We develop nonsmooth optimization techniques to solve H∞ synthesis problems under additional structural constraints on the controller. Our approach avoids the use of Lyapunov variables and therefore leads to moderate size optimization programs even for very large systems. The proposed framework is versatile and can accommodate a number of challenging design problems including static, fixed-order, fixed-structure, decentralized control, design of PID controllers and simultaneous design and stabilization problems. Our algorithmic strategy uses generalized gradients and bundling techniques suited for the H∞ norm and other nonsmooth performance criteria. We compute descent directions by solving quadratic programs and generate steps via line search. Convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods. The propose approach proves to be efficient even for systems with several hundreds of states.

Abstract. The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets A, B under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is O(k −ρ ) for some ρ ∈ (0, ∞).

We introduce regularity notions for averaged nonexpansive operators. Combined with regularity notions of their fixed point sets, we obtain linear and strong convergence results for quasicyclic, cyclic, and random iterations. New convergence results on the Borwein-Tam method (BTM) and on the cylically anchored Douglas-Rachford algorithm (CADRA) are also presented. Finally, we provide a numerical comparison of BTM, CADRA and the classical method of cyclic projections for solving convex feasibility problems.2010 Mathematics Subject Classification: Primary 65K05; Secondary 47H09, 90C25.

Abstract. We present a new approach to parametric robust controller design, where we compute controllers of arbitrary order and structure which minimize the worst-case H ∞ norm over a pre-specified set of uncertain parameters. At the core of our method is a nonsmooth minimization method tailored to functions which are semi-infinite minima of smooth functions. A rich test bench and a more detailed example illustrate the potential of the technique, which can deal with complex problems involving multiple possibly repeated uncertain parameters.