Probabilistic robust controller design: probable near minimax value and randomized algorithms

This paper presents a probabilistic approach to robust controller design. This design can be recast as a minimax problem with a cost function. In order to solve the problem efficiently, the definition of probable near minimax value is introduced. A probable near minimax value of the function can be calculated with a certain accuracy and a certain confidence by using a randomized algorithm, where independent identically distributed samples of optimized parameters are generated according to probability measures. It is shown that the necessary number of the samples depends on the accuracy and the confidence and is independent of the number of the parameters. Furthermore, a special case such that the cost function has a global saddle point is investigated. The definition of probable near saddle value, which is weaker than that of probable near minimax value, is introduced. Then, it is shown that the necessary number of samples is smaller in this case.