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PID equation
The Temperature Controller approximates the output M(t) using a discrete PID equation.
Let
Ts = Sample rate.
Kc = Proportional gain.
Ki = Kc * (Ts / Ti) = Coefficient of the integral term
Kr = Kc * (Td / Ts) = Coefficient of the derivative term
Ti = Reset or integral time.
Td = Derivative time or rate.
SP = SetPoint.
PV
= Process Variable (temperature) at nth sample.
n
e
= SP - PVn = Error at nth sample.
n
M
= Value to which the controller output has been initialized
O
Then
M
= Controller output at nth sample.
n
=
M
Kc e
*
n
The Temperature Controller modifies the standard equation slightly to use the derivative of the Process
Variable instead of the error as follows:
=
M
Kc e
*
n
These two forms are equivalent unless the SetPoint is changed. In the original equation, a large step
change in the SetPoint will cause a correspondingly large change in the error resulting in a bump to the
process due to derivative action. This bump is not present in the second form of the equation.
The Temperature Controller also combines the integral sum and the initial output into a single term
called the bias (Mx). This results in the following set of equations:
Mx0 = M0
Mx = Ki * eN + Mx
M
= Kc * e
- Kr (PV
n
n
The Temperature Controller will keep the normalized output M in the range 0.0 to 1.0. This is done by
clamping M to the nearer of 0.0 or 1.0 whenever the calculated output falls outside this range.
n
∑
+
Ki
e
n
=
i
1
n
∑
+
−
Ki
e
n
i
=
i
1
n-1
- PV
) + Mx
n
n-1
n
PID FUNDAMENTALS
−
−
Kr e
(
i
n
−
Kr PV
(
n
+
e
)
M
−
n
1
+
PV
)
M
−
n
1
0
0
5.3
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