Krasinskii

A method of stability investigation and stabilization of equilibrium of systems with geometrical constraints was elaborated and applied to real mechatronic device that is GBB1005 Ball & Beam. For constructing mathematical model Shulgin's equations with redundant coordinates were utilized. In order to perform stability investigation it’s necessary to combine Shulgin’s equations with kinematic constraint equations, derived by differentiating geometrical constraint equations with respect to time. First approximation equations in the neighborhood of equilibrium position have zero roots; the number of these roots is equal to the number of constraints. Asymptotic stability equilibrium in mechanical system with redundant coordinates is possible, despite formal reduction to Lyapunov's special case. This work presents a more complete non-linear model of the mechanical part of GBB1005 Ball & Beam bench. One more equilibrium position of the system was found during the investigation of complete non-linear cons. At the presence of one geometrical constraint between two coordinates there are two options for selecting a redundant coordinate. It was shown that the choice of linear control subsystem depends on the choice of redundant coordinate

For mechanical systems with geometric constraints application of the Shulgin method for obtaining motion equations without joining factors, which is based on a simple derivation of constraint equations, is considered in this article. Obtained motion equations for holonomic system in redundant coordinates are a special case of the Voronetz equations for non-holonomic systems in case of integrable constraints. The method for using these equations is developed for stability and stabilization problems of the equilibrium position of systems with redundant coordinates. The proposed approach allows to use previously obtained results based on the theory of critical cases. The problem of non-asymptotic stability was tackled by reducing it to the Lyapunov-Malkin theorem of stability for a special case of several zero roots. In this paper, the authors prove that consideration of constraints on initial perturbations asymptotically stabilizes the equilibrium state despite formal reduction to a special case, if the number of zero roots of the characteristic equation is equal to the number of constraints and real parts of other roots are negative. Efficiency of this approach was verified by investigating the stabilization problem of the equilibrium state for real mechatronic workbench GBB1005 Ball&Beam Educational Control System.