Closed-loop dynamics: a unity-feedback system has an integrating process Gp(s) = 1/(2 s). The controller considered is purely integral with Gc(s) = 1/(T1 s). For a step change in set point, what qualitative response will the controlled variable exhibit?

Introduction / Context:
Controller–process pairings profoundly affect closed-loop behavior. An integrating process already contributes one pole at the origin. An integral controller adds another pole at the origin. The combined loop dynamics can lead to borderline stability and sustained oscillations if not shaped by proportional or derivative action.

Given Data / Assumptions:

• Process: Gp(s) = 1/(2 s), unity feedback.
• Controller: Gc(s) = 1/(T1 s), integral only.
• Set-point (servo) step input.

Concept / Approach:
The open-loop transfer function L(s) = Gc(s)*Gp(s) = 1/(2 T1 s^2). The closed-loop characteristic equation is 1 + L(s) = 0 → 1 + 1/(2 T1 s^2) = 0. Multiplying by s^2 gives s^2 + 1/(2 T1) = 0. The roots are purely imaginary: s = ±j * sqrt(1/(2 T1)). Purely imaginary poles imply sustained oscillations with no decay or growth—an undamped response—under ideal linear assumptions.

Step-by-Step Solution:

Verification / Alternative check:
Root-locus from the origin for integral-only control on an integrator shows loci on the imaginary axis; adding proportional or derivative shifts poles to the left for damping.

Why Other Options Are Wrong:

Common Pitfalls:
Calling marginal (sustained) oscillation “unstable.” In linear theory, it is neutrally stable but practically unacceptable.

Final Answer:
Undamped response