Many industrial processes show step responses with pure aperiodic behaviour according to Figure 8.6. This S-shape curve is characteristic of many high-order systems and such plant transfer functions may be approximated by the mathematical model
element.
 |
Here the step response is characterised by constructing the
tangent at the turning point 
with the
following three values:
(
gain of the plant),
(rise time) and
(delay time). Then a rough approximation
according to Eq. (8.15) is to set
and
.
For a plant of the type described above a lot of tuning rules for standard controllers have been developed. These have been mostly developed empirically from simulation studies. The most famous empirical tuning rules are those of Ziegler and Nichols. These tuning rules have been derived to provide step responses for the closed loop, where the response shows a decrease of the amplitude of approx. 25% per period. For the application of these rules according to Ziegler and Nichols two different approaches can be used:
of the P controller is continuously increased until the
closed loop shows permanent oscillations. The value of the gain
at this state is denoted as the critical
controller gain
.
(critical period) of the
oscillations is measured.
and
one determines the
controller tuning values
,
and
using the formulas given in Table 8.1.
and
. Measuring the
step response of
the plant does not generally cause difficulties. Therefore, in
many cases the second form of the Ziegler-Nichols approach is more
expedient. The rules are based directly on the slope
of the tangent at the turning point and on the
delay
of the step response. One has to observe that the
measurement of the step response needs only to be taken at the turning
point T, as the slope of the tangent already describes the ratio
. Using the measured data
and
as well as the formula given in
Table 8.1 the controller tuning parameters can be
determined by simple calculations.
