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General note on optimization

Controller optimization (or tuning) is the adjustment of the controller to a given process or control loop. The control parameters have to be selected such that the most favorable response of the control loop is achieved under the given operating conditions. However, this optimum response can be defined in different ways, such as reaching the setpoint quickly, with a small overshoot, or a somewhat longer stabilization time with no overshoot. If all that is expected of the controller is a response such as for a limit contact (without pulsed operation), there is no need to find the optimum settings for proportional band, derivative time or reset time. Only the switching differential has to be predefined.
In most cases, the controller can itself determine the control parameters through the self-optimization (autotuning) facility, if the process permits self-optimization. Alternatively, the optimum parameter setting can be determined "manually", through experiments and empirical equations (see formulae in the appendix).
When controllers are swapped, or with identical control installations, control parameters can also be directly accepted or entered.
After a manual parameter setting, autotuning may no longer be started, since this would overwrite the settings.

Formulae for setting according to the oscillation method:

Controller action
P
  XP = XPk / 0,5
PI
  XP = XPk / 0,45
TP = 0,85 ·TK
PID XP = XPk / 0,6
Tn = 0,5 · TK
Tv = 0,12 · TK

Formulae for setting according to the step response:

Controller action Control loop Error
P XP = 3,3 · KS · (Tu/Tg) · 100 % XP = 3,3 · KS · (Tu/Tg) · 100 %
PI XP = 2,86 · KS · (Tu/Tg) · 100 %
Tn = 1,2 · Tg
XP = 1,66 · KS · (Tu/Tg) · 100 %
Tn = 4 · Tu
PID XP = 1,66 · KS · (Tu/Tg) · 100 %
Tn = 1 · Tg
Tv = 0,5 · Tu
XP = 1,05 · KS · (Tu/Tg) · 100 %
Tn = 2,4 · Tu
Tv = 0,42 · Tu