Control of a platform with oscillators under the action of dry friction

We study a local control problem for a system consisting of a solid carrier and several linear dissipative oscillators attached to it. The carrier moves along a horizontal straight line under a horizontal steering force. The system is a simple approximation of a model describing controlled motions of a vessel with a viscous fluid. Since the state of the fluid in the vessel is unknown at any specific time, the physical parameters of the oscillators and their phase states are also considered unknown. It is assumed that a dry fiction force acts between the carrier and the straight line, and the parameters of the dry friction are unknown and varying. It is required to bring the carrier to a stop at a given terminal position and to keep it at that position; no constraints are imposed on the behavior of the oscillators after the carrier stops. We propose a feedback control law with bounded absolute value that brings the carrier from a neighborhood of the terminal position to this position in a finite time. The control is given by a function that is smooth (analytic) everywhere except for the terminal position. This function can be interpreted as a linear feedback with coefficients depending on the state variables. Although the coefficients grow unboundedly as the carrier approaches the terminal position, the control remains bounded. The efficiency of the control is illustrated by means of numerical modelling.