Empirical differential Gramians for nonlinear model reduction

Yu Kawano*, Jacquelien M.A. Scherpen

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)
24 Downloads (Pure)


In this paper, we present an empirical balanced truncation method for nonlinear systems whose input vector fields are constants. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory), and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems.

Original languageEnglish
Article number109534
Early online date27-Feb-2021
Publication statusPublished - May-2021


  • Balanced truncation
  • Model reduction
  • Nonlinear systems
  • Proper orthogonal decomposition

Cite this