Nonsmooth H ∞ Synthesis

Pierre Apkarian (1) , Dominikus Noll (2)

1 ONERA - The French Aerospace Lab [Toulouse]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We develop nonsmooth optimization techniques to solve H∞ synthesis problems under additional structural constraints on the controller. Our approach avoids the use of Lya-punov 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 nons-mooth 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.

Mots clés: linear matrix inequality (LMI); H∞-synthesis; static output feedback; fixedorder synthesis; simultaneous stabilization; NP-hard problems; nonsmooth optimization; bundle methods; Clarke subdifferential; bilinear matrix inequality (BMI)

Domaines: Optimisation et contrôle [math.OC]

Pierre Apkarian, Dominikus Noll. Nonsmooth H ∞ Synthesis. IEEE Transactions on Automatic Control, 2006, 51 (1), pp.71-86. ⟨10.1109/TAC.2005.860290⟩. ⟨hal-01868781⟩