Question

Find a minimal representation for the system G (s) s2 = [ s+1 (s+2)(8+3) $2+38+2 1

Answer 1

$G(s)=[\frac{s^{2}}{s^{2}+3s+2} \ \ \ \frac{(s+1)}{(s+2)(s+3)}]$

G1(s) S2 + 3s + 2

$2 Gl(s) (s +1)(s +2)

$=\frac{A}{(s+1)}+\frac{B}{(s+2)}$

$A=\frac{s^{2}}{s+2},\ \ \ s=-1$

$A=1$

$B=\frac{s^{2}}{s+1},\ \ \ s=-2$

$B=-4$

$=\frac{1}{(s+1)}+\frac{-4}{(s+2)}$

$=e^{-t}-4e^{-2t}$

  

$G2(s)=\frac{(s+1)}{(s+2)(s+3)}$

$=\frac{A}{(s+2)}+\frac{B}{(s+3)}$

$A=\frac{s+1}{s+3},\ \ \ s=-2$

$A=-1$

$B=\frac{s+1}{s+2},\ \ \ s=-3$

$B=2$

$=\frac{-1}{(s+2)}+\frac{2}{(s+3)}$

$=-e^{-2t}+2e^{-3t}$

$G(s)=G1(s)*G2(s)$

$=(e^{-t}-4e^{-2t})(-e^{-2t}+2e^{-3t})$