my question does not contain any specific maths, but rather a general question about controller design using small signal models.
I'm familiar with the fact that it is often times hard to synthesize controllers for nonlinear systems, even when the system is described well from a mathematical viewpoint. The most common way of dealing with that problem is then to linearize the nonlinear system around an operating point (i.e. equilibrium point) and use the resulting linear model to derive a controller. This whole procedure makes sense to me - as long as the behaviour of the linearized system is approximately the same for the behaviour of that system being driven into this operating point.
As an example, imagine a simplified DC-DC converter with a constant load R and control input D (D for duty cycle of some switch in the circuit). My goal is now e.g. to drive the output of the converter from initially 0V / 0A to the equilibrium of 100V / 10mA. Say we'd need a duty cycle of D = 0.7 to reach that goal. We now need to design a controller that tries to do exactly that transition 0V / 0A -> 100V / 10mA using as little time as possible. Now, why does it make sense to linearize the plant about the operating point 100V / 10mA and use that as a basis for the controller design, when actually the resulting linear system dynamics covers the real, non-linear system dynamics only in close proximity of 100V / 10mA and nowhere near the whole transition 0V / 0A -> 100V / 10mA. I.e. the dynamics of the whole transition could be completely different from that in close to the equilibrium point 100V / 10mA. That means the controller would only work well if it e.g. should control the transition 100V / 10mA -> 110V / 10mA but behave completely different for much greater changes in equilibria.
So, how can the procedure be effective for the mentioned bigger transitions? Does one just assume that the system dynamics is approximately the same?