Transfer matrix and stability of MIMO state model

For a multi-input multi-output (MIMO) system the transfer function is a transfer matrix $G(s) = C(sI - A)^{-1}B + D$. Here there are 3 inputs and 3 outputs, so $G(s)$ is $3\times 3$.

Given

$$A = \begin{bmatrix}-1 & 1\\ 0 & -2\end{bmatrix},\; B = \begin{bmatrix}1 & 0 & 1\\ 0 & 1 & 1\end{bmatrix},\; C = \begin{bmatrix}1 & 2\\ 1 & 0\\ 1 & 1\end{bmatrix},\; D = 0$$

Step 1: $(sI - A)^{-1}$

$$sI - A = \begin{bmatrix}s+1 & -1\\ 0 & s+2\end{bmatrix}, \qquad |sI - A| = (s+1)(s+2)$$ $$(sI - A)^{-1} = \frac{1}{(s+1)(s+2)}\begin{bmatrix}s+2 & 1\\ 0 & s+1\end{bmatrix}$$

Step 2: $(sI - A)^{-1}B$

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

Step 3: Multiply by $C$

Row by row (common factor $\frac{1}{(s+1)(s+2)}$):

• Row 1, $C_1 = [1\;\;2]$: $[\,s+2,\;\; 1 + 2(s+1),\;\; (s+3) + 2(s+1)\,] = [\,s+2,\;\; 2s+3,\;\; 3s+5\,]$
• Row 2, $C_2 = [1\;\;0]$: $[\,s+2,\;\; 1,\;\; s+3\,]$
• Row 3, $C_3 = [1\;\;1]$: $[\,s+2,\;\; s+2,\;\; 2s+4\,]$

So

$$G(s) = \frac{1}{(s+1)(s+2)}\begin{bmatrix}s+2 & 2s+3 & 3s+5\\ s+2 & 1 & s+3\\ s+2 & s+2 & 2(s+2)\end{bmatrix}$$

After cancelling common factors:

$$G(s) = \begin{bmatrix} \frac{1}{s+1} & \frac{2s+3}{(s+1)(s+2)} & \frac{3s+5}{(s+1)(s+2)}\\[4pt] \frac{1}{s+1} & \frac{1}{(s+1)(s+2)} & \frac{s+3}{(s+1)(s+2)}\\[4pt] \frac{1}{s+1} & \frac{1}{s+1} & \frac{2}{s+1} \end{bmatrix}$$

Here $G_{ij}(s) = Y_i(s)/U_j(s)$ with the other inputs set to zero.

Step 4: Stability

Characteristic equation:

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

Since $A$ is upper triangular, its eigenvalues are simply the diagonal entries $-1$ and $-2$.

Answer: The transfer matrix is as given above. All eigenvalues of $A$ lie in the left half s-plane, so the system is asymptotically stable (and also BIBO stable).