Question

Let \( G(s) = \frac{1}{(s+1)(s+2)} \). Then the closed-loop system shown in the figure below is:

• A. stable for all K > 2
• B. unstable for all K ≥ 2
• C. unstable for all K > 1
• D. stable for all K > 1

Hint:

To assess closed-loop stability, derive the characteristic equation and apply the Routh-Hurwitz criterion. Ensure all coefficients are positive for stability.

Answer 1

The Correct Option is B

The open-loop transfer function is:
\[ G_{OL}(s) = K(s - 1) \cdot \frac{1}{(s+1)(s+2)} = \frac{K(s - 1)}{(s+1)(s+2)} \]

The characteristic equation for the closed-loop system is:
\[ 1 + G_{OL}(s) = 1 + \frac{K(s - 1)}{(s+1)(s+2)} = 0 \Rightarrow (s+1)(s+2) + K(s - 1) = 0 \]

Expand and simplify:

Apply the Routh-Hurwitz criterion for stability. The system will be stable if all coefficients are positive:
- \( 3 + K > 0 \) → always true for \( K > -3 \)
- \( 2 - K > 0 \) → \( K < 2 \)

So the system becomes unstable for \( K >= 2 \).