Page 178 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

An Underda01ped Process 153

Introduce a new independent variable t' = t I m, and a new
dependent variable y' = gy. Applying these substitutions gives

The scaled process has a natural frequency and a gain of unity.
Note that if the natural frequency is 1.0 rad/ sec then the natural fre-
quency will also be

1.0 rad/sec
f- = 0.159 cycles/sec
- 2n rad/ cycle
The equation yielding the eigenvalues becomes

and the eigenvalues or poles become

2
A,,~ =-{±~{ -1 =a±jb

The next batch of computations will deal with the scaled process.

6-3 PI Control of the MassfSpringfDashpot Process
You have been exposed to attempts to control the first-order process
(the single water tank) and the third-order process (the three-tank
process). The approach has been to feed the process output back and
subtract it from the set point and generate an error signal. Then an
adjustment to the process input was developed based on signals pro-
portional to the error and proportional to the integral of the error. The
time domain has been used to demonstrate the effectiveness (or lack
thereof) of these methods. The frequency domain has been used to
get an estimate for one of the control gains by making sure that the
open-loop combination of the process and controller, represented by
GPG" had sufficient gain margin when the phase was -180° (or suffi-
cient phase margin when the gain was unity).
For the mass I spring/ dashpot process with proportional-integral
control GpGc looks like

G G = 1 ks+l
2
P c s + 2{ s + 1 s