Unstable Oscillating Systems with Hysteresis: Problems of Stabilization and Control

The work is devoted to studying the dynamics of unstable oscillating systems (in the form of an inverted pendulum) controlled by the action of a hysteretic type. The results for different types of motion of a suspension point are presented, in particular, for vertical and horizontal motion. A mathematical model of the inverted pendulum with an oscillating suspension is considered. For this pendulum the explicit criteria of stability are obtained using the linearized equations of motion. The dependences between the initial conditions and the value of the control parameters providing periodic oscillations of the pendulum are described. A mathematical model of the inverted pendulum with feedback control is given under the conditions of the horizontal motion of the suspension point. The conditions that guarantee the stabilization of the considered system are obtained; the conditions are formulated in terms of constraints on the initial conditions. The solution to the problem of the optimal control of an oscillating system is presented in the sense of minimization of a quadratic goal functional. The stabilization problem for an unstable system with distributed parameters, the flexible inverted pendulum, is also considered, and the stabilization conditions are formulated. Fulfilment of these conditions ensures the boundedness of the phase coordinates in the infinite interval of time. The optimal parameters (in the sense of minimization of a quadratic goal functional) corresponding to stabilization of the distributed system are identified.