Nonlinear Optimal Control of the Ball and Plate Dynamical System

The ball and plate system exhibits strong nonlinearities and multi-variable dynamics and its stabilization and control is a non-trivial task. The related control problem becomes even more complicated because ofthe system's underactuation (the system has four degrees of freedom while receiving only two control inputs). In this article a novel nonlinear optimal (H-infinity) control approach is developed for this control problem. The dynamic model ofthe ball and plate system undergoes first approximate linearization around a temporary operating point which is updated at each iteration ofthe control algorithm. The linearization process makes use of first-order Taylor series expansion and of the Jacobian matrices of the state-space model of the system. For the approximately linearized description of the ball and plate dynamics a stabilizing H-infinity feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation is solved at each time-step ofthe control method. The stability properties ofthe control scheme are proven through Lyapunov analysis. The proposed control method retains the advantages of linear optimal control, that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs.