Question

The state variable description of a system is: \[ \dot{X} = AX + BU, \quad A = \begin{bmatrix} 0 & 3 \\ 3 & 0 \end{bmatrix} \] The poles of the system are located at:

• A. \( s = \pm 2 \)
• B. \( s = \pm j2 \)
• C. \( s = \pm j3 \)
• D. \( s = \pm 3 \)

Hint:

The eigenvalues of the state matrix \( A \) determine the system poles. If eigenvalues are purely imaginary \( \pm j\omega \), the system exhibits oscillatory behavior.

Answer 1

The Correct Option is C

Step 1: The poles of the system are given by the eigenvalues of the matrix \( A \). 
Step 2: The characteristic equation is found by solving: \[ \det(A - \lambda I) = 0 \] 
Step 3: Substituting \( A = \begin{bmatrix} 0 & 3 \\ 3 & 0 \end{bmatrix} \): \[ \begin{vmatrix} 0 - \lambda & 3 \\ 3 & 0 - \lambda \end{vmatrix} = 0 \] 
Step 4: Computing the determinant: \[ (-\lambda)(-\lambda) - (3 \times 3) = \lambda^2 - 9 = 0 \] 
Step 5: Solving for \( \lambda \): \[ \lambda^2 = 9 \] \[ \lambda = \pm j3 \] 
Step 6: The poles of the system are at \( s = \pm j3 \).