TITLE: How to control a nonlinear system with a PID controller
QUESTION [5 upvotes]: Is the design of a PID controller for a nonlinear system  different from for a linear system?
[I think math.stackexchange.com is the most suited SE.]

REPLY [6 votes]: I know its long time since the question was published. However, I would like to add a comment since it is well-referenced post on searching engines.

You can assume that the system is close to a linear one as previously mentioned. It does the job very well for many cases and this approach is used frequently.
You can use multiple PID or more complex control systems as previously mentioned and one again, it works very well in numerous circumstances (especially with systems that are characterised by very different behaviours depending on the state of the system).
A third solution and interesting one is to add a mathematics transformation on the input data. I like this one since you can use a classic PID on very non-linear systems.

For instance, if you want to control a reverse pendulum (the mass is above the pivot and so it is a meta-stable system) and you know the position of the mass in an Eulerian (x/y) referential, then you have different solutions.

Suppose that your system is linear. It works only if you are close to vertical position and variations around this point (due to perturbations) are minimal. The control will use the horizontal position (x-position) to determine the angular momentum to add to the pendulum. And once again, it works for little or slow perturbations.
You can use other types of control command systems as a state-space control command for non-linear systems, or multiple PIDs managed by a state machine.
Or simply, you add a transformation. Since, it is an inverse pendulum, you can compute whenever the angular position of the pendulum and add a control on it.

Obviously, this is a very simple example but it shows the general idea. Using a mathematical transformation on input data can simplify a lot the system and its control system. It is an elegant solution, however, depending on the hardware to control the system, it can be power/time consuming. The solution is really depending on your requirements.