2. Engineering background¶
2.1. PID controller¶
PID controller continuously calculates an error value e(t) as the difference between a desired setpoint (SP) and a measured process variable (PV) and applies a correction based on proportional, integral, and derivative terms (denoted P, I, and D respectively), hence the name.
2.1.1. Standard form¶
U(t) = MV(t) = K_{p}{e(t)}+ K_{i}\int _{0}^{t}{e(\tau )}{d\tau }+ K_{d}{\frac {d}{dt}}e(t)
where:
K_{p} the proportional gain, a tuning parameter,
K_{i} is the integral gain, a tuning parameter,
K_{d} is the derivative gain, a tuning parameter,
e(t)=SP-PV(t) is the error (SP is the setpoint, and PV(t) is the process variable),
t is the time or instantaneous time (the present), τ is the variable of integration (takes on values from time 0 to the present t.
2.1.2. Ideal form¶
K_{p} gain is applied to the I_{out}, and D_{out} terms, yielding:
MV(t) = K_{p}\left( e(t)+ \frac{1}{T_{i}}\int _{0}^{t}{e(\tau )}\,{d\tau }+ T_{d}{\frac {d}{dt}}e(t) \right)
where:
T_{i} is the integral time and T_{d} is the derivative time
the gain parameters are related to the parameters of the standard form through K_{i}={\frac {K_{p}}{T_{i}}} and K_{d}=K_{p}T_{d}
2.1.3. Algorithm¶
previous_error = 0
integral = 0
loop:
error = setpoint - measured_value
integral = integral + error * dt
derivative = (error - previous_error) / dt
output = Kp * error + Ki * integral + Kd * derivative
previous_error = error
wait(dt)
goto loop
