On control of motion of a parametric pendulum

The paper is devoted to a passive control problem. The problem of control of plane motions of a two-mass parametric pendulum in a uniform gravitational field is considered. The problem is important for and necessary in software design of automated systems for control of mechanisms. In particular, it can be applied to various modeling problems of pendulum motions of mechanical systems. The pendulum is modeled by two equivalent weightless rods with two equivalent point masses moving along the circle centered at the pivot. The control is carried out by varying continuously the angle between two rods. It is a function that depends on the representative point of the gravity center of pendulum in the phase plane. Two control processes of excitation and damping pendulum near the lower equilibrium position by swing principle are constructed. The problem is resolved by the method of Lyapunov's functions known from the classical theory of stability. The control is obtained in the form of closed form solution in the class of continuous functions. The obtained results are an important contribution to development of control mechanisms in engineering.