Robust, Potential Limited Control for an Indirectly Driven Saturated System

In this paper robust Variable Structure / Sliding Mode 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 4th order one because it is the 4th 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 (-90°,+90°). Furthermore, due to the limitations in the centripetal and vertical acceleration the rotational speed of the beam also is limited. In the present approach a feedback control is applied in which the above limitations are achieved by the application of an angular potential and an angular velocity potential. The here applied robust control is based on the traditional concept of "error metrics". The effects of friction at the beam's axle are also investigated.