Low nonconvexity-rank bilinear matrix inequalities: algorithms and applications in robust controller and structure designs

A branch and bound (BB) algorithm for solving a general class of bilinear matrix inequality (BMI) problems is proposed. First, linear matrix inequality (LMI) constraints are incorporated into BMI constraints in a special way to take advantage of useful information on nonconvex terms. Then, the nonconvexity of the BMI is centralized in coupling constraints so that when the latter are omitted, we get a relaxed LMI problem for computing lower bounds. As in our previous developments, the branching is performed in a reduced dimensional space of complicating variables. This makes the approach practical even with a large dimension of overall variables. Applications of the algorithm to several test problems of robust control are discussed.