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criteria. Fortunately, for this case and others there are simple rules for directly establishing values of the control parameters that usually give satisfactory gain and phase margins. These are the Ziegler-Nichols rules, which we develop in the next section.
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Consider selection of a controller G, for the general control system of Fig. 17.5. We first plot the Bode diagram for the final control element, the process, and the measuring element in series, GiG2H(ju). It should be emphasized that the controller is omitted from this plot. Suppose the diagram appears as in Fig. 17.6. As noted on the figure, the crossover frequency for these three components in . . senes is o,,. At the crossover frequency, the overall gain is A, as indicated. According to the Bode criterion, then, the gain of a proportional controller which would cause the system of Fig. 17.5 to be on the verge of instability is l/A. We define this quantity to be the ultimate gain K,. Thus
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K, = A'
(17.3)
The ultimate period P, is defined as the period of the sustained cycling that would occur if a proportional controller with gain K, were used. From the discussion of Fig. 17.3, we know this to be
p, = E
time/cycle
(17.3a)
The factor of 2~ appears, so that P, will be in units of time per cycle rather than time per radian. It should be emphasized that K, and P, are easily determined from the Bode diagram of Fig. 17.6. The Ziegler-Nichols settings for controllers are determined directly from K, and P, according to the rules summarized in Table 17.1. Unfortunately, specifications of K, and ro for PD control cannot be made using only K, and P,. In general, the values 0.6K, and PUB, which correspond to the limiting case of no integral action in a three-mode controller, are too conservative. That is, the
TABLE 17.1
Ziegler-Nichols Controller Settings
ljpe of control Proportional Proportional-integral (PI) (PID) G,(s) KC KC 71 7D
OJK, 0.45K,, p,, i
Proportional-integral-derivative
FREQUENCY
RESPONSE
resulting system will be too stable. There exist methods for this case which am in principle no more difficult to use than the Ziegler-Nichols rules. One of these is selection of ~0 for maximum K, at 30 phase margi , which was discussed above. Another method, which utilizes the step response L d avoids trial and error, is presented in Chap. 19. The reasoning behind the Ziegler-Nichols selection of values of K, is relatively clear. In the case of proportional control only, a gain margin of 2 is established. The addition of integral action introduces more phase lag at all frequencies (see Fig. 16.10); hence a lower value of Kc is required to maintain roughly the same gain margin. Adding derivative action introduces phase lead. Hence, more gain may be tolerated. This was demonstrated in Example 17.2. However, by and large the Ziegler-Nichols settings am based on experience with typical processes and should be regarded as first estimates.
Example 17.3. Using the Ziegler-Nichols rules, determine K, and ~1 for the control system shown in Fig. 17.10. For this problem, the computation will be done without plotting a Bode diagram; however, the reader may wish to do the problem with such a diagram. We first obtain the crossover frequency by applying the Bode stability criterion: -180 = - tan- (o) - 57.3(1.0;)(0) The value 57.3 converts radians to degrees. Solving this equation by trial and error gives for the crossover frequency, wCO = 2 rad/min. The amplitude ratio (AR) at the crossover frequency for the open loop can be written A R =
where we have used Eq. (16.16) for the first-order system and the fact that the amplitude ratio for a transport lag is one. According to the Bode criterion, the AR is 1.0 at the crossover frequency when the system is on the verge of instability. Inserting AR = 1 into the above equation and solving for K, gives K,, = 2.24. From the Ziegler-Nichols rules of Table 17.1, we obtain
K, = 0.45K,, = (0.45)(2.24) = 1.01
and 71 = PJ1.2 = [21rlw,,]/1.2 = [21r/2]/1.2 = 2.62 min. b :
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