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1 tS+1)4
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FIGURE 19-9
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Process for Example 19.2. From this result one can readily obtain the first and second derivatives; thus
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k(t) = $3,~t 6 r(t) = +-73t2 - t3)
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The location of the inflection point on the transient c(t) is obtained by setting the second derivative to zero:
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0 = je- (3t2 - t3>
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Solving for t gives as the root of interest in this problem t = 3. Knowing that the point of inflection occurs at t = 3, we can compute the slope of the tangent line through this point to be
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S = >(3) = ;(3)3,-3 = 0.224
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We can now determine Td as shown in Fig. 19.10. From the expression for
c(t), we obtain the value of c at the inflection point to be ~(3) = 0.353. The
value of t where the tangent line intersects the t-axis is obtained from the slope S, thus
0.353 - 0 = S = 0.224 3 - T,
solving for Td gives
Td = 1.42
Solving for T from Eq. (19.4) gives
T = B,IS = UO.224 = 4.46
Having found Td and T, we can apply the appropriate equations from Table 19.2 to get
K, = 2.91 rl = 2.86
The transient for these settings that was obtained by simulation is shown as curve C-Cl in Fig. 19.11. To our surprise, it is unstable.
Z-N method. When we apply the Z-N method for a PI controller, we obtain the following results: K,, = 4, P, = 2rr, K, = 1.8, and rt = 5.23.
CONTROLLER
TUNING
PROCESS
IDENTIFICATION
0.9 0.8 -
0.7 0.6 0.5 0.4 0.3-0.2 I
Point of inflection
FIGURE
I I 1 I
19-10
Process 19.2.
reaction curve for Example
The transient for this set of controller parameters is also shown in Fig. 19.11. We see that the response is stable and well damped. The lesson learned in this example is that the application of a tuning method may not produce a satisfactory transient. Fine tuning of these first guesses is usually needed. Before abandoning the C-C method for this example, the process reaction curve was fitted to a first-order with transport lag model by means of a least square fitting procedure. Applying the least square fit procedure out to t = 5 produced the following results
Td = 1.5
T = 3.0
Applying the C-C method for these values of Td and T gives
K, = 2.05
q = 2.49
Notice that the value of K, is now considerably less than the value obtained from the fitting procedure shown in Fig. 19.10. This leads to the expectation that the response will now be stable. This expectation is fulfilled as shown by the transient labeled C-C2 in Fig. 19.11. c
0.80 c Z;N C;Cl
0.60 0.40 0.20 0.00 - 0.20 - 0.40 - 0.60 - 0.80 - 1.00 0 I I I I I, I I I
FIGURE
19-11
Comparison of transients produced by different tuning methods for Example 19.2 (process shown in Fig. 19.9). Z-N: Ziegler-Nichols method; C-Cl: Cohen-Coon method based on tangent line through point of inflection; C-C2: Cohen-Coon method using model based on least square fit.
PROCESS APPLICATIONS
PROCESS
IDENTIFICATION
Up to this point, the processes used in our control systems have been described by transfer functions that were derived by applying fundamental principles of physics and chemical engineering (e.g., Newton s law, material balance, heat transfer, fluid mechanics, reaction kinetics, etc.) to well defined processes. In practice, many of the industrial processes to be controlled are too complex to be described by the application of fundamental principles. Either the task requires too much time and effort or the fundamentals of the process are not understood. By means of experimental tests, one can identify the dynamical nature of such processes and from the results obtain a process model which is at least satisfactory for use in designing control systems. The experimental determination of the dynamic behavior of a process is called process ident@cation. The need for process models arises in many control applications, as we have seen in the use of tuning methods. Process models are also needed in developing feedforward control algorithms, self-tuning algorithms, and internal model control algorithms. Some of these advanced control strategies were discussed in the previous chapter. Process identification provides several forms that are useful in process control; some of these forms are Process reaction curve (obtained by step input) Frequency response diagram (obtained by sinusoidal input) Pulse response (obtained by pulse input) We have already encountered the need for process identification in applying the tuning methods presented earlier in this chapter. In the case of the Z-N method, the procedure obtained one point on the open-loop frequency response diagram when the ultimate gain was found. (This point corresponds to a phase angle of - 180 and a process gain of l/K,, at the cross-over frequency w,,.) In the case of the C-C method, the process identification took the form of the process reaction curve.
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