A bundle method for solving the fixed order control problem

Bundle methods have been proposed in the literature for solving semi-definite programming (SDP) problems which consist of minimizing a linear objective subject to linear matrix inequality (LMI) constraints. These methods are appealing especially for large scale problems. Many control problems can be cast as SDPs. In contrast fixed order control problems in the LMI framework require an additional rank constraint on the LMIs which makes the problem insolvable with traditional SDP methods. One way of imposing the rank constraint is to formulate the fixed order control problem in terms of the eigenvalues of the LMIs. This makes the problem nondifferentiable, but amenable to bundle methods.