs(c' i)- O(c, C).lO0% O(c, c)
(35)
which shows how much worse are the results of static optimization than those of dynamic optimization. The results of that comparison and the sensitivity characteristics for a chosen value of c are presented in Fig. 5. It is worth mentioning that although the
(28) 20
.~ , , f
(34)
~d*=lnl_(l_i)2_xa"
l+c*+x/2c*+(c*)2+x a ]
18
a. . . . ~ - - - - ~ :;,. (29) ((1 - xd)[1 + c* + x/2c* + (c*)2]J
_ \
\ @,
16 14 12
It can be assumed c=c* if the inequality ma holds. If the inequality does not hold, the transcendental equation n~(c*) = m a should be solved and the necessary value c* < c determined. However, it is simpler to assume some values of c*~< c and determine the values ~a(c*), rh(c*) and q(c, c*) according to equations (27), (28), (29). Then, not only can the solution be obtained, but also the sensitivity characteristics of the solution with respect to the integral constraint and to the initial current t o = l - G 0 can be obtained also. It should be stressed that errors in determinat;on of that initial current can be caused by an incorrect estimation of such parameters as Cem, Ph or Ira. Therefore it is useful to present the results obtained as the values of the sensitivity measure [6]
to
I0 8 6 4 2 0
0 I
02
0'3
0.4
0"5
0-6
(30)
In order to compare the results of dynamic and traditional, static optimization, it should be
0.8
09
I0
zo 18 I6 14 12
o4
IO 8 6 4
:.,;7// I
2
sa(c, c*)- O(c, c*)- q(c, c)100%. O(c, c)
0.7
0
0"1
02
0'5
04
0'5
0"6
0.7
0"8
0.9
1-0
i, io
FIG. 5. Sensitivitycharacteristics of the optimal control, with respect to the chosen initial current value i or io, to the constant c* and to the integral constraint ma, for various c and xa.
774
A. GOSlEWSKIand A. WIF,RZBICKI
relative profits of the dynamical optimization are rather small, the sensitivity of the dynamic optimization with respect to the constraint of mean power supplied and to the choice of initial current is much lower than the sensitivity of the static optimization. It follows from this that the stronger the influence of the average power constraint, the more profitable is the application of dynamic optimization. In a similar way, [6], the sensitivity of the optimal solution with respect to the constant errors i, in the determination of optimal control t can be examined by assuming that
20 18
c=0"2 xd=0,6
16 14 12
S
I0 8 6
(o)
-02
!
:
-0"1
0
OI
0"2
i, 20
18
i = t + i,,= 1 + i,-~oe-"
(36)
16 14
or with respect to the amplitude i, of rapidly varying random disturbances ip of the arc current by assuming
=i+ip;
i=~;
ip=
t
+ i°,
8 6
P,(+ i,,)=
-
P,(0)=½
i.,
P , ( - i.)=
~h=ph(O--~(i.)2= 1--(1-- [)2__~(i.)2.
(37)
max F( i)= FO) .
-03
-o2
-ol
0
ol
o.z
0.3
i0 FxO.6. Sensitivitycharacteristicsof the optimalcontrol, with resl~et to the constant error ia (a), and to the amplitude & of random disturbances(b).
(38)
Some results of this sensitivity analysis are presented in Fig. 6. It should be emphasized that during the melting stage the disturbances of the are current are strong and the losses associated with them can amount to several percent of the optimal performance--as indicated in Fig. 6 either by dynamic or by static optimization. It is very important, therefore, to have a high performance of the arc controllers which counteract those disturbances. The sensitivity analysis presented above is related to the open-loop optimal control system. An application of the optimizing feedback corresponds, according to equation (14), to searching the optimal current f by the peak-holding control with the temporary index
F -p(i)-x c*+ i
S
2
(b) 0,
-
4
where [ is the mean value of the current, ip is the random component and P~ denotes probability. It is assumed here that the temperature x depends solely on the mean value of the heating power Ph which is in turn determined by changes of the mean current f= f(r). Then, according to equation (20),
i
12 S I0
(39)
Thus, the optimizing feedback results in full insensitivity of the system with respect to parameter variations in Ph,,, Ce,,, I,., k and T because the values p(i), x and i are determined by measurements. On the other hand, the sensitivity with respect to the setting the constant c*, the estimation of the mean power eonstraint's influence, and to the rapidly varying disturbances of the arc current, which cannot be suppressed by the rather slowly acting optimizing feedback, does not change. In order to estimate the influence of the measurement accuracy on the optimizing feedback performance, further sensitivity analysis has been carried out by means of digital modelling the cases when the state x, corresponding to the power of thermal losses Pt, is measured indirectly through an output signal y which is characterized by the time constant ® of the measurement inert[as and compensated by the forcing coefficient 6
Op + y=60:~ + x.
(40)
In determining the optimal control, the measured values of the signal y are put in place of the state x in the index F. The state x may be measured also with a relative error 2 y = ( l +2)x.
(41)
Another kind of the performance loss in the optimizing feedback system may be caused by
Dynamic optimization of a steel-making process in electric arc furnace finite searching steps of length Ai in the real peakholding controller of the index F. The sensitivity analysis results for these three cases are presented in Fig. 7a-c. It follows from them that the optimizing feedback system has a very low sensitivity to the measurement inaccuracies and a reasonable sensitivity to the searching step length since in a real system it can be assumed 2 < 0.1 and ® < 0.1 which results in s<0.5 ~o.
8:0
(a) 0 2
014
0,6
08
I0
8
S
0 I~
775
The results of those computations differed only slightly from the results presented above, rather quantitatively but not qualitatively. In order to verify experimentally the results achieved, the charges of 30 ton furnace in the Stalowa Wola Foundry were run during 6 months according to the optimal, or more correctly, the sub-optimal, program of arc current changes which were determined in advance. The program was conducted by a human operator who set the reference values of arc controllers. Then it was proved that the energy consumption for the melting stage decreased by about 5 per cent or by about 3 per cent with regard to the whole charge and 1 ton of steel, and the melting duration was shortened by about 11 per cent, or by about 5 per cent with regard to the whole charge and 1 ton of steel. The improvement of the cost index was about 8 per cent higher than that obtained in the theoretical research. This can be explained by the fact that the furnace had not previously been operated according to the static optimal conditions, as assumed in the theoretical studies.
O.I ,
(b) -- 0 'I0
_0.05
0
0.05
OIO
6. TECHNICAL IMPLEMENTATION
>,
For a technical implementation of the optimal control system for the process in an arc furnace the specialized control equipment has been designed and constructed. It consists of:
14l 12 l0 $
8
(i) modified arc current controllers and a remote controlled tap-setting device;
4 2
(c/
0
0.04
0100
0 112
O' 16
Ai
Fie. 7. Sensitivity characteristics of the optimizing feedback system, with respect to the measurements inertia O (a), to measurements error 2 (b), and to the step length Ai of the peak-holding controller (c). 5. ANALYSIS OF MORE ACCURATE MODELS A N D EXPERIMENTAL VERIFICATION OF RESULTS ACHIEVED
By applying more accurate models of the relationships (1) and the dynamics (2), several other computations have been carried out; the purposes for doing this were to obtain: (i) a verification of the modelling accuracy by computing the melting duration as well as the energy consumption when the usual are currents are applied and by comparing the results achieved to experimental data; (ii) a more accurate determination of the optimal current [(t) on a basis of the maximum principle and the numerical algorithms related to it [6]; (iii) a more accurate sensitivity analysis of various structures of the optimal control.
(ii) devices for automatic measuring and transducing the temperatures and electric power; Ctii) a device for generating the open-loop optimal control; (iv) an optimizer consisting of a computing device for the index F and the peak-holding controller; (v) a programming controller of the bath temperature that operates during the oxidizing and refining stages. A simplified block diagram of that control equipment is presented in Fig. 8. Application of a digital computer for the optimal control of the process in are furnace has also been considered. However, it has turned out that an application of a digital computer exclusively for that purpose is unprofitable. 7. POSSIBILITIES FOR F U R T H E R RESEARCH A N D APPLICATION
The application of a digital computer instead of the specialized control equipment described above may be profitable in an extended problem where the
776
A. GOSIEWSKIand A. WIERZBICKI
r--i !
l---i .... ....... t . . . . .
|u
Ai
L ~0
7
/
,-----,;
F ..........
! 1 Electrode I ]: ,
.
servo-
Arc
J l , ;:p[~J' L~-~-t 'I'-T----Icon'ro'er I !
}
I-
J
Controldesk
I
i TaP-' I sell - ' ng
i i
I
oev,ce
ui I
-
circuit
I I
I
j
_.m Ph Ph
Open-loop controller
Ai_o
I
i
E rx~
n IOp
1
IPho l'tlO Temperature controller
xI
'P
Measurement transducers numer j
Furnoce
X3 r ~
P~ ......
Cp
I
"u
g~ i
.
Optimizer
J
l X?
t FIG. 8. Block diagram of the control equipment of an arc furnace. Symbols: Pno, xlo, lo, reference values of the heating power Pn, the temperature xl and the arc current I; A/o, the step of Io.
optimal control would be determined for the whole complex of a number of furnaces in a foundry. This problem is directly associated with programming the optimal operation programs for such complex. In principle, optimal programs for each furnace could be determined independently. However, common constraints occur here which follow from: (i) admissible power supplied for the complex of furnaces; (ii) constrained transmittance of the loading devices for that complex; (iii) constrained transmittance of the casting devices for that complex. In order to consider those common constraints there is a possiblity of applying multilevel optimization methods. On a basis of these methods, the numerical algorithm for determining the optimal program of the operation of furnace's complex has been proposed [6]. This algorithm is of an iterative nature: in the first computation, the first "iteration", the optimal programs for each furnace are computed independently; afterwards, "the penalties" for violating the common dependent constraints are determined, the computations are repeated and so on. An example of the results obtained from computations using such algorithm are presented in Fig. 9 where two furnaces are
involved. In the first iteration (i=0) the optimal electric powers, loadings and castings for two furnaces, denoted c~ and /y, are determined; the variables, ~kt, denote the co-states adjoined to the time which, after a sign change, can be interpreted as the cost of usage time of a furnace. After completing the first iteration and finding the violation of the common constraints, the penalty coefficients rt are determined. These coefficients permit repeated computations which influence the nature of "the time cost" variation, ~t, and they result in a speeding up of the operations for the ~-furnace, but a slowing down of the operations for/Y-furnace. In the subsequent iterations, the operation programs for both furnaces separate appropriately and violations of the common constraints are reduced to minimum. 8. CONCLUSIONS
It was shown that the problem of dynamic optimization of the melting stage of steel-making process in an electric arc furnace can be formulated and solved by means of the Maximum Principle. Due to the global constraint which has the form of the ratio of two integral functionals, a generalized version of the Maximum Principle was applied to the stated task. It turned out that the Maximum Principle in the case considered can be transformed into a so called optimizing feedback principle which
Dynamic optimization of a steel-making process in electric arc furnace izO
i--0 -~= t p=
-*t
.... ' a
P
a U va
Ua v ° ~\~
- ~
777
I~\\\~I
E
t
pB
pB
B
U V0
U VB
t~
tt
t2
i=l
n
~,--,
QO
HI1
-~" t
t
pO
pa
el U
Ua Va
~ i
_~B
J-
V~ B
m
-~t t
t
p~ U
B
V
t)
te
t
tz t
tl
F[o. 9. A simplified example of the coordinating procedure of optimal solutions for two arc furnaces. Symbols: i, the number of coordinating iteration; a, B, the furnace's number; n, the coordinating variables; ~t, the equivalent cost of time; p, the heating power; u, the casting; v, the loading.
permits synthesis of the optimal control system without a priori knowledge of the mathematical model of the process when it is of the first order. When the model is of higher order the same optimizing feedback can also be applied, but for proper operation, some additional correction is necessary. In a general case, the optimal control can be determined only by numerical computations. An analytic solution of the problem can be obtained for the simplified linear model of the process, and this solution makes it possible to analyse the sensitivity of the optimal control with respect to various errors in determining process parameters, to the global constraint and, what is particularly important, to the results of dynamic optimization with static solutions. The conclusions from the theoretical analysis have been applied to the effective control of the process in an electric arc furnace. The results of this application have confirmed the theoretical analysis and economic purposefulness of the research. Further investigation related to the optimal programming of a complex of arc furnaces shows that there is a possibility of applying multilevel optimization algorithms in order to solve this important problem.
REFERENCES [1] J. E. YEFRIMOVITCH" Automatic control of steelmaking are furnace (in Russian). Metallurgizdat, Moscow (1958). [2] L. McGE~ and A. METCALF: Controlling arc-furnace power. J. Control, 91-96 (1962). [3] A. N. SOKOLOV: Reasonable operation conditions of steel-making arc furnaces (in Russian). Metallurgizdat, Moscow (1960). [4] A. GOSIEWSKI and A. WmRZmCKi et aL: Automatic optimization of steel-making process in arc furnace (in Polish). Proc. 4th National Control Conference, Vol. 7, 73-80, Krak6w, 1967. [5] L. S. PONTRYAOIN, V. BOLTYANSKIJ, R. GAMKRELIDZE and E. MISHCHENKO: The Mathematical Theory of Optimal Process. Interscience, New York (1962). [6] A. WIERZBICKI: Maximum principle and synthesis of optimal controllers. Pt. III: Sensitivity and structure of optimal controllers. Pt. IV: Example of industrial application (in Polish). Archiwum Automatyki i Telemechaniki 14, 75-101, 187-213 (1969). R6sum6--L'article pr6sente la base th6orique, la r6alisation technique, et les r6sultats exp6rimentaux pr61iminaircs de la commande dynamique optimale d'un proccssus d'61aboration d'acier dans un four 61ectrique ~ arc. A titre de mesure des performances du proc6d6, le cofit unitaire de production est suppos6 ~tre constitu6 par une combinaison du cofit de l'energi¢ 61ectrique et de celui du temps. Un syst~me d'6quations differentielles ordinaires est pris comme mod~le math6matique du proc6d6. Le probl~me th6oriques d'optimalisation de proc6d6s et de synth~s¢ de syst~mes de commande sent r6solus ~t l'aide du Pfincipe du Maximum. II est montr6 qu'avec certaines hypoth/~ses la commande optimale peut 6tre r6alis6e au moyen d'un r6gulateur oxtromal.
778
A. GOSIEWSKI a n d A. WIERZBICKI
Le probl6me d'optimalisation du procddd d'61aboration d'acier clans un four /t arc pr6sent6 dans cet article a 6t6 6tudi6 en collaboration entre la Chaire d'Automatique de I'universit6 de Varsovie et la Division d'Electrothermie de l'Institut Electrotechnique de Varsovie-Miedzylesie, de m6me que la Fonderie de Stalowa Wola qui a financ6 la r6alisation. La r6alisation technique des syst6mes de commande optimale aussi bien en chaine ouverte qu'en boucle ferm6e, actuellement en voie d'applieation, est principalement bas6e sur des techniques analogiques en ce qui conceme aussi bien l'optimalit6 de la commande que le programme de temperatures. Les r~ultats de la commande optimale rbadis6e sur la base d'algorithmes pr6par6s et pr~sent6s darts l'article eonfirment les estimations th6oriques. La possibilit6 d'envisager un probl6me 61argie se rapportant/t l'optimalisation d'un groupe de fours/t arc par rapport /t des contraintes communes est 6galement diseut6e Ala fin de l'article.
Hinblick auf eine Optimalwertregelung und -temperaturprogramm aufgebaut. Die experimentellen Ergebnisse der Regelung auf der Basis vorbereitete und in der Arbeit optimalen mitgeteilter Algorithmen best~itigen die theoretischen Sch~.tzungen. Die M6glichkeit der Betrachtung der Problems einer Optimierung einer Reihe yon LichtbogenSfen unter Beriicksichtigung gemeinsamer Beschr~inkungen wird am Ende der Arbeit diskutiert. Pe31oMe--CTaTbg IlpC,R~l'aB.rl~leT TeOpCTHtI~Kyto OCHOBy, TeXHHqecKoe ocymecTn.qeHHe H HpeRBapMTeJH~H~e 3gcuepHMeHTaJlbHble pe3yBbTaTI,I ,RHHaMH'~OFO OnTHMaJIbHOFO yHpaBHeHHg npou~-,C~M BHpa6oTKH ffraHH B 3HegTpHq~KO~ ~yroBoil nCqH. B KaqecTBC Mep~ pa6oTH ~po~ecca, e~HHHtIHa~I CTOHMOCTb IIpo~yKl~H n p e ~ n o ~ a r a e T c a COCTOaI/~ei~ H3 KOM6HHaIIHH C°I'OHNIOCTH3nezTpO~HeprHH H CTOHMCTH BpeMeHH. CHCTCMa O 6 b l K H O B C ~ ~IH~pCHIIHabHblX ypaBHOHI~ B3HTa KaK MaTeMaTHqCCKag MOReJIb npoIlecca.
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Zm~mmenfs__~mng--Behandelt werden die theoretische Basis, der technische Entwuff und die vorl~iutigen experimentellen Resultate einer dynamischen optimalen Regelung der Stahlherstellung in einem Lichtbogenofen. Als MaB f'tir die LeistungsfS.higk¢it der Prozesse wird annommen, dab sich die Kosten ffir die Produktionseinheit aus denen ffir die der Elektroenergie und dem Zeitaufwand zusammensetzen. Als mathematisches Modell des Prozesses wird ein System gew~hniicher Diiferentialgleichungen angesetzt. Die theoretischen Probleme tier ProzeBoptimiemng und der Synthese des Regelungssystems werden mittels der Maximum-Prinzips gelfst. Gezeigt wird unter Benutzung einiger Annahmen, dal~ die optimale Reg¢lung mittels eines Optimalwertreglers durchgeffihrt werden kann. Das hier behandelte Problem in einem Lichtbogenofen wurde in Zusammenarbeit zwischen dem Lehrstuhl ffir Regelungstechnik der Technischen Universitiit Warschau, der Abteilung fiir Elektrow~irme des Elektro-Ingenieur-Instituts Warschau-Mi¢~lzylesie und der Stalowa-Wola-GieBerei, durch die das Projekt unterstiitzt wurde, untersucht. Der nun ausgef'fihrte technische Entwurf sowohl yon offenen als auch yon geschlossenen optimalen Kontrollsystemen ist hauptsiichlich auf der Analogtechnik im
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