According to the page above, an LTI MIMO system could be affected by many interactions. I have two questions about this page:
- Is it described the system using differential equations (as well as it would be done with a spring-mass system)?
- Why does it say that the MIMO system is described by differential equations if the set of equations (8.11) is in the complex variable 's'? I guess here it was applied a Laplace transformation.
Yes the system is described with differential equations, i.e:
$$ \begin{align} \dot{x} &= A x + B u \\ y &= C x + D u \end{align} $$
MIMO for the LTI system just mean that $B$ has $\geq 2$ columns (multiple inputs) and $C$ has $\geq 2$ rows (multiple outputs). And yes the 8.11 equation is the Laplace transformation.
For example:
$$ A = \begin{pmatrix} 0&0&1 \\0&-1&0 \\-5&-10&-15 \end{pmatrix},B = \begin{pmatrix}0 & 0 \\ 1 & 0 \\ 0 & 1 \end{pmatrix},C =\begin{pmatrix}1&0&0\\0&1&0\\0&0&1 \end{pmatrix},D=\begin{pmatrix}0&0\\0&0\\0&0\end{pmatrix} $$
Laplace transform:
$$ \begin{pmatrix} y_1(s)\\y_2(s)\\y_3(s) \end{pmatrix} = \begin{pmatrix}\frac{-10}{s^3 + 16 s^2 + 20 s + 5} & \frac{1}{s^2 + 15 s + 5} \\ \frac{1}{s + 1} & 0 \\ \frac{-10s}{s^3 + 16 s^2 + 20 s + 5} & \frac{s}{s^2 + 15 s + 5} \end{pmatrix} \begin{pmatrix} u_1(s) \\ u_2(s) \end{pmatrix} $$
You can see that $u_1$ influences all outputs but $u_2$ influences only $y_1$ and $y_3$ but not $y_2$.