A New Efficient Method for Optimization of a Class of Nonlinear Dynamic Systems Without Lagrange Multipliers

摘要

This paper deals with optimization of a class of nonlinear dynamic systems with n states and m control inputs commanded to move between two fixed states in a prescribed final time. Using conventional procedures with Lagrange multipliers, it is well known that the optimal trajectory is the solution of a two-point boundary value problem. The solution is obtained using multiple shooting methods which are computation intensive. The procedure also requires a good guess for Lagrange multipliers. Unfortunately, Lagrange multipliers can not be guessed since they are artificial variables.In this paper, a new procedure for dynamic optimization of this problem is presented that does not use Lagrange multipliers. The procedure relies on the tools of feedback linearization where it is possible to transform classes of nonlinear dynamic systems into linear systems through nonlinear change of coordinates and control. The new form is such that the states and control can be written as higher derivatives of a subset of the states. Using this new form, it is possible to change the constrained dynamic optimization problem into an unconstrained dynamic optimization problem. The fundamentals of calculus of variations are then applied to this new cost functional and the necessary conditions for optimality are obtained. These necessary conditions are then efficiently solved using weighted residual methods. The computational efficiency of this procedure makes it attractive for realtime implementation.