Controller design basics: for a proportional–derivative (P–D) controller, what is the standard transfer function form in the Laplace domain?

Introduction / Context:
Classical controllers—P, PI, PD, and PID—are building blocks of feedback control. Recognizing their transfer functions is essential for frequency response design, time-domain tuning, and implementation in analog or digital form.

Given Data / Assumptions:

• P–D controller includes proportional and derivative actions only.
• Laplace transform conventions are standard.
• Kc is controller gain; τD is derivative time (seconds).

Concept / Approach:
The ideal PD controller has transfer function Gc(s) = Kc(1 + τD s). The proportional term Kc sets overall gain. The derivative term Kc τD s anticipates changes by acting on the slope of the error, adding phase lead and improving transient performance. Practical implementations include a small high-frequency roll-off (a filter), but the canonical ideal form remains Kc(1 + τD s).

Step-by-Step Solution:
Write the ideal PD action: u(t) = Kc[e(t) + τD de/dt].Take Laplace transform: U(s)/E(s) = Kc(1 + τD s).Match to the option list and select Kc(1 + τD s).

Verification / Alternative check:
Tuning correlations (e.g., Ziegler–Nichols variants) and frequency response analysis use the same PD transfer form; hardware realizations often add a derivative filter term 1/(1 + τf s) to limit noise amplification.

Why Other Options Are Wrong:
Kc(1 + 1/τD s): Dimensionally inconsistent; additive term should be τD s.Kc τD s: Pure derivative, missing proportional path.Kc/τD s or Kc/(1 + τD s): Resemble integrator or lag forms, not PD.

Common Pitfalls:
Using an “ideal” derivative without filtering causes noise amplification; always include a small roll-off in practice.

Final Answer:
Kc(1+τD s)