I'm trying to think of limitations or disadvantages of LTI systems in control theory, however, I can't think of many. The only one I can think of is that not every system is an LTI system.
Linear time-invariant systems (LTI systems) are a class of systems used in signals and systems that are both linear and time-invariant. Linear systems are systems whose outputs for a linear combination of inputs are the same as a linear combination of individual responses to those inputs. Time-invariant systems are systems where the output does not depend on when an input was applied. These properties make LTI systems easy to represent and understand graphically.