Henry J. Kelley

A second variation test for singular extremals

The second variation for the Mayer problem is examined with reference to singular test extremals and evaluated for the case of a special variation in a control variable. By means of a limiting process, a necessary condition for a minimum is obtained from the requirement of positive semidefiniteness of the second variation. Some applications of the test to singular extremals arising in flight mechanics problems are presented.

An optimal guidance approximation theory

Synthesis of optimal guidance approximations is undertaken by means of a perturbation theory approach and a scheme for obtaining linear, quadratic and higher order feedback approximations to the optimal control presented. Mechanization is carried out in terms of state deviations from an optimal reference trajectory with the comparison point along the reference selected so that the state comparison is approximately transverse. A simple example is treated analytically, linear and quadratic feedback expressions obtained, and some digital computer simulation results presented which permit a comparison of system performance and accuracy between various approximations.

Flight path optimization with multiple time scales

Flight path optimization with multiple time scales, discussing decoupling of high order three dimensional aircraft flight problem into several low order problems

Aircraft maneuver optimization by reduced-order approximation

Publisher Summary This chapter discusses the principles of aircraft maneuver optimization by reduced-order approximation. The essential feature of the energy type of approximation is reduction in the order of the state differential system. The chapter describes craft flight in terms of singular perturbation theory and in extending it to three-dimensional maneuvers. It examines singular perturbations for differential equations arising in optimal control for a system of fairly general form but low order. Optimal aircraft flight in various reduced-order approximations are investigated, and results of some numerical computations reviewed. The chapter focuses on singular perturbations of the state-Euler system of differential equations and on the optimal aircraft flight application. The computation of higher-order corrections in the expansion parameter should be investigated and carried out for suitable examples, as a matter of research interest. The complexity of the computations would seem comparable to that of second-order variational methods, but the problems of ill-conditioning and numerical-error propagation ought to be mitigated by the time-scale separation. A higher order correction process for turn-dash modeled aircraft flight has been examined.

Energy management of three-dimensional minimum-time intercept

A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission. The proposed scheme has roots in two well known techniques: singular perturbations and neighboring-optimal guidance. Use of singular-perturbation ideas is made in terms of the assumed trajectory-family structure. A heading/energy family of prestored point-mass-model state-Euler solutions is used as the baseline in this scheme. The next step is to generate a near-optimal guidance law that will transfer the aircraft to the vicinity of this reference family. The control commands fed to the autopilot (bank angle and load factor) consist of the reference controls plus correction terms which are linear combinations of the altitude and path-angle deviations from reference values, weighted by a set of precalculated gains. In this respect the proposed scheme resembles neighboring-optimal guidance. However, in contrast to the neighboring-optimal guidance scheme, the reference control and state variables as well as the feedback gains are stored as functions of energy and heading in the present approach. Some numerical results comparing open-loop optimal and approximate feedback solutions are presented.

SINGULAR EXTREMALS IN LAWDEN'S PROBLEM OF OPTIMAL ROCKET FLIGHT

Derivation of the singular extremals of lawdens problem of optimal rocket flight by use of the legendre-clebsch necessary condition applied in a transformed system of state and control variables

Singular perturbations for a Mayer variational problem

Optimal trajectory and control approximate solution by asymptotic expansion, considering Mayer variational problem solution

Fixed-trim re-entry guidance analysis

The terminal guidance problem for a fixed-trim re-entry body is formulated with the objective of synthesizing a closed-loop steering law. A transformation of variables and subsequent linearization of the motion, with the sight-line to the target as a reference, reduces the order of the state system for the guidance problem. The reduced order system, although nonlinear and time-varying, is simple enough to lend itself to synthesis of a class of guidance laws. A generalization of the feedforward device of classical control theory is successfully employed for compensation of roll autopilot lags. The proposed steering law exhibits superior miss-distance performance in a computational comparison with existing fixed-trim guidance laws.

Reprisal Strategies in Pursuit Games

A preference-ordered discrete-gaming model, developed for pursuit-evasion studies in an earlier paper, is first reviewed and some refinements are described: graduation of draw outcomes via a threat-reciprocity concept and an improved cell buildup technique. A reprisal-strat egy scheme, which exploits opponents errors by extrapolation, is described and illustrated in an air-to-air combat example. The results suggest that the approach is quite promising.

Singular Optimal Atmospheric Rocket Trajectories

Presently studied is the problem of the ascent and acceleration of a vehicle in atmospheric flight in which a variable-thrust arc forms a part of the optimal trajectory. A two-parameter family of singular arcs was generated for time-range-fuel problems of an ascending rocket, using the modeling of Zlatskiy and Kiforenko. The shortterm optimality of singular subarcs has been checked in terms of certain necessary conditions: the classical Clebsch condition, the Kelley condition or the Generalized Legendre-Clebsch condition, and the Goh condition. All of these are found to be satisfied computationally for all the candidates. The calculations were repeated for simplified thrust-along-the path modeling and similar results on optimality were obtained.