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Comparator 2
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I B B t ' I 1 I.------------------------I . - - - - - - - - - - - - - - - - - - - - - - - - (4 (4
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FIGURE 18-30 Internal model control structures: (a) basic structure, (b) alternate structure, (c) structure equivalent to conventional control.
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loop control structure, we can identify the single controller block as G,. After one designs the IMC controller (GI) by the method to be described, one can determine the equivalent conventional controller G, by the relation (18.14) G, = G1/(l - GIG,) For the structure shown in Fig. 18.30a, one can show that
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c = u1+ GGr
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1 + G,(G - G,)
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(18.15)
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If the model exactly matches the process (i.e., G, = G), the only signal entering comparator 1 in Fig. 18.30~ is Ut . ( The signals from G and G, are equal and cancel each other in going through comparator 2.) Since Ut is not the result of any processing by the transfer functions in the forward loop, Ut is not a feedback signal but an independent signal that is equivalent to R in its effect on the output C. In fact, there is no feedback when G = G, and we have an open-loop system as shown in Fig. 18.3 1, In this case the stability of the control system depends only on GI and G,. If GI and G, are stable, the control system is stable. Ideally, we should like to have C track R without lag when only a set-point change occurs (i.e., Ut = 0). In order for this to occur, we see from Fig. 18.3 1 or Eq. (18.15) that GIG = 1 or since G = Gmr we may write GIG, = 1. Solving for GI gives GI = l/G, (18.16) Equation (18.16) simply states that the IMC controller should be the inverse of the transfer function of the process model. Keep in mind that Eq. (18.16) is based on the assumption that the model exactly matches the process. For the case of only a change in load lJ1 (i.e., R = O), we should like to have the output C remain unchanged (i.e., C = 0). In order for this to occur, we see again from Fig. 18.31 or Eq. (18.15) that GIG, = 1; this leads to the same result as given by Eq. (18.16). Even if there is no mismatch between the model and the process, the application of Eq. (18.16) will usually lead to a transfer function that cannot be implemented because it will be unstable, requires prediction, or requires pure differentiation. For example, if G, = l/(rs -t l), the application of Eq. (18.16) gives GI = TS + 1 This result is equivalent to an ideal PD controller, which cannot be implemented because of the derivative term. If G, = e - /(rts + l), we obtain Gr = (~1s + l)e7 The term ers, which represents prediction, cannot be implemented. If G, = (1 - s)/[(l + s)(rs + l)] GI = [(l + s)(rs + l)]/( 1 - s)
FIGURE 18-31 IMC structure when model matches Drocess (G, = G).
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The term 1 - s in the denominator means that a pole is in the right half plane, which leads to an unstable controller. With such difficulties of implementation of the internal model controller, one might ask if any practical result can be obtained. These difficulties can be overcome by application of the following simplified procedure.
Design of IMC Controllers
In using these rules, only a step change in disturbance is considered. The procedure for disturbances other than a step response is more complicated and beyond the scope of the limited discussion presented here.
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