Synthesis of a control in a non-linear dynamical system based on decomposition

Abstract

A non-linear controllable dynamical system with many degrees of freedom, described by Lagrange equations of the second kind, is considered. Geometric constraints are imposed on the magnitudes of the controls. It is assumed that, in the equations of motion, the kinetic energy matrix is close to a certain constant diagonal matrix. It is possible, for example, to reduce the equations of motion of robots, the drives of which have large gear ratios, to a system of this kind. A problem is formulated on the transfer of a system in a finite time from a specified initial state to a final state with zero velocities. The method of decomposition [1] is used to construct the equations. Sufficient conditions are found subject to which the maximum values of the non-linear terms in the equations of motion do not exceed the permissible magnitudes of the controls. In this case, non-linearities are treated as limited perturbations and the system is decomposed into independent, linear, second-order subsystems. A feedback control is specified for these subsystems which guarantees that each of them is brought into the final state for any permissible perturbations. The control has a simple structure. Applications of the proposed approach to problems in the control of manipulating robots are considered.