A scenario approach to non-convex control design: set-based probabilistic guarantees

S. Grammatico, X. Zhang, K. Margellos, P. J. Goulart and J. Lygeros

in American Control Conference, Portland, OR, USA, pp. 3431-3436, June 2014.
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  author = {S. Grammatico and X. Zhang and K. Margellos and P. J. Goulart and J. Lygeros},
  title = {A scenario approach to non-convex control design: set-based probabilistic guarantees},
  booktitle = {American Control Conference},
  year = {2014},
  pages = {3431-3436},
  url = {http://dx.doi.org/10.1109/ACC.2014.6859142},
  doi = {10.1109/ACC.2014.6859142}

Randomized optimization is a recently established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable for a certain class of non-convex problems, but they usually are conservative in terms of performance and are computationally demanding. In this paper, we present a novel scenario approach for a wide class of random non-convex programs. We provide a sample complexity similar to the one for uncertain convex programs, but valid for all feasible solutions inside a set of a-priori chosen complexity. Our scenario approach applies to many non-convex control-design problems, for instance control synthesis based on uncertain bilinear matrix inequalities.