This note shows examples of how to generate states space \(A,B,C,D\) from diﬀerential equations. The state
space will be in the controllable form.

Every transfer function which is proper is realizable. Which means the transfer function \(G(s)=\frac {N(s)}{D(s)}\) has its
numerator polynomial \(N(s)\) of at most the same order as the numerator \(D(s)\). Therefore \(G(s)=\frac {s^2}{s^2+s+1}\) is proper
but \(G(s)=\frac {s^3}{s^2+s+1}\) is not. To use this method, we start by writing \[ G(s)= k + \tilde {G}(s) \] Where \(\tilde {G}(s)\) is strict proper transfer
function. A strict proper transfer function is one which has \(N(s)\) polynomial of order at most
one less than \(D(s)\). If \(G(s)\) was already a strict proper transfer function, then \(k\) above will be
zero.

Converting a proper \(G(s)\) to strict proper is done using long division. Then the result of the division is
moved directly to \(A,B,C,D\) in some speciﬁc manner. If \(G(s)\) was already strict proper then of course the long
division is not needed.

The following two examples illustrate this method.