T H E INVARIANCE PRINCIPLE OF CONTROL FOR CHEMICAL PROCESSES D. E. HASKINS' AND C. M . SLlEPCEVlCH University of Oklahoma, Norman, Okla.
The application of the invariance principle of control for chemical processes is discussed. The results from theoretical analysis, analog simulation, and experimental measurements of heat transfer in a jacketed, stirred-tank reactor are presented, and from these a number of generalized conclusions are drawn. An important generalization of Petrov's dual channel concept is proposed for nonlinear systems. The three requirements that must be satisfied in order to attain invariance are given.
for a general control theory which fits chemical problems, the theory of invariance provides an effective approach to the design of process control systems. Briefly, the realization of invariance requires that at least one of the system output variables remains invariant for any disturbance in a specified input variable. In this respect the theory encompasses a broad class of control techniques, of which feedforward is only one type. Complete invariance represents an ideal toward which the designer strives. The quality of control achieved depends on the relevance of the limitations of the theory as discussed in this paper. The principle of invariance has been developed in Russia during the past 25 years, primarily along theoretical lines and with little regard for applications. Most of the attention has been directed to linear systems except for a few very specific nonlinear system examples. T h e treatment of practical application has not been supported by experimental evidence. Even though invariance theory has been reviewed and restated in several ways, there still exists a general lack of understanding regarding its usefulness. Presumably, it is this scarcity of information, rather than inherent limitations in the practical realization of invariance, that has delayed more widespread application of the theory. Therefore this paper delineates the problems encountered in the practical application of invariance theory and presents a number of the experimental results that have been obtained (2). Based on this study the general applicability of invariance theory can be inferred. N THE SEARCH
I process
Heat was transferred from the mixed oil temperature to a temperature which was the arithmetic mean of the coolant inlet and outlet values. The thermal capacitance of' the reactor wall was too large to be neglected. Coolant and oil inlet temperatures were constant. Heat losses were lumped and included in the oil energy balance. Heat capacities and densities of the oil, coolant, and wall metal were constant. The wall temperature was proportional to the average e.m.f. produced by four thermocouples embedded in the wall. T h e heat transfer coefficients were constant with respect to operating temperatures and flow rates. The model equations were derived by writing energy balances on the oil, coolant, and reactor wall in the manner of Stewart (,5), as can be seen from Figure 1.
Experimental System
The physical system employed was a completely mixed, stirred tank chemical reactor. However, since no chemical reaction took place, the actual simulation was a heat transfer process. In reference to Figure 1, the hot oil which entered the reactor was held at constant temperature; likewise the coolant, composed of a solution of ethylene glycol and water, was maintained a t a constant temperature at the inlet of the reactor jacket. The system disturbance was the coolant flow rate, and the controlling variable was the oil flow rate. The following assumptions were made in deriving the mathematical model of the system:
Coolant
WC
T h e oil was perfectly mixed in the reactor 1
La.
Present address, Esso Research Laboratories, Baton Rouge,
W
Figure 1.
Schematic of process
VOL. 4
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AUGUST
1965
241
Taylor series expansion method of linearization for transient variable equations is equivalent to deleting the product terms in the model. Therefore, the linearized form of Equations 3 is the following :
r------!
1
i I
I I
I
d7’,*=
I
dt
I I
-136.3 T f *
dT,* _ _ _ - -83.3 T,* dt
I
+ 31.7 T,* + 10.6 M’*
+ 41.1 T f * + 21.0 Tc,*
(4)
+ 82.8 T,* - 67.6 W,*
-~ dTcu* - -171 Tcu* dt
I I
T
Linear invariance theory has been presented by Petrov (4) and Kulebakin (3) and restated by Bollinger and Lamb (7). T h e usual approach is the transformation of the equations using either Laplacian or differential operators. T h e Laplace transforms of Equations 4 can be written in matrix form.
I
I
Figure 2.
Block diagram
T h e constants were evaluated from available data from the literature or from laboratory measurements.
dT, - - -31.7 T f dt
dT, - -83.3 Z’w dt
dTcu - _- -41.4 T,, dt
+ 41.1 TI + 21.0 Tc, + 658 -
3 TI*
+ 31.7 T,* + 10.6 W* 0.666 W*Tf*
dTW*- -83.3 T,* dt ~~
~~
dt
= -171
+ 41.1 T,* + 21.0 T,,*
+ 82.8 T,*
Tco*
-
(3)
67.6 w,* - 2.4 Wc*Tco*
1-he specific problem chosen was the control of the system such that the mixed oil temperature, T,, was invariant with respect to changes in the coolant flow rate. T h e theory of invariance \vas used for the design of a control system for this purpose. linear System Analysis
In this paper a linear system is rigorously defined as one in ivhich the principle of superposition holds. Superposition means that the system’s total response is the sum of all the responses of all applied forces individually. For convenience linearity can be assumed if the expression relating system input and output involves only the first powers of the input and output and their derivatives. Nonlinearity refers specifically to the loss of the superposition property. T h e existence of the product terms in Equation 3 makes this model nonlinear. l’herefore. in order to use linear systems analysis these equations must be linearized. Stewart (5) has shown that the 242
l&EC FUNDAMENTALS
L
1296
+
d Z c u*
-10.61
rTf*(r:
1
(2)
T h e variables in Equations 2 are usually called “total variables” because each of them can be assumed to consist of a Tf*]. steady-state plus a transient portion [ T , = Equations 2 can also be written in terms of the transient variables only. = -136
o
+ 82.8 T , + 75.2 W C2.4 MIcT,,
d7;* dt
-31.7
+ 31.7 T , + 102.5 111’ - 0.666 W T f - 186
.~
~
p + 136.3
I
In addition to Equation 4 or 5, the linearized form of the general control equation is given by
W*(s) = Kf(s)T,*(s)
+ K , ( J ) T ~ * (+~ )Kc,(s)z’,,*(~) + Kc($)bL,*(J)
(6)
where each of the K ( s ) terms is an undetermined function of s corresponding to the controller transfer function. At this point clarification of the terms appearing in Equation 6 is warranted, since the distinction between feedforward and feedback control is somekvhat arbitrary. In this paper the distinction is based on which system variables are measured. Feedforward involves the measurement of a process disturbance (input) variable, whereas feedback constitutes the measurement of a process output variable. Thus, the transfer function, K f ( s ) , is that of a feedback controller because a system output variable is measured. This type of controller is normally the one used for process control. T h e transfer functions, K,(s) and Kc,(s), are those of feedback controllers also because they involve measurement of system output variables. However, these controllers are different in several respects from customary feedback systems. They d o not require a measurement of the output variable which is to be controlled. They allow permanent error in the controlled variable if the controllers are not properly adjusted. They may allow better control than customary feedback because of anticipatory action. T h e transfer function, K,(s), is that of a feedforward controller because a system disturbance (input) variable is measured. Equation 6 is combined with Equation 5 to form the matrix equation :
s
+ 136.3 -41.1
0
-31.7 83 3
s+
0
-82
-K/(s)
- T,*is)
1
_Tt.*(S)
J
-21
8
-K,(s)
-10.6
0
171
s+
-K,,(s)
r
1
O
l
Since T,* was chosen to be invariant, this set of equations is solved for 7‘,*(s)
r o
0
-31.7
-10.61
Any controller which is designed to these transfer function specifications will make 7‘,* invariant to any change in PV,*. T h e transient variable equations do not necessarily have to be used for the synthesis of invariance principle controllers. I n fact. it i? often preferable to use the total variable model because it gives rise to controller specifications which are based on the more commonly measured total variables. T h e transfer functions based on transient models require the measurement (or calculation) of the transient portions of variables, Ivhich is not always convenient. IVhen the total variable model (Equations 2) is used, T,(s) is equated to 7.f38instead of zero. All other steps in the procedure are identical to those for the transient model equations. The resulting controller equations based on total variables are the following:
PI7
+ 12.76 T,, - 1333 -2.99(41.1 TI,, + 21.0 T,, + 658) + D + 83.3
= -2.99
617 =
T,
~~~
~~
~- -~
12.76 T J S s- 1333
s
[4J
+ 136.3 -41.1
where [ A ] =
0
-31.7
0
-K,(J)
-21
0
+ 171
0
-Krofi)
1
83.3
S+
-82.8
-K,(J)
s
-
=
J17,*(s) { - (31.7)(21)(67.6) - (10.6)(21)(67.6)K,(~)
+
(10.6)(67.6)(~ 83.3)KC,(s) (10.6)[(s
+ 8 3 . 3 ) ( ~f
+ 1
171) - (21)(82.8)]KC(s)
-.
(9)
PI T h e first invariance condition is that T,*(s) equals zero. T h e second invariance condition is that det [ $ ] is not equal to zero when all off-diagonal elements in the first column equal zero. There are infinitely many ways to satisfy the first condition, because all rhe K ( s ) terms are arbitrary. The simplest three examples are
K,(s) = K,,(J) = 0 ; K,(s) = -2.99
=
K,,(s)
=
0 ; K,(s)
=
4245/(?
+
+ 254.3 D + 12,506
(18)
T h e dynamic portions of Equations 16 through 18 are identical to Equations 1 3 through 15. T h e difference bet\veen the two controller types is that Equations 16 through 18 have provided for a set-point value of T,; consequently these equations are more useful for the construction of actual controllers. The foregoing treatment of a linear system model is similar to the usual manner in which invariance theory is presented. Most papers on this subject imply that invariance theory is primarily a linear system technique. The emphasis on linear systems is due to several reasons. Most papers are tutorial in nature and the theory is more simply first presented for linear systems. Multivariable systems are very conveniently handled by using linear operators and matrix equations which are applicable only for linear systems. If models are determined by existing dynamic testing methods. such as frequency response, instead of using material or energy balances. only linear models are found. l h i s fact is a direct result of the lack of nonlinear model identification techniques.
(10)
+ 83.3) (11) + 254.3 s + 12,506)
K,(s) = K,(J) = 0; K,,(J) = - 6 2 . 8 / ( ~
K,(s)
(17)
4245 11’ - 20.900 TI,,$- 734.800 ~12.76 TtS8- 1333
Expanding the numerator of Equation 8 gives T,*(S)
=
D2
-10.6-
(16)
(12)
[A]:
When Equation 10 is substituted into it can be shown that det [ A ] does not equal zero; this is also true for substitution of Equations 11 or 12. Thus the second condition of invariance is satisfied. ‘Three choices of invariance controller transfer functions result from substituting Equation 10: 11: or 12 in Equation 6.
There is an equivalent method of obtaining the controller equations without resorting to a linear system method such as the Laplace transformation. Simply set the variable which is to be controlled equal to its steady-state value (or zero for transient variables) and set the derivative of the controlled variable equal to zero. Then the unnecessary variables are eliminated from the set of equations which form the model to give the desired controller equations. This method is also applicable to nonlinear systems. Nonlinear System Analysis
I n Equations 3. 7,* and d T f * ,dt are equated to zero.
0 Lt7*(s) -
T,,*(s)
-62.8 s 83.3
+
=
31.7 T,*
=
(14)
dt
dTco* __- -171 T,,* dt
-83.3
+ 10.6 TL7*
T,* f 21.0 T,,*
+ 82.8 T,*
(19) (20)
-
67.6 M*,* - 2.4 W7,*T,,* (21) VOL. 4
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AUGUST 1965
243
Figure 3.
Simulated system response for various disturbances
W -
Equation 19 gives one controller equatim directly.
'w* = -2.99 Tu*
(22)
-2.99(41.1
D
This equation is the same as the linear model result, Equation 13. Substituting Equation 20 into 22 gives the second controller equation
w* = D
+ 83.3
T m
*
+ 254.3 DW* + 12,506 W* - 4245 W,* + 2.4 W,*(D
+ 83.3)W* = 0
(24)
which is similar, but not identical, to the linear model result, Equation 15. T h e difference between the two equations is the last term on the left-hand side of Equation 24. As shown in the linear system analysis, the total variable model can also be used. In this case the controlled variable T , is equated to T,,, and dT,/di is set equal to zero in Equations 2. After unnecessary equations are eliminated in the same manner as for the transient equations, the following controller equations arise:
W =
244
-2.99
T,
+ 2.99 TI,, + 17.43
9.61 - 0.0624 T f 8 ,
l&EC FUNDAMENTALS
(26)
+
+
+
-(SO40 TI,, 219 293 T+W, 9396 W,) (0% 124.6 D 1705) 2.4 W,(D 83.3)
which is likewise identical to Equation 14. Substituting Equations 22 and 23 into 21 gives the third controller equation
Daw*
+ 83.3
9.61 - 0.0624 T,,
W =
-62.8
+ 21.0 T,, + 658) + 2.99 T,, + 17.43
TrS,
(25)
+
+
2.99 T,,, ~~
+
+ 17.43
9.61 - 0.0624 T,',
+
t (27)
These equations are dynamically similar to the linear model results (Equations 16 through 18), but each is different from its corresponding linear model result because of the way in which the set point ( T f S 8is) introduced. This difference between linear and nonlinear model controllers indicates that the total variable models should be used if the correct set-point determination is to he included in the controller specification. In the linear system analysis the second condition for invariance is verified by examining det [A] in Equation 9 with first column, off-diagonal terms equal to zero. For nonlinear systems there is no such simple method of verifying the second condition. T h e most convenient check is provided by the dual channel concept of Petrav (4). The system, including the controller, is represented in block diagram form, such as
Figure 4.
Process laboratory and t(
Figure 2, and the second condition is visually checked to determine if the following requirements are satisfied. There must be two channels of disturbance information which independently reach the output variable being held invariant, in this case T,. Neither of these channels of disturbance information can be dependent a n any variations which might occur in T,. In other words, if these two conditions are satisfied, the circuit can be broken at the break point denoted by bin Figure 2 without producing any change in the output, T,. Even under the conditions of the broken circuit there has to be one path in the block diagram which conveys the disturbance and another one which provides far the compensating signal. A more complete discussion of the dual channel concept, including its mathematical and physical ramifications, is given by Haskins (2). 'The block diagram representation of the system is also useful for determining whether the controller for the nonlinear system will be linear or nonlinear. An important generalization for nonlinear systems, proposed by Haskins ( Z ) , is the following :
If either dual channel of information from the measured variable to the variable which is to be made invariant contains a nonlinearity, the controller to be synthesized must be nonlinear also. However, if both channels are linear, the controller will also he linear, irrespective of other nonlinearities in the system. Simulation Studies
In order to study the feasibility of the designed controllers and to test the controller equations, the process was first simulated with an analog computer. T h e variables were magnitude-scaled to increase the computer accuracy and timescaled to decrease the computation time. T h e process model was programmed on an analog computer. T h e transient model was used to increase analog computer accuracy. The controllers were programmed according to the control Equations 13 through IS, or 22 through 24. T h e disturbances, which were completely arbitrary, were chosen to be sine, square, and triangular waves of varying frequency. T h e magnitude of the disturbance was *16%. Figure 3 gives the response of the three output variables, T,, 7,, T,, to various
disturbances In lhe coolant now rate, W,, while controlling the oil flow rate, W . I n this case, it can be seen that T,, was held invariant by the nonlinear controller irrespective of whether the disturbance was sine, triangular, or square wave. (Throughout the res1 of the paper, T,, rather than T,,, was held invariant. Figure 3, based on T , being held invariant, simply gives a qualitative demonstration of a typical, invariant analog system.) This selection of disturbances provided information about the efficiency of each controller for various shapes and frequencies of disturbance. The efficiency of a controller was defined to be
where E is the controller efficiency; I., is the integrated, absolute value of the output variable which is to be held invariant but under conditions of no control; and I, is the integrated, absolute value of this same output variable under conditions of control. In other words, if the output variable is truly invariant, then I , = 0. T h e results far some of the analog simulation studies are given in Table I. Experimental Work
T h e controllers specified by invariance theory were also tested on the physical system in the laboratory. T h e Process Dynamics Laboratory at the University of Oklahoma has experimental equipment which has been assembled for general investigations of process identification and control problems. T h e components of the complex system could be combined in the manner described above. Figure 4 shows the process laboratory and control panel. Behind the control panel is the analog computer laboratory shown in Figure 5. T h e two laboratories are connected by telemetry lines through the control panel. This arrangcment allowed all system variables to be measured and transduced at the control panel and resulted in voltages which were proportional to system variables and were available for analog computation. T h e controllers specified by Equations 16 through 18, or 25 through 27, were programmed on the analog computer. T h e controller output voltages were transduced to signals which were usable in final control elements. VOL. 4
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245
and ouxiliary equipment
l n e experimental testmg manner as the simulation stuaies. u m c r a u r r emucncy was defined in the same manner. The results for some experimental tests are given in Table 11. Conclusions
T h e results of the analog computer simulation studies, the experimental testing, and the additional results reported by Haskins (2) have led to several conclusions ahout the practical application of the invariance theory. Whencrer invariance is very close to 100% efficient (98% or better), the quality of control is independent of disturbance magnitude and frequency.
Table 1. Run No.
A2 A4 A6 A8 B2 B4 B6 B8 C107 (2109
CllO C118 c120 ClZl c112 C113 C115 C116
Controller Equation9
23 23 23 23 22 22 22 13 or 22 15 15 24 15 15 24
14 or 14 or 14 or I 4 or 13 or 13 or 13 or
15
24 15 24
~~
246
l&EC FUNDAMENTALS
No particular problems result from using ConrroucrS m d contain a single differentiator (2). The use of a double differentiator proved feasible in the analog simulation studies, although the controllers dependent on the second derivative of a variable are not always reliable. T h e double differentiator is not recommended for real systems unless some way is found for increasing the reliability. T h e linear model proved to be most valuable for design
Simulation Stvdie PIOCtfS
Sirnulatiorz Linear Nonlinear NO"li".Z'W
Nonlinear Linear Nonlinear Nonlinear Nonlinear Linear Nonlinear Nonlinear Linear
Nonlinear Nonlinear Nonlinear Nonlinear Nonlinear Nonlinear
alne"
"."3
Sine' Sine Sine
0.05 0.10 0.10
Sine
0.10
Triang. Triang.
0.05 0.05 0.05 0.05
Souare
02
100 100 81 100 86 100
73 100
could have been designed from the linearized model, provided practically l O O ~ ,efficiency. For accurately modeled systems, typified by the simulation studies: linear system theory alone gave high quality control even for severe disturbances. Nonlinear theory may improve the efficiency somewhat, but the more complicated nonlinear controller may not be economically justified. If the model used for the synthesis procedure is not very accurate, the use of nonlinear theory may not provide any improvement over linear theory. Nonlinear theory becomes important when the nonlinear model is an accurate representation of the system and when nonlinearities are severe. If the ability to change the set point and to retain the invariance quality is required, the nonlinear theory becomes important If the system parameters, such as thermodynamic properties, are not accurately defined, some optimization of controller settings may be necessary, which can be tedious. Particular attention should be given to the procurement of controller components that can execute the controller equations accurately and as slokvly as the process dynamics dictate. Total variable models are recommended for determining controller equations for real systems because of the more useful set-point specifications: Nonlinear systems can be treated if the additional system variables can be eliminated from the model equations. This elimination can be done analytically as shown in this article or by using analog computation with the invariance condition form of the system model. Double-valued functions. such as the hysteresis phenomenon, will cause difficulties in the elimination of variables. For linear systems, stability can be examined by conventional methods. The roots of the characteristic equation, det [ A ] = 0, where [A]is given by an equation similar to Equation 9, determine stability. Feedforward controllers do not affect stability because their transfer functions do not appear in [A]. Any feedback controller will affect stability through and the roots of the characteristic equation have to be examined. Systems which are only marginally stable, or unstable, will have to be examined for stability after the controllers are chosen. For nonlinear systems there are some techniques available for verifying stability, but perhaps the
most useful method will be provided by analog computer simulations of the controlled system. T h e dynamics of measuring and transducing devices and final control elements (valves) can be ignored if their actions are fast in comparison to the process rates. Some care should be exercised in minimizing the dynamic effects of auxiliary equipment. Although a pure dead time was not encountered in this work, such a dynamic effect could cause trouble in synthesizing controllers. Whether it does or not depends on where in the total system the dead time is. Sometimes it can be beneficial; a t other times it is detrimental, which is the usual assumption of customary feedback control theory. A delay between the occurrence of a disturbance and the observation of its effect would necessitate introducing a pure lead time of the same magnitude as the dead time into the required controller transfer function--a requirement which is physically impossible to realize. Nevertheless, the general control equation permits the selection of the least troublesome controller which can be better than conventional feedback controllers. The attainment of invariance is fundamentally based on three requirements. The mathematical model adequately describes the system behavior, including the important process disturbances. The control system is allowable by satisfaction of the dual channel conditions. The theoretically derived controller equations can be accurately executed except as noted above, where pure dead time is encountered. If these conditions can be met, invariance theory is a methodical design procedure for chemical process control systems and represents a definitive statement of an optimum control system. The quality of control obtained in practice depends on how well these conditions are achieved.
[e] Acknowledgment
The authors gratefully acknowledge the support of the National Science Foundation and the Phillips Petroleum Co. Nomenclature
Ai A, Table II. Experimental Results on Process Equipment EfiController ___ Disturbance _ _ _~~ czency, Equations" Wave Cycleslmin. % 0.192 88 X2 17 or 26 Square 0.384 88 x4 17 or 26 Square 0.192 89 X6 17 or 26 Sine 0.384 88 X8 17 o r 26 Sine 0.192 96 16 or 25 Square XI 0 Square 0.384 83 16 or 25 x12 0.192 86 16 or 25 Sine X14 Sine 0,384 86 X16 16 or 25 0,192 67 X25 18 Square 21 Square 0.192 43 X27 0,384 52 X29 18 Square 27 Square 0.384 57 X31 0,600 62 X33 18 Square 0.600 20 x35 27 Square 18 Sine 0.192 74 X37 0.192 82 X39 27 Sine 18 Sine 0.384 51 X41 27 Sine 0.384 58 x43 18 Sine 0.600 44 x45 21 Sine 0.600 46 X4? First 4 rows refer to feedback controller 1 in Figure 2 , next 4 rows to feedback controller 2, and remaining rows to feedforward controller. Rut1 .Yo I
(1
Cp, Cp, Cp,
det
=
=
E hi
= = = =
h,
=
I,
= =
D
Z, K,(s) K,,(s) K,(s) K,(s) QL S
t
T,, T,, TI T,, 7,
V, V, V,
reactor wall inside area, sq. ft.
= reactor wall outside area, sq. ft. = coolant heat capacity, B.t.u./lb.-" F. = oil heat capacity, B.t.u./lb.-" F.
=
= = = = = = =
= =
= = = = =
wall metal heat capacity, B.t.u./lb.-" F. determinant differential operator controller efficiency reactor wall inside heat transfer coefficient, B.t.u.,/ hr.-sq. ft. " F. reactor wall outside heat transfer coefficient, B.t.u.:/ hr.-sq. ft. O F. integral absolute value for controlled system integral absolute value for uncontrolled system transfer function of the feedforward controller on FP'? transfer function of feedback controller on T,, transfer function of feedback controller on T I transfer function of feedback controller on T, heat loss term: B.t.u./hr. Laplacian operator process time coolant inlet temperature, O F. coolant outlet temperature, O F. oil bulk temperature, " F. oil inlet temperature, " F. wall temperature, O F. reactor coolant volume, cu. ft. reactor oil volume, cu. ft. reactor wall volume: cu. ft. VOL. 4
NO. 3
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247
W’ kV,
= oil flow rate, Ib.;hr. = coolant flow rate. lb.,’hr.
Literature Cited
GREEKSYMBOLS = a characteristic matrix [ 41 = density of coolant, lb./cu. ft. pc = density of oil, Ib.icu. ft. p, = density of wall metal, Ib./cu. ft. pLD SUBSCRIPTS = coolant variable = oil variable
C
f zu
= =
ss
wall variable steady-state variable
(1) Bollinger, R. E., Lamb, D. E., IXD. ENG. CHEM.FUNDAM E N T A L S 1, 245-52 (1962). (2) Haskins. D. E., ”Synthesis of Invariance Principle Control
Systems for Chemical Processes,” Ph.D. dissertation, University of Oklahorna. 1964. ( 3 ) Kulebakin. V. S., .‘The Theory of Invariance of Regulating
and Control Systems.” “’Automatic and Remote Control,” Proceedings of the First Interriational Congress of the International Federation of Autoinatic Control, Moscow, CSSR, 1960. Vol. I. pp. 106-16. Butterworths. London, 1961. (4) Petrov, B. N.. “The In\-ariance Principle and the Conditions for Its Application during the Calculation of Linear and Nonlinear Systems.” Ibid.. pp. 117-25. (5) Stewart. IV. S.. “Dynamics of Heat Removal from a Jacketed, Agitated Vessel.” Ph.D. dissertation, University of Oklahoma, 1960.
SUPERSCRIPT
*
RECEIVED for review June 26, 1964 ACCEPTED .January 15, 1965
= transient variable
GREEN’S FUNCTIONS AND OPTIMAL SYSTEMS Complex Interconnected Structures M . M. DEN”
AND
RUTHERFORD A R l S
Cnireisitj of .Iriinnesota. dlznnrapoiis, .Mznn.
The solution of first-order variational equations b y means of Green’s functions i s utilized to develop necessary conditions and computational methods for the optimization of plants with complex interconnections. A concrete example of a staged system and a system with both discrete and continuous elements i s completely solved.
has recently been focused on the opof large systems whose units are interconnected in a complex fashion. ‘The system of units linked in a straight chain or sequence has long been studied and is amenable to optimization in a variety of ways--notably. by calculus with the use of Lagrange multipliers, Pontryagin’s principle, and dynamic programming. These methods have been extended to chains with feedback and to more general systems ( 7 , 2, 779, 72--7R). I n some cases the adaptation of Pontryagin’s methods and of dynamic programming has been incorrect and the work of Jackson (76) and Jackson and Horn (7,l. 7.5) has been noteworthy in discovering both gross and subtle fallacies. We are not aware of any published calculations 011 complex systems that have completely foundered on these shoals. but this is in part due to the scant attention that has been paid to real computational feasibility. The only systems that have been studied in detail are t ~ v o -and three-stage co- and countercurrent autothermic reactors (27) and systems for which the policy is disjoint or equivalent to a straight-chain optimization and a one-dimensional search ( J , 72. 27). I n this paper we use the solution of the linear variational equations by Green‘s functions to obtain both the necessary conditions and computational techniques for systems of arbitrary topology containing both discrete and continuous elements. .A concrete numerical example is included to demonstrate the workings of the computational algorithm. UCH ATTENTION
M timization
Topological Structure
By a serial or “straight-chain“ process we mean a sequence of stages such as that shobvn in Figure 1. Its stages may be numbered consecutively in the direction of flow of information, the input to the nth stage consists solely of the entire effluent of the ( n - 1)st stage. and no component of the decision vector a t one stage appears in the transformation equation for any other stage. A continuous process of duration 0 whose finite difference approximations form straight chains will also be called a serial process. The recycle systems considered previously (7, g! 12) are examples of nonserial processes, and other basic structures have been discussed ( 7 , 2, 73, 77). Every complex system can be broken down into a number of subunits, each a serial process, together with matching conditions imposed by continuity relations. An optimal system will not, in general, be made u p of the set of optimal serial processes which satisfies the continuity relations. This fact has been dramatically demonstrated by .Jackson ( 7 , 76). and is clear from the form of the new boundary value problem caused by recycle ( 7 , 9 ) . \Ve can, however. extend all of the results which we have developed for straight-chain systems to complex rystcms of general topological structure ( 7 . 9. 70).
STAGE STATE DEClSlO N
Present address. Department of University of Delaware, Newark. Del. I
248
I&EC FUNDAMENTALS
Chemical Engineering,
p,
- r k ,---pFll ---iFk qn
41
Figure 1 .
Serial process
qH
