Optimal Control of Two-Stream Plug-Flow Processes Jprrgen LZvIand Department of Chemical Engineering, Norwegian Institute of Technology, Trondheim, Norway
Necessary conditions have been derived for optimum control of two-stream plug-flow processes. The derivations are based on elementary calculus only. Although the derivations are simple, it is clear that actual computation of optimal control policies will b e quite complicated for all but the simplest problems.
A
number of authors have treated the problem of finding necessary conditions for optimal control of processes described by partial differential equations. -4brief review of much of this work is given by Seinfeld and Lapidus (1968). I n particular, results for systems described by one set of linear, hyberbolic partial differential equations (one-stream, plugflow systems) have been given by Koppel and Shih (1968) when the control is time- and space-distributed, and by Russell (1966) when the control is time-distributed only. Optimal velocity control of plug flow processes has been treated by Shih and Wang (1970) and by Seinfeld, et al. (1970). Only Jackson (1966) seems to have treated systems described by two sets of linear hyperbolic partial differential equations (two-stream, plug-flow systems). He gives a maximum principle for the case of constant velocities, arbitrary boundary, and time- and space-distributed control. His results are the same as those derived here by a different approach. I n this paper, necessary conditions for optimal control of eo-current and counter-current plug-flow processes are derived in a simple manner, using only elementary calculus. T i m e and spacedistributed control as well as time-distributed control are treated. The necessary conditions are “strong” maximum principles. I n addition, first-order conditions are derived for the optimal choice of initial values, equipment length, and total operating time and, for co-current flow, equipment length.
(Hildebrand, 1962) one gets equations holding along each stream (3) dl -dt -
(4)
VI
(5) dl ds
- = FV*
where t and s denote the times of residence along the characteristics or trajectories of the fluid elements in each stream. The initial conditions are assumed to be known. The functions f and g are assumed to be a t least once continuously differentiable with respect to x and y. The control u is assumed to be bounded and piecewise continuous. Furthermore, eq 3 and 5 must be such that an infinitesimal change in x and y remains infinitesimal as it is propagated through the system. This will hold for all physical systems. If time, 7, should enter any of the functionsf and g they are modified in the usual manner by defining time as another state variable. For a characteristic beginning a t T = 70 this new state variable becomes
System Equations
Two-stream plug-flow systems are described by two sets of linear hyperbolic partial differential equations of the form ax,
+
ax, VI
= f,(x,y,u)
(i
=
1, 2, . . ,, n)
(7)
A performance index of the following form is assumed for the
(1)
r=T
where x and y are the state variables of streams 1 and 2, respectively, 7 is time, and 1 is distance. The velocities of streams 1 and 2 are V I and V Z , and they may be functions of time. The control u may be a function of time and distance or a function of time only. I n eq 2, and throughout, the upper sign will refer to the counter-current case when it is necessary to distinguish between the two cases. For the counter-current case, the system is illustrated in Figure 1, where L is the total equipment length and T the total operating time. Equations 1 and 2 are relative to a fixed coordinate system. Transforming the equations according to the method of characteristics 566 Ind.
Eng. Chem.
Fundam., Vol. 1 1 , No. 4, 1972
(8)
T h a t is, the performance index is a function of the output of the system and of the remaining part of the streams in the equipment a t the final time T. Equation 8 is for the countercurrent case; for the co-current case the second integral is taken at 2 = L. The functions fo,go, and ho are also assumed to be continuously differentiable with respect to their arguments. The form of the performance index is completely general. If the interior value of any of the variables should enter the expression for P , this part of the performance index can be transformed to a final value criterion by defining new state variables just as is done for lumped-parameter systems.
For co-current streams it would perhaps be more natural to choose a performance index where the functionsfo(z) and ga(y) are substituted by a single function ko(x,y). The modifications which would have been caused by this choice will be obvious throughout. The problem is then to choose u everywhere so as to make P a maximum. To derive the necessary conditions for this optimal choice of u,a variation in u is considered over a small time element. However, this element of perturbed control is different for the two types of control considered.
AT1
L
/
I i C
8
.-n n
\
Time- and Space-Distributed Control
The control u is assumed to be optimal everywhere. Then in a n arbitary point (so,to),see Figure 1, the control is changed from u = u0 to u = u1over a small element with edges As and At along the characteristics through (so,to). The change in u may be finite, but As and At must be infinitesimal. The change in u will introduce changes in x and y out of the element, and these changes will propagate throughout the region of influence of the point (so,to). It follows from the process equations that the change in x and y out of the element will be of order As or At. The resulting change in P will be of order As. At or A2. For the times AT^ and A72 shown in Figure 1, the change of order A in x or y is integrated over an in. finitesimal time to give the change in P over A71 and A T Z For the rest of the time it follows from the linearized process equations that the changes in x and y are of order A2, and these changes are integrated over a finite time to give the rest of the change in P. The necessary condition for optimal u is then that the resulting change in P must be nonpositive
This change AP may be expressed by a first-order approximation
/
0
\,
-
4 Stream
r
OT2
Time,T
2
Figure 1 , Counter-current system with element of perturbed time- and space-distributed control
Figure 2. Infinitesimal element used to derive equations for z and w. Counter-current streams
To determine the value of z , and w zto be used in eq 10, equations for the change of z and w along the characteristics are first developed. Consider another element As by At as shown in Figure 2. For this element one has
where summation over repeated indices is implied here and throughout. The change Ax, in eq 10 is the change in z i because of the change in u over the interval At. From eq 3 one therefore gets Linearizing eq 3 and 5 gives xj,t+At
and similarly from eq 5
= xj,t
+ .fj(z,Y,u)&
Yj,s+As
= Yj,s
=
xj,z
+ fjAt
+ gjAs
(17)
(18)
Performing the derivations in eq 15and 16 on eq 17 and 18 gives where o(Az) means terms of second and higher order in As and At. The values of x and y used in fi and gi are taken a t (so,to) *
The change in z out of the element a t (so,to) occurs over a "width" As, and it follows that the resulting change in P must be proportional to As in the first order. Similarly, the change in P due to the change in y must be proportional to At. This proportionality in AP to A s and At is expressed by defining new finite variables by where atj is the Kronecker 6. Substituting eq 13,14,19, and 21 into eq 15 gives
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 4, 1972
567
I
1.0
I
t ot
t-
Figure 4. Element with perturbed time-distributed control. Counter-current streams (d)
(C)
Figure 3. Determination of initial values of ziand wi on I = L and on T = T; a and c, counter-current streams; b and d, co-current streams
Canceling As, rearranging, dividing by At, letting At and omitting the position index gives
3 --
-2.-
dt
&fj
bx,
h l - w.-
bx,
-+
I n the same way one obtains the initial value for w ion the boundary 2 = 0 or 1 = L. The case of co-current streams, with v2 > VI, is shown in Figure 3b where
0
aT
-
AT,
bt
=
01 1--
At
02
I n general, the boundary condition for w ibecomes (24)
Combining eq 16, 13,14,20, and 22 in the same way gives The boundary conditions for z i and w ion same way from Figure 3c and d Equations 24 and 25 for z and w are integrated in the opposite direction of the respective streams. z and w have their initial values where x and y have their final values; i.e., z has initial values on the boundaries 1 = L and 7 = T (final time), and w has initial values on the boundaries I = 0 (counter-current streams) or I = L (co-current streams) and on T = T . To determine the integration times for eq 24 and 25 through the equipment, eq 4 and 6 have to be integrated backwards simultaneously. To determine the initial value of 2 %on the boundary 1 = L the element with perturbed control is placed along the boundary as shown in Figure 3a. The change of x out of the elemeht will now influence only that part of the performance index lying between rl and r1 AT,. Thus to first order in A
z,(T)
bho 91 bxias
= -- =
bho
--
ax,
(VI
T
=
T follows in the
i uz)
(32) (33)
I n terms of the new variables z and w eq 10 may now be written A p = 2t,to+4tAski f
wt,so+~sAtAYi
(34)
Also by eq 24 and 25 =
Zi,to+At
+
zt,zo
Wt,so+As = wt,s
+o(W + o(As)
(35) (36)
Substituting eq 11, 12, 35, and 36 into eq 34 and omitting higher-order terms in As and At gives AP =
From Figure 3a it also follows that for counter-current streams 3s
-
AT, As
For co-current streams with that
u2
'I+&
02
> vl, it follows from Figure 3b
If 01 > 02,s and t are interchanged in Figure 3b. The sign of the second term in eq 28 is also changed, so that the same expression is obtained. Hence, the general expression for zi(L) becomes
(
::>
:z,(L) = bXi - 1 i
568
Ind. Eng. Chern. Fundam., Vol. 1 1 , No.
4, 1972
[zt(fr'
- fzO)
+ wt(gi' - gtO)Iso,roAsAt
(37)
Combination of eq 37 and 9 gives, since (so,to) is an arbitrary point, that for u to be optimal everywhere t h e function HI defined by
HI
= Z*f,
+ wigt
(38)
must everywhere have a global maximum with regard to u. The corresponding equations for a single stream are obtained by setting 02 = wi = gt = 0. The result above is seen to be analogous to the well-known result for lumped-parameter systems. Also, as in lumpedparameter systems, it follows from eq 38 that if the process equations, eq 3 and 5, are linear in u the optimal control is either bang-bang or singular. For lumped-parameter systems the function H1 is constant along the path. This is not the case for two-stream processes in general, but for a single stream process it will a t least hold along each characteristic. I n this case, since there is no inter-
action between the characteristics, the system is nothing more than a number of lumped-parameter processes, and the optimal u will maximize the integrands fo and ho in eq 8 a t each instant. If the maximal value of jo and ho is constant, then HI 1%-illalso have constant value in each of the two areas influencingj, and ho. Time-Distributed Control
For time-distributed control, u = u ( t ) , the control is perturbed from its optimal value over an infinitesimal time A T , as shown in Figure 4 for the counter-current case. Again, the perturbation in u may be large. If the total length L is divided into increments Al, as shown in Figure 4, the resulting change in profit may be written, analogous to eq 10
AP
(” bxi
Axi
= AZ
+
For a single stream the second term above vanishes as does the second term in the integrand. Optimal Equipment Length L. F o r co-current streams, a small change in the equipment length will result in just a corresponding change in the residence times of the two streams. For counter-current streams, however, a change in L will also shift t h e trajectories relative to each other. This change will introduce changes in the state variables throughout the whole time-space domain, and the expression for bP/bL will become very complex. For co-current streams the expression for bP/bL is given by
(39)
where Axi =
(Si1 - f 2 ) A ~+ o ( A T )
+
(40)
Ayi = (gL’ - g$)AT o(AT) (41) From Figure 4, from the definition of z in eq 13 and from the expression for bl/bs and bljbt in eq 32 and 33 it follows
For a single stream the last term is omitted. Optimal Initial Values of x and y. It follows from t h e definition of z and w, eq 13 and 14, t h a t the optimal initial values are determined by
Zi(0) = w,(O)
=
0
(48)
for interior values of t,(O).If in the counter-current case the functionf, should also depend explicitly on y,(O), the condition for interior values of y,(O) is modified to
and in the same way (43)
(49)
Substituting eq 40-43 back into eq 39 and letting A1 + 0 gives AP =
S,z
-1
L [zi(fi’
- fi”)
- wi(gi’
- g?)ldlAT (44)
Since A P 5 0 for optimal u and arbitrary A T it follows that for all T it is necessary to have a global maximum with regard to u of the function H2 defined by
He
=
~
211
-I
J L ( z i j , - UJl9i)dl
f 212
(45)
-4gain, the corresponding result for a single stream is obtained by everywhere setting ve = g i = w i = 0. Also, t,he optimal control is bang-bang iff and g are linear in u.Even for a single stream the function H 2 will not be constant with time, nor will zifi be constant along each characteristic in general, as is seen by differentiation. Optimal Total Time, Length, and Initial Values
First-order necessary conditions for optimal total operating time T , equipment length L , and initial values of x and y are obtained by setting the appropriate partial derivative equal to zero for interior values, positive for values a t their upper boundaries, and negative for values a t their lower boundaries. Optimal Final Time T. The expression for b P / b T is obtained by considering the change in P resulting from changing the total time a n infinitestimal time A T . The result is
Conclusion
Secessary conditions for optimal control of eo-current and counter-current plug-flow processes without diffusion have been proved in the form of maximum principles, using only elementary calculus, for time- and space-distributed control. First-order conditions for optimal choices of initial values, equipment length, and total time have been found. Acknowledgment
The author gratefully acknowledges many helpful discussions with 0. A. Asbjdrnsen and financial assistance from Yorsk Hydro Co. Literature Cited
Hildebrand, F. B., “Advanced Calculus for Applications,” Prentice-Hall, Englewood Cliffs, X. J., 1962. Jackson, R., In!. J . Contr. 4, 585 (1966). Koppel, L. B., Shih, Y . P., IND. ENG.CHEM.,FUNDAM. 7, 414 (1968).
Russell, D. L., S.I.A.M. J . Contr. 4, 276 (1966). Seinfeld, J. H., Lapidus, L., Chem. Eng. Sci. 23, 1461 (1968). Seinfeld, J. H., Garalas, G. R., Hwang, M., IXD. ENG.CHEX, FUNDAX 9, 651 (1970). Shih, Y. P., Wang, 11. L., Chem. Eng. Sci. 2 5 , 997 (1970). RECEIVED for review July 8, 1971 ACCEPTED July 7, 1972
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