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Feedfotward -controller
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FIGURE 18-12 Control system with feedback and feedforward controllers.
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Rather than use the Gf(s) of Eq. (18.5) in the feedforward controller, one can try using only the constant term of Gf(s), that is, Gf(s) = -1 The response for Gf = - 1 gives Curve III in Fig. 18.11; this response has a very large undershoot before the feedback controller returns C to the set point. If we try using Gf(s) = -0.5, we obtain Curve IV of Fig. 18.11; the undershoot is less in this case, but the response is still unsatisfactory. As shown by Curves III and IV, omitting the dynamic part of Go can give very poor results. The success of using a feedforward controller depends on accurate knowledge of the process model, a luxury that may not be available in many applications.
Implementing Feedforward lkansfer
Functions
In applications of feedforward control, Gf(s) may take the form of a lead expression, such as Gf(s) = 1 + rfs. When this occurs, it is necessary to approximate 1 + rfs by a lead-lag expression, such as Gf(s) = (1 + rfs)/(l + &s) where p << 1. To see how Gf(s) takes the form of a lead exptission, consider the load disturbance, ci, of Fig. 18.9 to enter tank 2. Since no change in concentration occurs in the stream entering the preconditioning tank, we may eliminate it from the diagram for the case under consideration to obtain the diagram in Fig. 18.13. Adding feedforward control and feedback control to the system in Fig. 18.13 gives the block diagram of Fig. 18.14. The diagram shown in Fig. 18.14 is the same as that in Fig. 18.12 with the exception that the disturbance Ci enters tank 2 instead of the preconditioning tank. As shown previously, the response of C to a change in Ci and R can be written directly from Fig. 18.14 as follows: (18.6) C(S) = Gl(s)Ci(s) + Gf(S)Gp(s)Ci(s) + Gc(s)Gp(s)E(s) where E(s) = R(s) - C(s)
tank
tank 2
tank 3
FIGURE 18-13 Composition control with disturbance to second tank.
ADVANCED CO NTRO L STRATEGIES
FIGURE
18-14
Feedforward-feedback control for system in Fig. 18.13.
In order for C not to change from the set point R, which is 0, we solve Eq.( 18.6) for Gf(s) with C = 0 and R = 0 to obtain:
Gf(s) = G(s) G&)
(18.7)
Introducing the expressions for Gt(s) and Gp(s) from Fig. 18.14 into Eq. (18.7) gives Gf(s) = -(s + 1) (18.8) It is not practical to implement -(s + 1). To see this, consider the response of -(s + 1) to a step change as shown in Fig. 18.15. There is no hardware that will produce an impulse as shown in Fig. 18.15; however, one can approximate -(s + 1) by means of a lead-lag transfer function of the form.
y(S) -=- TfS + 1 prfs + 1 X(s)
(18.9)
If we iet p = 0.1 and rf = 1 for the control system under consideration, we obtain as an approximation to Eq. ( 18 3)
$0) = -,;, I I
(18.10)
FIGURE 18-15
Step response for -(s + 1).
PROCESS
APPLICATIONS
-1 Y -Y=(lO-1)C"O.l +l -10
--_-_-_.
FIGURE 18-16 Step response for -(s + l)/ (0.1s + 1).
The response of this transfer function to a step input is shown in Fig. 18.16. The effect of this transfer function, -(s + l)/(O.ls + l), on the output of the feedforward controller for a step change in load is to give a sudden drop in flow followed by a fast exponential increase in the flow to a steady-state flow of - 1. Note that for the parameters chosen for the transfer functions in Fig. 18.14, a unit increase in Ci must eventually be compensated by a unit decrease in the signal from the feedforward controller if there is to be no change in the process output. The sudden, initial drop in flow may be too abrupt for the control hardware, in which case the output would saturate. In practice, /3 can be increased (perhaps to 0.5) in order to reduce the magnitude of the initial drop. The effect of using Gf(s) = -(s + l)/(O. 1s + 1) with feedback control is shown in Fig. 18.17. The responses shown, which were obtained by simulation, are for a unit-step change in Ci. Curve I is for the case of feedback control only with K, = 2.84 and rI = 5 .O. Curve II is for feedforward-feedback control using Eq. (18.10) for Gf(s) and K, = 2.84 and 71 = 5.0. One can see that the overshoot for the feedforward-feedback response has been reduced significantly.
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