Znd. Eng. Chem. Res. 1991,30, 1205-1212
1205
Time-Optimal Startup Control Algorithm for Batch Processes? Eckart von Westerholt,*John N. Beard,* and Stephen S.Melsheimer Department of Chemical Engineering, Clemson University, Clemson, South Carolina 29634-0909
Startup control using a novel bang-bang (time-optimal or near-optimal) algorithm that is insensitive to measurement noise, requires little or no tuning, and does not require an unsteady-state model of the process is described. A second-order dynamic model of the system can be determined from the bang-bang setpoint change for use in tuning a regulatory controller, if desired. The algorithm, which can be used to control any system that can be modeled as second-order plus dead time, uses a generalized switching curve in a modified phase plane. The only restrictions are that the system must be initially a t steady state, the process steady-state gain must be known, and the limits of the control input must be specified. A priori knowledge of the time constants is not required. The startup of a batch experimental laboratory extruder, a laboratory sandbath, and a simulated nonlinear batch exothermic chemical reactor are described.
Introduction In starting up a process, it is desirable to move quickly and smoothly from an initial steady state at ambient conditions to the operating setpoint. A time-optimal or near-optimal policy can achieve these ends for many applications. Throughout this paper the term time-optimal is used in a generic sense to include true time-optimal, near-optimal, and other bang-bang controls that approximate true time-optimal control. The time-optimal policy for a linear second-order system or a system that can be modeled as second-order plus dead time is shown in Figure 1. The upper graph in Figure 1 shows the time-optimal response of the normalized process output, Y, to the forcing of the control input, U, shown in the lower graph. Simple open-loop step forcing and its response are also shown for comparison. For timeoptimal control, full positive forcing, U1,is applied from time zero until time t,. At time t,, the control input is switched to full negative forcing, U,. In the case of a heater, this could be the “off” position. At time t2, the control input is switched to the new steadystate value or to a regulatory controller. As shown in Figure 1, the process output rises rapidly to the new steady state, where it lines out with no overshoot. The time-optimal control problem is to determine the switching times, tl and t,. Many chemical operations including cracking furnaces (Lapse, 1956), large distillation columns (Lupfer and Parsons, 19621, heat exchangers (Hougen, 19641, liquidliquid extraction columns (Biery and Boylan, 1963), extruders (Lin, 1988), and exothermic batch reactors (von Westerholt, 19891, can be approximated by a second-order plus dead time model. However, processes do exist that cannot adequately be represented by this model, and therefore the algorithm described herein is not applicable to those processes. Although the control action shown in Figure 1 is timeoptimal for a linear second-order overdamped system, it is not necessarily time-optimal for systems described by higher order, nonlinear, or distributed parameter equations. A systematic theory of time-optimal control for such systems is not well developed, and each design tends to be unique to its process. For these systems, the rise time with forcing sustained at U,,instead of switching to U2 at time t l , provides a useful indicator; it is the fastest conceivable time for the system to reach the new setpoint Presented at the San Francisco AIChE Meeting, Nov 5-11, 1989;Session on Batch Process Control/Session 143. ? Present address: ENSEEIHTLEEI, 2 rue Charles-Camichel, 31071 Toulouse Cedex, France.
if the maximum forcing available is U,. Of course, except for fmt-order systems, the response would then necessarily overshoot the setpoint, so this would not be a viable mode of control. Still, when an algorithm achieves the new setpoint at a time close to the sustained U , forcing rise time, it must be providing control very close to time optimal. Time-optimal control has been demonstrated by controlling a variety of systems, including stirred-tank reactors (Nyquist and Ramirez, 1971), a batch distillation column (Robinson, 1970), a batch reactor (Eckman and Lefkowitz, 1957), the velocity control of a step motor (Miyamoto and Goeldel, 1982), the control of a batch thermal system (Law and Robinson, 1981), and a double-effect evaporator (Fisher and Seborg, 1976). Many other experimental and computer simulation studies have been performed, indicating the wide range of possible applications of time-optimal control. Yet, to date, there me few applications where timeoptimal control is actually used. One of the primary reasons for not using time-optimal control is that conventional time-optimal control algorithms require a reasonably accurate dynamic model of the system being controlled. In most studies, either this model is known before the timeoptimal change is initiated or it is determined during the time-optimal change by curve fitting. A second reason is that in real systems the dynamics often change so that a dynamic model may have to be updated frequently. For example, in a chemical plant running batch polymerization reactions, there might be 100 or more “recipes”,each of which would have different dynamics and therefore would require a different model. To develop a model for each recipe would be prohibitively costly. This paper describes startup control using a novel time-optimal algorithm that does not require a dynamic model of the process and does not use curve-fitting techniques.
Conventional Implementation of Time-Optimal Control If an accurate dynamic second-order model of the system is available and if U1and U, are specified, the switching times, t , and t 2 , can be calculated a priori (Koppel, 1968). However, open-loop time-optimal control based on precalculated switching times is very sensitive to modeling errors and process disturbances since it does not take advantage of process variable feedback. Feedback can be obtained by using a traditional phase plane switching curve, such as that shown in Figure 2 (Koppel and Latour, 1965). Two-state variables are required to define the state of a second-order system; however, in most applications
0888-5885/91/26301205$02.50/0 0 1991 American Chemical Society
1206 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 12
- TIME-OPTIMAL __ 02
---
/’
,
STEP SETPOINT SUSTAINED U1 FORCiNG
1 ~
/
’
00
10
0
-1
-
-
t, 2
20
30
40
--
t2
50
, 60
Generation of the Switching Curve The differential equation describing a linear, overdamped second-order system in terms of deviation variables is d2Y*(t) Y*(t) = KU* (1) 7172+ (71 + 7 dY*(t) 2 1 7 dt2 In terms of deviation variables normalized about the new setpoint, Y*,, the equation is
+
STEP
. , .
,
_I
,,,\ - SWiTCHlNG CURVE
where Y = Y*/Y*, and U = KU*/Y*,. During time-optimal control a t t = ~ Y=O, Y = o att=t2 Y=I, Y = o
PROCESS
__ dt dY
-
0 .2 1 20 15 5 0-I ,/
/
0.05. I
,’
00
the elapsed time multiplied by the derivative of the process output. Lin (1986) demonstrated Beard’s algorithm in controlling a simulation of a double-effect evaporator, Ornitz (1980) used the algorithm to demonstrate the time-optimal control of simulated open-loop underdamped systems, and Minnick (1984) studied the extension of the algorithm to multiinput multioutput systems. Lin (1988,1989) modified the Beard algorithm so that the phase plane ordinate is (l/t)SY dt, the area under the process output curve divided by the elapsed time. This time-optimal control algorithm (TOCA) is the basis for the work described herein. It is relatively insensitive to measurement noise because it uses the integral of the process output instead of the derivative.
.-
02
04
___----
06
08
10
12
Y
Figure 2. Traditional phase plane switching curve.
only a single output variable is measurable. A suitable second state variable is thus obtained by differentiating the output. The switching curve is thus based on the values of the process output, Y, and ita derivative, dY/dt, at the first switching time. To use this switching curve, Y is measured and dY/dt is estimated so that the Y vs dY/dt trajectory can be plotted in the phase plane. The process trajectory will intersect the Switching curve at time tl. At time tl the forcing is switched from U1to U2,and the process trajectory moves down the switching curve to the new steady-state value of Y, 1.0. The phase plane in Figure 2 can be used to control systems regardless of whether they are initially at steady state. Figure 2 is based on specific values of the system dynamic parameters, 71 and 72, as well as specific forcing values Ul and U, and the process steady-state gain, K. Thus, this method provides feedback, but an a priori dynamic model is still required. Further, estimation of the derivative can be difficult and inaccurate, especially in the presence of noise.
Model-Independent Time-Optimal Control Beard (1971, 1974) developed a time-optimal control algorithm that requires that the system be initially at steady state, that the steady-state process gain, K, be known,and that the forcing values, U, and U2,be specified. However, a priori knowledge of the system dynamic parameters, T , and 7 2 , is not required. This algorithm uses a modified phase plane in which the ordinate is t dY/dt,
Athans and Falb (1966) show that for this system, the control action to minimize J4dt should be bang-bang with U, and U2at the allowable extremes of the control input and that this system is normal, so that singular control is not a problem. Thus for 0 C t It,, U = U, for tl
C
t 5 t2, U = U2
Because of the manner in which U and Yare normalized, the initial control is always Ul, U , is always positive, and Y in eq 2 always goes from 0 to 1. The corresponding nonnormalized control input, U*l, may be either positive or negative depending on the signs of the gain K,and the setpoint change, Y*,. When eq 2 is solved from t = 0 to t = tl with U = Ul, the result is eq 3.
Minnick (1984) solved eq 2 from t = 0 to t = tl with U = Ul and then solved it backward from t = t2to t = tl with
U = U2to obtain four equations, two for Y and two for Y at tl. These were rearranged to get eqs 4 and 5.
t2 =
7
Ind. Eng. Chem. Res., Vol. 30, No. 6,1991 1207 1 .o
10
- SWITCHING CURVE 0.8
-
.
I 1 1 I I I 1
- - PROCESS
0.5
i -I-
0.8 .
I
""""
OVERDAMPED EXTENSION CRITICALLY DAMPED
__-
L
05-
i
0-0.01
1 -
-I-
,; 0 4 -
,; 0 4
_ _0 0
0.0
B 0.99 -0
,r ' 0 s
02-
0.2
0c
- -0.99-
02
/*
/ -
c
04
05
08
10
12
Y
Y
Figure 3. TOCA phase plane overdamped switching curve.
Figure 4. Complete TOCA Switching curve.
Equation 6 is obtained by integrating eq 3 from t = 0 to t = tl and dividing the result by tl.
stant, T* The second switching time, t2,is then calculated by using eq 9. Thus when the process trajectory first intersects the switching curve at t,, the time constants and t2 can be determined. The remainder of the process trajectory and any subsequent intersections with the switching curve have no meaning other than to show the behavior of Y after the switch.
(
72't1 71/72
-
1)(1 - e-t1/T2)
+ I)
(6)
If the parameters in eq 2 are known, then eq 4 can be solved by iteration for the first switching time, tl, and eq 5 can be used to calculate tP. Equations 3 and 6 then give the values of the system state at the first switching time. A switching curve in the ( Y , n phase plane can be generated by carrying out this calculation for a sequence of parameter values. The development of the switching curve is more easily understood if the substitutions A = t1/71 and B = 71/72 are made in eqs 3-6, yielding eqs 7-10, respectively.
Systems Other Than Second-Order If the system being controlled is other than second-order overdamped, it is possible that the process trajectory will pass above or below the switching curve shown in Figure 3. To avoid this, another section of switching curve is generated to connect the lower point of the curve shown in Figure 3 with the origin. This curve is based on a second-order critically damped plus dead time system:
-
(11)
A derivation completely analogous to that for the overdamped curve yields eqs 12-15, where A, = t1/7 and B, Y(tl) = Ul(l - (1 + A, - Bc)e(-Ac+BC)} (12)
1-
If K, Ul, U,, and B are specified, then the only unknown in eq 8 is A. One point on the switching curve is generated by choosing a value for B and solving eq 8 for the corresponding value of A. With these values, eqs 7 and 10 can be solved for the switching curve abscissa and ordinate, respectively. This process is repeated for values of B from 0.01 to 1 to complete the portion of the switching curve based on eqs 7,8, and 10. This curve is referred to aa the overdamped switching curve and is shown in Figure 3. The switching curve in Figure 3 has two important properties: (1)Each point on the switching curve corresponds to a specific value of B = r1/r2,the ratio of the system time constants. (2) Each point on the switching curve corresponds to a specific value of A = tl/rl. Figure 3 also shows a typical process trajectory. The trajectory intersects the switching curve at time tl. Then, property 2 is used to determine the first time constant, rl, and property 1is used to determine the second time con-
(A,)e-Ac
(1-
e
-Ac
- u_2 = 0 (13) u,
= 9f 7. These equations are used to generate the critically damped portion of the switching curve shown in Figure 4. The overdamped and critically damped switching curves intersect where the two transfer functions are ~ T~ in eq 1,and B = 0 in eq 11; therefore B identical ( T = = 1.0 and B, = 0). Since B, does not appear in eq 13, it is solved only once in generating the switching curve. Finally, the upper point on the overdamped curve is connected to point (1,l) with a straight line. Trajectories striking this part of the switching curve would be essentially first order. In this case, the upper point on the overdamped switching curve (B = 0.01), is used for determining t2, for obtaining model parameters, and for tuning the regulatory controller. This method of handling essentially first-order systems, treats them as second-order with one dominant time constant and one very small time constant. This is necessary since eqs 7 and 10 cannot be
1208 Ind. Eng. Chem. Res., Vol. 30, No. 6 , 1991
Table I. Model Parameters, Values of KO,and Switching Times model
extruder run 1 extruder run 4 lab sandbath "reactor" in sandbath "reactor" and measurement noise simulated exothermic reactor third-order syst
1.3 2.3 1.9 12.4
134 114 185 207
95 72 145 163
20 30 50 15
16.4 183
163
15
67.0
88.4
20
22.3
27.6
tl 7.9
tz 13
6.1
40.6
7 8.4
0 2.7
1.6
106.3 107.2 36.7 39.7 45.0 45.7 72.3" 88.4O
(a = 1.25)
"Time after initial forcing, which occurred at time = 10 min.
solved for B = 0 due to the 1 / B terms.
Tuning the TOCA The algorithm as discussed thus far has no provision for tuning. In fact, however, many processes may be nonlinear, or higher than second-order, or otherwise deviate from the models shown in eqs 1 and 11. It has been found that these deviations can usually be dealt with satisfactorily by introducing a tuning parameter, a. The phase plane ordinate for the system trajectory is divided by the tuning parameter, a, which ranges from about 0.5 to 1.5, so that the ordinate for the system trajectory is (l/at)J"Ydt. Only the system trajectory is changed; the switching curve itself is not affected by a. A value of a = 1 is satisfactory for many systems, but the empirical introduction of a has been found to be a reasonable solution for exceptions. Regulatory Controllers In the experimental studies, control was switched to a regulatory controller following completion of startup using the TOCA. The regulatory controllers used in this study were digital PI or PID controllers designed by von Westerholt (1989) using direct synthesis techniques (Seborg et al., 1989). The controller settings were determined by using the model parameters obtained during startup and eqs 16-18. It should be noted that KOis the sole desired open-loop gain - KO _ K, = (16) process steady-state gain K 71
=
71
+
Tp
(17)
7172
=(18) 71 + 72 tuning parameter of the regulatory controller. The model dynamic parameters determined by the TOCA, the values of KOused for the various runs, and the switching times are shown in Table I. The estimated values of K used with the TOCA and the actual values of process gain observed during startup are shown in Table 11. Table I1 also shows the units of K. 7D
Experimental Testing The TOCA was demonstrated in controlling and modeling several pieces of laboratory equipment. The algorithm was programmed in MACBASIC on an Analog Devices pMAC 5000 data acquisition and control system (DACS). Before time = 0, the estimated process steady-
Table 11. Estimated and Actual Process Steady-State Gains estd actual steady-state steady-state gain used in gain during startup ( K ) startup 95 89 extruder run 1, OC/mA 72 62 extruder run 4, "C/mA 145 138 lab sandbath, OC/V "reactor" in sandbath, 163 167 "C/V "reactor" and measure163 193 ment noise, OC/V simulated exothermic 1.6 variable reactor, OC/OC
state gain and the extremes of the control input were entered into the DACS. At time = 0, the DACS began reading the process variables and calculating the values of Y and Y for the process and began computing the switching curve values of Y and Y. When the switching curve calculations had been completed, the DACS began comparing the values of P for the process and the switching curve at successive measured values of the process Y. The intersection of the process trajectory and the_ switching curve occurred when the value of the process Y was equal to or less than the switching curve 7at the same value of Y.
Control of an Extruder The first piece of equipment tested was a small batch carbon fiber extruder used by the fiber group in the Department of Chemical Engineering at Clemson University. The pitch precursor for the fibers is temperature sensitive, so it is desirable to heat to the spinning temperature as fast as possible. Using conventional controllers, it took researchers over 100 min to reach steady-state and begin spinning. Figure 5 shows the temperature responses, input forcings, and modified phase planes of two experimental runs using the TOCA. In run 1 the initial forcing was 2 times the final steady-state forcing, whereas in run 4 the initial forcing was over 3 times the final steady-state forcing. In both runs the TOCA performed well, as did the regulatory controller which was tuned by using the model parameters obtained during startup. For each run, the melt temperature response curve moved at its maximum rate to the new setpoint, where it lined out with little overshoot, indicating near time-optimal control. In run 4 the temperature reached steady state in about 50 min (less than half the time required when using a conventional controller), the startup was smooth, and there was little overshoot. No tuning of the TOCA (a = 1) was used for these runs. The model parameters and switching times obtained during startup are shown in Table I. Physical changes were made in the extruder between runs 1 and 4. The difference in the model parameters obtained during run 1 (71 = 1.3 min, T~ = 134 min) and run 4 (7*= 2.3 min, T~ = 114 min) and the difference in the steady-state control forcing for run 1 (about 3.5 mA) and run 4 (about 5.0 mA) are indications of the insensitivity of the TOCA with respect to changes in system parameters. Control of a Laboratory Sandbath Figure 6 shows the startup of a laboratory sandbath. The sandbath, Model SBL-2D manufactured by Techne Ltd., had a working space 216 mm in diameter and 305 mm deep. It contained approximately 32 kg of sand and was heated by two 1-kW electric heating elements immersed in the bottom of the bath. The top graph in Figure 6 shows a smooth, rapid startup temperature response that lines out at the setpoint with no overshoot. The bottom graph
Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 1209
150 -
RUN 1 RUN 4 SETPOINT
100
-
50
SETPOINT
O
~
0
-24 0
- . , . . . . , . 50
100
.
.
,
-
.
150
.
.
I
200
.
.
.
,
250
V
_
.
.
I
500
"
.
.
.
50 50
.
,
.
.
. 100 100
50
100
.
,
.
~ 150 150
"
~
~
"
'
"
'
"
~
'
~
"
200
250
300
200
250
300
1
0
150
TIME [min]
'
0.8 -
- - SWITCHING CURVE - TRAJECTORY
Y
Y
Figure 5. Control of a batch carbon fiber extruder.
Figure 6. Control of a sandbath.
shows the process trajectory and switching curve in the modified phase plane. The trajectory intersects the switching curve along the extension indicating a nearly first-order system. A near first-order system is also indicated by the shape of the temperature response and, in the middle graph, by the short time that the heater was turned off. The temperature response is very similar to that reported by Phillips et al. (1988) for the startup of a sandbath with heaters on the outside of the bath wall. The degree of initial forcing for their base run was about 1.5 times that of the new steady-state value. The degree of forcing for Figure 6 was much higher, about 4 times the new steady-state value, yet good results were obtained and no tuning of the TOCA was required. Although the switching curve was intersected on the extension, the temperature response indicates that very near time-optimal control was obtained. When the temperature reached 100 "C during the startup, the fluidizing air rate was reduced by about 20%, introducing a disturbance into the system. The effect can be seen by close inspection of the temperature curve. The TOCA performed well in spite of this disturbance. The model obtained during startup was used to tune the PID controller for control at the new setpoint. At about 108 min, the sandbath fluidizing air rate was decreased further by about 20% to introduce a disturbance into the system during regulator control at the setpoint. The controller took action to decrease the amount of heating, as seen in the middle graph; however, the change is barely discernible in the temperature graph.
and was 180 mm long. It was filled with cracking catalyst, and a 1.6-mm-0.d. resistance temperature detector (RTD) was inserted in the catalyst bed. The dummy reactor was immersed in the sandbath. Figure 7 shows the temperature and control input profiles and the modified phase plane for startup of this system. A PI controller was used at the new setpoint. As before, the model parameters that were obtained were used to tune the regulatory controller. Again the algorithm performed well, yielding a smooth rapid startup. No tuning of the TOCA (a= 1)was used for this run. It should be noted that the initial heating was not started until time = 10 min. Figure 8 compares step responses of the system and the model. The agreement is very good.
Control of a Laboratory "Reactor" To have a system that was more difficult to control than the sandbath, a dummy reactor was constructed. The reactor was made of 1.5-in. schedule 40 pipe (41-mm 0.d.)
Control of a Simulation of an Exothermic Batch Reactor Work is proceeding toward demonstrating the algorithm in starting up commercial exothermic batch reactors.
Control of a Laboratory "Reactor"with Measurement Noise To demonstrate the TOCA when measurement noise is present, another run was made with the dummy reactor in the sandbath and a random temperature increment with a range of i 5 "C was added to the measured temperature. While the integral on the ordinate inherently smooths the noise, the measurement of the process output itself is used as the abscissa. Therefore the value of the process output must be smoothed in some manner. To provide a simple data smoothing mechanism, 10 temperature measurements were made and averaged at each measurement time. The temperature response and the phase plane in Figure 9 show that the TOCA performs well in the presence of measurement noise.
Ind. Eng. Chem.Res., Vol. 30, No. 6, 1991
-i I
. . . . . . . . . . . . . . . . . . . , . . . . . . . . .
0
0
loo
50
150
200
250
300
'1 '
..-
a - .1
!
I
I 160
0
.
160 TIME
260
200
[min]
I
I
'
0.80.81
0.6
0.4
I
I
I
I
- - SWITCHING CURVE - TRAJECTORY
I
3
-
Y
Figure 9. Control of a reactor with measurement noise.
-
0.2
0.0
0.2
0.4
0.6
Y--+
controller
-
1 .z
1 .o
0.8
Y
Figure 7. Control of a reactor in a sandbath.
oil out
150 /
50
O
500
- REACTOR - - MODEL
IN SANDBATH
1000
1500
I
\
/
metal wall
100
d i x i n g valve
iacket
A k ' B Figure 10. Exothermic batch reactor configuration. 2000 F
TIME [min]
Figure 8. Step responses of reactor in sandbath and model.
,,...,....,...,,./-._.,..,.. I ?............ ...... ........... I
0
Y
Y
...
I
W
3
Exothermic reactions are challenging because heat must be applied to raise the batch toward the reaction temperature; then as the reaction begins to produce heat, the reactor must be cooled to remove the excess heat in order to maintain the reaction temperature. Thus,the gain from heat input to reactor temperature not only changes in a very nonlinear fashion but in fact also changes sign. As a preliminary step, the TOCA was tested in controlling a reactor simulation presented by Luyben (1973). Luyben's volumes, flow rates, reaction rates, and other simulation parameters were used. The simulation was modified somewhat in that Luyben used the reaction A B C, whereas in this case it was A B, and that he used steam and water for heating and cooling, whereas in this case hot and cold oil were used. A mixing valve maintained the inlet oil temperature at Tht as shown in Figure 10. The process gain was defined at T/TFtto avoid the sign change that would occur if the heat input were taken as the control input. To introduce more realism into the demonstration, the reactor simulation was implemented in real-time on an
--
-.
$
\
I
- 4
..... Tref (Reactor Setpoint)
I
-T
I-
-- TM
04 0
.
.
.
.
.
.
.
.
.
50
(Reactor) (Metal Wall)
.
loo
.
.
.
.
I
I
150
I
Y
Y
W
a
$
.....T J S ~(Jacket Setpoint)
H
-TOJ -- TJ
I-
(Jacket Inlet) (Jacket Outlet)
-20
0
loo
50
I
150
TIME [min]
Figure 11. Startup of batch reactor simulation using proportional controller.
Analog Devices MACSYM 150 data acquisition and control system and the controllers were implemented on a pMAC 5000. The measured temperature and the control signal
Ind. Eng. Chem. Res., Vol. 30,No.6,1991 1211 1.2-
i
,-.--/
.... ........ , 1. ' A. ..*
loo
.-
1.0-
A
-- - - - _ _ _
.
v
0.8-
e-
a
e 0.6 -
..... T,,f
-T - - TM
0
(Reactor Setpoint) (Reactor)
,
V, 0.4v,
-a _- a
W
0
(Metol Wall)
E
.
0.2
-1
20
10
0
30
= 1.25 = 1.00
50
40
60
TIME [min]
.
..'..T J ~ E T(Jacket Setpoint) -TOJ (Jocket Inlet) - - TJ (Jocket Outlet)
o,8
-
- TRAJECTORY, a = 1.25 - - TRAJECTORY, a = 1 .OO SWITCHING CURVE
\ \ \ \
I I
-20
50
0
100
150
TIME [min] 1.0, '
0.8-
0.6
-
0.4
-
0.0
I
- - SWITCHING CURVE - TRAJECTORY
I
0.2
0.0
0.6
0.4
0.8
1.2
1.0
Y
Figure 13. Tuning example.
0.2
0.4
0.6
0.8
1 .o
1.2
Y
Figure 12. Startup of batch reactor simulation using TOCA.
-
1 .o
v d .
0.8
e 3
2 a
0.6
0
were,sent as 0-10-V signals between the pMAC 5000 and the MACSYM 150. Luyben's controller settings and a proportional controller gave the responses shown in Figure 11, which are similar to those reported in his text. The upper graph shows the temperature responses of the reactor and the lower graph shows the responses of the reactor jacket, or essentially the control action. Details of the simulation, including a listing of the computer program, are given by von Westerholt (1989). . Figure 12 shows a startup of the reactor simulation using the TOCA. The startup is smooth, and the oscillations are less pronounced than with the conventional controller. Again the model parameters were used to tune the regulatory controller. No tuning of the TOCA was required. As noted above, the process gain changes rather dramatically from the initial heating stages to the reactor operating condition where cooling is applied. For the TOCA, the appropriate "steady-state gain" is that which applies at the end of the startup, when control is transferred to the regulator. The steady-state gain used in the TOCA for the startup shown in Figure 12 was obtained from Figure 11. The steady-state gain is the setpoint from the upper graph (93.3 "C) divided by an "eyeball fit" through the regulator controlled part of the lower graph (56 "C). Figures 11 and 12 clearly show that the steady-state gain for this sytem continues to change as the reaction proceeds even after completion of the startup. The ability of the TOCA to handle this severely nonlinear reactor simulation is strong evidence of its applicability to a wide range of startup situations.
Sensitivity of the TOCA to Process Gain Table I1 compares the estimated steady-state gains used in startup with the actual steady-state gains measured during startup. The errors in estimating the gains range from +16% for extruder run 4 to -16% for "reactor" and measurement noise. These numbers indicate that the TOCA is not very sensitive to the estimated steady-state gain.
Tuning Example Neither the available experimental equipment nor the exothermic batch reactor simulation required tuning of the TOCA, nor was the critically damped part of the switching
Concluding Remarks The TOCA performed exceptionally well on all of the physical systems on which it was tested: a batch carbon fiber extruder before and after modifications to the ex-
m V, 0.4 W
- - MODEL - SYSTEM
0
0.2 0.0
10
20
30
40
50
60
TIME [min]
Figure 14. Step responses for tuning example.
curve intersected. To demonstrate this feature of the TOCA, the control and modeling of a linear third-order simulation (rl = 10,r2 = 5, r3 = 5) is shown in Figure 13. Figure 13 shows the temperature responses and phase plane trajectories of the system with tuning (a= 1.25) and without tuning (a= 1.0). The untuned system overshoots by about 12%;however, the tuned system lines out at the setpoint with no overshoot. The TOCA is rather insensitive to the exact value of a chosen. In this example, the control input was set to the new steady-state at t2 rather than switched to a regulatory controller. Figure 14 shows good agreement between the step response of the system and that of the model obtained during the startup using the tuned TOCA ( r = 8.4, 8 = 2.7).
1212 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
truder, a sandbath, and a d u m m y reactor in the sandbath with and without measurement noise. It also performed very well in starting up the nonlinear simulation of a batch exothermic reactor. Tuning of the TOCA was not required in a n y of these cases. The TOCA was provided with inputs of U1, U,, and an e s t i m a t e of the new steady-state U (or the process gain, K ) and the TOCA carried out the startup and the t u n i n g of the regulatory controller. One t u n i n g parameter for the regulatory controller, the desired open-loop gain, KO,was also i n p u t i n t o the system. Acknowledgment
Partial financial support provided by the South Carolina Energy Research and Development Center is greatly appreciated. MACBASIC, MMAC5000, and MACSYM 150 are registered trademarks of Analog Devices, Inc. Nomenclature A = parameter of overdamped switching curve, tl/rl A, = parameter of critically damped switching curve, t l / r B = parameter of overdamped switching curve, r l / r 2 B, = parameter of critically damped switching curve, O/r K = process steady-state gain (see Table I1 for units) K, = regulatory controller gain (units the same as for 1/K) KO = desired open-loop gain, dimensionless U = normalized control variable U 1= normalized control variable, 0 < t 5 tl U, = normalized control variable, tl < t It2 U* = control variable in deviation variables Y = normalized process output Y = dY/dt Y* = process output in deviation variables ?**= process setpoint in deviation variables Y = ordinate of modified phase plane s = Laplace operator t = time tl, t 2 = first and second switching times a = TOCA tuning parameter O = dead time in critically damped model r = time constant in critically damped model rl, r2 = time constants in overdamped model 71 = regulatory controller integral time, min T D = regulatory controller derivative time, min
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Receiued for review February 5, 1990 Revised manuscript receioed November 5, 1990 Accepted November 23, 1990
