0.19 Defining the state for an n-th order differential equation

Consider the n-th order linear differential equation:

One way to define state variables is by introducing the auxiliary variable $w$ which satisfies the differentialequation:

The state variables can then be chosen as derivatives of $w$ . Furthermore the output is related to this auxiliary variable as follows:

The proof in the next three equations shows that the introduction of this variable $w$ does not change the system in any way. The first equation uses a simple substition based on the differential equation . Then the order of $p(s)$ and $q(s)$ are interchanged. Lastly, $y$ is substituted in place of $p(s)w(t)$ (using output equation ). The result is the original equation describing our system.

Using this auxillary variable, we can directly write the $A$ , $B$ and $C$ matrices. $A$ is the companion-form matrix; its last row (except for a $0$ in the first position) contains the alpha coefficients from the $q(s)$ :

The $B$ vector has zeros except for the $n$ -th row which is a $1$ .

$C$ can be expressed as

When all of these conditions are met, the state is

In conclusion, if the degree of $p$ is less than that of $q$ , we can obtain a state-space representation by insertingthe coefficcients of $p$ and $q$ in the matrices $A$ , $B$ and $C$ as shown above.