We were talking about realization. Every proper transfer function \(H\left ( s\right ) \) is a realization of some \({\displaystyle \sum } \left ( A,B,C,D\right ) \). We saw the recipe before, but not the justification.
Reader: Consider \(H\left ( s\right ) =\frac{s^{5}+7s^{4}+19s^{3}+25s^{2}+16s+4}{s^{5}+12s^{4}+56s^{3}+12s^{2}+125s+54}\), is there a realization with \(n=5\) states? Yes. Now, do the realization. What about with \(n=6\) states? How about with \(n<5\) states? If there is zero/pole cancellation. So do factorization first.
If we have a transfer function, then minimal realization is one with no cancellations. If the system is uncontrollable or unobservable, there will always be some cancellation. Any system that is controllable and observable is minimal.
Mason rule:
For multi-input, start by zeroing out all input except for one that is of interest. Example given of using Mason rule now. See handout of Mason that was given during the class. Example now given for electrical network. The first step is to find the equations. Once the equations are found, then Mason rule is used to find the transfer function between one input and the output.
Reader: Solve \(\begin{pmatrix} 2 & 1 & 3\\ 4 & 0 & -2\\ 1 & 2 & -1 \end{pmatrix}\begin{pmatrix} x_{1}\\ x_{2}\\ x_{3}\end{pmatrix} =\begin{pmatrix} 1\\ 2\\ 0 \end{pmatrix} \) using Mason rule. I did this, see note on my pages.
HW 2 assigned.