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## Generating state space in controllable form from diﬀerential equations

July 2, 2015   Compiled on September 7, 2023 at 9:49pm

### Contents

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

$y''(t)+3y'(t)+2y(t)=u(t)$

$y'''(t)+6y''(t)-2y'(t)-7y(t)=4u'''(t)+3u''(t)+2u'(t)+4u(t)$