barcode reading in asp.net For the control system shown in Fig. 19.7, determine controller in Software

Paint USS Code 128 in Software For the control system shown in Fig. 19.7, determine controller

settings for a PI controller using the Z-N method and the C-C method. This problem will be instructive because the transfer function of the model is already in the form of first-order with transport lag, which is the form used by Cohen and Coon to derive their tuning rules. C-C method. Since the transfer function of the plant is in the form of Eq. (19.6), we obtain T and Td immediately without having to draw a tangent line through the point of inflection, i.e., T = 1 and Td = 1. We also observe from the block

Closed-loop method (Z-N method)

6.4 5.8 5.6 11.4* 1 .o* 1.1 3.4 1.6

Open-loop method (C-C method)

8.1 7.0 4.4 9.8 0.43 10.5 3.9 0.59

, 10

Pr;porJion;

0 FIGURE 19-6

- O.lOB 0 10 20 (4

Comparison of load responses for the system of Fig. 19.1 using controller settings obtained by the Ziegler-Nichols method (Z-N) and the Cohen and Coon method (C-C). diagram that Kp = 1. Substituting these values into the appropriate equations of Table 19.2 gives

Kc = &i,.,+ g) = ;jo.9+ &)= 0.983

Using these values for K, and 71, the step response shown in Fig. 19.8 was obtained by simulation. Z-N Method. Application of the Bode criterion from Chap. 17 gives the following results w co = 2.03 or

R = u(t)

FIGURE 19-7

FIGURE 19-8 Response to unit step in set point for the system in Fig. 19.7 (Example 19 . 1).

The details for obtaining these results will not be given here since this type calculation was covered in depth in Chap. 17. Applying the Z-N rules for PI control from Table 19.1 gives: K, = 0.45K,, = (0.45)(2.26) = 1.02 and TI = PJ1.2 = 3.0911.2 = 2.58 The step response for these controller settings is shown in Fig. 19.8. The ISE value for each response was calculated out to a sufficiently long time (10 units of time) for the integral to converge; the results are as follows: C-C response: ISE = 1.54 Z-N response: ISE = 1.49 at t = 10 at t = 10

Although the ISE values are nearly the same, the transient for the Z-N settings is better than the transient for the C-C settings. The Z-N transient has much less overshoot. The lesson to be learned from this example is that the comparison of two transients based on only one criterion (in this case, the ISE) may be misleading in the selection of the best transient. It is also important to judge the quality of a transient by its actual appearance. It should be noted that for this example, in which there is a relatively large transport lag (Td = l), much of the contribution to the ISE occurs from f = 0 to t = 1, during which time the ISE reaches 1.0. This value of the ISE at t = 1 is the same, regardless of the tuning method used because the transport lag causes error to be constant from t = 0 to t = 1. Example 19.2. For the control system shown in Fig. 19.9, determine the controller settings for a PI controller using the Z-N method and the C-C method. In this problem, the process reaction curve must be modeled by the method shown in Fig. 19.4. C-C method. Since the transfer function of the plant is given as l/(s + 1)4, we can obtain the value of 7 d and T for use in the C-C method analytically. A unit-step response for the plant transfer function is 13 c(t) = 1- -6t + 12+ t + 1 eWr Zt l 1