Introduction

The ball and plate system is a device found in experimental benches at some universities around the world\cite{Zhao_2014}, where researches can propose different approaches to define a control model that would be more effective and require less energy. Those strategies are slightly distinct one from another not because of the complexity of the mechanical system which is actually quite simple but because of the control variables involved.
The response of controlled variables behaves according to its nonlinear governing equations making it a bit harder to predict a trajectory without any simplification and it is also possible to increase the complexity if we include the effect of viscous damping, loss of contact, slip or other features besides external disturbances. Otherwise the unconsidered effects would act in a black box mode or simply would take the model away of a real situation.
Wherefore many controllers have been developed and investigated as options to optimize few parameters and more recently those that are based on evolutionary training algorithms for neural networks \cite{Fei_2011} have allowed us to reach even better solutions for the path. It means we are searching for an equilibrium among lowing the energetic cost and to execute a quick and stable movement.
Before we start the comparison between these two control strategies we should have a brief overview on dynamical models in order to understand the physics that governs our simulations.
Having the set of equations the next step is to linearize and rewrite it in a state-space representation then proportional–integral–derivative (PID) controller can be set up and provide us the first data to use in comparison with a smarter control.
Finally we leave the linear approximation and start to design the artificial neurons of the neural network (NN) to low energetic cost without giving up the stability and speed of the control getting functions for the actuation and more data to compare.