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.
\[ y''(t)+3y'(t)+2y(t)=u(t) \]
\[ y'''(t)+6y''(t)-2y'(t)-7y(t)=4u'''(t)+3u''(t)+2u'(t)+4u(t) \]