The most general nonlinear dynamical model for which control design methods exist are feedback-linearizable systems for which control designs through feedback linearization/dynamic inversion or backstepping are available. Given that most kinematic chains or dynamics governed by Euler-Lagrange equations fall into this category (with appropriate actuation and measurement of course), this is the most fruitful class of nonlinear systems studied so far.
Have a look at this paper for an example of a nonlinear benchmark problem. Article Alstom Benchmark Challenge II on Gasifier Control
Many authors have attempted this one. If you wish to try it, you can download the model (and control requirements) for simulink here: Deleted research itemThe research item mentioned here has been deleted
Actually I am working on fault detection using e-learning machine techniques. I will be honored by a future cooperation between our research team. We need real normal and fault Data on a benchmark nonlinear plant.