Page 127 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

102 Cha11ter Fonr

4-4·1 Proportional-Only Control of a Pure Dead-nme
Process
Based on Chap. 3, we can guess that there will be an offset between the
set point and the process output. The open-loop transfer function is

where k is the proportional control gain that we wish to estimate.
When s is replaced by jro, the open-loop transfer function becomes

GpGc(jro) = e-it»Dk = 1Giei 6
IGI=k (4-14)
6=-roD

The Bode plot for k = 1 and D = 8 is the same as that in Fig. 4-24.
Find the point where the phase lag reaches 180°. The amplitude
remains unity at all frequencies so a control gain of unity will cause
problems. When the control gain k is decreased to 0.7 the whole ampli-
tude line is shifted down to a value of 0.7 and the phase margin
requirement is satisfied. Using this control gain, a simulation in the
time domain shows the response to a step in the set point (Fig. 4-26).

1.5
1 ... ""· ............. .
.
.
.
:::s 0.5
•
0
•
0
0
0 ·.· .. · ..... : .... · ... ·.· ... ·.· ..
--0.5
0
1.5
1
V)
1 0.5
>-
0
--0.5 0
10 20 30 40 50 60 70 80 90 100
Time
F1auRE 4-28 Proportionak>nly control of pure dead-time process, k = 0. 7.