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FIGURE 18-9 Composition control system: (a) physical process; (b) block diagram.
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FIGURE 18-10 Responses to a step change in set point for PI control. Curve 1: Ziegler-Nichols settings: K, = 3.65, = 3.0. Curve II: Settings for improved te&nse:K: = 284t q = 50 . . .
a signal that opens or closes the control valve, which in turn supplies concentrated reagent to the first tank. The block diagram corresponding to the control system of Fig. 18.9~ is shown in Fig. 18.9b.* To obtain some specific control system responses, numerical values of the time constants of the tanks have been chosen as shown in Fig. 18.9b. To study the response of this control system, the block diagram shown in Fig. 18.9b was simulated on a computer. The values of Kc and ~1 were chosen by trial and error to give the response to a step change in set point shown in Curve II of Fig. 18.10; this response, which has a decay ratio of about i, was obtained with K, = 2.84 and ~1 = 5.0. The Ziegler-Nichols settings (K, = 3.65 and 71 = 3.0) give a set-point response shown as Curve I of Fig. 18.10, which is too oscillatory. Having obtained satisfactory settings for the controller (K, = 2.84, 71 = 5.0), the response of the system to a step change in Ci of 10 units was obtained and is shown as Curve I in Fig. 18.11. Note that the response is oscillatory and has a long tail. This response illustrates the fact that the feedback control system does not begin to respond until the load disturbance has worked its way through the forward loop and reaches the measuring element, with the result that the composition can move far from the set point during the transient.
*In Figure 18.9u, concentration is denoted by c (lower-case letter). In the block diagram of the process in Fig. 18.9b, the symbol for concentration is denoted by C (capital letter) to denote a deviation variable. This use of symbols follows the procedure established in Chap. 5.
FIGURE 18-11 Responses to a step change in load for feedforward-feedback control. Curve I: PI control with K, = 2.84, 71 = 5.0 Curve II: FF control with K, = 2.84, ~1 = 5.0, Gf = -1/(5s + 1) Curve III: FF control with K, = 2.84, q = 5.0, Gf = -1 Curve IV FF control with K, = 2.84, q =
5.0, Gf = -0.5
ADVANCEDCONTROLSTRA"GIES
If the change in load disturbance (Ci) can be detected as soon as it occurs in the inlet stream, this information can be fed forward to a second controller that adjusts the control valve in such a way as to prevent any change in the outlet composition from the set point. A controller that uses information fed forward from the source of the load disturbance is called a feedforward controller. The block diagram that includes the feedforward controller (Gf) as well as the feedback controller (G,) is shown in Fig. 18.12.
Analysis of Feedforward
Control
The response of C to changes in Ci and R can be written from Fig. 18.12 as follows: (18.3) C(S) = Gl(s)Gp(s)Ci(s) + Gf(s)Gp(s)Ci(s) + Gc(s)Gp(sW(s) where E(s) = R(s) - C(s) In order to determine the transfer function of Gf(s) that will prevent any change in the control variable C from its set point R, which is 0, we solve Eq. (18.3) for Gf(s) with C = 0, R = 0. The result is (18.4) Gjb) = -G(s) For the example under consideration in Fig. 18.12. (18.5) Gj(s) = -1/(5s + 1) This transfer function can be implemented easily with control hardware now available. If the load response of the control system in Fig. 18.12, with G&) given by Eq. (18.5), were obtained for a step change in C i , there would be no deviation of C from the set point (i.e., perfect control). This response is shown as Curve II in Fig. 18.11, which, of course is a horizontal line at C = 0.
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