In many areas of engineering and sciences, decision rules and control strategies are usually designed based on nominal values of relevant system parameters. To ensure that a control strategy or decision rule will work properly when the relevant parameters vary within certain range, it is crucial to investigate how the performance measure is affected by the variation of system parameters. In this paper, we demonstrate that such issue boils down to the study of the variation of functions of uncertainty. Motivated by this vision, we propose a general theory for inferring function of uncertainties. By virtue of such theory, we investigate concentration phenomenon of random vectors. We derive uniform exponential inequalities and multidimensional probabilistic inequalities for random vectors, which are substantially tighter as compared to existing ones. The probabilistic inequalities are applied to investigate the performance of control systems with real parametric uncertainty. It is demonstrated much more useful insights of control systems can be obtained. Moreover, the probabilistic inequalities offer performance analysis in a significantly less conservative way as compared to the classical deterministic worst-case method.
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