Controller design – ideal proportional plus reset (PI) controller For an ideal proportional-plus-reset (integral) controller with reset time T, what is the transfer function Gc(s)?

Introduction / Context:
PI (proportional–integral) controllers are ubiquitous in process control, combining proportional action for immediate response with integral action to eliminate steady-state offset. Understanding the correct transfer function form is fundamental for tuning and stability analysis.

Given Data / Assumptions:

• Proportional gain is Kc.
• Reset time (integral time constant) is T.
• Ideal PI form is intended (no derivative or filters).

Concept / Approach:
The time-domain ideal PI law is u(t) = Kc [ e(t) + (1/T) ∫ e(t) dt ]. Taking Laplace transforms yields Gc(s) = U(s)/E(s) = Kc [1 + 1/(T s)]. This distinguishes PI from PD (Kc(1 + T s)), pure integral (Kc/(T s)), and pure proportional (Kc).

Step-by-Step Solution:

Verification / Alternative check:
Block-diagram realisations of PI show a proportional branch in parallel with an integrator scaled by Kc/T, summing to the same transfer function.

Why Other Options Are Wrong:

• Kc(1 + T s): This is PD, not PI.
• Kc/(T s): Pure integral action only.
• Kc: Pure proportional action only.

Common Pitfalls:
Mixing “reset time T” with “integral gain Ki”; note Ki = Kc/T so Gc(s) = Kc + Ki/s.

Final Answer:
Kc * (1 + 1/(T s))