Regularization for design

Nikolai Matni, Venkat Chandrasekaran

Abstract: An algorithmic bridge is starting to be established between sparse reconstruction theory and distributed control theory. For example, ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -regularization has been suggested as an appropriate means for co-designing sparse feedback gains and consensus topologies subject to performance bounds. In recent work, we showed that ideas from atomic norm minimization could be used to simultaneously co-design a distributed optimal controller and the communication delay structure on which it is to be implemented. While promising and successful, these results lack the same theoretical support that their sparse reconstruction counterparts enjoy - as things stand, these methods are at best viewed as principled heuristics. In this paper, we describe theoretical connections between sparse reconstruction and systems design by developing approximation bounds for control co-design problems via convex optimization.