Transfer function from state equations, output x2

The transfer function of a state model is $G(s) = C(sI - A)^{-1}B + D$, and the system is stable if all roots of $|sI - A| = 0$ (the eigenvalues of $A$) lie in the left half of the s-plane.

Step 1: Write the state model

$$\dot{x}_1 = -3x_1 + x_2, \qquad \dot{x}_2 = -2x_1 + u, \qquad y = x_2$$ $$A = \begin{bmatrix}-3 & 1\\ -2 & 0\end{bmatrix},\; B = \begin{bmatrix}0\\ 1\end{bmatrix},\; C = \begin{bmatrix}0 & 1\end{bmatrix},\; D = 0$$

Step 2: Find $(sI - A)$ and its determinant

$$sI - A = \begin{bmatrix}s+3 & -1\\ 2 & s\end{bmatrix}$$ $$|sI - A| = s(s+3) - (-1)(2) = s^2 + 3s + 2 = (s+1)(s+2)$$

Step 3: Inverse

$$(sI - A)^{-1} = \frac{1}{(s+1)(s+2)}\begin{bmatrix}s & 1\\ -2 & s+3\end{bmatrix}$$

Step 4: Transfer function

$$\begin{aligned} (sI - A)^{-1}B &= \frac{1}{(s+1)(s+2)}\begin{bmatrix}1\\ s+3\end{bmatrix} \\ G(s) = C(sI - A)^{-1}B &= \begin{bmatrix}0 & 1\end{bmatrix}\frac{1}{(s+1)(s+2)}\begin{bmatrix}1\\ s+3\end{bmatrix} = \frac{s+3}{(s+1)(s+2)} \end{aligned}$$

Transfer function:

$$\frac{Y(s)}{U(s)} = \frac{s+3}{s^2 + 3s + 2}$$

Cross-check by Laplace transform

With zero initial conditions: $sX_1 = -3X_1 + X_2$ gives $X_1 = X_2/(s+3)$. Then $sX_2 = -2X_1 + U = -\frac{2X_2}{s+3} + U$, so

$$X_2\left(s + \frac{2}{s+3}\right) = U \;\Rightarrow\; \frac{X_2}{U} = \frac{s+3}{s^2 + 3s + 2}$$

Same result.

Step 5: Stability check

Characteristic equation:

$$|sI - A| = s^2 + 3s + 2 = 0 \;\Rightarrow\; s = -1,\; -2$$

Routh array for $s^2 + 3s + 2$: first column $1, 3, 2$, all positive, no sign change.

Answer: $G(s) = \dfrac{s+3}{(s+1)(s+2)}$. Both poles ($-1$ and $-2$) are real and negative, so the system is stable (asymptotically stable, overdamped response).