First-order corrections in optimal feedback control of singularly perturbed nonlinear systems

In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications. This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer. Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized. The new method is applied to an example of a constant speed minimum-time interception problem. Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods.