Semi-Analytical Closed-Form Solutions of the Ball–Plate Problem

Mathematical models and numerical simulations are necessary to understand the dynamical behaviors of complex systems. The aim of this work is to investigate closed-form solutions for the ball–plate problem considering a system derived from an optimal control problem for ball–plate dynamics. The nonlinear properties of ball and plate control system are presented in this work. To semi-analytically solve this system, we explored a second-order nonlinear differential equation. Consequently, we obtained the approximate closed-form solutions by the Optimal Parametric Iteration Method (OPIM) using only one iteration. A comparison between the analytical and corresponding numerical procedures reflects the advantages of the first one. The accordance between the obtained results and the numerical ones highlights that the procedure used is accurate, effective, and good to implement in applications such as sliding mode control to the ball-and-plate problem.