Robust control via concave minimization local and global algorithms

This paper is concerned with the robust control problem of linear fractional representation (LFT) 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 nonconvex algebraic constraint. The nonconvexity 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, but also more generally rank inequalities and bilinear constraints, can be formulated as the minimization of a concave functional subject to LMI constraints. First of all, a local Frank and Wolfe (1956) feasible direction algorithm is introduced in this context to tackle this hard optimization problem. Exploiting the attractive concavity structure of the problem, several efficient global concave programming methods are then introduced and combined with the local feasible direction method to secure and certify global optimality of the solutions. Computational experiments indicate the viability of our algorithms, and in the worst case, they require the solution of a few LMI programs.