Page 132 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

A New Do11ain and More Process Models 107

4-5 A First-Order with Dead-Time (FOWDT) Process
Consider Fig. 4-33 where the tank of liquid has been placed upstream of
the pure dead-time process. The placement of the buckets-on-the-belt
ahead of the tank suggests a dead time in series with a first-order pro-
cess. Please do not be confused by the length of the pipe at the outlet of
the tank. Let's assume that it is actually relatively short and that the pipe
diameter is small so that the transit time of the liquid spent in the pipe is
negligible compared to the time spent in the buckets on the belt
Figure 4-34 shows the open-loop step-change response of the pro-
cess for the case of g = 2.5, -r= 10, D = 8. This is the first example pro-
cess in this chapter that has had a nonunity gain.
In the continuous time domain, this model would be described
by an extension of the first-order model:

-r~~ +y=gU(t-D) (4-16)

In the Laplace domain, the open-loop transfer function is
G (s) = e-sD_g_ (4-17)
P t'S+ 1
After applying s = jro, the magnitude and phase can be found as
follows:
G
(jro) = e-it»D -. _g_ = IG lei'
p ]t'OJ+ 1 p

I GI- g (4-18)
P- J<-rrof + 1
1
8= -tan- (-rro)- roD

Valve position (U)
J;
I

'

F1auRE 4-33 First-order process with dead time.