Finite representations and solutions of continuous minimax decision problems with applications to control of hybrid systems
Many problems in Robotics involve Machine - Real World interactions. We focus on two important aspects of such problems: the interplay between continuous and discrete phenomena, and the inherent uncertainty of the environment. Frequently, such problems can be cast as Statistical Decision Problems. In these problems minimax solutions are often the most robust and, therefore, the most desirable. Furthermore, minimax solutions provide performance guarantees, facilitating analysis and prediction of system behaviors under real-world uncertainty. However, with few exceptions, minimax solutions have been computed one specific problem at a time, making it difficult to apply them to complicated systems. We develop a compact finite natural representation of continuous minimax problems, which can be used to solve such problems under a wide variety of conditions. Since the representation is compact, it affords fast and efficient computation with a potential to simplify analysis of hybrid systems. We demonstrate the application of this method by computing solutions of previously unsolved problems. In particular, we solve a long-standing problem of finding a minimax estimator for a restricted location parameter problem under Cauchy noise and 0–1 loss. We also solve this minimax parameter estimation problem for location mixtures of Cauchy distributions. Furthermore, we apply our method to minimax decision problems with multidimensional parameter spaces and noise distributions with dependent components. Finally, as an important application of these ideas, we develop the beginnings of a framework for the design and implementation of control laws for hybrid systems under uncertainty, and illustrate this approach through examples.
Cherkassky, Dmitry, "Finite representations and solutions of continuous minimax decision problems with applications to control of hybrid systems" (2001). Dissertations available from ProQuest. AAI3031648.