Control engineering: for a proportional–integral–derivative (PID) controller, which transfer function in the parallel (non-interacting) form is correct (τi = integral/reset time, τD = derivative time)?

Introduction / Context:
The PID controller combines proportional, integral, and derivative actions to shape closed-loop dynamics. Its standard parallel-form transfer function is a bedrock concept in process control, robotics, and motion systems.

Given Data / Assumptions:

• We use Laplace-domain representation with zero manual reset bias for simplicity.
• τi is the integral (reset) time, τD is the derivative time, Kc is proportional gain.
• Parallel (non-interacting) form is intended.

Concept / Approach:
A parallel PID controller sums three terms: proportional Kc, integral Kc/τi * 1/s, and derivative KcτD * s. Thus Gc(s) = Kc [ 1 + 1/(τi s) + τD s ]. Other forms (series/ideal or velocity form) are algebraically equivalent but patterned differently.

Step-by-Step Solution:
Write proportional term: Kc.Write integral term: Kc * (1/τi) * (1/s).Write derivative term: Kc * τD * s.Sum: Gc(s) = Kc * [ 1 + 1/(τis) + τDs ].

Verification / Alternative check:
Time-domain controller: u(t) = Kc [ e(t) + (1/τi) ∫ e(t) dt + τD de(t)/dt ]. Laplace transform with zero initial conditions gives the same Gc(s).

Why Other Options Are Wrong:

• Option b: integral written as τis (incorrect dimensionally); derivative constant as τD (missing s).
• Option c: lacks proportional term and misrepresents derivative as 1/(τDs).
• Option d: represents a lead-lag, not a PID in parallel form.

Common Pitfalls:
Confusing “reset time” τi with reset rate (1/τi); mixing interacting vs. non-interacting forms; forgetting derivative filtering in practical implementations.

Final Answer:
Gc(s) = Kc * [ 1 + 1/(τis) + τDs ]