Control theory: control loop PID tuning with step response
For a stable control loop the control parameters have to be set that no oscillations will occur.
There are several methods to estimate the parameters for a stable control loop.
In this application I use the step response method.
Control loop
First you have to define the control loop. This is done by selecting the control element types in the table below.
The control elements time and gain parameters can be changed in the columns next to the control element type.
The Start button is used to draw the control loop step response.
Control path parameters
Number
Control element
Parameter K
Parameter T
Parameter D
Transfer function
1
2
3
4
Control path step response
The step response is used to estimate the PID controller parameters.
For this purpose the inflection tangent of the step response is calculated and drawn.
The intersection points with the diagram axes are used to calculate the parameters
Ks, Tu and Tg. The parameter Ks is calculated from
the step response amplitude: $$K_s = \frac{\Delta y}{\Delta u} $$
The control path parameters Ks, Tu and Tg are used to calculate the PID controller
parameters Kp, Ki und Kd.
The most common calculation rules are these from Ziegler/Nichols and Chien/Hrones/Reswick.
Calculation rules according to Ziegler/Nichols
Controller
Kp
Ki
Kd
P
$$ \frac{T_g}{K_s \cdot T_u} $$
PI
$$ 0.9 \cdot \frac{T_g}{K_s \cdot T_u} $$
$$ 0.273 \cdot \frac{T_g}{K_s \cdot {T_u}^2} $$
PID
$$ 1.2 \cdot \frac{T_g}{K_s \cdot T_u} $$
$$ 0.6 \cdot \frac{T_g}{K_s \cdot {T_u}^2} $$
$$ 0.6 \cdot \frac{T_g}{K_s} $$
Calculation rules according to Chien/Hrones/Reswick
