Some Approaches to Optimal Pursuit Evasion Air Combat Games
by
Fumiaki Imado
Mechanical Systems Engineering Shinshu University

Some Approaches to Optimal Pursuit Evasion Air Combat Games Fumiaki IMADO Professor, Department of Mechanical Systems Engineering, Shinshu University 4-17-1 Wakasato Nagano, Nagano 380-8553 JAPAN Extended Abstract A two player game is the problem where there is a common performance index and one player strives to maximize it, while the other to minimize it. If the whole process is dynamic, and expressed by a set of ordinary differential equations,we call the problem as "differential games". The kind of problems have attracted considerable interest in recent years, and many studies have appeared in the literatures. However, their results still seem to be difficult to apply to actual dynamic games. That is, most studies have been devoted to obtaining precise mini-max solutions for very simplified problems, which are practically of no use in engineering point of view, and the obtained solutions are often trivial. The early studies of Isaacs and Merz showed the existence of a large number of different kinds of solutions for a very simple problem called as "Homicidal chauffeur".Actually an enormous number of solutions may exist for large-dimensional nonlinear differential games, and it is almost impossible to know all the different kinds of solutions. As our purpose is to supply some sophisticated control algorithms for practical engineering systems, it will be helpful to us to find some features of the solutions under some assumptions, even if they are not precisely derived by a differential game formulation.

We have been studying to obtain game solutions of practical engineering meanings for realistic pursuit evasion games. One of the idea is, by giving apriori optimal or suboptimal feedback strategies to both players, then conducting massive simulations in the parameter space of initial geometries and guidance law parameters, and analysing the results. Although statistically very small probability, however,we find sometimes very skillful strategies for both players in the results. We will call the method as the "method I".

If one of the player has perfect informations about the strategy of the other, he can calculate the nonlinear optimal control against the former. Although it may be time consuming, but at least possible in principle, and recent development of high speed computer has enabled its onboard calculation. The strategy of the latter may be improved, however, the optimal control of the former is also adapted because this is a perfect information game. There is a possibility that by successively improving the strategy of the former, the game approaches to a true differential game, however, as the nonlinear optimal control of the latter must be calculated only by numerically, and not expressed by an analytical form, it is usually very difficult. We will call the metod as the "method II".In the methods I and II, we can treat multiple players' games. A new algorithm of our own development to treat them, and some results are also shown.

In the method I, if the pre-given suboptimal controls of both players are expressed as parameter dependent feedback strategies, then by solving numerically a set of one-sided optimal control for each player against a set of parameters of the opponent, the optimal values of the parameters are found. If the difference between the outcomes of the two parameter optimization problems is small, the feedback strategies using the optimal parameters represent an approximation of the saddle point strategies of the game. We will call the method as the "method III". Perhaps this is one of the most hopeful methods to solve an exact differential game problem, however, the difficulty exists in finding the form of the suboptimal control with a parameter dependent feedback strategy. The form must approximate the real solution farely well, otherwise the method will be failed. One of our successful examples will be shown.

Date received: January 10, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cago-21.