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 Quote by Mårten Sorry for late reply, been away for a while...
 Hm... Still something here that confuses me. Imagine that you have a system described by the ODE $y'' + 4y' + 2y = u$. It's possible to describe this system with a transfer function $G(s) = \frac{1}{s^2+4s+2}$. It's also possible to describe this system with a state space representation, constructing the A-matrix and so on.
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.

 Then the following obscurities: 1) With the state space representation the system is described by N2 numbers in the A-matrix. But most of the numbers in the A-matrix are just zeros and ones. So it seems that the number of significant numbers in the A-matrix are just 2N+1. So then the state space representation is not so much more complicated?
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.

 2) If a system is described by this ODE above, how could there be any internal information that is being revealed if putting up this system on a state space representation, compared to when using the transfer function to describe it?
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.