Example

Consider a nth order differential equation

$\displaystyle \frac{d^ny}{dt^n}+a_1\frac{d^{n-1}y}{dt^{n-1}}+\ldots+a_ny=u$

Define following variables,

The nth order differential equation may be written in the form of n first order differential equations as

or in matrix form as,

$\displaystyle \dot{\boldsymbol{x}}=A\boldsymbol{x}+Bu$

where

The output can be one of states or a combination of many states. Since, ,

$\displaystyle y=[1\; 0\; 0\; 0 \;\ldots\; 0]\boldsymbol{x}$