Quantified multivariate polynomial inequalities. The mathematics of practical control design problems

This article describes how a large number of practical feedback design problems can be reduced to the study of quantified multivariate polynomial inequalities (MPIs). However, the computation required to solve quantified MPI problems is very intensive. As defined here, most practical control problems do not have analytical solutions. Three approaches for the study of this class of mathematical problems are reviewed: symbolic quantifier elimination methods, Bernstein branch-and-bound methods, and probabilistic (Monte Carlo) methods. The three approaches are listed in order of computational complexity required for a solution, with symbolic computation the most computationally complex and probabilistic methods the least.