Global optimization for robust control synthesis based on the Matrix Product Eigenvalue Problem

Abstract

AbstractIn this paper, we propose a new formulation for a class of optimization problems which occur in general robust control synthesis, called the Matrix Product Eigenvalue Problem (MPEP): Minimize the maximum eigenvalue of the product of two block‐diagonal positive‐definite symmetric matrices under convex constraints. This optimization class falls between methods of guaranteed low complexity such as the linear matrix inequality (LMI) optimization and methods known to be NP‐hard such as the bilinear matrix inequality (BMI) formulation, while still addressing most robust control synthesis problems involving BMIs encountered in applications.The objective of this paper is to provide an algorithm to find a global solution within any specified tolerance ε for the MPEP. We show that a finite number of LMI problems suffice to find the global solution and analyse its computational complexity in terms of the iteration number. We prove that the worst‐case iteration number grows no faster than a polynomial of the inverse of the tolerance given a fixed size of the block‐diagonal matrices in the eigenvalue condition. Copyright 2001 © John Wiley & Sons, Ltd.