A controller can be simply portrayed as follows:
The comparing element and the controller function describe the conduct of the controller.
The following describes the most important types of controller. A controllerstep response tells us a lot about its conduct. The step response describes how a controller reacts to the erratic change in the process value.
There are 3 important basic types of controller:
Proportional-action controller (P controller)
Integral-action controller (I controller)
Differential-action controller (D controller – wenot touching on this here)
These are combined for a real controller. For instance, the PI controller:
P Controller
A proportional-action controller (P controller) changes the manipulated variable M proportional to the loop error. The P controller works immediately. By itself it cannot drive the loop error to zero.
MPn = kP × en
MPn: Manipulated variable of the P controller at the time n
kP: Gain of the P controller
en: Loop error at the time n
The following picture shows a jump in process value and step response of the controller:
Summary
The P controller has the following characteristics:
It cannot correct faults with the control process > lasting loop error.
It reacts immediately to a change in the process value.
It is stable.
I Controller
An integral-action controller (I controller) changes the manipulated variable M proportional to the loop error and to the time. The I controller works by delayed action. It completely remedies a loop error.
In order to calculate the value of the manipulated variable at a period of time n, the time up until this period of time must be divided into small time slices. The loop errors at the end of each time slice must be added up (integrated) and they are then entered in the calculation.
MIn = kI × (TS / TI) × (en + en-1 + en-2 + en-3 + … + e0) = kI × (TS / TI) × en + MIn-1
MIn: Manipulated variable of the I controller at the time n
MIn-1: Manipulated variable of the I controller at the time n-1; also called integral sum
kI: Gain of the I controller
TS: Sampling time, duration of a time slice
TI: Integral time: by means of this time, the influence of the integral part is controlled on the manipulated variable, also known as integral-action time
en: Loop error at the time n
en-1: Loop error at the time n-1; etc.
e0: Loop error at the beginning of the calculations
The following picture shows a jump in process value and step response of the controller:
Summary
The I controller has the following characteristics:
It sets the process value exactly to the command variable.
By so doing, it tends to oscillate and is unstable.
It requires more time to carry out the control action than the P controller .
PI controller
A PI controller reduces the loop error immediately and will eventually drive the loop error to zero.
Mn = MPn + MIn = kP × en + kI × (TS / TI) × en + MIn-1
Mn: Manipulated variable at the time n
MPn: Proportional part of the manipulated variable
MIn: Integral part of the manipulated variable
MIn-1: Manipulated variable of the I controller at the time n-1; also called integral sum
kP: Gain of the P controller
kI: Gain of the I controller
TS: Sampling time, duration of a time slice
TI: Integral time; by means of this time the influence of the integral part is controlled on the manipulated variable, also known as the integral-action time
en: Loop error at the time n
The following picture shows a jump in process value and step response of the controller:
Summary
The PI controller has the following characteristics:
The P controller components quickly intercept an occurring loop error.
The I controller components can then remedy the remaining loop error.
The controller components supplement each other so that the PI controller works quickly and precisely.