Electrical & Computer Engineering Faculty Publications

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Many control problems are so complex that analytic techniques fail to solve them [2]. Furthermore, even when analytic solutions are available, they may be computationally costly [2] and generally result in very high-order compensators [3]. Due to these reasons, we tend to accept approximate answers which provide us with certain performance guarantees for such problems. Sampling methods thus come into the picture to try and remedy the “cost of solution” problem by drawing samples from an appropriate space, and providing an approximate answer. For many years, random sampling has dominated the afore mentioned arena [8, 11, 4]. Random sample generation, with a uniform underlying distribution, however tends to cluster the samples on the boundary of the sample space in higher dimensions. It is for this reason that we are interested in presenting a method that distributes the points regularly in the sample space while providing deterministic guarantees on the error involved. Recently, deterministic or quasi-Monte Carlo (QMC) methods have proven superior to random methods in several applications such as the calculation of certain integrals [6], financial derivatives [7] and motion planning in robotics [10]. They have also been used for stability analysis of high speed networks [9]. In this work, we provide an overview of such deterministic quasi-Monte Carlo method of sampling, and their applications to control systems analysis and design. We present the basic concepts pertaining to quasi-Monte Carlo deterministic sampling. Such concepts include the following: Indicator functions, performance objective, generation of point sets, total variation, and error bounds.

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