Page 422 - Mechanical Engineers' Handbook (Volume 2)

7 Further Criteria for Gain Selection 413

Figure 33 Process signature for a unit step input. 1

The ultimate-cycle method uses experiments with the controller in place. All control
modes except proportional are turned off, and the process is started with the proportional
gain K set at a low value. The gain is slowly increased until the process begins to exhibit
P
sustained oscillations. Denote the period of this oscillation by P and the corresponding
u
ultimate gain by K . The Ziegler–Nichols recommendations are given in Table 2 in terms
Pu
of these parameters. The proportional gain is lower for PI control than for proportional
control and is higher for PID control because integral action increases the order of the system
and thus tends to destabilize it; thus, a lower gain is needed. On the other hand, derivative
action tends to stabilize the system; hence, the proportional gain can be increased without
degrading the stability characteristics. Because the rules were developed for a typical case
out of many types of processes, final tuning of the gains in the field is usually necessary.

7.4 Nonlinearities and Controller Performance
All physical systems have nonlinear characteristics of some sort, although they can often be
modeled as linear systems provided the deviations from the linearization reference condition
are not too great. Under certain conditions, however, the nonlinearities have significant effects

Table 2 The Ziegler–Nichols Rules
Controller transfer function G(s) K 1 Ts

p 1 D
Ts
I
Control Mode Process Reaction Method Ultimate-Cycle Method
1
P control K K 0.5K pu
p p
RL
0.9
PI control K K 0.45K pu
p
p
RL
T 3.3L
I T 0.83P u
I
1.2
PID control K K 0.6K pu
p
RL p
T 2L T 0.5P u
I
I
T 0.5L T 0.125P u
D
D