this is true, but while there is really only one G(s)
that accurately represents what the ODE says, there are many (and infinite number) of state space representations that will have the same transfer function, G(s)
. some are completely controllable, some are completely observable, some are both, some are neither
completely controllable nor completely observable.
it's that there are many different state space representations (all with different A, B, and C matrices, but i think the D matrix is the same) that have the same effect as far as input and output are concerned. put these different state-variable systems (with identical transfer functions) in a box and draw out the inputs and outputs to the rest of the world and the rest of the world could not tell the difference between them. at least as long as they stayed linear inside.
if it stays linear inside and you put it in a black box, you can't tell everything about the internal structure. but if some internal state goes non-linear (perhaps because this internal state blew up because it was unstable), then you can tell something is different on the outside. if the state-variable system is completely observable and you have information about the internal structure (essentially the A, B, C, D matrices), you can determine (or "reveal") what the states must be from looking only at the output (and its derivatives). if it's completely controllable, you can manipulate the input so that the states take on whatever values you want.
this "controllable" and "observable" stuff is in sorta advanced control theory (maybe grad level, even though i first heard about it in undergrad when i learned about the state space representation). i dunno if i can give it justice in this forum. i would certainly have to dig out my textbooks and re-learn some of this again so i don't lead you astray with bu11sh1t. so far, i'm pretty sure i'm accurate about what i said, but i don't remember everything else about the topic.