Fractional Robust Control of a Ball-Beam System

In this paper the robust control of a 2 degrees of freedom (DOF) classical mechanical system, a ball-beam system is considered. The control task has the interesting feature that only one of the DOFs of the system, i.e. the linear position of the ball along the beam is controlled via controlling the other axis, the tilting angle of the beam. Since the acceleration of the ball rolling on the beam depends on the gravitation and on this tilting angle, and that the directly controllable physical quantity is the rotational acceleration of the beam, this system is a 4^th order one because only the 4^th time-derivative of the ball's position can directly be influenced by the torque rotationally accelerating the beam. This system has a "saturation" since the rotational angle of the beam must be limited within the interval (-90deg, +90deg) that also sets limits to the available acceleration of the ball. In the present approach a feedback control is applied in which the above limitation is taken into account by the application of angular and angular velocity potentials. The method is based on a standard error metrics that has to converge to zero during finite time according to a fractional order differential equation in discrete time approximation. It is shown that little reduction of the order of differentiation from 1 improves the robustness of the control against the measurement noises. The control is illustrated via simulation