Robust choice of process parameters for conservative proportional gain: given Gp(s) = 10 * exp(−0.1 s) / (0.5 s + 1) with ±20% uncertainty, which (K, τ, θ) set should be used to compute the largest safe Kc?

Introduction / Context:
For a first-order-plus-dead-time (FOPDT) process controlled by a proportional controller, model uncertainty can severely reduce stability margins. A conservative design chooses the parameter combination that makes the loop “most difficult to control,” then computes a safe controller gain Kc using that worst-case set. Here we must choose the proper triplet (process gain K, time constant τ, and dead time θ) from the ±20% bounds of the nominal model Gp(s) = 10 * exp(−0.1 s) / (0.5 s + 1).

Given Data / Assumptions:

• Nominal values: K = 10, τ = 0.5 min, θ = 0.1 min.
• Uncertainty: ±20% on each parameter ⇒ K ∈ [8, 12], τ ∈ [0.4, 0.6], θ ∈ [0.08, 0.12].
• Controller: proportional only, Kc to be set conservatively.

Concept / Approach:
Stability robustness for proportional control worsens when the open-loop gain is larger and the apparent phase lag is greater. Larger K increases loop gain directly, and larger θ reduces phase margin. The ratio θ/τ captures “dead-time severity”; larger θ and smaller τ are most challenging. Among the listed options, the set with simultaneously high K and high θ is the most conservative choice available. While a smaller τ would be even more conservative, we must choose from the provided combinations.

Step-by-Step Solution:

Verification / Alternative check:
Most FOPDT tuning correlations (e.g., robustness-oriented rules) reduce Kc as K increases or θ/τ increases. Thus the combination with largest K and largest θ will yield the smallest stability-limited Kc — a conservative basis.

Why Other Options Are Wrong:

Common Pitfalls:
Treating each parameter independently rather than choosing a consistent worst-case triplet for gain calculation.

Final Answer:
12, 0.6, 0.12