Linear stabilization of programmed motions of non-linear controlled dynamical systems

A non-linear controlled dynamical system (NCDS), describing the dynamics of a wide class of non-linear mechanical and electromechanical systems, is considered. A technique is proposed for synthesizing control laws with linear feedback according to state, which describe the stabilization of programmed motions of such systems. A non-singular linear transformation of the state space is constructed, bringing the initial NCDS in deviations (from its programmed motion and programmed control) to a certain NCDS of a special form, which is convenient for analysing and synthesizing control laws governing the motion of the system. A NCDS of canonical form is separated out from the initial NCDS in deviations. The aforementioned non-singular linear transformation of coordinates of the state space and the method of Lyapunov functions are used to synthesize control laws with linear state feedback, which guarantee global symptotic stability of an equilibrium position of a NCDS of canonical form and asymptotic stability in the large of a NCDS of special form and of the initial NCDS in deviations. Estimates are given for the domain of asymptotic stability in the large of the equilibrium positions of a NCDS of special form, of the initial NCDS in deviations, and of programmed motions of the initial NCDS, closed by the synthesized stabilizing controls.