{
  "id": "2206.01782",
  "version": "v1",
  "published": "2022-06-03T19:01:07.000Z",
  "updated": "2022-06-03T19:01:07.000Z",
  "title": "Optimal Competitive-Ratio Control",
  "authors": [
    "Oron Sabag",
    "Sahin Lale",
    "Babak Hassibi"
  ],
  "categories": [
    "math.OC",
    "cs.LG",
    "cs.SY",
    "eess.SY"
  ],
  "abstract": "Inspired by competitive policy designs approaches in online learning, new control paradigms such as competitive-ratio and regret-optimal control have been recently proposed as alternatives to the classical $\\mathcal{H}_2$ and $\\mathcal{H}_\\infty$ approaches. These competitive metrics compare the control cost of the designed controller against the cost of a clairvoyant controller, which has access to past, present, and future disturbances in terms of ratio and difference, respectively. While prior work provided the optimal solution for the regret-optimal control problem, in competitive-ratio control, the solution is only provided for the sub-optimal problem. In this work, we derive the optimal solution to the competitive-ratio control problem. We show that the optimal competitive ratio formula can be computed as the maximal eigenvalue of a simple matrix, and provide a state-space controller that achieves the optimal competitive ratio. We conduct an extensive numerical study to verify this analytical solution, and demonstrate that the optimal competitive-ratio controller outperforms other controllers on several large scale practical systems. The key techniques that underpin our explicit solution is a reduction of the control problem to a Nehari problem, along with a novel factorization of the clairvoyant controller's cost. We reveal an interesting relation between the explicit solutions that now exist for both competitive control paradigms by formulating a regret-optimal control framework with weight functions that can also be utilized for practical purposes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-03T19:01:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal solution",
      "clairvoyant controller",
      "control paradigms",
      "optimal competitive-ratio controller outperforms",
      "explicit solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}