On applications of singular perturbation techniques in nonlinear optimal control

The technique of singular perturbations (SPT) has been applied with considerable success in several nonlinear optimal control problems. In many cases the zero-order approximation of the optimal control function has been expressed in a feedback form. This paper deals with topics involved in such closed-loop application, which seem to merit further discussion. It is formally demonstrated that a ‘forced’ singular perturbation model (obtained by artificial insertion of the perturbation parameter) results in the same zero-order composite feedback control solution as a classical singularly perturbed model (where a small parameter of physical significance appears as a consequence of a scaling transformation). The accuracy of the zero-order feedback approximation depends in both cases on the actual time scale separation of the variables. Two inherent limitations of the feedback solution are also pointed out: (1) first and higher-order correction terms of the zero-order approximation have to be computed by a predictive or off-line integration; (2) on-line implementation of SPT control strategy in a terminal boundary layer requires iterative computations. A simple pursuit problem serves as an illustrative example.