Page 143 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

118 Chapter Four

Answer The equations used to solve for kc and me for the case of proportional-
only control of a pure dead-time process are

0
8
G,Gc(jwc) = e-J"', k = 1Gie' = -1
IGI=kc = 1

By inspection kc = 1 and CIJc = tr I D or fc = CIJc I (2tr) = (tr I D) I (2tr) = 1 I (2D).
Since D = 8, it follows that fc = 0.125.
For integral-only control the equations are

Therefore, I, = m, and CtJc = tr I (2D) or fc = 1 I (4D) = .0625.
The reader might conclude that the Bode plot is effectively a
graphical solution of Eq. (4-25). If the reader is interested in this
approach, conventional textbooks on control usually contain many
methods for quickly constructing Bode plots by hand. With the
incredible access to computers and software like Matlab, these graph-
ical techniques have become less attractive to some (especially mor)
and will not be covered here.

4-8 Summary
The frequency domain was introduced by means of the substitution
s ~ jm into the Laplace transform. A stability requirement for sinu-
soidal forcing was developed in terms of the amplitude ratio or mag-
nitude and the phase lag of the open-loop transfer function G,GP.
The phase of G,GP should not equal-180° when the magnitude of
G,G is unity.
When the amplitude ratio is unity the phase margin should be on
the order of 30° to 45°. That is, the phase lag should be less than 150°
when the magnitude is unity. Alternatively, when the phase is -180°
the gain margin should be on the order of 0.5.
The Bode plot of the open-loop amplitude ratio and phase ver-
sus frequency provided a graphical means of checking the stability
of the candidate process and controller. Bode plots were constructed
for the first-order process presented in Chap. 3. An auxiliary curve
of the magnitude of the transfer function, E/N = -1/(1 + G,Gl'),
called the error transmission curve, provided insight into the abihty