Discrete mechanics and its application to vibration suppression control for 1-dimensional distributed parameter mechanical systems

This paper develops a new discretizing method for distributed parameter mechanical systems, called “discrete mechanics,” and proposes a new control method by blending discrete mechanics and nonlinear optimization. First, some fundamental concepts of discrete mechanics are shown and discrete Euler-Lagrange equations are derived based on discrete Hamilton's principle. Next, for the discrete Euler-Lagrange equations with control inputs, we formulate a nonlinear optimal control problem with constraints by setting an objective function and initial/boundary conditions. Then, it is shown that the problem can be represented as a finite-dimensional nonlinear optimal problem with constraints and it can be solved by the sequential quadratic programming method. After that, vibration suppression control problems for strings and Euler-Bernoulli beams are solved by the proposed method. As a result, simulation results show that vibration of the string and the Euler-Bernoulli beam is suppressed and the whole of these systems is stabilized.