Robust control via concave minimization local and global algorithms

This paper is concerned with the robust control problem of LFT (Linear Fractional Representation) uncertain systems depending on a time-varying parameter uncertainty. Our main result exploits an LMI (Linear Matrix Inequality) characterization involving scalings and Lya-punov variables subject to an additional essentially non-convex algebraic constraint. The non-convexity enters the problem in the form of a rank deeciency 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 Linear Matrix Inequality constraints. First of all, a local Frank and Wolfe feasible direction algorithm is introduced in this context to tackle this hard optimization problem. Exploiting the attractive concavity structure of the problem, several eecient global concave programming methods are then introduced and combined with the local feasible direction method to secure and certify global optimality of the solutions. Convergence and practical implementation details of the algorithms are covered. Stopping criteria are introduced in order to reduce the overall computational overhead. Computational experiments indicate the viability of our algorithms, and that in the worst case they require the solution of a few LMI programs. Power and eeciency of the algorithms are demonstrated through realistic and randomized numerical experiments. Frank and Wolfe algorithms.