44 SISO (A,B,C,D)

Let $A,B,C,D$ be constant complex matrices with shapes:

$$A\mathit{\hspace{1em}}(n\times n);B\mathit{\hspace{1em}}(n\times 1);C\mathit{\hspace{1em}}(1\times n);D\mathit{\hspace{1em}}(1\times 1).$$

Then the linear system with one input, $n$ states and one output is

$$\frac{dX}{dt}=AX+Bu$$

$$y=CX+Du$$

Here $t$ is time, $u$ is the input, $X$ is the state, and $y$ is the output. We call the system single-input single-output or SISO.

Examples. (i) An electrical fan is a SISO system. The input is electricity, and the output is moving air. The states can involve the speed of rotation, voltage, current and so on.

(ii) A wind turbine has input moving air and output electricity.