Computational Complexity in Robustness Analysis and Design

Abstract

For many engineering analysis and design problems, one can think about various different types of algorithms with quite different features. Depending on the specific application, “general”, “practical”, and even “optimal” may have different meanings. In this chapter, we will be following the standards defined in the theory of computation, (Garey, Johnson (1979) Computers and intractability, a guide to the theory of NP-completeness. W. H. Freeman, San Francisco; Papadimitriou (1995) Computational complexity. Addison-Wesley/Longman, Reading), but we also would like to present an engineering interpretation of these concepts. First of all, an algorithm is considered as “general”, if it really provides a correct answer for all cases of the problem. Although this makes perfect sense from a scientific perspective, it may be too restrictive from an engineering/economical viewpoint. For example, a technique which can be used for most cases may also be considered as “general” enough. An algorithm is considered as “practical” if its execution time is reasonable. From a scientific perspective, the worst case execution time looks more appealing, but for many engineering projects average execution time may be more meaningful. Similarly, there is no universally definition for reasonable execution time, but most scientists and engineers agree that polynomial execution time is reasonable, and anything beyond that is not. In this entry, we will look at some of the classical robust control problems, and try to present an engineering interpretation of the relevant computational complexity results.