So what this describes is it actually describes the system, what relates the inputs to the outputs under static conditions. That is exactly what this does, so if you have static conditions, that means u, y, and x are all constant. Then of course, you have x dot and in that case is zero; x is constant. See if zero = A + Bu, y = Cx + Du and if you eliminate x from these equations by solving for x = minus A inverse Bu here and you plug that in here, you get this. Okay? So this is assuming A is invertible here, so this is what it describes. ...

See the whole transcript at Introduction to Linear Dynamical Systems - Lecture 15