A Novel Modal Series Representation Approach to Solve a Class of Nonlinear Optimal Control Problems

This paper presents a new approach to solve a class of nonlinear optimal control problems which have a quadratic performance index. In this approach, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s maximum principle, is transformed into a sequence of linear time-invariant TPBVP’s. Solving the proposed linear TPBVP sequence in a recursive manner leads to the optimal control law and the optimal trajectory in the form of uniformly convergent series. Hence, to obtain the optimal solution, only the techniques of solving linear ordinary differential equations are employed. In order to use the proposed method in practice, a control design algorithm with low computational complexity and fast convergence rate is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Finally, numerical examples are included to demonstrate efficiency, simplicity and high accuracy of the proposed method.