On the Prevention of Vibrations in the Problem of the Time-Optimal Control of a System with Two Degrees of Freedom

We study a mechanical system with two degrees of freedom, consisting of two absolutely rigid bodies (material points) connected to each other by a weightless rectilinear viscoelastic rod that can be stretched or compressed. The bodies can move translationally along a fixed straight line. A control force limited in absolute value is applied to one of them, whose vector is directed along the rod. A continuous piecewise-linear control in time, which transfers the system from one equilibrium position to another in a length of time close to the minimum possible time, is constructed. In the absence of viscosity, with the constructed quasi time-optimal control, unwanted vibrations of the bodies are not excited either during the transition process or when it ends. In contrast to the time-optimal relay control, the constructed continuous control is also robust with respect to the uncertainty of the design parameters.