Transfer function to state space manually

for a 1^st order system, there about only one state space representation. but for higher order, there will be many different state space representations for the same transfer function. some will be completely observable, some completely controllable. if there are no hidden pole/zero cancellations, then i think you can have it both completely observable and controllable.

anyway (your textbook should say this):

[tex] Y(s)/U(s) = G(s) = \frac{10}{s+10} = \frac{10 s^{-1}}{1+10 s^(-1)} [/tex]

[tex] 10 s^{-1} Y(s) + Y(s) = 10 s^{-1} U(s) [/tex]

[tex] Y(s) = -10 s^{-1} Y(s) + 10 s^{-1} U(s) [/tex]

state:

[tex] X(s) = Y(s) [/tex]

[tex] sX(s) = -10 X(s) + 10 U(s) [/tex]

[tex] x'(t) = -10 x(t) + 10 u(t) [/tex]

[tex] y(t)= x(t) [/tex]