Quote by Mårten
Before we go on  thank you very much for this personal class I get here in control theory! It helps me a lot!
Okey, sorry. That was to signal that they could be matrices or vectors. Bold face may be better...
Okey, I think I got it there. y(0) would just have been a transformation (made by the Cmatrix) of the vector x(0).

yeah, and it would be nice if i practice what i preach. i should have said
x(t) for the state vector instead of
X(t).
Then comes the ultimate question: Why don't always use the statespace model in preference to the Laplace transform model? What benefits does the Laplace transform model have, which the statespace model doesn't have?

simplicity. if a system is known to be linear and timeinvariant,
and if you don't care about what's going on inside the black box, but only on how the system interacts (via its inputs and outputs) with the rest of the world that it is connected to, the inputoutput transfer function description is all that you need. if you have an
N^{th}order, singleinput, singleoutput system, then 2
N+1 numbers fully describe your system, from an inputoutput POV. with the state variable description, an
N^{th}order system (single input and output) has
N^{2} numbers, just for the
A matrix. and 2
N+1 numbers for the
B,
C, and
D matrices. so there are many different statevariable systems that have the same transfer function. all of these different systems behave identically with the outside world
until some internal state saturates or goes nonlinear and that's where they are modeled differently in practice. you can even have internal state(s) blow up and not even know it (if the system is "not completely observable") until the state(s) that are unstable hit the "rails", the maximum values they can take before clipping or some other nonlinearity. when something like this happens, there is polezero cancellation, as far as the inputoutput description is concerned. so maybe some zero killed the pole and they both disappeared in the transfer function,
G(s), but inside that bad pole still exists.