Question

A closed-loop system has the characteristic equation given by: \[ s^3 + k s^2 + (k+2) s + 3 = 0 \] For the system to be stable, the value of \( k \) is:

• A. \( k>1 \)
• B. \( 0.5<k<1 \)
• C. \( 0<k<1 \)
• D. \( 0<k<0.5 \)

Hint:

The Routh-Hurwitz criterion helps determine the stability of a control system by ensuring no sign changes in the first column of the Routh array.

Answer 1

The Correct Option is C

Step 1: A system is stable if all the roots of the characteristic equation have negative real parts. We use the Routh-Hurwitz criterion to determine stability. 
Step 2: Constructing the Routh array for: \[ s^3 + k s^2 + (k+2) s + 3 = 0 \] \[ \begin{array}{c|cc} s^3 & 1 & k+2 \\ s^2 & k & 3 \\ s^1 & \frac{k(3) - (k+2)(k)}{k} & 0 \\ s^0 & 3 & \end{array} \] 
Step 3: For stability, all first-column elements must be positive. - \( k > 0 \) (ensures the second row is positive) - The third-row term must be positive: \[ \frac{3k - k^2 - 2k}{k} > 0 \] \[ \frac{k - k^2}{k} > 0 \] \[ k - k^2 > 0 \] \[ k (1 - k) > 0 \] 
Step 4: The inequality holds for: \[ 0 < k < 1 \] 
Step 5: Thus, the system remains stable when \( 0 < k < 1 \).