Learning Bilinear Models of Actuated Koopman Generators from Partially Observed Trajectories

Samuel Otto, Sebastian Peitz, Clarence Rowley

Research output: Contribution to journalArticlepeer-review


Data-driven models for nonlinear dynamical systems based on approximating the underlying Koopman operator or generator have proven to be successful tools for forecasting, feature learning, state estimation, and control. It has become well known that the Koopman generators for control-affine systems also have affine dependence on the input, leading to convenient finite-dimensional bilinear approximations of the dynamics. Yet there are still two main obstacles that limit the scope of current approaches for approximating the Koopman generators of systems with actuation. First, the performance of existing methods depends heavily on the choice of basis functions over which the Koopman generator is to be approximated; and there is currently no universal way to choose them for systems that are not measure preserving. Second, if we do not observe the full state, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address these issues, we write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model and determine the model parameters using the expectation-maximization algorithm. The E step involves a standard Kalman filter and smoother, while the M step resembles control-affine dynamic mode decomposition for the generator. We demonstrate the performance of this method on three examples, including recovery of a finite-dimensional Koopman-invariant subspace for an actuated system with a slow manifold; estimation of Koopman eigenfunctions for the unforced Duffing equation; and model-predictive control of a fluidic pinball system based only on noisy observations of lift and drag.

Original languageEnglish (US)
Pages (from-to)885-923
Number of pages39
JournalSIAM Journal on Applied Dynamical Systems
Issue number1
StatePublished - 2024

All Science Journal Classification (ASJC) codes

  • Analysis
  • Modeling and Simulation


  • Koopman generator
  • bilinear system
  • control-affine system
  • data-driven system identification
  • expectationmaximization algorithm
  • hidden Markov model


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