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FIGURE 17-11
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Block diagram for two-tank chemicalreactor system.
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Example 17.4. Using the Ziegler-Nichols rules, determine controller settings for various modes of control of the two-tank chemical-reactor system of Chap. 11. The block diagram is reproduced in Fig. 17.11. For convenience, the process gain K and the controller gain K, are combined into an overall gain K1. The equivalent controller transfer function is regarded as
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where K1 (as well as 71 and 7~) is to be selected by the Ziegler-Nichols rules. The requited value of KC is then easily determined as
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where K = 0.09 for the present case (see Chap. 11.) The Bode diagram for the transfer function without the controller
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(s + 1)(2s + 1) is prepared by the usual pmcedums is found that
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and is shown in Fig. 17.12. From this figure, it
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= 1.56 rad/min (17.4)
1 KLd = - = 6 . 9 0.145
Pu = f& = 4.0 mm/cycle
Hence, the Ziegler-Nichols control constants determined from Table 17.1 and Eq. (17.4) are given in Table 17.2. A plot comparing the open-loop frequency responses including the controller for the three cases, using the controller constants of Table 17.2, is given in Fig. 17.13. This figure shows quite clearly the effect of the phase lead due to the derivative action. The resulting gain and phase margins am listed in Table 17.3. From this table it may be seen that the margins are adequate and generally conservative.
236 F R E Q U E N C Y R E S P O N S E
1 0.5
0 % 0.145 5 + 0.1
9 0.05
.J -\;
. l\ I I
-I-\I 1
FIGURE 17-12 Bode diagram for e- .5J/[(s + 1)(2s + I)].
Note that to obtain the Bode diagram for systems including the PID controller, the controller transfer function is rewritten as (17.5) This is second-order in the numerator and has integral action in the denominator. In general, the numerator factors into first-order factors; hence it contributes two
TABLE 17.2 Control P PI PID 4 3.5 3.1 4.2 71 3.3 2.0 v
CWTBOL SYSTEM DESIGN BY PREQuwcr
RESFONSE
2 3 7
1 0.5 t jk f 1 0.1 0.05
0.01 0
-45 p-90 I &I35 -180 FIGURE 17-13 -225 5 IO
Open-loop Bode diigrams for varifor various cc41troIters wifb system of Fig. ow 17.11.
curves simii to that of Fig. 16. I1 to the overaIl diagram. Far the Ziegler-Nichols settings it is seen from TabIe 17.1 that rf = 4~. Making this substitution into Eq. (17.5) (17.6) shows that the numerator is equivalent to two PD components in series. This AR is repreSented by a high-frequency asymptote of slope +2 passing through the freTxlsLE 17.3 COIltlVl P Pl PID Gpinmargin 2.0 1.9 2.6 ==msrgia 450 33 34
FREQUENCY
RJ .SPONSE
quency o = 1%~ and a low-frequency asymptote on the line AR = 1. It should be emphasized that these special considerations apply only to the Ziegler-Nichols settings. In the general case, the two times constants obtained by factoring the numerator of Fq. (17.5) will be different. The Bode plot of the denominator follows from
The gain is a straight line of slope - 1 passing through the point (AR = 1, w = l/~f). The phase lag is 90 at all frequencies. Plotting of ihe overall Bode diagram for the PID case to check the results of Fig. 17.13 is recommended as an exercise for the reader.
lhnsient
Responses
For instructive purposes, the two-tank reactor system of Fig. 17.11 was simulated on a computer. Responses of C(t) to a unit-step change in R(t) am shown in Fig. 17.14. These responses were obtained using the Ziegler-Nichols controller
settings determined in Example 17.4.
The responses to a step load change were also obtained on a computer. These am the curves of Fig. 10.7 that were discussed in Chap. 10 to illustrate the function of the various modes of control. A load change for this system corresponds to a change in the inlet concentration of reactant to tank 1 (refer to Fig. 11.1). As process control engineers, we would be more interested in controlling against this kind of disturbance than against a set-point change because the set point or desired product concentration is likely to remain relatively fixed. In other words, this is a regulator problem and the curves of Fig. 10.7 am those we would use to determine the quality of control. However, the step change in set point is frequently used to test control systems despite the fact that the system will be primarily subject to load changes during actual operation. The reason for this is the existence of well-established terminology used to describe the step response of the underdamped second-order system. This terminology, which was presented in Chap. 8, is used to assign quantitative measure to responses that am not truly second-order, such as those
Time -
FIGURJI 17-14 Closed-loop response to step change in R(t) for control system of Fig. 17.11, using various control modes (obtained by computer).
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