Deferred-Correction Optimal Control with Applications to Inverse Problems in Flight Mechanics

Abstract

Wedevelopanumericalprocedurefortheinversesimulationofe ight,formulatedasanoptimal-controlproblem. The resulting method is general and applicable to both e xed and rotating wing vehicles. We use the p version of the e nite-element-in-time method, equipped with a deferred-correction acceleration technique for achieving the desired level of accuracy in the response at a low computational cost. We test our numerical procedures with the aid of some representative simulations. E study the inverse simulation of e ight for both e xed and rotating wing aircrafts. The problem is formulated by using optimal-control theory. Numerical procedures for inverse problems in e ight mechanics have been presented in Refs. 1 -8. Among these, Refs. 3 and 8 use optimal-control theory for studying the problem of inverse simulation, and they are therefore closely related to the present work. Our numerical approach is based on a direct, e nite-element dis- cretization of the classic variational principle of optimal control. This idea was e rst proposed in Ref. 9. References 10 -15 detail the application of this technique to all the major aspects of optimal control, including control inequality constraints, state constraints, unknown e nal time, and multiphase problems, and they present in- teresting applications of this technique. We proved in Ref. 16 that this procedure is equivalent to the use of a certain class of global, implicit Runge -Kutta (RK) schemes. This result is valid for the p version of the method, that is, for arbi- trarily high order. Similar results were derived in Ref. 17 for initial value problems. The analysis shows that the quadrature rule used for evaluating the integrals in the weak form plays a major role, in practice determining the algorithmic properties of the resulting e nite-element-in-time (FET) scheme. Using the Gauss -Legendre, Lobatto, and Radau -Left quadrature rules, one obtains FETs that correspond to the Kuntzmann -Butcher (Gauss), Lobatto IIIB, and Radau IA RK schemes, respectively. The e rst two are effective solvers for boundary-value problems. In fact, the publicly available codes COLNEW 18 and ACDC 19 are respectively based on Gauss and Lobatto IIIA RK schemes. Notwithstandingtheequivalenceofthemethods,webelievethere is still a motivated desire to look at FETs. For example, some new developments such as a posteriori error estimation might be easier to accomplish in oneframework ratherthan another. 20 Furthermore, additional insight can usually be obtained by a unie ed view of any given problem. Thesolutionofaninverseproblembytheoptimal-controlmethod implies having to deal with the solution of a differential algebraic boundary-value problem dee ned over the whole maneuver interval. Discretization by FETs in turn requires the solution of a nonlinear global problem on a given mesh. The system of discrete equations is obtained by assembling the contributions from each element and appending the other boundary and constraint conditions. Although the resulting matrices are banded and highly sparse, the desire for very quick response times and high numericalperformance calls for efe cient acceleration techniques.