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I'm trying to find tandem/cascade and parallel realization for the following transfer function: $$G(s)=\frac{(s+6)^2}{(s+2)^2(s^2+4s+6)}$$

For Tandem/cascade realization, I draw the following block diagram:

enter image description here

And there is my resolution attempt:

$$X_1=\frac{1}{s+2}U\Leftrightarrow (s+2)X_1=U\Leftrightarrow sX_1+2X_1=U\xrightarrow {\mathcal{L}} \dot{x_1}+2x_1=u\Leftrightarrow \dot{x_1}=2x_1+u$$

$$W_2=\frac{s^2+12s+36}{s^2+4s+6}X_1$$

$$W_2=\bigg(1+\frac{8s+30}{s^2+4s+6}\bigg)X_1$$

I don't know how to obtain the matrix and the final result. What are the steps that are missing?

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  • \$\begingroup\$ For parallel realization why don't you use partial fractions to deliver an A + B answer? \$\endgroup\$ – Andy aka Jan 8 at 12:15
  • \$\begingroup\$ @Andyaka Thanks for the hint, but my doubt is related with tandem/cascade realization. The parallel realization I know how to do. \$\endgroup\$ – Carmen González Jan 8 at 12:20
  • \$\begingroup\$ Looking at your block diagram, what else would you need to prove? The block diagram is self evidently a tandem/cascade realization or maybe I'm missing something? \$\endgroup\$ – Andy aka Jan 8 at 12:24
  • \$\begingroup\$ @Andyaka I want the matrix with a column $\dot{x_1} ; \dot{x_2} ; \dot{x_3} ; \dot{x_4}$ equals to a matrix with coefficients multiplied for a column with $x_1 ; x_2 ; x_3 ; x_4$ and sum with a column multiplied for the input signal u. \$\endgroup\$ – Carmen González Jan 8 at 12:36
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    \$\begingroup\$ Also, your last formula for \$W_2\$ seems wrong compared the the one directly before. \$\endgroup\$ – Andy aka Jan 8 at 12:58
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Looks like you are trying to find a state-space representation of a transfer function. There are many ways of doing this and there are a few standard methods called canonical realizations. The Wikipedia article on the subject provides a couple such canonical realizations.

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