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 12:00–12:30, Room AV 02.17
Computing Gramians of Quadratic-Bilinear Systems from Data
Peter Benner (Max-Planck-Institute for Dynamics of Complex Technical Systems)
Joint work with Pawan Goyal (Max Planck Institute for Dynamics of Complex Technical Systems)

We discuss balanced truncation model reduction for continuous time quadratic-bilinear systems (QBDAEs). Balanced truncation for linear systems mainly involves the computation of the Gramians of the system, namely controllability and observability Gramians. These Gramians are extended to the general nonlinear setting in Scherpen (1993), where it is shown that Gramians for nonlinear systems are solutions to Hamilton-Jacobi equations, which, in general, depend on the state vector. Therefore, they are not only difficult to solve for large-scale systems, but also hard to utilize in the model reduction framework. In this talk, we aim to derive algebraic Gramians for quadratic-bilinear systems based on Volterra series of QBDAEs, which are solutions to generalized quadratic Lyapunov equations. These algebraic Gramians, in quadratic forms, approximate controllability and observability energy functionals for QBDAEs. Moreover, we give a characterization of input and output energies based on the proposed algebraic Gramians for QBDAEs. Furthermore, we make use of these Gramians for balancing of quadratic-bilinear systems in order to determine reduced-order models (ROMs). Additionally, we come up with truncated Gramians for QBDAEs and present their advantages in model reduction. Finally, we discuss how these Gramians can be computed using simulation data only.