the next step after PID control is state-space based control.
it's not a single tool but more of a framework. it goes beyond frequency based representations and can also model non-linear control problems.
the kalman filter is an example of the use of this framework where it's combined with statistics.
the problem in control theory though is that once you want to go beyond linear control things get very difficult, most of the literature seems to be about finding clever ways of approximating your problem into something linear.
Check out model predictive control. And LQR if you haven’t learned that yet. And the bazillion state estimation techniques that start with Kalman filtering variants (eg UKF) and progressively account more and more for nonlinearities.
Kalman filters and variations are covered in Ch. 19. It can be viewed as a special case of belief updating in a partially observable Markov decision process.
