{
  "id": "1404.1972",
  "version": "v2",
  "published": "2014-04-08T00:00:54.000Z",
  "updated": "2015-01-18T18:51:17.000Z",
  "title": "Regularization for Design",
  "authors": [
    "Nikolai Matni",
    "Venkat Chandrasekaran"
  ],
  "comment": "Submitted to the IEEE Transactions on Automatic Control",
  "categories": [
    "math.OC",
    "cs.SY"
  ],
  "abstract": "When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given. The task of designing this architecture is now as important as the design of the control laws themselves. By interpreting controller synthesis (in a model matching setup) as the solution of a particular linear inverse problem, we view the challenge of obtaining a controller with a desired architecture as one of finding a structured solution to an inverse problem. Building on this conceptual connection, we formulate and analyze a framework called $\\textit{Regularization for Design (RFD)}$, in which we augment the variational formulations of controller synthesis problems with convex penalty functions that induce a desired controller architecture. The resulting regularized formulations are convex optimization problems that can be solved efficiently, and these problems are natural control-theoretic analogs of prominent approaches such as the Lasso, the Group Lasso, the Elastic Net, and others that are employed in statistical modeling. In analogy to that literature, we show that our approach identifies optimally structured controllers under a suitable condition on a \"signal-to-noise ratio\" type quantity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-08T00:00:54.000Z",
      "abstract": "An algorithmic bridge is starting to be established between sparse reconstruction theory and distributed control theory. For example, $\\ell_1$-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. We also give a concrete example of a design problem for which our approach provides approximation guarantees.",
      "comment": "Extended CDC'14 submission -- includes proofs and additional discussion not included in the submitted version due to space constraints",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-18T18:51:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regularization",
      "sparse reconstruction counterparts enjoy",
      "co-designing sparse feedback gains",
      "control co-design problems",
      "communication delay structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.1972M"
    }
  }
}