## Is

en O en shown in Figure 21.5. Theoretically any number of feedforward algorithms Df(z) can be added. DF(z) is derived in the following manner.

A control-loop equation for C(z) is written from Figure 21.5.

_ L(z)Df(Z)GpH(Z) + GlL{Z) C{Z) ~ 1 + Dl(Z)GpH(Z)--<2L12>

C(z) is set equal to zero, which says that there will be zero error at each sampling point. DF(z) is solved from equation (21.12).

Df(z) can be determined from equation (21.13) once the nature of the load and the process model are known. Let us calculate the feedforward algorithm for a feed-composition disturbance from the information in Table 21.1 for a step-load change.

_ 49.3(1 + 0.65 ,-')(! - 0.368O FK ' (1 — 0.48 z )

The time output DF(t) from equation (21.14) is the required feedforward change in the manipulative variable after a change in the feed composition is detected.

DF{t) = 49.3 [Aj:y(i) + 0.28 Axf(t - T) - 0.24 Axf(t - 2T)]

If the load form is not known exactly, L(s) may be approximated by a staircase function L*(s)H(s).31 The asterisk denotes that L(s) is a sampled variable. This simply says that the load is fictitiously sampled and the value is held constant for the sampling period. The numerator of equation (21.13) becomes:

Substitution of equation (21.16) into equation (21.13) gives a different form for DF(z).

Equation (21.17) can be solved for the feedforward algorithm without knowing the actual form of the load. If L(s) is a step function, DF(z), from equation (21.17), will be equal to equation (21.13).

The nonlinear simulation was used to illustrate the closed-loop response of the controlled variable x2 following a 30 percent increase in feed composition. The results are shown in Figure 21.4b with the feedback-only dual and PID algorithms. Control is immensely improved with the feedforward action. The slight deviation in x2 with feedforward control is due to inaccuracies in the linear model and the long sampling time relative to the process dead time. The 