Modified Quasflinearization Methods,
- J. F. Andrus
- Rept. No. 69, Math. Dept., Uoiv. of New Orleans,
of Technical Progress In earlier work the necessary conditions of optimality were derived for a problem of minimum miss-distance guidance of air-to-air missiles. The mdei was based upon nonlinear translational equations of moticn. The solution of the necessary conditions requires a solution of a two-point boundary-condition problem. Two methods proposed for the latter solution, an elliptic integral method and a series technique, were studied and both methods were rejected in favor of a procedure based upon the quasilinearization method. The latter requires fewer assumptions and exhibits excellent convergence prooerties. order to rer'ove the numerical integration problem and to simplify the linear two-point boundary-condition problem associated with quasilinearization, the regular method was modified, three alternative techniques being deriyed, and a technical report was wTitten which discusses the convergence properties and accuracy of the t'.ree mod:fied quasilinearization methods applied to two-point b-undarv-ccnditicn prcciemsin general. , One of the latter metiods was applied to the missile guidance problem and compared to a linear method of guidance. The comparison was favorable in the c¢ses of missile encounters considered. A second technical report was prepared which describes the application and the numerical results. Aoce131on For NTIS ;A& I By_ DI s t rbit Ion/ Availability Codes 7~vall and/or Disut SpeCAaI Technical Report The principal objective was the development of an optimal guidance method fo; air-to-air missiles. The proposed model was based upon nonlinear translational equations of motion and a quadratic performance index. Availability of a predicted time-history of the target's position was assum,d. In an earlier investigation the necessary conditions of optimality were derived and two methods were proposed for solving these conditions. They were: An elliptic integral method and a series method. These were to be placed in a form suitable for compur-tion and applied to the solution of the necessary conditions, which reduce to a two-point Doundary-condition (TPBC) problem associated .ith a system of nonlinear ordinary differential equations. As far as possible the following extensions (among others) to the ellipt. intr:egral aad series methods were to be made: (a) Introduction of variable missile velocity magnitude; (b) Extension into three dimensions; (C) Allowance for variations of more than 90 in the flight direction angies, (d> incusion of the case of control angles which do not vary monotonicallv with time; (e) Inclusion of an improved method for calculating time-to-go; (f) Placing bounds upon the control variables. After a study of the methods based upon elliptic integrals and series, it was decided that a third meLhod based upon quasilinearization (or the generalized Nr ,tonRaphson method) would be a more powerful computational device (as far as numerical convergence to a solution to the 7PBC problem is concerned) and a more flexible tool in regard to the implementation of extensions (a) (f) given above. On consulting with the ?ro'ram Manager, it was decided thdt the quasilinearization method would be used rather than the methods originally proposed for study.