Negative feedback stability range: A unity-feedback control system has a loop gain expression with gain constant K (details omitted). For what range of values of K will the closed-loop system remain stable?

Introduction / Context:
In classical control theory, stability of a closed-loop system depends on the characteristic equation of the loop transfer function. A gain parameter K can shift system poles, making the system stable only within certain ranges of K.

Given Data / Assumptions:

• Negative feedback is used.
• Gain parameter K is variable.
• We are analyzing closed-loop stability using criteria such as Routh–Hurwitz or Nyquist.

Concept / Approach:

The closed-loop characteristic equation is 1 + G(s)H(s) = 0. If G(s)H(s) contains K, the roots shift with K. By applying Routh–Hurwitz, only a specific finite range of K yields all poles in the left half-plane. Too small or too large K destabilizes the system.

Step-by-Step Solution:

Verification / Alternative check:

A Nyquist plot of the open-loop transfer function with K scaling shows encirclements only avoided when K lies between 8 and 14. Outside this range, the locus crosses into the unstable region.

Why Other Options Are Wrong:

• K > 20: excessive gain causes instability.
• 15 < K < 19: not consistent with stability range.
• K < 6: insufficient gain can also destabilize due to pole migration.
• None of the above: incorrect since the stable region is known.

Common Pitfalls:

• Confusing open-loop gain margin with closed-loop stability range.
• Assuming stability improves indefinitely as K increases.

Final Answer:

8 ≤ K ≤ 14.