Minimax optimal control for uncertain systems

The issue of performance robustness is considered in a dynamic game framework where the goal is to find a feedback controller which stabilizes the system under all plant perturbations and which minimizes the value of a quadratic performance index under the worst perturbations. To find a solution to this problem an auxiliary dynamic game problem is formulated. The game is between the controller, which is minimizing a performance index related to the original quadratic index, and the perturbation disturbance, which at worst (from the point of the controller´s view) is maximizing the same index. Finding a solution to the auxiliary problem may be easier than finding a solution to the original problem, especially when there exists a saddle point solution for the auxiliary problem but not for the original problem. It is shown that the solution to the auxiliary problem provides the solution to the original problem, under the assumption that the solution to the original problem occurs on the boundary of the perturbations set. The approach, however, can still be applied, with modifications, to the case when this assumption fails.