A GENERAL PROCESS CONTROLLER L O W E L L B.
K O P P E L
A N D
P H I L I P
M.
A I K E N
School of Chemical Engineering, Purdue University, Lafayette, I n d . 47907
A controller is derived to minimize a time-weighted integral-square-error criterion for process models containing time constants and time delays. Performance as tested on a real process is superior to that of conventional controllers tuned for the same model.
MODERN optimal control theory
has made significant progress in designing feedback controllers for linear processes. An excellent recent textbook by Athans and Falb (1966) describes these controllers, which have not received widespread acceptance in the process industries. There are a t least three reasons for this: 1. The optimal controller utilizes state variables defined about final steady-state values, and assumes accurate knowledge of the process dynamics, particularly of the process gain. I n practical terms, this means that offset (permanent discrepancy between actual process conditions and desired or set-point values) will always result from the inevitable discrepancy between the dynamics of the actual process and its model, Qince the optimal controller is based on the untrue assumption that a t steady state a zero manipulated input will cause a zero value of controlled output. 2 . The optimal controller cannot deal with unmeasured zhanges in process load conditions. An unmeasured change in load variable corresponds to a discrepancy between plant and model dynamics, and results not only in the offset described above, but also in deterioration of the closed-loop response. I t is possible to deal with this problem in a statistical sense, choosing a control which minimizes an averaged response, but this requires assigning a statistical distribution to load changes. Further, though the averaged offset may be zero, nonzero offsets may occur over relatively long periods of time, a condition not likely to be regarded with favor by management. 3. Optimal feedback controllers require measurement and/or calculation of all state variables. For a process with nth order dynamics, this often means measurement of the process output and calculation of its first ( n 1) derivatives. This becomes impractical if n > 2 . In this paper we use a modification of the usual approach to optimal control theory to derive a controller which overcomes these difficulties, and is suitable for application in the process industries. We restrict attention to processes with a single manipulated variable, and derive controllers for processes in which the dynamics between the manipulated variable, rn(t), and the controlled output signal, c ( t ) , can be adequately modeled by a transfer function of the form
Throughout this paper, an overbar distinguishes a model quantity from the actual quantity. Thus, G,(s) is the 174
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
model transfer function and c(s)is its output. Equation 1 is a partial-fraction expansion of an N,-th order transfer function, with real roots and with a delay time d,. We ignore the case of repeated roots; extension of the approach to this case is obvious. Latour et al. (1967) give references to the process dynamics literature to show that the secondorder version
is adequate to model a variety of processes. The transfer function in Equation 2 is a special case of Equation 1 w i t h p , = -p2 = K , / ( T -~ T J , rl = l / ~r2 ~ = 1, / ~ Adequate ~ . modeling is interpreted to mean that a controller designed for Equation 1 performs satisfactorily on the actual process; this concept is investigated below. The controller designed herein includes feedforward action for application in circumstances in which a load variable ub(t) can be measured, and its effect on the process output can be modeled by a transfer function of the form
If more than one load variable is to be measured and fed forward, a different transfer function of the form of Equation 3 is used for each such variable. We treat explicitly only the case of a single feedforward load variable. Predictor Equation
According to Equations 1 and 3, the process output should obey the equation
where S ( t ) is the unit step function. However, a t each instant of time to, we instead choose to predict future values of the process output from the equation
realize an optimal controller for a linear process with a quadratic performance criterion (Athans and Falb, 1966). Performance Criterion
We choose rn(to) such that, when Equation 5 is used to predict future output, we obtain a minimum value of the performance index
where u=t-to
J ii
where r ( t J is the present desired value of process outputLe., r ( t ) is the set point signal. Equation 1 is a weighted integral-square-error criterion. The integration is begun a t u = d, because the input cannot affect the output, and hence P , until the elapse of one delay time. The exponential weighting factor, with b 2 0, weights most heavily deviations occurring for u just after d,, because Equation 5 is accurate for only such values of U . As 6 + 0, all weight is placed on the deviation a t u = d,, a t which time Equation 5 is accurate, and no weight is placed on deviations a t other times. Thus b + 0 represents a limiting ideal case. As b 0, m ( t o )is chosen only with regard to c ( t , + d,); this is repeated a t each to t o generate the "optimal" control.
-
e(t) = c(t)
-
E(t)
and c ( t ) is the actual process output as measured and fed back by the instrumentation. Equation 5 will be used as the basis for making the instantaneous control decision-Le., for choosing m (t,,). The method, rationale, and assumptions used to obtain Equation 5, which applies only for u 2 d,, are as follows: to and 0 represent present and future times, respectively, so that t(0; i,,) is the approximate prediction made at t, of the output u units of time later. Assume that m ( t ) and u. ( t ) will be constant a t m(t,) and ub(t,) in the future. Break the integrals in Equation 4 into a past contribution from 0 = 0 to 0 = t,,, and a future contribution from 0 = t,, to 0 = t - d. I n so doing, we write the result in a form which is correct only if d,, 2 d h , as evidenced by the fact that S ( a - d,) multiplies the terms involving ut rather than S ( U - d - ) . This is done for convenience in implementation; its effect will be discussed later. Ideally, we want the terms multiplied by S ( U - d,) to agree with for u 2 d,. The last summation in Equation 5 uses e, the difference between c and 7,to improve this agreement. Only the first term of this summation, which is a Taylor series predictor, is physically realizable. The second term, which predicts future errors from a linear extrapolation may be approximated by constructing phase-lead (diferentiator) circuits. I n practice, it will usually not be feasible t o use J > 1, since this requires a t least two differentiations of a process signal. Clearly. Equation 5 will be accurate only for u close to dn7.This is due to the assumptions of constant m and ui. This will be adequate for our purposes for the performance criterion used, as described below. The error term e(t,) compensates for errors in the model. Equation 4, and for unmeasured load disturbances. If neither modeling error nor unmeasured load disturbances exist, this term is not needed (and is inoperative), as demonstrated below. Therefore, the optional differentiations of e,(t) in Equation 5 should not be confused with the essential differentiations of the output variable required to
Control Law
To find the minimizing rn ( t o ) ,differentiate P with respect to m(to)and set the result to zero, t o obtain the control algorithm
where t, has been replaced by t , the general time variable, and the constants are defined as follows:
c
j = 0,1,2,. . .J
Lvm
A,= b'x -1
P
(1 + br,)(1+ br, + b r , )
i = 1,2,. . .N,
VOL. 8 NO. 2 APRIL 1969
175
where m ” ( t ) is the value computed from Equation 11. This is implemented in a most natural manner; the value generated by Equation 11 is applied to the valve (or other final control element) which is allowed to saturate. The control law of Equation 11 is linear. A general block diagram for its realization is given in Figure 1. (A specific example for a first-order model is shown in Figure 5, A , ) All components in Figure 1 are physically realizable, except those which take derivatives of e ( t ) .An unmeasured load disturbance, uh(t), is also shown entering the process in Figure 1. To show that m ( t ) in Equation 11 minimizes the performance index we take the second derivative of P to obtain
Theoretical Performance
From Figure 1 we derive, using standard block diagram algebra, the relationship
J
[ HI(s)+ Gm(s) J
=o
+is’]C(s) = +oGm(S)R(S)
+
where G,(s), G~(s),and Gh(s) are the actual process transfer functions relating output C(s) to M ( s ) , Ua(s), and Uh(s), respectively, SO that
C(S) = Gm(s)M(S)+ G ~ ( sUk(s) ) + Gh(s) u h ( s ) Since this is necessarily positive, we have P as a function of m ( t J concave upwards. Therefore, if the manipulated variable is constrained by saturation limits, mL I m ( t ) I, m H , as is always true in process applications, the “optimal” control is
where Uh(s)is the Laplace transform of an unmeasured or hidden load change, and
I I
-
r---i
I
I
I
I
& ‘-
--I----
I
~~
Figure 1. Block diagram of controller realization 176
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Here, i s J } is used to denote an operator which takes the j " derivative, as indicated in Figure 1. I t is different from simply s' unless all initial conditions are zero. Offset. We first consider the final value produced by the control algorithm when the system is subjected to step changes in R ( s ) , C,(s), or Uh(s). Equations 14, 16, = H?(O)= 0. Then application 20, and 2 1 show that H1(0) of the final-value theorem to Equation 19, for R ( s ) = U h ( s )= C c ( s ) = l i s shows that the control completely eliminates offset, provided that G,(O), Gh(O), and G,(O) are finite-i.e., the process has self-regulation. (These are the same conditions under which conventional reset action can eliminate offset.) Elimination of offset occurs no matter how much modeling error exists. I n practice the delay elements in Figure 1 must be approximated, and to have H1(0) = H L ( 0 ) = 0 we require an approximation such as the Pade circuit; e-dms
1-
---
so that by choosing a small value of brl, the response of the closed loop can be made very rapid. No controller can reduce the delay to less than d,, so these considerations show that the closed-loop response approaches the ideal. The gain required to achieve the response of Equation 24 is independent of the delay time, a significant advantage of the present controller over a conventional proportionalintegral-derivative (PID) controller. Using Equation 23 we next compute the effective controller transfer function G,(s)
to produce the result in Equation 23. Combining these with Equation 23 Pives
d,s
-
2
dms
1+2 which has unity again. This and higher-order Pade approximations are simple to construct (Coughanowr and Koppel, 1965~1. Set Point Response for Perfect Model. Here we assume a perfect model, G,(s) = G,(s), and take U i ( s ) = U h ( s ) = 0. Then using Equations 1 and 14 in Equation 19 yields
From Equations 12 and 13 it follows that as + 1. Therefore, it follows from Equation
A,/$(>
precisely the ideal controller transfer function proposed by van der Grinten (1963). However, Equation 25 is unrealizable because of the factor G,(s) in the denominator. Figure 1 thus gives a diagram which ran approximate the ideal transfer function as closely as desired, by choosing a small value of b. Feedforward Load Response for Perfect Model. We assume Gh(s) = Gh(s),G,(s) = G,(s) and use Equations 3, 14, and 16 in Equation 19 with R ( s ) = l i , ( s ) = 0 to obtain
+
-
Obviously, there is no physically realizable control system which can produce a better set point response. However, as b + 0, it is also true that $ " / ? a. Therefore realization of Figure 1 requires infinite gain. Thus in practice, the extent of approach of the actual response to the ideal response of Equation 23 will be determined by saturation. T o consider the effect of nonzero b we choose a firstorder process, N , = 1. Using Equations 12 and 13 reduces Equation 22 to
-
As b 0, Equations 12 and 15 show that u ~ / $ ~exp [ - - ~ ~ ( d-, d i ) ] while Equations 12 and 13 show that A,/ 4" 1. Then
-
+
(, 1 2brl
S+l
This shows that the actual closed loop responds as a first-order system with delay. However, the time constant for the closed-loop response is much smaller than that of the process; in fact 'closed 'opw
=
2brl 1 + 2brl
I t follows from Equation 2'7 that if the delays in the manipulated and measured load variables are identical, d,, = d,, the limiting control is perfect, c ( t ) = 0. N o response can occur regardless of the load disturbance. If d, # d,*, then we consider the specific case of a step disturbance, Cb(s) = l i s . Equation 2 i yields
lim
c"(t) =
b-0
The summation may be recognized as the undelayed openloop step response of Gh(s). If d, > d i the response is as sketched in Figure 2, A , while if d, < da, the response is as shown in Figure 2, B. Figure 2, A , shows that when there is more delay in the manipulated variable VOL. 8 NO. 2 APRIL 1 9 6 9
177
and 28 to assess the value of feedforward action. Since the second term in Equation 30 is delayed by d,, the unmeasured load response will be uncontrolled until a time t = d, + di after the disturbance has occurred. Thus for a unit step load disturbance, L'h = l / s , the peak value of the closed-loop response will be a t least as great as the value of the open-loop step response a t t = d,, + d,.
We now compare Equation 31 with the peak values reached by Equation 28 for d, > di
B
and for d, < db
Figure 2. Ideal feedforward response A. 8.
d, > d6 d, > d,
than in the disturbance variable, the limiting response is the best possible response, reducing the output precisely to zero a t the earliest possible time. Obviously such a response is not physically possible, requiring an infinite m ( t ) ; this results because 6 0 implies infinite gain. When there is more delay in the disturbance than in the manipulated variable, Figure 2 , B , shows that the control acts too soon, because the approximation made in Equation 5 is true only if d, > d,. T o overcome this one need only delay by d k - d, the entry of u , ( t ) into the control circuit. Because of various limitations in practice, such as saturation and modeling error, the ideal responses in Figure 2 cannot be achieved. However, the experimental results presented show that acceptable responses can be achieved. Unmeasured Load Response for Perfect Model. We assume G,(s) = G,(s) and use Equations 1 and 14 in Equation 19 with R ( s ) = Uh(s) = 0, to obtain +
I n both cases the peak value is smaller than the value given by Equation 31, which shows that the feedforward action reduces the peak value. Further, the step response o f Equation 28 returns to zero a t max { d,, d k } , while the response of Equation 30 does not reach a peak value before t = d m + d l . Thus feedforward action also reduces the settling time. To assess the value of the predictor summation in Equation 5 , assume Gii(s) = G , ( s ) ,Uh(s) = l : ~R, ( s ) = Uc(s) = 0. Then from Equation 30
(34) Until t = d, + d,,, the response proceeds uncontrolled. At t = ( d l + d,) , the value of the response is
while a t t = (db + d,)
As 6
- 0, Equation
1 2 shows that a /$,,
-
d , / I ! Therefore,
Comparison of Equations 35 and 36 shows that for J > 0-Le., the use of prediction on the error-there is a significant reduction in the response a t the earliest possi(clearly not a physically approachable ble time. As J condition) Equation 34 shows that the response approaches zero for all t > d, + d,. For example, the response of the first-order with delay system to a unit-step change in unmeasured load is sketched in Figure 3 for J = 0, 1, +
We also assume that G ~ ( s =) f%(s) but that we choose not to use feedforward action-Le., we do not measure the load variable. Then we can compare Equations 30 178
I B E C PROCESS DESIGN A N D DEVELOPMENT
J=2 dk
1
dk+ dmJ/
Figure 3. Ideal response for step change in unmeasured load, first-order plus delay system
and 2 . Since no controller can affect the unmeasured load response before t = d,, + d i , Figure 3 shows t h a t the control effected by Equation 11 approaches the best possible control for b + 0, J m , and a perfect model. Control of Processes Modeled as Pure Delay. If the process is modeled by G,(s) = exp(-d,s), Figure 1 reduces m , J = 0. to Figure 4 by taking N , = 1, p , = r, The effective controller transfer function is G,(s) = l / [ l - exp(-d,s)], identical to van der Grinten's (1963) ideal controller. There are no considerations of saturation for b = 0 in this special case because the ideal controller is physically realizable. If the model is perfect, the closedloop transfer functions are C ( s ) / R ( s )= exp(-d,s) and, -d,s if G ~ ( s )= G,(s) = e , C(s)/Crh(s) = exp(-d,s) exp(-2dms). Therefore, the set-point response is perfect but delayed, and the response to an unmeasured load returns to zero after t = 2d,. These responses are the best possible. In practice the block diagram of Figure 4 should be useful for any process dominated by delay time.
-
-
Control of Processes Modeled as First-Order with Delay. If the process is modeled by G',(s) = p l exp(-d,s)/(s + r l ) , Figure 1 reduces to Figure 5, A , if J = 0, which in turn may be reduced to Figure 5, B. As b -+ 0 , Figure 5 , B , shows that the controller approaches the ideal controller of van der Grinten (1963). However, Figure 5 , A , shows that an infinite gain is required to meet this limiting condition. Figure 5, A , points out some similarity between the present controller and that suggested by Smith (1957, 1959). However, Smith suggested use of a conventional controller, with the process time constants and a delay element in a feedback loop around this controller, to make the conventional controller input identical to that which would exist if there were no delay in the process. Figure 1 does not utilize a conventional controller, but rather an alternative realizable control structure which can approach ideal performance, as demonstrated above. A significantly superior feature of the present controller over the Smith controller is demonstrated by Figure 5, C, a rearranged version of Figure 5 , A . The Smith controller replaces the l / ( l + 2brl) gain in the inner feedback loop with unity, and uses a conventional controller in place of the (1 + 2 b r l ) / 2 b p l gain element in the forward path. In practical terms, this results in a requirement for integral action (with its attendant deterioration of dynamic performance) in the forward path of the Smith controller in order to eliminate offset. The controller in Equation 11 does not require this integral action, and Figure 5, C, shows only a proportional gain (1 + 2 b r l ) / 2bpl in the forward path. Nevertheless, offset will be eliminated, provided only that whatever approximation is used for the feedback delay element has unity gain, as discussed above. T o demonstrate the effect of nonzero b, we present some analytical responses derived from Figure 5 when the model is perfect, G,(s) = C,(s). We utilize the results later for experimental comparisons. Consider a unit-step
I
, I
U
Figure 4. Control of process modeled as pure delay
VOL. 8 NO. 2 A P R I L 1 9 6 9
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I
e-r,d,
I
]1
+ 2brl (39)
1 + 2br1
-
PROCESS
I
I
PI
strl
m ( t ) = -S(t
I
I
A
L T
1
L CONTROLLER
-
As b 0, c(tp) p , / r l [ l - exp(-rid,)] and t, 2dm as suggested by Figure 3. If b # 0 the peak value is increased and occurs a t a later time. We calculate the control signal m ( t ) necessary to obtain the response given by Equation 37 as - d,)
+
[ I - e- (1 + 2brl)(t- d m ) / 2 b
I
(40)
Thus m ( t ) decays exponentially to the correct final value, m = -1. As b --t 0 , the rate of decay becomes infinite and velocity limitations in the control will have effects. However, the value of m ( t ) itself will not saturate during a (controllable) load change regardless of the value of b. Next consider a unit-step change in set point, r ( t ) = S ( t ) . Then we obtain
and
PROCESS
I
I B
The response rises exponentially to the correct value of unity. The manipulated variable has a maximum value att=O-of
T
T
I
I
C
I
PROCESS
The ratio of this peak value to the final value mi = r l / p i is
I
I
1
(43)
J
C Figure 5 . Control of process modeled as first-order with delay A. 6. C.
Realization diagram Reduced diagram Diagram rearranged for comparison with Smith controller
Therefore to avoid saturation effects, the requested set point change cannot exceed in magnitude the fraction F of the total available set point range in the desired direction, where F is given by
F= change in U h ( t ) with G h ( S ) = G,(s). Then from Figure 5, using conventional Laplace transform techniques,
1
___
1 +-
(44)
L
2brl
F is the fraction of the effective set point range between
The peak value of this response is found by differentiating Equation 37, setting the derivative to zero, and solving for the time a t which the peak occurs as
and the peak value as 180
l & E C PROCESS D E S I G N A N D DEVELOPMENT
the present set point and a set point value which would require the maximum value mH to maintain. For example, experience suggests that bri = 0.1 is an acceptable value for many applications. Thus set point changes smaller than 1 7 5 of the available range in the desired direction of change will not cause saturation in the loop, and should cause responses which follow Equation 41. This response, a delayed exponential rise, is clearly superior to the oscillatory set point response with significant overshoot, normally obtained when a PID controller is used on the first-order with delay process. Modeling Error. When G,(s) # G,(S) it is difficult to make general statements about the nature of the response.
Experience indicates that if br! 2 0.1, where r! (reciprocal of the dominant time constant) is the smallest of the r,, the controller will not be excessively sensitive to modeling error; however, smaller values of b may lead to undesirable effects such as oscillation. I t has also been observed that if linear error prediction ( J = 1) is to be used, and significant modeling error is present, the value actually used for $1, the coefficient of the error derivative, should be approximately one half the value obtained from Equation 12. If higher values are used, an oscillatory m ( t ) may result. As an example, consider a second-order with delay process
ACTUAL b = O . l , J - 1
-
ACTUAL b = O . l , J = O
IDEAL J = b - 0
-
,-0.5s
G,(s) =
-
(s + 1)(0.1s+ 1) 0
I
3
2
4
5
t b
modeled by ,-0.6s
G,(s) =
Figure 7. Responses to unit step change in load with modeling error
~
s+l
Figure 6 shows the response of the system controlled by Equation 11 to a unit-step change in set point, for b = 0.1, and J = 0. Shown for comparison are the ideal response for b = 0: and the response obtained using a P I D controller with settings given by the Cohen-Coon (1953) method for the process model. Figure 'i shows responses to a unit step change in u , ( t ) , with GI,(s) = G,(s). The cases without error prediction (6 = 0.1, J = 0) and with error prediction (6 = 0.1, J = 1) are compared with the response of the P I D controller, and with the ideal response for J = b = 0. As suggested above, the value of @1 used in Figure 7 for the case 6 = 0.1, J = 1, is one half the value obtained from Equation 12. The conclusions to be drawn are that Equation 11 is significantly superior to a P I D controller for set point response even without error prediction ( J = 0 ) , and for unmeasured load responses, linear error prediction ( J = 1) renders Equation 11 slightly superior to a PID controller. Of course, the response obtained from Equation 11 can be further improved by using the exact second-order with delay model. Figures 6 and 7 demonstrate that a 2 0 5 error
in the model delay time does not cause a significant deterioration in the controller performance. Experimental Performance
The control of Equation 11 was tested on a physical system which consists of two constant-volume stirred tanks separated by a connecting tube. Water flows through the system, and the control involves electronic manipulation of the heat input to the first (upstream) tank to regulate (1) the temperature of the fluid immediately before it leaves the connecting tube and enters the second tank, thus providing a system with first-order plus delay dynamics, or (2) the temperature of the fluid in the second tank, thus providing a system with second-order plus delay dynamics. Details of the equipment are given by Aiken (Aiken, 1966; Aiken et al., 1968). The residence times of the fluid in the first tank, second tank, and delay tube are 2.5, 1.3, and 0.2 minutes, respectively. Therefore the thoretical transfer function is
K Gmis)=
,-0.2S p1
(2.5s + 1)(1.3s
+ 1)
(45)
where the process gain K , is 13 F' per kw. For tests with the one-tank first-order plus delay system, the model transfer function chosen was
"t I 0
I 2
I 3
I I
I 4
I 6
t b Figure 6. Set point responses with modeling error
11
The additional 0.2 minute of delay time corresponds to the suggestion by Hougen (1964) that 5 to 10% of the time constant be added as model delay to account for imperfect mixing. For operation of the two-tank system, modeling error was purposely introduced by using the firstorder plus delay model transfer function
7
(47) for selection of the control parameters in Equation 11. VOL. 8 NO. 2 APRIL 1 9 6 9
181
1
1
4.46 C O
4.48F0
T
T
Figure 8. Response of one-tank heating process to a step change in unmeasured load variable A.
No error prediction (J = 0)
6. linear error prediction (J
=
1)
This transfer function was obtained by a least-squares fit of the actual two-tank process open-loop step response. For tests involving linear error prediction ( J = l), a lag-lead differentiating network was used to obtain the required error derivative. In all tests the value of b was 0.1.For the one-tank system, since chosen so that br, r1 = 1 / 2 5 = 0.4 min. ', b = 0.25 minute; for the twotank system r1 = 0.3 min. ', b = 0.3 minute. Load Responses. Figure 8 shows responses of the onetank system to a step change in the unmeasured load variable; Figure 8, A , is the response without error prediction ( J = 0), and Figure 8:B , is the response with linear error prediction ( J = 1). Physically, the load disturbance was caused by a sudden change in an auxiliary heat input to the first tank a t the instant indicated by the vertical arrow. The magnitude of the load input in Figure 8, A and B , is 1.1 kw.. which corresponds to an ultimate uncontrolled response of 14.3F". The peak heights of the controlled responses in Figure 8, A and B , expressed as fractions of the ultimate uncontrolled response, are 0.22 and 0.20, respectively. Thus the theoretical indication of Figure 3, that error prediction does not alter the peak value but only the response after the peak value, is supported by Figure 8. However, the response of Figure 8, B , after the peak does not drop vertically, as suggested by the theoretical results for error prediction, but rather falls rapidly by comparison with the results for J = 0. This is due to saturation of the manipulated variable in the physical system, which cannot produce the infinite values required to achieve the results of Figure 3 for J > 0. Figure 8 also shows that the peak time is reduced by adding error prediction, in contradiction to the results of Figure 3. This occurs partly because the model delay time does not totally correspond to pure delay in the process; the linear error prediction is also able to overcome some of the dynamic effects of nonideal mixing. Further, the results of Figure 3 are for the case b = 0. Since no saturation occurs during the response of Figure 8, A , it can be compared with the theory in Equations 38 and 39. These relations predict tp = 1.4 minutes and c(t,) = 0.23, while the experimental values are t, = 1.5 minutes and c ( t D ) = 0.22. Here the values of c(t,) are expressed as fractions of the uncontrolled response, 14.3F" = p i .1.1kw. i r,. Figure 9 presents load responses for the two-tank system, with the same scaling and disturbance size as are 182
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Figure 9. Response of two-tank heating process to a step change in unmeasured load variable A.
No feedforward control or error prediction ( J = 0)
6. No feedforword control, linear error prediction (J C. D.
=
1)
Feedforward control, no error prediction (J = 0) Feedforword control, linear error prediction (J = 1)
used in Figure 8. Figure 9, A , gives the response to a step change in unmeasured load without error prediction, and Figure 9, B , gives the response when linear error prediction ( J = 1) is used. Figure 9, C, and D , gives responses without and with linear error prediction, respectively, when the step load disturbance is measured, and feedforward action with e b ( s ) = G,,(s) is used. As expected, error prediction provides significant improvement in the absence of feedforward information but does not materially improve the response when feedforward action is used. Set Point Responses. Figure 10 shows responses of the one-tank plus delay system to a 7.2F" step change in set point. The magnitude of change, 7 . 2 F 0 , corresponds to approximately 50% of the available range, while Equation 44 shows that changes greater than 17% will saturate the system. Therefore the responses shown in Figure 10 were obtained under saturated conditions. While Equation 41 suggests that 63.25 of the total rise will occur in 2 b / ( l + 2br,) = 0.42 minute plus one delay time after introduction of the set point change, 0.7 minute plus one delay time is required for Figure 10 due to saturation. (On both Figures 10 and 11 the vertical arrows represent the point a t which one delay time has elapsed.) Kevertheless, the responses resemble the theoretical exponential rise and are significantly superior to those of a PID controller. When used on a time-constant plus delay time process, PID controllers yield set point responses which show overshoot and oscillation, or else are slow by virtue of being tuned to avoid the overshoot. The responses are free of offset despite the absence of direct reset action. In Figure 10, A , no prediction is used, so J = 0, while J = 1 in Figure 10, B. As expected from the theoretical considerations, addition of error predictior has little effect on the set point response.
2.23F0
Figure 11 gives the responses of the two-tank system to a 7.2F" step change in set point. The response in Figure 11, A , was obtained using a P I D controller tuned by the Cohen and Coon method (Coughanowr and Koppel, 1963b), and that in Figure 11, B , was obtained using the same controller tuned by the Ziegler and Xichols method (Coughanowr and Koppel, 19658, b ) . Figure 11,C, is the response of the present controller with settings based on the first-order plus delay model Gm,(s) of Equation 47. The superiority of the present controller is evident; it can be further improved by using a second-order controller based on Equation 45. Discussion
Figure 10. Response of one-tank heating process to a 7 . 2 ~ 0 set point change A. B.
No error prediction (J = 0) Linear error prediction ( J = 1)
i
The physically realizable controller presented here shows good performance for both set point and unmeasured load disturbances. I t allows easy incorporation of feedforward action in a systematic fashion. Controller settings are computed directly from the process model, which is of the form usually used to model chemical processes. Derivative predictor action may be added to improve the response in the presence of the modeling error and/ or unmeasured load disturbances. Excellent response Occurs even in the presence of significant delay time. Experiments on the temperature control system show that the controller will perform satisfactorily in the presence of significant modeling error, including error in the delay time. The controller can approach the (unrealizable) ideal transfer functions suggested by van der Grinten, the degree of approach being limited by saturation effects. Although all tests reported in the present work utilized analog realization of the controller, digital realization is actually simpler (Orent, 1965).
I
2.23P0
t
Acknowledgment
Herbert H. Orent assisted in the development of an early form of the control algorithm in his Ph.D. thesis. F. D. Schwab performed the experimental transients as part of his M.S. thesis. Nomenclature
Figure 1 1 . Response of two-tank heating process to a 7.2F" set point change A. B.
C.
Cohen and Coon PID settings Ziegler and Nichols PID settings Proposed controller
K, = J = m, A4 = Nh =
exponential weighting factor in performance index as defined by Equation 10 process output delay in model transfer function G,,(s) delay in model transfer function Gm(s) error between process output and model output temperature difference in Fahrenheit degrees process transfer function between the measured load variable u p and output c model transfer function as defined by Equation 3 process transfer function between measured manipulated variable u, and output c model transfer function as defined by Equation l transfer function as defined by Equation 20 transfer function as defined by Equation 2 1 indices of summation heating process gain F c per kw. order of Taylor series error predictor as defined by Equation 3 manipulated variable order of model load transfer function G, as defined by Equation 3 VOL. 8 NO. 2 APRIL 1 9 6 9
183
N,
pt =
P = PID = q8 =
r, R = r, = s =
S = t = to = uh, cTh
=
u i , CP =
ut = x, = J’,
L = refers to lower saturation limit of manipulated
= order of model manipulated variable transfer
=
function G, as defined by Equation 1 constants in partial fraction expansion of G, index of performance as defined by Equation 10 refers to the three modes (proportional, integral, derivative) of a conventional controller constants in partial fraction expansion of GI set point of process output c roots of transfer function G, Laplace transform variable unit step function time present time unmeasured or hidden load variable measured load variable used in feedforward control roots of transfer function G, functions as defined by Equation 8 functions as defined by Equation 7
GREEKLETTERS constant in control law as defined by Equation 14 constant in control law as defined by Equation 16 dummy time variable of integration constant in control law as defined by Equation 13 future time = t - t, major time constant of a second-order process minor time constant of a second-order process constant in control law as defined by Equation 12 constant in control law as defined by Equation 15
variable SUPERSCRIPTS = future value of a variable = variable predicted by process model -’ = optimal control
~
Literature Cited
Aiken, P. M., “A General Experiment in Automatic Process Control,” M. s. thesis in chemical engineering, Purdue University, Lafayette, Ind., 1966. Aiken, P. M., Koppel, L. B., Coughanowr, D. R.. “A General Process for Control Experiments,” Inst. Tech. 15 ( 4 ) . 63 (1968). Athans, Michael, Falb, P. L., “Optimal Control,” pp. 788-93, McGraw-Hill, New Y ork, 1966. Cohen, G. H., Coon, G. A., Trans. A S M E . 75, 827 (1953). Coughanowr. D. R., Koppel, L. B., “Process Systems Analysis and Control,“ p. 241, McGraw-Hill, Xew York, 19658. Coughanowr, D. R., Koppel, L. B., “Process Systems Analysis and Control,“ p. 314, McGraw-Hill, New York, 196513. Coughanowr, D. R., Koppel, L. E . , “Process Systems Analysis and Control,“ p. 465, McGraw-Hill, New York, 1965~. Hougen, J . 0.. Chem Eng. Progr. Monograph Ser. 60 (41, 73 (1964). Latour, P. R., Koppel, L. B., Coughanowr. D. R., IND. DESIGNDEVELOP. 6, 452 (1967). ENG. CHEM.PROCESS Orent, H. H., Ph. D. thesis in chemical engineering, Purdue University, Lafayette, Ind., 1965. Smith. 0 . J. M.. Chem. En,?.Progr, 53, 217 (1957). Smith, 0. J . >I,. I S A Journal 6 ( 2 ) , 2 8 (1959). van der Grinten, P. M. E. M., Control Eng. 10 (12),51(1963).
SUBSCRIPTS o = variable defined a t time to H = refers to upper saturation limit of manipulated variable
RECEIVED for review March 18, 1968 A C C E P T E D January ‘i.1969
NOVEL ELECTROLYTIC CELLS Sodium Dithionite Production at a Swept Mercury Cathode M . S .
S P E N C E R
A N D
W .
J .
S K I N N E R
Agricultural Dicision, Imperial Chemical Industries, Ltd., Billingham, Co. Durham, England
IN
MOST industrial electrochemical processes a single product is formed by the electrode reaction and so no problem of isolating a reactive intermediate arises. Many raw materials, however, give mixtures of products on oxidation or reduction. If the selectivity of electrode reactions can be improved, the scope for the use of electrochemical processes is increased considerably, particularly if yields are better than an equivalent catalytic oxidation or reduction. Control of electrode potential by a potentiostat is a prerequisite of high yields in many instances, but actual reaction rates are a function of reactant concentrations a t the double layer as well as electrode potential. Diffusion
184
I & E C PROCESS D E S I G N A N D DEVELOPMENT
limitations then become of critical importance in determining yields of intermediate products. We can take the general reaction scheme
A+ne+B B+me-C
(1)
(2)
where B is the desired product and the electrode overpotential is sufficiently large for reverse reactions to be neglected. Then to obtain a high yield of B from A a high concentration of A and a low concentration of B are necessary a t the electrode surface. On the start of electrolysis the concentration of A will fall and that of
