On applications of Attractive Ellipsoid Method to dynamic processes governed by implicit differential equations

This paper deals with the application of the attractive (invariant) ellipsoid method for stabilization of class of the, so-called, implicit systems whose dynamics cannot be represented in the standard Cauchy form given by some ODE resolved with respect to the states of derivates. This class of dynamics systems includes, as a particular case, the models whose part of state-components is given in ODE-format while the rest of them represent only some algebraic nonlinear relations of states. To design a stabilizer as a linear state-feedback we suggest to apply the descriptive method with vector Lagrange multipliers in the Lyapunov stability analysis. The suggested technique leads to the sufficient conditions of the global practical stability which are shown to be expressed in BMI (bilinear matrix inequality) form. The last, after some coordinate transformation, can be converted to LMI (linear matrix inequalities) under fixed scalar parameters arising during the Lyapunov function construction. Results of numerical simulation realized by the standard MATLAB packages application illustrates the effectiveness of the suggested approach.