Robust control via concave minimization local and global algorithms

This paper deals with the robust control problem of linear fractional representation uncertain systems depending on a time-varying parameter uncertainty. Our main result exploits a linear matrix inequality (LMI) characterization involving scalings and Lyapunov variables subject to an additional essentially non-convex algebraic constraint. The non-convexity enters the problem in the form of a rank deficiency condition or matrix inverse relation on the scalings only. It is shown that such problems and many others can be formulated as the concave minimization of a nonlinear functional subject to LMI constraints. The local Frank and Wolfe feasible direction algorithm is introduced. Several efficient global concave minimization programming methods are also introduced and combined with the local feasible direction method to secure and certify global optimality of the solutions. The implementation details of the algorithms are covered.