Probabilistic Methods for Uncertain Systems

This chapter introduces the key concepts for the probabilistic approach to analysis and design of uncertain systems, such as the performance function, the good and the bad sets, and the probabilistic degradation function. As pointed out in Chap. 5, many pessimistic results on the complexity-theoretic barriers of classical robust control have stimulated research in the direction of finding alternative approaches. One of these approaches, which constitutes the main subject treated in this book, proceeds by first shifting the concept of robustness from its usual deterministic sense to a probabilistic one. In this respect, we claim that a certain system property is “almost robustly” satisfied if it holds for “most” of the instances of the uncertainty. In other words, one accepts the risk of this property being violated by a set of uncertainties having small probability measure. Such systems may be viewed as being practically robust. Various examples are introduced to illustrate these concepts.