ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Data-Driven Model Reduction

Thu 16:30–17:00, Room AV 02.17
A Linear Cross Operator for Nonlinear Model Reduction
Christian Himpe (University of Muenster)
Joint work with Mario Ohlberger (University of Münster)

Input-output systems are a popular class of models mapping an input to an output function via a state, which is the solution to a differential equation and may be high-dimensional. Model reduction for such systems aims to reduce this intermediary state-space dimension.

The cross operator encodes the input-output coherence of the underlying system’s states [2]. For square linear systems this operator is known as the cross gramian matrix and computable as the solution to a Sylvester equation [3]. Alternatively, a cross gramian can be obtained empirically by accumulated inner products of discrete state and output trajectory components [1]. This empirical cross gramian relies purely on simulated or measured trajectory data, and beyond linear systems it is also applicable to nonlinear systems.

We illustrate the empirical cross operator’s background in linear system theory and present numerical properties and results as well as variants for non-square and parametric systems, which for the latter can additionally convey identifiability information on the parameters.

  1. C. Himpe and M. Ohlberger. Cross-Gramian Based Combined State and Parameter Reduction for Large-Scale Control Systems. Mathematical Problems in Engineering, 2014:1–13, 2014.
  2. T. Ionescu, K. Fujimoto, and J. Scherpen. The Cross Operator and the Singular Value Analysis for Nonlinear Symmetric Systems. In Proceedings of the European Control Conference 2009, pages 1565–1570, 2009.
  3. D. Sorensen and A. Antoulas. The Sylvester equation and approximate balanced reduction. Linear Algebra and its Applications, 351–352:671–700, 2002.