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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.

1 Example 1

$$y^{″}(t)+3y^{\prime }(t)+2y(t)=u(t)$$

2 Example 2

$$y^{‴}(t)+6y^{″}(t)-2y^{\prime }(t)-7y(t)=4u^{‴}(t)+3u^{″}(t)+2u^{\prime }(t)+4u(t)$$

3 References

1. Lecture notes, ECE 717 Linear systems, Fall 2014, University of Wisconsin, Madison by Professor B. Ross Barmish
2. Linear system theory and design, Chi-Tsong Chen.