Any order approximate solution of the state equation for controlled nonlinear systems

Based on the solution of the linear state equation, the mean square envelope matrix can be obtained for a linear approximation to the controlled nonlinear system, and the transfer rule of mean square envelope matrix for the nonlinear system is discussed. By making of the Taylor expansion, the nonlinear state equation of controlled systems under ideal state is transformed to a set of ordinary differential equations with infinite series expression. Based on the solution of this set of linear state equation, the integral equation of the nonlinear state equation is also obtained by utilizing constant variation method. Finally, any order approximate solution of the nonlinear state equation is given by utilizing successive approximation method.