Question

Consider the state-space model 
\[ \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B r(t), \quad y(t) = C \mathbf{x}(t) \]
where \( \mathbf{x}(t) \), \( r(t) \), and \( y(t) \) are the state, input, and output, respectively. The matrices \( A \), \( B \), and \( C \) are given below:
\[ A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \end{bmatrix} \]
The sum of the magnitudes of the poles is __________ (round off to the nearest integer).

Hint:

The poles of a state-space system are the eigenvalues of matrix \( A \), obtained by solving \( \det(sI - A) = 0 \). Their magnitudes can be added directly when asked for total damping or system decay rate.

Answer 1

To find the poles, we compute the eigenvalues of matrix \( A \).
\[ \text{Characteristic equation: } \det(sI - A) = 0 \]
\[ \det\left( \begin{bmatrix} s & 0 \\ 0 & s \end{bmatrix} - \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix} \right) = \det\left( \begin{bmatrix} s & -1 \\ 2 & s + 3 \end{bmatrix} \right) \]
\[ = s(s + 3) - (-1)(2) = s^2 + 3s + 2 \]
\[ \Rightarrow s^2 + 3s + 2 = 0 \Rightarrow s = -1, -2 \]
Sum of magnitudes of poles:
\[ |{-1}| + |{-2}| = 1 + 2 = \boxed{3} \]