14. Bilinearity and Complementarity in Robust Control
Mehran Mesbahi, Michael G. Safonov, George P. Papavassilopoulos · Society for Industrial and Applied Mathematics eBooks · 1999
14.1 Introduction In this chapter, we present an overview of the key developments in the methodological, structural, and computational aspects of the bilinear matrix inequality (BMI) feasibility problem. In this direction, we present the connections of the BMI with robust control theory and its geometric properties, including interpretations of the BMI as a rank-constrained linear matrix inequality (LMI), as an extreme form problem (EFP), and as a semidefinite complementarity problem (SDCP). Computational implications and algorithms are also discussed. The simultaneous appearance of the (unknown) variables x and y in the matrix in-equality ∑ i=1 n ∑ j=1 m xi yj Fij >0, 14.1 for a given set of symmetric matrices not only provides an unexpectedly powerful formulation for a wide range of robust control problems, but it also introduces new and elegant structural and computational questions. A possible initial attempt to rename the products as in (14.1) and rewriting it in terms of 's as an LMI [64], ∑ i,j zij Fij >0, 14.2 introduces yet another twist to this problem, since the unknown variables 's are now constrained to be related in a rather peculiar manner, for example, zij z (i+1) (j+1) = xi yi xi+1 yi+1 = zi (j+1) z (i+1) j.