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I'm new in Control system and in Mathematica too.

res1 is the state space as follows

res1 = StateSpaceModel[{m1 y''[t] + b1 y'[t] + c1 y[t] == 
F[t]}, {{y[t], 0}, {y'[t], 0}}, {{F[t], 0}}, {y[t]}, t]

the result is correct, then it switch to tranfer function with

x1 = TransferFunctionModel[res1, s]

This is OK too. But when i try switch it back to space state with

StateSpaceModel[x1]

the result it not the same as res1. Thank you for explanation(sorry for bad english..)

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The state-space representation is not unique, because the choice of states are not unique. (This is like saying that 3 could be expressed as 1+1+1, 2+1, etc). You could try to force the state-space realization to be the same by explicitly specifying it.

The following will give the same realization:

StateSpaceModel[{m1 y''[t] + b1 y'[t] + c1 y[t] == 
F[t]}, {{y[t], 0}, {y'[t], 0}}, {{F[t], 0}}, {y[t]}, t, 
StateSpaceRealization -> "Controllable"]

StateSpaceModel[TransferFunctionModel[res1, s], 
StateSpaceRealization -> "Controllable"]

enter image description here

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