Siemens SIMATIC S7 System Manual page 314

Program instructions
7.9 PID
Since the digital computer must calculate the output value each time the error is sampled
beginning with the first sample, it is only necessary to store the previous value of the error
and the previous value of the integral term. As a result of the repetitive nature of the digital
computer solution, a simplification in the equation that must be solved at any sample time
can be made. The simplified equation is:
Output = Proportional term + Integral term + Differential term
M = K
n
where:
M
n
K
c
e
n
e
n-1
K
I
MX
K
D
The CPU uses a modified form of the above simplified equation when calculating the loop
output value. This modified equation is:
Output = Proportional term + Integral term + Differential term
M = MP
n
where:
M
n
MP
n
MI
n
MD
n
314
* e + K * e + MX + K
C n I n
Calculated value of the loop output at sample time n
Loop gain
Value of the loop error at sample time n
Previous value of the loop error (at sample time n - 1)
Proportional constant of the integral term
Previous value of the integral term (at sample time n - 1)
Proportional constant of the differential term
+ MI + MD
n n n
Calculated value of the loop output at sample time n
Value of the proportional term of the loop output at sample time n
Value of the integral term of the loop output at sample time n
Value of the differential term of the loop output at sample time n
* (e - e )
D n n-1
System Manual, V2.3, 07/2017, A5E03822230-AF
S7-200 SMART