Nonlinear Optimal Control Analysis Using State-Dependent Matrix Exponential and Its Integrals

THERE exist various strategies [1,2] for reaching numerical 1 solutions to the nonlinear optimal control problem. The directmultiple-shooting (DMS) method is usually preferred for analyzing general nonlinear optimal control problems due to its convenience in handling system constraints and a large convergence radius compared with other methods. However, these advantages can be decreased in case the estimation of the related Karush-Kuhn-Tucker (KKT) system is not accurate enough to guarantee robust analyses. The related estimation errors are originated from time integration and finite difference approximation in the standard DMS method. The present work intends to propose a new method of accurately estimating the KKT system. For this purpose, the state-dependent coefficient (SDC) factorization method, which has been successfully implemented in the state-dependent Riccati equation (SDRE) technique [3-5], is used to derive a linear system structure from the nonlinear motion equation. Applying the linear system theory to the resultant SDC form of equations, the KKT system can be built without resorting to any time integration and finite difference formula to calculate of gradients and Hessian matrices, as opposed to the standard DMS method. The present paper proves that the convergence and the accuracy of the DMS method can be greatly enhanced through the applications of the new method to the rotorcraft trajectory tracking problem.