By now it is known that several problems in the robustness analysis and synthesis of control systems are NP-complete or NP-hard. These negative results force us to modify our notion of “solving” a given problem. An approach that is recently gaining popularity is that of using
randomized algorithms. In this paper, it is argued that many problems in robust control can be formulated as the minimization of the expected value of an objective function with respect to the controller parameters. Then it is shown that, whenever a property from statistical learning theory known as uniform convergence of empirical means (UCEM) holds, there exists an efficient (i. e., polynomial-time) randomized algorithm. Using recent results in VC-dimension theory, it is shown that the UCEM property holds in several problems such as robust stabilization and weighted
H∞-norm minimization. Hence it is possible to solve such problems efficiently, in a probabilistic sense, using randomized algorithms. A complete version of the contents of the present paper can be found in 14).
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