ESE 499 Senior Design

 Jim Brockman   Dr. Hiro Mukai (Faculty Advisor)

Optimum Assignment Problem Extension of "Sequential Linear-Quadratic Method for Differential Games with Air Combat Applications"

April 15, 2006


In June of 2003, the paper "Sequential Linear-Quadratic Method for Differential Games with Air Combat Applications" presented software modeling air combat situations alongside a method for computing optimal control inputs and trajectories for different forces based on the current state of the combat environment.  The software envisions two forces: a Red Force and a Blue Force.  Each force consists of various units of vehicles (fighters, bombers, missile launchers, etc.), each of which serve a particular purpose.  In situations involving only two units, one Red and one Blue, the solution converged quickly to a local saddle solution to the original game problem.  However, when multiple units were introduced on each side, a solution did not converge and was not always evident.  The purpose of this research project is to introduce a superstructure to the original program, which would analyze the current situation for possible assignment patterns within each force before proceeding with an air combat simulation. 


       A Summary

     Having only two units in the environment, an optimal state trajectory can be calculated iteratively, to a local saddle point solution to the original game problem.  The Red bomber unit approaches its final target attempting to avoid the Blue unit, and the Blue interceptor unit pursues the Red unit, and then eventually approaches its final target.  Below is a screen shot of a simulation of one on one combat.

Figure 1:  1 on 1 combat simulation

     But when multiple units are introduced into the environment, the program breaks down and a solution does not converge.  Therefore, the purpose of this project is to introduce a superstructure to the original software, that will assign units to each other.  The pairs of units engage in combat, utilizing the original program.  The purpose of the superstructure is to analyze the environment at each time step in order to reassign units based on attributes of each unit. 

    The attributes analyzed in making pair assignments are unit strength and distance between units.  Each force always wants to maintain a strength advantage, and each force wants to engage with unit close by, so as not to waste resources traversing the environment.  The following is a display of a two on two combat simulation.  Due mainly to proximity, since all four units begin with the same strength, Blue 1 is assigned to Red1 and Blue 2 is assigned to Red2.  Although each craft does not complete its course, that is, finish at its target, its objective (Blue intercept Red units, Red avoid Blue units) is evident.

Figure 2: 2 on 2 combat simulation


I conclude from this project that deterministic situations are difficult to simulate due to the inherent complexities and nonlinearities that exist in making such decisions.  Having only two attributes available, there are many possibilities that require different actions to take place.  The situation becomes more complex as more units are introduced into the environment.  Although distances are relatively simple to analyze, incorporating the strength differences with the units' distances becomes a complex problem.  Analyzing strength alone can be difficult, for what makes a good assignment?  Units close in strength to each other, or a unit being far superior in strength than its opponent?  Add more units into the simulation, and then there is also the question of a greater good or how the assignment can help the overall cost for the team. 

It should also be noted that in real combat situations, many of the variables analyzed in this simulation would be unknown.  Hence, the decision process here could be viewed stochastically by taking expectations of attributes.