Nonsmooth H_∞Synthesis

Abstract

We develop nonsmooth optimization techniques to solve H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> 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 proposed approach proves to be efficient even for systems with several hundreds of states.