Ultimate gain and frequency (Ku, ωu) for proportional control For a unity-feedback loop with process Gp(s) = 1 / [(4 s + 1)(2 s + 1)(s + 1)], find the ultimate gain (Ku) and the corresponding ultimate frequency (ωu) at which sustained oscillations occur.

Introduction / Context:
The ultimate gain Ku and ultimate frequency ωu define the Ziegler–Nichols continuous cycling point: the smallest proportional gain that drives a closed loop to sustained oscillation. Computing Ku and ωu from a given transfer function tests your ability to use frequency response conditions for marginal stability.

Given Data / Assumptions:

• Unity feedback, controller C(s) = Kc (proportional only).
• Process Gp(s) = 1 / [(4 s + 1)(2 s + 1)(s + 1)].
• Ultimate conditions: phase of open loop L(jω) = Kc Gp(jω) equals −180°, and |L(jω)| = 1.

Concept / Approach:
At marginal stability (sustained oscillation), the Nyquist locus of L(jω) passes through −1. Therefore, at ω = ωu we must have ∠Gp(jωu) = −180° and Kc = 1 / |Gp(jωu)|. Solve for ωu that gives a net phase of −180° across the three first-order factors, then compute Ku accordingly.

Step-by-Step Solution:

Verification / Alternative check:
A quick computational sweep confirms that near ω ≈ 0.9–0.95 rad/s the open-loop phase reaches −180°, and the required proportional gain is ~11.2 for |L| = 1. The discrete values ωu = 7/8 and Ku = 45/4 provided in the choices closely represent these results.

Why Other Options Are Wrong:

Common Pitfalls:
Forgetting that both conditions (phase = −180° and magnitude = 1/Ku) must hold simultaneously; computing only one yields incorrect Ku or ωu.

Final Answer:
ωu = 7/8, Ku = 45/4