Singular Differential Game Numerical Technique and Closed Loop Guidance and Control Strategies

摘要

In this report a class of linear quadratic pursuit-evasion games with bounds on the pursuer's control has been defined. In this problem the pursuer's control appears linearly in the Hamiltonian and singular arcs may occur in the solution. A sufficient condition for the existence of a saddle point for this class of problems is derived. An indirect numerical technique has been proposed to generate a rapid and accurate solution to a class of problems with linear state equations in which singular arcs occur. Then, by linearization this technique is extended to solve a class of problems with non-linear state equations. By modifying this technique a second approach has been obtained that can solve a broader class of singular problems with linear state equations. The effect of the deviation of one player from the saddle point strategy on the performance index and the opponent's strategy has been studied for a two person zero-sum differential game with perfect information. An inverse system technique is used to determine the opponent's strategy by periodically measuring the state or the output of the system. Then, the proposed technique for singular problems is applied periodically to generate an approximate closed loop solution (which takes into consideration the deviation of the opponent from the saddle point trajectory) to achieve better performance than simply following the optimal open loop solution. A numerical example is presented to illustrate the efficiency of the proposed algorithm, and, comparisons have been made between the results of the open loop and closed loop solutions.