Minimal realizations

From Canonical Decompositions we saw that unobservable and uncontrollable subsystems are removed when going to a transfer function/matrix. As learned in Realizations, we have infinitly many realizations of a proper rational transfer matrix $G(s)$. This is why we want to create a Minimal Realization.

Minimal Realization

We do not create redundant unobservable and uncontrollable states. The resulting State Space Representation model will have the same dimensions as

\dot{\bar{x}}_{co}&=\bar{A}_{co}\bar{x}_{co}+\bar{B}_{co}u \\ y &= \bar{C}_{co}\bar{x}_{co}+Du \end{align}

which is minimal

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Criterias

Comprime

If the order of the transfer function after removing common factors, the system is comprime, and we have a comprime fraction.

$$\frac{N(s)}{D(s)}$$

and $n=dim(A)=deg(\hat{g}(s))$

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After cancelling the terms, we can see that the denominator has a degree of $2$. Since the dimension of $A$ is $3$, we do not have a minimal realization. This implies that the system is controllable and observable.

Through a similarity transform, $x=T\bar{x}$, we can see that all minimal realizations are equivalent.

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