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1. After the process reaches steady state at the normal level of operation, switch
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the controller to manual. In a modem controller, the controller output will remain at the same value after switching as it had before switching. (This is called bumpless transfer.) 2. With the controller in manual, introduce a small step change in the controller output that goes to the valve and record the transient, which is the process reaction curve (Fig. 19.4). 3. Draw a straight line tangent to the curve at the point of inflection, as shown in Fig. 19.4. The intersection of the tangent line with the time axis is the apparent transport lag (Td); the apparent first-order time constant (7) is obtained from
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T = B,IS
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where B, is the ultimate value of B at large t and S is the slope of the tangent line. The steady-state gain that relates B to M in Fig. 19.3 is given by
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K, = B,IM MIS + R=O +!p G,. l+J+i$-t
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FIGURE 19-3 Block diagram of a control loop for measurement of the process reaction curve.
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CONTROLLER
TUNING
PROCESS
IDENTIFICATION
FIGURE 19-4 Ifrpical process reaction curve showing graphical construction to determine first-order with transport lag model.
input
4. Using the values of K,, T, and Td from step 3, the controller settings are found from the relations given in Table 19.2. Notice in Table 19.2 that all of the controller settings are a function of the dimensionless group Td/T, the ratio of the apparent transport lag to the apparent time constant. Also K, is inversely proportional to K,.
TABLE 19.2
Cohen-Coon controller settings
Qpe of control Proportional (P) Proportional-integral (PI) Parameter setting
71 = Td
Proportional-derivative (PD)
30+ 3TdfT 9+2OTd/T
70 = Td
Proportional-integral-derivative (PID)
6 -2TJT 22+ 3Td/T
71 = Td TD = T,
32+6Td/T 13 +&'-d/T 4 11 + Z Td/T
PROCESS
APPLICATIONS
The rationale for the C-C tuning method begins with the representation of the S-shaped process reaction curve by a first-order with transport lag model; thus
(19.6)
Using the system expressed by Eq. (19.6)) Cohen and Coon obtained by theoretical means the controller settings given in Table 19.2. Their computations required that the response have $ decay ratio, minimum offset, minimum area under the loadresponse curve, and other favorable properties. In applying the C-C tuning method, an important task is the graphical construction, shown in Fig. 19.4, which reduces the process reaction curve to the first-order with the transport lag model given by Eq. (19.6). To understand the basis for the graphical procedure, consider the response of the transfer function of Eq. (19.6) to a step change in input; the resulting transient is shown in Fig. 19.5. After t = Td, the response is a first-order response. The point of inflection of the curve in Fig. 19.5 occurs at t = Td and the slope of the tangent line at this point is related to the time constant by the relation:
S = B,IT
Solving for T gives the expression in Eq. (19.4). The response after t = Td, shown in Fig. 19.5, was also presented in Fig. 5.6. The attempt to model the process reaction curve by the method shown in Fig. 19.4 is crude and does not give a very good fit. Finding the point of inflection and drawing a tangent line at this point is quite difficult, especially if the data for the process reaction curve are not accurate and if they scatter. A better method for fitting the process reaction curve to a first-order with transport lag model is to perform a least-square fit of the data. The disadvantage to this fitting procedure is the time and effort required. An example to be presented later will study the effect of the type of model fitting procedure on the selection of controller parameters.
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Tangent line, slope = !$
FIGURE 19-5 Step response for a first-order with transport lag model.
CONTROLLER TUNING AND PROCESS IDENTIFICATION
More recently, Lopez et al. (1967) studied the tuning of controllers with error-integral criteria for the first-order with transport lag model of Eq. (19.6). The error-integral criteria that they considered were ISE, IAE, and ITAE. In their work, a search procedure was used to find the controller parameters that minimized each particular figure of merit. Their results, developed for Td/T varying from 0 to 1.0, were presented in graphical form and as empirical equations that were fitted to their graphical results. Their results, which can be considered as a variation of the C-C tuning method, were not compared with the C-C method. The interested reader may wish to compare the method of Cohen and Coon and the method of Lopez et al. as a project. To illustrate the two methods of controller tuning just presented, the system shown in Fig. 19.1 was simulated by use of a computer program called TUTSIM. (This simulation software is described in Chap. 35.) Table 19.3 gives the values of the controller parameters obtained by applying each tuning method; Figure 19.6 shows the resulting transients. Since the Z-N method does not give a rule for a PD controller, the settings listed for a PD controller udder the Z-N heading of Table 19.3 were obtained by using a theoretical frequency response calculation in which the design was based on 30 phase margin and a maximum K,. No general conclusions can be made about the relative merits of the two tuning methods from the results shown in Fig. 19.6, since these results apply to one specific example. About all that can be said is that for this specific example, both methods give reasonable first guesses of the control parameters.
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