45 MIMO (A,B,C,D)

Let $A,B,C,D$ be constant complex matrices with shapes: $A$ $(n\times n)$; $B$ $(n\times k)$; $C$ $(m\times n)$; $D$ $(m\times k)$. Then the linear system with $k$ inputs, $n$ states and $m$ outputs is

$$\frac{dX}{dt}=AX+BU,$$

$$Y=CX+DU.$$

Here $U$ is the input, $X$ is the state, and $Y$ is the output.

i) If $k=1$, then we say the system is single input; if $m=1$, then the system is single output. If $k=m=1$, then the system is SISO.

ii) If $k>1$, then we say the system is multi input; if $m>1$ then the system is multi output.

iii) If $k\ge 1$ and $m\ge 1$, then we call the system MIMO. (SISO is a special case of MIMO)