A stabilization method for steady motions with zero roots in the closed system

Abstract

Based on previous results, we consider stabilization problems for both non-asymptotic stability and asymptotic stability with respect to all variables for equilibrium positions and stationary motions of mechanical systems with redundant coordinates. The linear stabilizing control is defined by the solution of a linear–quadratic stabilization problem for an allocated linear controllable subsystem of as small dimension as possible. We find sufficient conditions under which a complete nonlinear system closed by this control is ensured asymptotic stability despite the presence of at least as many zero roots of the characteristic equation as the number of geometric relations. We prove a theorem on the stabilization of the control equilibrium applied only with respect to redundant coordinates and constructed from the estimate of the phase state vector obtained by a measurement of as small dimension as possible.