Koopman Operators for Estimation and Control of Dynamical Systems

Samuel E. Otto, Clarence W. Rowley

Research output: Contribution to journalReview articlepeer-review

46 Scopus citations


A common way to represent a system's dynamics is to specify how the state evolves in time. An alternative viewpoint is to specify how functions of the state evolve in time. This evolution of functions is governed by a linear operator called the Koopman operator, whose spectral properties reveal intrinsic features of a system. For instance, its eigenfunctions determine coordinates in which the dynamics evolve linearly. This review discusses the theoretical foundations of Koopman operator methods, as well as numerical methods developed over the past two decades to approximate the Koopman operator from data, for systems both with and without actuation. We pay special attention to ergodic systems, for which especially effective numerical methods are available. For nonlinear systems with an affine control input, the Koopman formalism leads naturally to systems that are bilinear in the state and the input, and this structure can be leveraged for the design of controllers and estimators.

Original languageEnglish (US)
Pages (from-to)59-87
Number of pages29
JournalAnnual Review of Control, Robotics, and Autonomous Systems
StatePublished - May 3 2021

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Engineering (miscellaneous)
  • Human-Computer Interaction
  • Control and Systems Engineering


  • dynamic mode decomposition
  • ergodic theory
  • kernel methods
  • neural networks
  • stochastic systems
  • system identification
  • transfer operator


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