Optimization in control system design

This paper reviews i) the background for the modern H^∞ control theory, and then ii) how it led to the modern optimization theory such as those using LMIs (linear matrix inequalities) and BMIs (bilinear matrix inequalities). Starting from the simplest sensitivity minimization problem, we give solutions via the Nevalinna–Pick interpolation and the Nehari theorem. The latter leads to a Riccati equation on which most of the H^∞ solutions are based. This in turn leads to the modern approach using mathematical programming such as LMIs and BMIs. Two types of global optimization algorithms to solve BMIs are introduced.