Publication Date: August 1966. ACS Legacy Archive. Cite this:Ind. Eng. Chem. Fundamen. 5, 3, 403-413. Note: In lieu of an abstract, this is the articl...
makes the closed-loop stabilization difficult. Although a number of significant time-delay compensation techniques have been proposed to improve the control ...
The major advantage of the proposed control system design is to track the set point without producing a vigorous control action and requiring the exact ...
With this model structure, it is shown that a simple analytical predictive control law can be formulated to control a nonlinear process. Simulation examples are ...
Department of Chemical Engineering, The Hong Kong University of Science and ... it is shown that a simple analytical predictive control law can be formulated to ...
Jun 1, 1995 - Nonlinear Control of a CSTR Disturbance Rejection Using Sliding. Mode Control. Maria C. Colantonio. PWIQUI-UNS-CONICET, 12 de Octubre ...
Apr 28, 2009 - voltage, current, and fuel-cell-tube temperature responses of the cell to step .... used a PID controller to regulate the SOFC voltage across a.
DOI: 10.1021/ie50649a603. Publication Date: January 1964. Note: In lieu of an abstract, this is the article's first page. Click to increase image size Free first page.
File failed to load: https://cdn.mathjax.org/mathjax/contrib/a11y/accessibility-menu.js .... Previous studies have explored the design and control of processes with ...
Fax: +1-215-895-5837. E-mail ... This paper presents a study of the dynamic behavior and control of a tubular solid oxide fuel cell system. ... Dynamic outlet voltage, current, and fuel-cellâtube temperature responses of the cell to step changes in
Shaanxi Key Laboratory for Advanced Energy Devices, Key Laboratory for ... Science of Shaanxi Province, School of Materials Science and Engineering, ...
DYNAMICS AND CONTROL OF A CLASS OF NONLINEAR, TUBULAR, PARAMETRICALLY FORCED HEAT EXCHANGERS AND CHEMICAL REACTORS L0W ELL B
.
K0
P P E L , School of Chemical Engineering, Purdue University, Lafayette, Ind
Earlier work oin the dynamics of distributed parameter models of heat exchangers and chemical reactors is extended by dlefining a more accurate type of linearization. An improved method for controlling distributedparameter systems involving nonlinear feedback is illustrated for proportional control.
HE dynamics of diistributed parameter models of heat Texchangers and chemical reactors have been studied, without recourse to linearization (2, 3 ) . In the present paper, this earlier work is extended by defining a more accurate type of linearization which can be used to represent the dynamics, and studying a control system design problem resulting from this linearization. A new and improved method for controlling such disiributed parameter systems, involving nonlinear feedback, is developed and illustrated for proportional control.
Introduction to Processes The entire class of processes to be studied has dynamics represented by the equation
be -
at
+ [l + r(t:l]-ae ax
= -P[I
+ b r ( t ) ] en
(11
with boundary conditions
The dependent variable,, 8, represents normalized temperature in the case of a heat exchanger, and normalized concentration in the case of a reactor. Theforcing function, r ( t ) , is a normalized disturbance in flow rate, and the parameter, P, is the ratio of heat exchange to heat capacity for the case of a heat exchanger or, in the case of a chemical reactor, is the ratio of reactive to convective rates of transport of the reactant. Parameter b, different from zero only in the case of a heat exchanger, is the linearized rate of change of over-all heat transfer coefficient, with flow rate (2). For n = 1, Equation 1 represents the dynamics. of a constant wall temperature heat exchanger or of a reactor with a first-order irreversible decomposition of reactant; for other values of n the dynamics are those of a chemical reactor with an nth-order irreversible decomposition. For n := 1, we normally refer to the process as a heat exchanger, and to 0 as temperature, although it is clear that the results are directly applicable to a first-order reactor.
The assumptions upon which Equation 1 is based are those of plug flow, constant physical properties, perfect radial mixing, and negligible back mixing. Experimental verification of Equation 1, as a good representation for the dynamics of a tubular heat exchanger (for the case of constant wall temperature and negligible wall capacitance, as required by restrictions on this equation), has recently been provided (I). Equation l a , a boundary condition, indicates that one assumes a specified form of disturbance in the process load variable, either the entering temperature or the entering reactant concentration. Equation 1b is the steady-state solution of Equation 1, and is an initial condition. This is obtained by setting the time derivatives and the flow rate disturbance, r ( t ) , to zero in Equation 1 and solving the resulting equation, also imposing the condition given by Equation IC. The concept of disturbance from an initial “normal” steady state is the common one for control system analysis, and occurs with no loss of generality. A major objective of this study is the design of control systems to manipulate the flow rate r(t) in order to control the exit process-dependent variable, e( 1,t), despite load disturbances caused by departures in &(t) from unity. I n the chemical reactor, for example, this means control of conversion by manipulation of residence time. This design must be based upon the dynamic responses of the process to changes in these manipulated and load variables, which are developed below.
Dynamic Analysis Definition of Auxiliary Variable 4.
The transformation
4(.,t> = -
(2)
together with the definition of a deviation variable 4(x,t) = d x , t )
-x
(3)
reduces Equation 1 and its initial and boundary conditions to
at
+ [l +Y(t)]G a4 = -(1 - b)r(t)
(4)
= 0
(5)
B(0,t) = B l ( t )
(6)
B(X,O)
where r$l(t) is the transformed load disturbance, derived VOL. 5
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AUGUST 1966
403
directly from & ( t ) through the transformations given by Equations 2 and 3. The advantage of Equation 4 over Equation 1 is that Equation 4 is linear; however, Equation 4 is still parametrically forced and hence the principle of superposition with respect to the forcing function r ( t ) does not apply. Solution of Transformed Equation. The solution of Equation 4 with boundary conditions 5 and 6, obtained by the method of characteristics ( 2 , 3 ) ,is
6(x,t)
=
f
I
t
t
t
of the disturbance, r ( t ) , through the auxiliary function, g(x,t). I n particular, this accuracy in no way depends upon the process parameter P, as contrasted to the result of direct linearization of Equation 1 . I t was demonstrated previously ( 2 ) that, for sufficiently high values of P, even small flow rate disturbances can render inaccurate the solution obtained by direct linearization. I t is clear that ignoring the term rb$/dx is tantamount to assuming that g(x,t) is small with respect to typical values of the quantity ( t - x ) . This provides a convenient method of checking upon the likely accuracy of dynamic responses predicted by Equation 1 1 . Step Response. Although the step response of this class of processes has been compared with the result of direct linearization earlier, we here compare it with the result of linearization in terms of the auxiliary variable, B(x,t). Thus, we take
(7)
(1 2)
r ( t ) = au(t)
for which Equation 9 shows that A(?)
=
-
11
(1 3)
~
l + a
Using this result in Equation 8 yields
(9)
The auxiliary functions g and X are determined solely by the velocity disturbance, r ( t ) . Several examples of the responses, which may be deduced from Equations 7, 8, and 9, have been presented (2, 3 ) . If in EquaLinearization Based on Auxiliary Variable tion 4 we ignore the term r?@/bx, the resulting equation is no longer parametrically forced, and may be solved by standard techniques of Laplace transforms. In the transformed variable ~ ( X , S the ) solution may be written
and it is clear that g will be small, provided that the amplitude, a, of the step disturbance is small. For the exact response, Equation 7 shows that
I-a(l - b ) t ;
d(x,t) =
4.
-
X
1 + a
while Equation 11 gives for the linearized response
- b)t; -a(l - b)x; -a(l
B(x,t)
=
t
x
(1 6)
Comparison of Equations 15 and 16 shows exact agreement until a time given by t = min { x , x/(l a) ) . Furthermore, the extent of disagreement beyond this time depends solely upon the amplitude, a, of the disturbance, as expected. This result should be compared with the results of direct linearization ( 2 ) . The direct linearization has the effect, in the heat exchanger case, of replacing the actual exponential form of the temperature response with a linear form, so that the only real agreement obtained is in the value and slope at time zero. Furthermore, for high values of the ratio of heat exchange to heat capacity (say 50), it was shown that even a step disturbance as small as a = 0.1 is inadequately represented by directly linearized response. Pulse a n d Impulse Response. To illustrate further the accurate dynamic results which can be obtained from the linearization in B ( x , t ) , we consider here a square pulse disturbance
+
where r(s) and &(s) are the transforms of r ( t ) and & ( t ) , respectively. The inverse of Equation 1Oi s obtained by standa r d inversion techniques, and results in t
I-X
Comparison of Equations 7 and 11 shows that: (1) the linearized solution given by Equation 11 agrees exactly with the true solution until a time given by tl = min {x,x-l"rd.) This agreement even as to the form of the solution is entirely different from the results obtained by direct linearization of Equation 1 in terms of the original dependent variable, O(x,t), as is pointed out ( 2 , 3 ) . (2) The accuracy of the approximation given by Equation 11 is dependent only upon the magnitude 404
l&EC FUNDAMENTALS
A r(t) = - [ ~ ( t ) ~ ( -t d ) ] d
(17)
By letting d go to zero, we can also derive from this the impulse response. The linearized and exact $ ( x , t ) responses to this disturbance
are derived and presented in Appendix I, and compared graphically in Figure 1. Figure 1, a and b, shows the exact pulse and impulse response, respectively, while Figure 1, c and d, shows the linearized pulse and impulse responses. From Figure 1, it ma.y be concluded that the disagreement between the exact and linearized responses depends, in both the pulse and impulse cases, only upon the area, A , of the disturbance compared with the position of observation, x . Furthermore, the two responses are identical up to a time t = x A . The total area under the response is independent of d and is always given by the expression A / 2 x . Further-
-
-
pulse response, more, it may beinseen the exact from Equations case, is the15 derivative and 33 that of the theexact im-
-
step response u p to t = :c A . These last two properties are a direct carryover from the convenient properties of systems which are linear and are not parametrically forced, and are therefore regarded as favorable properties. Again, the linearization based upon B ( x , t ) gives initial responses which are accurate with regard - to form when compared with the exact responses. This property provides the control system designer with added confidence in the use of a linearized transfer function, confidence which may be unwarranted if the linearized transfer function is ‘based upon direct linearization in 0. If one proceeds to callculate the function g ( x , t ) , as was done in the case of the step response, it is found that this function is directly proportional to the amplitude of the pulse disturbance. This further bears out the contention that the accuracy will be dependent only upon the disturbance amplitude. All other dynamic responses for this class of processes may now be obtained with reasonable confidence from the new linearization, provided one restricts attention to reasonably small disturbance amplitudes. The two cases given here illustrate the results; we proceed below to use this linearization for control system desig:n.
d
X
X+d
&‘K/’Tl ;I/ 1
I
IiAtI
l
l
_i_
A(I-b)
I--
I-
Figure 1 * Typical Pulse and impulse responses, x a. b.
Exact pulse response Exact impulse response
E.
Linearized pulse response
> (A
+ d)
d. Linearized impulse response
Figure 2. Block diagram for simple proportional feedback control
Feedback Proportional Control
The use of linearized transfer functions for d ( x , t ) having been justified by the dynamic analyses above, we now assume that the flow rate is to ‘be manipulated to control the value of the response B ( x , t ) a t some observation point x within the process. Thus, for the case of proportional control,
r ( t ) == K’[&x,t)
- 61
The sign adopted in Equation 18 is necessitated by the fact that Equation 11 indicates that an increase in r ( t ) will decrease i ( x , t ) , and vice versa. The set point value, 6, is the level a t which it is desired to control d ( x , t ) ; presumably this value will normally be zero because i ( x , t ) is a deviation variable. When the Laplace transform given by Equation 10 is used, the block diagram of Figure 2 results. The load disturbance, d~(s),represents the transformed fluctuation in the inlet value of the dependent varia.ble. Furthermore, the effective loop gain, K , has been defined by the equation
K = K’(l
- b)
(19)
where K’ is the gain set in the proportional controller. Equation 19 shows that this effective gain is reduced by any dependence of parameter P upon the flow rate. Such dependence will always be present in the case of heat exchangers, but is not likely in the case of chemical reactors. In the former case, it may be advisable to make parameter b as small as possible, to obtain the maximum effective gain within the capacity of the given proportional controller. This may be done by ensuring that the major resistance to heat transfer does not
reside in the controlled fluid side of the heat exchanger. Typical values for b, reported in previous experimental work, are in the range 0 . 2 to 0.4 (7,4). Specification of r(t) as a function of 4 ( x , t ) , regardless of the particular control algorithm, renders Equation 4 nonlinear. This motivates preference for use of Equation 10, which results from dropping the nonlinear term, in control system design because of its linearity. The new significance of Equation 18 and the block diagram of Figure 2 is that nonlinear feedback is incorporated. That is, the direct process temperature or concentration, 8, is no longer measured and provided to the controller for appropriate action, but rather the logarithm of normalized temperature divided by a constant, P, or a quantity involving some power of the normalized concentration, is measured and used to make the control decision. The motivation for use of nonlinear feedback is primarily provided by the fact that, to an approximation whose accuracy was demonstrated in the section “Dynamic Analysis,” the combination of nonlinear process and nonlinear feedback results in an over-all linear dependence of the control system output variable, d ( x , t ) , on the manipulated variable, r ( t ) , and hence enables use of linear analysis for control system design. Below we discuss specific examples of such design, and show from some readily conducted analyses that there is every reason to expect comparable, if not better, performance from the nonlinear feedback system as from a system involving the usual feedback of 8. The present section specifically studies the proportional control algorithm of Equation 18. VOL. 5
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AUGUST 1966
405
Closed Loop Response to Step Changes in Inlet Conditions and Set Point. In Appendix I1 it is shown that a step change of M in the inlet variable 41 results in a closed loop response in the controlled variable given by
where for later convenience we have included K as a parametric argument in the response. I t is also shown that the final value of this response, reached after a sufficiently long time, is
Offset is defined as the absolute difference between the desired final value, in this case zero, and the actual final value, given in this case by Equation 21. Hence, the right-hand side of Equation 21 gives the offset for a unit-step change in inlet conditions. The response indicated by Equation 20 has been plotted for the exchanger exit, x = 1, for the cases K = 2 and K = 5 in Figures 8 and 9, respectively. These responses are shown by dashed lines. Although the response indicated by Equation 20 is stable for all values of K , the responses show that increasing K causes increasing oscillation and less desirable response. However, one would like to have high values of K in order to reduce offset as shown by Equation 21. Hence, it is likely that reasonable values of loop gain lie somewhere between the values 2 and 5 for which the responses are shown. The indicated stability for all values of K should also be viewed with caution because of the neglect of valve lags, etc., and the other assumptions made in the analysis. By formal procedures identical to those of Appendix 11, it is found that the response to a set point step change of amplitude M is given by
change in the normalized temperature, 0, through the exchanger. This is equivalent to reducing the exchanger length to zero (or changing it by the value of M ) . Conversely, the choice of M = +1 corresponds, according to Equation 24, to a steady-state condition at the exit given by 0 = e-2p, in the absence of control. However, this is nothing more than the result which would be obtained by adding a second identical exchanger in series with the first, and we can say that the length of the exchanger has been doubled by this change. In general, for all processes described by Equation 1, the value of M is the change in the length of the tubular process which, for r ( t ) = 41(t)= 0, will result in the same ultimate change in B(1,t) as results (for the actual unity process length) from the step change of M in the inlet variable. Accuracy of Closed Loop Responses. From the dynamic analysis of the previous sections, we are assured that the closed loop responses given by Equations 20 and 22 are exact u p to some finite time. A further assessment regarding the accuracy of these responses is available because the final values may be calculated for the exact response and compared with those predicted by the linearized response. In the case of the load response, it is clear that ultimately the flow rate will come to some new constant value, and that the ultimate response of $ ( x , t ) , to whatever is the complete form of the flow rate manipulation, is identical with the ultimate response to a hypothetical flow rate manipulation consisting of a sudden step change to this ultimate flow rate. Hence, it follows from Equations 7 , 15, and 18 that
where M is the amplitude of the step change in the load &(s), and b has been taken as zero for convenience. (Similar results are obtained for nonzero b.) Solution of this equation for the final value results in
4(x,t;K) ~M
+ K(x -
1
2 MK
{d1
-k [I
+
4 MK K(x - M ) ] *
1--x
with final value
These set point responses are not shown graphically, but will be used later to obtain the initial slope of the response and thus the speed of this type of control. The desired final value for the step change in set point is clearly M , and hence the offset for a unit-step change is still as given by the right-hand side of Equation 21. I t is pertinent at this point to discuss the significance of the value of M . This parameter really has the units of fraction of full process length. T o illustrate this, consider the case of a heat exchanger for which = -(In O)/P. At the exit of this exchanger, I#J
Therefore, if we choose M = -1, it follows that, in the absence of control, the exchanger exit temperature would tend to reach unity. Physically, this corresponds to having the inlet temperature drop so very low that there is essentially no 406
l&EC FUNDAMENTALS
Equation 26 is obtained without linearization and hence is the exact final value. The limit as M goes to zero of Equation 26 yields a final value in precise agreement with that given by Equation 21, obtained by linearization. For disturbances of magnitudes other than zero, the final value given by Equation 26 will not agree with that given by Equation 21. The exact and linearized offsets are compared in Figure 3, which shows that the exact offset is greater than the linearized prediction for positive inlet step disturbances, and vice versa. Also shown on Figure 3 are 10% error bands, indicating the region of magnitudes of disturbances, for the various values of gain K , for which the exact offset does not differ from the linearized offset by more than 107,. As the gain approaches infinity, this band contains only disturbances of magnitude loyo,but widens for lower gains. The conclusion to be reached from this figure is that, for reasonable sizes of step disturbance, the offset predicted by the linearized response is in agreement with the exact offset. Similar considerations will yield a result corresponding to Figure 3 for set-point changes. However, it will be found on such analysis that the true offset in response to a step change of magnitude M will be identical with that for a load change of
for any type of disturbance. Control system design in terms of 0 rather than cannot provide any of these advantages. I n addition, design for nonlinear systems must be conducted on a case study basis. Thus, nonlinear feedback can be recommended simply for the significant improvement in design methods and in confidence in control system performance it provides. However, the performance of the nonlinear feedback system may also be superior to that of the comparable linear feedback system, and this offers even stronger motivation for use of nonlinear feedback.
4
LINEARIZED OFFSET: IIWK
Comparison of Performance
-1.0 - 0 . 0
-0.4
0
0.4
0.8
I:O
M-MAGNITUDE OF INLET STEP DISTURBANCE Figure 3. True offset for feedback control Stsep change at inlet
the same magnitude but with opposite sign. There is no need to plot these results since the analysis of error due to use of the linearized offset will therefore yield the same result as the error band of Figure 3. Thus, we can in general expect reasonable accuracy from the responses predicted using the linearized Equation 10 to represent the process transfer function. Other Feedback Controllers
If it is desired to consider controllers other than proportional, one substitutes G(s) for K i n Equation 38 to obtain
T o obtain, for example, the step response, one then needs to be able to invert
for all n. However, only the terms corresponding to the first few values of n contribute to the response. Therefore, if one can obtain the inversion for the case n = 0, a few of the subsequent inversions may be obtained by successive convolutions. The importance of the result in Equation 27 is that only linear design methods need be used. One can, if desired, bypass the time domain and conduct the design in the frequency domain, using the Nyquist diagram. Alternatively, one may use an analog simulation of the process and controller transfer functions, and adjust the parameters for good response. However, caution must be exercised in the choice of a simulating circuit for the process transfer function. If simple Pad6 approximants are used Tor the dead time, the responses may not be too accurate because of the lack of filtering in the loop. Mechanical or tape simulations may be preferable. If straight electronic simulations are necessary, one would have to use high order approximations. For any control algorithm G(s), the over-all linear nature of the system employing nonlinear feedback provides all the advantages of linear system design to the problem of selection of control parameters. Furthermore, the linearity ensures that examination of, for example, the step response only gives complete information regarding the control system performance
We compare the performance of linear feedback and nonlinear feedback systems, using proportional control, for the case n = I , b = 0 (heat exchanger with negligible flow dependence of over-all coefficient or first-order reactor), with regard to two characteristics which may be readily calculated from the response equations in both cases: speed of response and offset. Speed of response will be measured by the initial slope of the response to a step change in set point. I n Appendix I11 we calculate the ratio, R, of the initial slope for the control system using nonlinear feedback to that of the system using linear feedback for identical changes, M , in set point and identical gains, K. The results are plotted against parameter P i n Figure 4. I t may be concluded from Figure 4 that the heat exchanger response for nonlinear feedback is always faster than for linear feedback. T o verify this result for other than the heat exchanger case, such as for a reactor with order other than 1, requires machine computation because the equations corresponding to Equation 40 are not easily soluble. Although this has not yet been investigated, the qualitative result should be no different from that suggested by Figure 4. I n Appendix IV we show that use of linear feedback results in an offset, for a small step change in set point, given by:
for the identical circumstances which result in Equation 21
P-
Figure 4.
Slope ratio at time zero
Heat exchanger using logarithmic feedback compared to one using ordinary temperature feedback
VOL. 5
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when nonlinear feedback is used. Comparison of Equations 21 and 28 shows that even in the most favorable case, P = 1, the effective loop gain for determining offset is reduced by 63% in the linear feedback system over that using nonlinear feedback. Thus, in both speed and offset, the nonlinear feedback system shows superior performance for the situation investigated. Feedforward Proportional Control
The simple feedback responses shown by the dashed lines in Figures 8 and 9 suffer from the defect that the control system has no inkling of the disturbance until its full effect is felt on the process outlet, at which point control is desired. This situation may be improved by measuring the process variable ahead of the process outlet, thus anticipating the disturbance before it reaches the process outlet. The block diagram of Figure 5 indicates the arrangement which may be used to control the outlet variable, designated as &(l,r), by manipulations of flow rate based upon the dependent variable a t location X.
I n Appendix V we show that the response of the control system of Figure 5 may be expressed in terms of the response of the simple loop of Figure 2 to an identical disturbance $l(s) by the relation &(l,t) = B ( x , t
+ x - 1;K) + K
[&x,7
+ x - 1;K) -
which has a final value for a unit-step change, given by : q3z("m)
-M
-
1
dl(t) = Mu(t),
- K(l - x) 1
+ Kx
(30)
Equation 30 shows that, if we choose
K=-
1 1 - x
for the gain, the offset will be zero. However, because the block diagram of Figure 5 is based upon a linearized response, the offset will not be identically zero for all disturbances but rather will be reduced from the offset realized with a simple proportional feedback system using the same gain. The magnitude of the reduction may be assessed because the exact offset of the nonlinearized model may be computed, as is done in Appendix VI. The exact offset is plotted for various values of gain on Figure 6 . As the disturbance goes to zero, the
q 1,s) (
Figure 5. control 408
Block diagram for proportional feedforward
l&EC FUNDAMENTALS
Figure 6.
True offset for feedforward control Step change at inlet
offset goes to zero for all values of gain. However, nonzero disturbances yield nonzero offsets. Comparison of Figures G and 3 shows that the anticipated reduction in offset is realized, the magnitude of the reduction being substantial, with the exception of high values of proportional gain where substantial reduction is not really needed. Hence, the value of gain suggested by Equation 31 is satisfactory with regard to offset. In Appendix VI1 we derive expressions for the response of the loop of Figure 5 to a step change in the load variable, 41. These responses are plotted for three cases in Figures 7, 8, and 9. In all cases, we use the gain given by Equation 31, In the simplest case, we may measure the value of d ; ( x , t ) a t the process inlet, and use this value to determine the flow rate. I t may be expxted that this case ( x = 0, K = 1) will be the most rapid anticipator of process disturbances, and should therefore give some advantages. The response is illustrated in Figure 7. I t is easy to see what is happening; as the step change in the process inlet is detected, the flow rate, which is directly proportional to the response at x = 0 shown on Figure 7, is given a step change by the proportional controller to compensate. Therefore, the response at x = 1 undergoes the path given by Equation 15, reaching the value = -1 at unit time. Precisely at this time, the effect of the inlet variable change reaches the exit of the process, and exactly compensates for (or is compensated for by) the response to the imposed flow rate change. Hindsight shows that the response of Figure 7 could have been sketched merely on the basis of what was already determined in the section on dynamic9 leading to Equation 15. It is desirable to investigate other values of gain because Figure 6 shows that the true offset will not be zero, except for small disturbances, and the use of unit gain yields relatively high offsets. There is motivation for looking to higher gains in order further to reduce the offset. Furthermore, although the fastest return to "normalcy" is to be obtained by the control used to generate Figure 7, the maximum deviation of the process outlet variable from the desired condition is still precisely equal to the magnitude of the disturbance at the inlet, and it is desirable to see if this maximum deviation can be reduced.
"M, RESPONSE AT X.0
I
0.8
- 1.0
1
I\
K= 2
0.6
RESPONSE AT X z l
0.4
0.2
I Figure 7. Response of controlled process to step change at inlet
K =
1 . Control point at X =
0 (inlet)
0
u ! o
51 X
e 0 u
P
1.S
Figures 8 and 9 show the responses computed from Equation 69 for the cases K = 2 and K = 5, respectively. The time scale on Figure 8 is somewhat expanded with respect to the time scale on Figure 9, for clarity. Three quantities are plotted on each chart: the process exit response, the response of the process a t the control point (which is proportional to the flow rate manipulation and therefore effectively gives a plot of the manipulated variable), and (in dashed lines) the response which would be realized by the use of simple feedback control with the same value of gain. The effect of this type of control is to prelower the process outlet variable, in anticipation of the inlet change which will eventually reach the process outlet. The responses of Figures 8 and 9 show some advantage over the response of Figure 7, in that the maximum deviation of the response from its desired value is reduced, the total deviation being divided roughly equally in both directions. While Figure 6 suggests that, as we increase the gain from Figure 8 to Figme 9, we should expect a reduced offset, it is clear that the reduction comes at the expense of a more oscillatory response. There is no need to search beyond the value K = 5 used in Figure 9, and it is reasonable to suggest that the design value of the loop gain might be somewhere between the values used on Figures 8 and 9, unless the speedy return to normalcy of Figure 7 is desirable. T o give some measure of the increased oscillatory tendency as the gain is increased, settling times for gains 1, 2, and 5 are 1, 1.6, and 2.8 uni1.s of dimensionless time, respectively. For most process applications, the response of Figure 8 would be completely satisfactory-that is, the total of the maximum and minimum values of &(I. t ) / M can never be reduced below the value of unity because the inlet disturbance passes through the process without disiortion. Hence, the best that can be done is to divide this disturbance equally between values above and below the drsired level. After the disturbance has reached the process exit, Figure 8 shows that the proportional control action quickly and directly brings the exit variable back to the desired level, without excessive oscillation. Although there is some motivation for further control studies for more sensitive applications, wherein even the deviations after t = 1 suggested in Figure 8 are unacceptable, the problem may be solved by this simple proportional feedforward system in a great many process applications. I n some situations, it may be desirable to add integral control to remove completely the offset which, in a practical situation, will arise not only from the effects studied on Figure 6, but also from the fact that Equation 1 is an inexact process model. However, addition of integral mode generally causes a
Figure 8. Response of controlled process to step change at inlet
5'z
Figure 9. Response of controlled process to step change at inlet VOL. 5
NO. 3 A U G U S T 1 9 6 6
409
more oscillatory response. Because of the already reduced offset realizable from the feedforward proportional mode, a compromise is available. A small amount of integral action, say 0.1 to 0.3 repeat per residence time, based upon the value of $(1, t ) may be added to the controller equation. This integral action will be so slow that its effect on the early part of the transient responses shown in Figures 7, 8, and 9 will be negligible, but it will eventually reduce offset to zero. With the feedforward system we can afford the luxury of this long wait because the offset remaining from the proportional action is small. Conclusions
Process Dynamics. Exact dynamics of a broad class of distributed parameter processes, modeled by Equation 1, are given by Equation 7. Linearization of the exact dynamics in terms of the auxiliary variable 6 provides a linear response which shows agreement with the exact response significantly improved over the agreement obtained from a directly &linearized response. The agreement is exact during initial portions of the response, is independent of the process parameter P, and depends only on the magnitude of the disturbance. Use of the auxiliary variable 6 provides a consistent measure of process disturbances, given in terms of fraction of full process length. Process Feedback Control. The linearization in $ allows use of linear methods of control system design for a loop containing nonlinear feedback of the measured variable. Exact formulas for the closed loop response may be developed. For proportional control, the use of nonlinear feedback results in reduced offset, and faster response (shown rigorously only for n = 1) then is achieved using linear feedback in the otherwise identical control system. The effective gain in the linear feedback system is reduced Pe-‘), or at least 63%, over that of the by the factor (1 nonlinear feedback system. This applies to small disturbances in the case n = 1. For both systems, the effective gains for heat exchanger control are reduced by the factor (1 - b ) . The offset predicted by the linearized model of the nonlinear feedback system, 1/(1 f K x ) , is reasonably close to the true offset. For nonlinear feedback control a t the process exit, a good compromise between considerations of offset and oscillation is obtained for gains in the range 2 < K < 5. Process Feedforward Control. Improved control at the process exit may be obtained by basing the proportional control action upon the value of 6 at some point x upstream of the exit. The linearized model of the nonlinear feedback system predicts that combinations of K and x satisfying K = 1/(1 - x ) will give zero offset a t the process exit. This prediction is accurate for small disturbances. For all disturbances, the offset from this combination is significantly reduced over that realized by control using the same gain but based upon the value of 6 at the process exit. The combination K = 2, x = 0.5 gave good compromise for considerations of offset, speed, and oscillation. A formula was developed for predicting the response of the feedforward system in terms of that of the feedback system. Nonlinear feedback offers significant promise in the analysis and design of control systems for tubular processes described by Equation 1, and in improvement of control system performance. Further research may be profitably directed a t establishing techniques for finding transformations similar to
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410
I&EC FUNDAMENTALS
Equation 2 for tubular processes more complex than those described by Equation 1. Appendix 1.
Derivation of Pulse and Impulse Responses
Substitution of Equation 17 into Equation 9 yields
-(A f d)
I -7-A;
11
< -(A
< 11 < 0
(32)
+ d)
I n proceeding to obtain expressions for $ ( x , t ) from Equations 7, 8, and 32 it is found that there are two possible cases, given by x > ( A f d ) or x < ( A d ) . Of these two, the more important is the former; the latter corresponds to the case where the step-decrease portion of the pulse is not introduced until such a time that a new steady state, in response to the first step, has already been established at location x. This latter case is really nothing but the response to two essentially separate and distinct step functions, equal in amplitude but each in a different direction. I n the former case, the resulting expression for 6 ( x , t ) is given by:
+
O
O = normal, steady-state velocity = velocity disturbance = 0 f i ( t ) , total velocity = wall temperature = mass flow rate, massltime = dimensionless distance along tubular process, x’/L = distance along tubular process = normalized set point disturbance for heat exchanger or first-order reactor, defined by Equation 45 = normalized gain for heat exchanger or first-order reactor, defined by Equation 45 = dimensionless magnitude of set point disturbance for heat exchanger or first-order reactor = dimensionless set point value, normally zero = dummy variable of integration = dummy argument of h = dimensionless temperature for heat exchanger, ( W - T ) / ( W - T o ) ; dimensionless concentration for chemical reactor, CA/CA, = dimensionless disturbance in inlet process variable = functional of r ( t ) , defined by Equation 9 = dummy time variable of integration = transformed process variable, defined by Equation = =
Substitution of Equation 70 into Equation 65 yields, after simplification,
+
40(1, t ) --- t [ u ( t - 1) - u ( t ) ] (71) M Equation 71 is plotted in Figure 7, while Equation 69 is plotted in Figures 8 and 9, for .the cases (x = 0.5, K = 2) and (x = 0.8, K = 5), respectively. Nomenclature = dimensionless area under pulse or impulse flow = = = = = = =
= = = = = =
= = =
= = = = = = =
rate disturbance total heat exchange area dimensionless amplitude of step flow rate disturbance dimemionless linear rate of change of over-all heat transfer coefficient with flow rate concentration of reactant normal, steady-state inlet reactant concentration specific heat dimensionless time of duration of pulse controller transfer function functional of r ( t ) , defined by Equation 8 dimensionless loop gain dimension less proportional controller gain reaction velocity constant length of tubular process dimension less amplitude of inlet disturbance, expressed in fraction of full process range order of reaction heat exchange to heat capacity ratio UAh/iwCp in the case of a heat exchanger; ratio of convective to reactive rates of transport of reactant kCAOn-l.L/a in the case of a chemical reactor; dimensionless ratio of initial slopes of set point responses, nonlinear feedback to linear feedback dimensionless flow rate disturbance, C/O dimension1 ess Laplace transform variable temperature normal, steady-state inlet temperature dimensionless time, ot’/L time
9 P
i(x, t )
= deviation in transformed process variable, de-
gl(t)
= transformed inlet disturbance
a,(x) $(x, t ;
fined by Equation 3
K)
&(l,t )
final value of +(x, t ) step response of closed loop system when proportional gain is K = exit process response when proportional control is based upon value of 6 a t location x , using gain given by Equation 31
= =
Literature Cited
(1) Kamman, D. T., Koppel, L. B., IND.ENG. CHEM.FUNDAMENTALS
5. 208 (1966).
(2) Koppe1,L. B.,~Zbid.,’l, 131 (1962). ( 3 ) Zbid., 4, 269 (1965). (4) Weber, T. W., Harriott, P., Zbid., 4,155 (1965). RECEIVED for review June 21, 1965 ACCEPTED December 17, 1965
CONTROL OF TUBULAR HEAT EXCHANGERS AND CHEIMICAL REACTORS BY PIECEWISE CONSTANT MANIPULATION OF FLOW RATE L0W ELL B
. K0PPEL,
class of chemical reactors and heat exchangers, when subject to changes in flow rate (4) shows a transfer function of the form ABROAD
(1)
where B(x,t) == -
(2)
School of Chemical Engineering, Purdue University, Lafayette, Znd.
The dependent variable b’(x,t) representsn ormalized temperature in the case of a heat exchanger, and normalized concentration in the case of an isothermal chemical reactor. The forcing function r(s) is a Laplace-transformed, normalized disturbance in flow rate, and parameter POis the ratio of heat exchange to heat capacity for the case of a heat exchanger, or, in the case of a chemical reactor, is the ratio of reactive to convective rates of transport of the reactant. Parameter b, different from zero only in the case of a heat exchanger, is the linearized rate of change of over-all heat transfer coefficient VOL. 5
