Stabilization of the Adaptive Control of a 4th Order System Using Coordinate and Velocity Potentials

In this paper the adaptive 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 position of the ball 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 the tilting angle of the beam, and due to the phenomenology of classical mechanical systems the directly controllable physical quantity is the rotational acceleration of the beam, this system is a 4^th order one because it is the 4^th time-derivative of the ball\´s position that can directly be influenced by the control. Another interesting feature of this system is its "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 achieved by the application of an angular potential and an angular velocity potential. Utilizing the fact that the partial derivative of the 4^th time-derivative of the ball\´s position according to the angular acceleration of the beam has a well defined sign, a single tuneable adaptive parameter is introduced that does not represent the parameters of the system under control. The control is illustrated via simulation results