Balanced truncation model order reduction in limited time intervals for large systems
For engineers and scientists simulating large-scale dynamical systems, this work provides a more accurate model reduction method when only a specific time interval is of interest, though it is an incremental extension of existing balanced truncation techniques.
This paper extends balanced truncation model order reduction to finite time intervals, using rational Krylov subspace methods to efficiently compute low-rank approximations of matrix exponentials and Lyapunov equations. Numerical experiments show that time-limited balanced truncation yields reduced models with higher accuracy in the specified time region and that the numerical rank of Lyapunov solutions can be smaller than in standard balanced truncation.
In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.