Page 141 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

116 Chapter Fonr

Now, the problem is to find n such that the right-hand side of
Eq. (4-24) equals L. This can be done quickly with an iterative method
of the engineer's choosing.
This situation has actually occurred in my experience and this
algorithm does in fact work. I suggest that this is a good example of
using simple calculus to solve a problem that is often overlooked.

4-7 Partial Summary and a Slight Modification
of the Rule of Thumb
Our approach has been to find a proportional control gain, called the
critical gain, that makes the open-loop amplitude ratio unity or the
open-loop phase lag 180°. In the former case we reduce the critical
proportional control gain to make the phase lag less than 180° by
about 45°. In the latter case we reduce the critical proportional control
gain by a factor of 0.5.
Therefore, as a starting point we are trying the find the critical
values of m and k, namely m, and k,, such that the open-loop gain has
a magnitude of unity and a phase of -180°. The App. B shows that the
complex number -1.0 has a magnitude of unity and a phase of -180°,
so we are really trying to find values of m, and k, that satisfy the fol-
lowing equation:

or

(4-25)

or
IG(j mc>l = 1

9(jm,) = -n

Since several of the closed-loop transfer functions have a denom-
inator of 1 + G,G,, it follows that finding the poles of these transfer
functions is equivalent to solving Eq. (4-25).
If the proportional gain is set equal to k, the performance should
be on a cusp between instability and stability. That is, the process
with the controller should experience sustained oscillations.
The critical values for proportional-only control of the FOWDT
process would be the solution of the following two equations that
come directly from Eq. ( 4-25):