The mathematical modelling of the dynamics of systems with redundant coordinates in the neighborhood of steady motions

This paper describes a method of use of equations in M. F. Shul'gin's form in Lagrangian variables for steady motion stability and stabilization problems of systems with geometric constraints. These equations of motion are free from Lagrange multipliers; we substantiate their advantage for solving stability and stabilization problems. Depended coordinates corresponding to zero solutions of characteristic equation are allocated in the disturbed equations of motion. These variables are necessarily present in systems with geometric constraints for any control method. It is suggested to present equations of motion in Routh variables for finding stabilizing control coefficients; Lagrangian variables are more useful for constructing an estimation system of object state. In addition to previous results, we evaluate the ability to reduce the dimension of measured output signal obtained in conformity with the chosen modelling method. Suppose the state of system is under observations and the dimension of measurement vector is as little as possible. Stabilizing linear control law is fulfilled as feedback by the estimation of state. We can determine uniquely the coefficients of linear control law and estimation system can be determined uniquely by solving of the corresponding linear-quadratic problems for the separated controllable subsystems using the method of N. N. Krasovsky. The valid conclusion about asymptotical stability of the original equations is deduced using the previously proved theorem. This theorem is based on the nonlinear stability theory methods and analysis of limitations imposed by the geometric constraints on the initial disturbances.