Excitation and stabilization of equilibrium by parametrical constrained control of a two-mass pendulum

Theoretical mechanics


Аuthors

Mukhametzyanova A. A.

Samara National Research University named after Academician S.P. Korolev, 34, Moskovskoye shosse, Samara, 443086, Russia

e-mail: alain.20@mail.ru

Abstract

The paper considers the task of plane motion control of a two-mass parametric pendulum. The purpose of the work is to build new control laws with specified properties that implement processes of excitation and damping the pendulum near the lower equilibrium position. Precisely, the possibility of a pendulum excitation and swing damping parametric control near the lower equilibrium position are discussed. The problem is solved by the method of Lyapunov’s functions of the classical theory of stability.

Two equivalent weightless rods pin-edge fixed at the suspension generic point model the pendulum. Two equal point masses are attached to the ends of each rod. The pendulum allows rotational or oscillatory movements in a vertical plane around the suspension point. Along with these movements in homogenous gravitational field, the pendulum has two equilibrium positions. As is known, the lower position is steady and upper the position is unsteady.

Let us consider the possibility of control realization by implementation of continuous variation of the angle between the two rods. In this case the control law can be set by the function depended on the representative point of motion of the gravity center of pendulum. Natural restriction on the gravity center movement along the bisectrix of angle between the two rods is assumed. Let us consider that the control law has continuous derivative.

In this paper two control laws of processes of excitation and damping pendulum are developed. The corresponding Lyapunov’s functions proving the asymptotic pendulum lower position stability and instability in cases of the pendulum damping and excitation are sel ected. It is shown, that with controlled movements according to the first law the asymptotic oscillations amplitude damping of pendulum occurs for any initial conditions of the movement. When implementing control according to the second law the increase of the amplitude and the transition fr om oscillatory to rotational motions takes place. The theoretical results are confirmed by numerical modeling.

The results of this work may present applied technical interest and be applied while flat pendulum movement of various mechanical systems modeling and control.

Keywords:

two-mass pendulum , Lyapunov's function, constrained control, asymptotical stability, the swing principle

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