# Preprint No. MPIMD/13-17

#### Author(s): Peter Benner, André Schneider

#### Email: andre.schneider@mpi-magdeburg.mpg.de

#### Date: 2013-10-07

#### Abstract:

The model reduction method introduced in [Benner, P. and Schneider, A.;
*Balanced Truncation Model Order Reduction for LTI Systems with many Inputs
or Outputs*, in A. Edelmayer: Proceedings of the 19th International Symposium on
Mathematical Theory of Networks and Systems, 2010, ISBN/ISSN: 978-963-311-370-7]
shows how to reduce linear time-invariant (LTI) continuous-time state space
systems with either many inputs or many outputs using the well-known balanced
truncation approach. We call this method balanced truncation for many terminals
(BTMT). In this work we generalize BTMT to descriptor systems of the form

Eẋ(t) = Ax(t) + Bu(t), A, E ∈ ℝ^{n×n},B ∈ ℝ^{n×m}

y(t) = Cx(t) + Du(t), C ∈ ℝ^{p×n}, D∈ ℝ^{p×m},

where m ∈ 𝒪(n) and p ≪ n, or vice versa. We show how to
obtain a reduced order model by solving one Lyapunov equation and using the
Gauss-Kronrod quadrature to compute the needed projection matrices. In
particular, we discuss the case when E is singular and show numerical results.

#### BibTeX:

@TECHREPORT{MPIMD13-17,

author = {Peter Benner and André Schneider},

title = {Balanced Truncation for Descriptor Systems with Many Terminals},

number = {MPIMD/13-17},

month = oct,

year = 2013,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}