Simplification of $H(s)=(4+2/s)(3/(1+2s))$

Question

I had the following problem: We combine a parallel PI-control system ($H_{1}(s)=P+\frac{I}{s}$ with $P=4, I=2$) with a 1st order process ($H_{2}(s)=\frac{3}{1+\tau s}$ with $K=3$, $\tau=2$) This leads to the following expression: $$H(s)=(4+\frac{2}{s})(\frac{3}{1+2s})$$ which can be simplified to $H(s)=\frac{6}{s}$ (this obviously has an effect on the domain). The form I ended up with was: $$H(s)=\frac{6+9s}{s+2s^2}$$ Which is more correct because the domain stays the same. Now, I wonder about the stept which lead to the $H(s)=\frac{6}{s}$ expression. I hope someone can help me out.

Answer 1

Does this answer your question? It is just plain algebra really.

$$ \left(4 + \frac{2}{s}\right)\left(\frac{3}{1 + 2s}\right) = \frac{12}{2s + 1} + \frac{\frac{6}{s}}{2s + 1} = \frac{12 + \frac{6}{s}}{2s + 1} = \frac{12 + \frac{6}{s}}{2s + 1} \frac{\frac{s}{6}}{\frac{s}{6}} = \frac{2s + 1}{2s + 1}\frac{1}{\frac{s}{6}} = \frac{6}{s} $$