Ideal, parallel and series PID forms

In short. The same setting can be written in three ways, with different numbers. Before entering parameters into a PLC block you need to know which form it computes — otherwise Kc, Ti and Td do not mean what you calculated. The parameter converter does the conversion.

Equations

Error e = SP − PV, time in seconds.

Ideal (ISA, standard, "dependent"):
    u = Kc · [ e + (1/Ti)·∫e dt + Td·de/dt ]

Parallel (independent):
    u = Kp·e + Ki·∫e dt + Kd·de/dt

Series (interacting):
    U(s) = Kc′ · (1 + 1/(Ti′·s)) · (1 + Td′·s) · E(s)

NI describes exactly these three classes and stresses that there is no industry-wide standard. MathWorks calls the two forms of its block "ideal" and "parallel"; in the MathWorks ideal form the parameter I is 1/Ti, not Ti.

Conversion

Ideal ↔ parallel (always possible when Ti > 0):

Kp = Kc        Ki = Kc / Ti        Kd = Kc · Td
Kc = Kp        Ti = Kp / Ki        Td = Kd / Kp

Series → ideal (always possible):

Kc = Kc′ · (Ti′ + Td′) / Ti′        Ti = Ti′ + Td′        Td = Ti′·Td′ / (Ti′ + Td′)

Ideal → series — only when Ti ≥ 4·Td:

r = √(1 − 4·Td/Ti)
Kc′ = Kc·(1 + r)/2        Ti′ = Ti·(1 + r)/2        Td′ = Ti·(1 − r)/2

With Ti < 4·Td the ideal controller has complex zeros, which a series controller cannot reproduce. This is a structural limit, not a numerical one.

Example

Ideal form: Kc = 2.5, Ti = 40 s, Td = 4 s.

Open the example in the converter

Check back: Kc = 2.218·(35.49 + 4.508)/35.49 = 2.5; Ti = 35.49 + 4.508 = 40.0 s.

When conversion is impossible

Typical mistakes

• Entering the ideal Kc as the parallel Kp without converting Ki, Kd. With Kc = 2.5 and Ti = 40 s the parallel Ki is 0.0625 1/s; entering "Ki = 40" makes the integral action 640 times stronger.
• Confusing Ti and 1/Ti (reset rate) or seconds and minutes — see Kp, Ki, Kd, Ti, Td and PB.
• Missing that the block filters D and sets it through N. The meaning of N depends on the form: in the ideal form Tf = Td/N, in the MathWorks parallel form Tf = 1/N — see D filter.

Limits

The formulas describe a continuous controller without filter, limits or set-point weighting. Real blocks add sampling, a D filter, P and D on measurement; this changes the behaviour but not the relations between Kc, Ti, Td and Kp, Ki, Kd. Check the correspondence to a particular block (Siemens, CODESYS, OWEN, …) against the documentation of its version.