What are the differences between ideal, series, and parallel PID forms?

PID controller forms differ in how P, I, D interact: Ideal (ISA) form - Kp multiplies all terms: Kp*(e + 1/Ti*integral(e) + Td*de/dt). Series (interacting) form - terms multiply: Kp*(1 + 1/Ti*s)*(1 + Td*s). Parallel (independent) form - each term independent: Kp*e + Ki*integral(e) + Kd*de/dt. Conversions between forms required when moving tuning parameters between different controller types. Series form was common in pneumatic controllers; ideal form is standard in DCS/PLC.