This note shows examples of how to generate states space 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 has its numerator polynomial of at most the same order as the numerator . Therefore is proper but is not. To use this method, we start by writing
Where is strict proper transfer function. A strict proper transfer function is one which has polynomial of order at most one less than . If was already a strict proper transfer function, then above will be zero.
Converting a proper to strict proper is done using long division. Then the result of the division is moved directly to in some speciﬁc manner. If was already strict proper then of course the long division is not needed.
The following two examples illustrate this method. The ﬁrst one uses the diﬀerential equation
And the second example uses