Stabilization of steady motions of systems with geometric constraints and cyclic coordinates

Abstract

We consider mechatronic systems with nonlinear geometric constraints, cyclic coordinates and one or more electric drives with DC motors. It is assumed that control is carried out by changing the voltage on the anchor windings of the motors. The solvability condition for the nonlinear stabilization problem of possible stationary motions and a method for determining control actions are investigated. A mathematical model of the system dynamics is constructed using the Lagrange variables. The nonlinearity of the constraints was taking into account by means of the vector-matrix equations in the form of Shulgin with redundant coordinates. The equations of actuators dynamics were explicitly included into the constructed mathematical model. It is shown that the stability of stationary motions of such systems is possible only in critical cases. The number of zero roots of the characteristic equation is not less than the number of constraints. We analyze the structure of the obtained vector-matrix equations using the results of the theory of critical cases and then formulate a theorem on a sufficient condition for the stabilization of stationary motions. Coefficients of stabilizing effects can be found by the method of N. N. Krasovsky by solving the linear-quadratic stabilization problem for a controlled subsystem of lower dimension.