ИСТИНА

We consider here the problem of stabilization of a ball that can roll without slipping on the straight or curvilinear beam. The beam may turn about its pivot point that is located below it. Thus the beam is similar to an inverted pendulum. In the equilibrium the beam is located horizontally, and the ball is in the middle of the beam. Torque developed by an electric DC motor is applied in the pivot. The considered system has two degrees of freedom and it is controlled by only single torque, thus it is under-actuated. Two angles are generalized coordinates of the system. Voltage supplied to the motor is assumed limited in absolute value. The system has an unstable (when no control torque is applied) equilibrium that is to be stabilized by means of the motor. In the case of the straight beam, the linearized near unstable equilibrium system has one positive eigenvalue; if the curvature of the curvilinear beam is sufficiently large, then the corresponding system linearized near unstable equilibrium has two positive eigenvalues; all the other eigenvalues have negative real parts. Thus, the system with the curvilinear beam is more difficult to stabilize than with straight one. Using a non-degenerate linear transformation, linearized system can be reduced to Jordan form. After we separate unstable modes from this system in Jordan form – for the ball on the straight beam it is only one mode, for the ball on the curvilinear beam there are two unstable modes. We design control law using “unstable variables” in the feedback loop taking into account saturations imposed on the control signal. By this way a large basin of attraction of the desired equilibrium can be ensured. So, if we want to ensure large basin of attraction of the desired equilibrium it is necessary to use all resources of control for suppressing the unstable modes. When we suppress unstable mode for the straight beam, the basin of attraction coincides with the controllability domain [1]. And it is maximal as possible basin of attraction for this case. If we suppress the both unstable modes for the ball on the curvilinear beam, then the basin of attraction can be made arbitrarily close to the controllability domain [1]. In the case of the curvilinear beam, the boundary of the basin of attraction is unstable periodical cycle. It can be calculated solving equations of motion in the inverse time [2]. References 1. A.M. Formalskii. Controllability and Stability of Systems with Limited Resources. Moscow, “Nauka”, 1974, 368 p. (In Russian). 2. Alexander M. Formalskii. Stabilization and Motion Control of Unstable Objects. Walter de Gruyter, Berlin/Boston, 2015, 250 p.