{
  "id": "2303.11015",
  "version": "v1",
  "published": "2023-03-20T10:49:50.000Z",
  "updated": "2023-03-20T10:49:50.000Z",
  "title": "A Notion of System Comparison",
  "authors": [
    "Armin Pirastehzad",
    "Arjan van der Schaft",
    "Bart Besselink"
  ],
  "categories": [
    "math.OC",
    "math.DS"
  ],
  "abstract": "We introduce $(\\gamma,\\delta)$-similarity, a notion of system comparison that measures to what extent two dynamical systems behave similarly in an input-output sense. This behavioral similarity is characterized by measuring the sensitivity of the difference between the two output trajectories in terms of the external inputs to the two potentially non-deterministic systems. As such, $(\\gamma,\\delta)$-similarity is a notion that characterizes \\emph{approximation} of input-output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the $L_2$ signal norm, $(\\gamma,\\delta)$-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of $(\\gamma,\\delta)$-similarity as a linear matrix inequality feasibility problem and derive its interpretations in terms of transfer matrices. Our study on the compositional properties of $(\\gamma,\\delta)$-similarity shows that the notion is preserved through series and feedback interconnections. This highlights its applicability in compositional reasoning, namely abstraction and modular synthesis of large-scale interconnected dynamical systems. We further illustrate our results in an electrical network example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-20T10:49:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "system comparison",
      "similarity",
      "linear matrix inequality feasibility problem",
      "dynamical systems",
      "robust control techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}