A linear programming approach to approximating the infinite time reachable set of strictly stable linear control systems

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Abstract

The infinite time reachable set of a strictly stable linear control system is the Hausdorff limit of the finite time reachable set of the origin as time tends to infinity. By definition, it encodes useful information on the long-term behavior of the control system. Its characterization as a limit set gives rise to numerical methods for its computation that are based on forward iteration of approximate finite time reachable sets. These methods tend to be computationally expensive, because they essentially perform a Minkowski sum in every single forward step. We develop a new approach to computing the infinite time reachable set that is based on the invariance properties of the control system and the desired set. These allow us to characterize a polyhedral outer approximation as the unique solution to a linear program with constraints that incorporate the system dynamics. In particular, this approach does not rely on forward iteration of finite time reachable sets.

Original languageEnglish
Pages (from-to)521–543
Number of pages23
JournalJournal of Global Optimization
Volume86
Issue number2
DOIs
Publication statusPublished - 2 Dec 2022

Keywords

  • Discrete-time linear systems
  • Disjunctive program
  • Linear optimization
  • Numerical approximation
  • Polytopes
  • Reachable set

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