MOR for Differential Algebraic Equations

Differential algebraic equations (DAEs) or descriptor systems are dynamical systems with algebraic constraints. Model order reduction (MOR) of such systems is extremely usefull for their simulation, control and optimization. Most of the model reduction techniques are based on projection of the original states of the system in a lower dimensional subspace. Standard projection of DAEs however has the issue that the approximation error between the original and the reduced systems may increases (unboundedly) with the increase in frequency. Modified projection techniques are therefore used to ensure bounded approximation error.

The focus of our work is on the following three projects:
1. MOR for (non-)linear DAEs
2. MOR for flow problems
3. MOR for bilinear ODEs.

MOR for (non-)linear DAEs: To ensure bounded approximation error, MOR techniques for linear DAEs require the following:

Introduce left and right spectral projectors to decompose linear DAEs into strictly proper ODEs and polynomial parts.
Reduce the ODE part and retain the polynomial part to obtain the required reduced system.

Explictly identifying spectral projectors is known to be computationally complex. However, it is shown in the literature that for some special linear DAEs, one can achieve the splitting of the DAE with out explicitly identifying the spectral projectors. We observed that this is also possible for second order linear DAEs [8].

Recently in [3], we extended this idea to some nonlinear DAEs having special structured linear part. The reduction scheme ensure bounded approximation error in contrast to the standard projection approach.

MOR for flow problems: Flow problems, in particular, those modelled by Navier-Stokes equation have been considered in linear as well as nonlinear settings. We extended the MOR idea in [7] to unstable linear DAEs and tested the reduction scheme on linearized Navier-Stokes equations with high Reynolds number. For nonlinear systems, two-sided moment-matching technique has been proposed [1] for MOR of Stokes-type quadratic-bilinear systems.

MOR for bilinear ODEs: Bilinear systems are weakly nonlinear systems that are widely used in the literature for analysis and control. They are also closely related to linear parametric systems [5]. Unlike linear systems, the input output repesentation (Volterra series) of bilinear systems in the frequency domain have an infinite series of generalized multivariate transfer functions which are related to the Laplace transform of the kernels in the associated Volterra series. Interpolatory techniques for MOR of bilinear systems require interpolation of the first K generalized transfer functions. Recently in [4] the bilinear iterative rational Kryloav algorithm (B-IRKA) has been proposed, which on convergence ensure that the first order necessary conditions for H₂ optimality are satisfied by the reduced bilinear system. We also implemented the B-IRKA method for some actual examples from industry [2a].

References

[1] M. I. Ahmad, P. Benner, P. Goyal, J. Heiland, Moment-Matching Based Model Reduction for Stokes-Type Quadratic-Bilinear Descriptor Systems, Preprint MPIMD/15-18, Max Planck Institute Magdeburg, 2015.

[2] P. Benner and T. Breiten, Two-sided projection methods for nonlinear model order reduction, SIAM J. Sci. Comput. 37(2), B239–B260 (2015).

[3] P. Benner and P. Goyal, Multipoint Interpolation of Volterra Series and H₂-Model Reduction for a Family of Bilinear Descriptor Systems, Preprint MPIMD/15-16, Max Planck Institute Magdeburg, 2015.

[4] P. Benner and T. Breiten, Interpolation-based H₂-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 859–885.

[5] P. Benner and T. Breiten, On H₂-model reduction of linear parameter-varying systems, Proc. Appl. Math. Mech., 11 (2011), pp. 805–806.

[6] C. Gu, QLMOR: A projection-based nonlinear model order reduction approach using quadraticlinear representation of nonlinear systems, IEEE Trans. Computer-Aided Design Integrated Circuits Systems, 30 (2011), pp. 1307–1320.

[7] M. Heinkenschloss, D. C. Sorensen, and K. Sun, Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations, SIAM J. Sci. Comput., 30 (2008), pp. 1038-1063.

[8] M. Uddin, J. Saak, B. Kranz and P. Benner, Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation, Prod. Eng., 6 (2012), pp. 577–586.

PhD Thesis

[1a] T. Breiten, Interpolatory Methods for Model Reduction of Large-Scale Dynamical Systems, PhD thesis, Otto-von-Guericke Universität Magdeburg, 2013.

[2a] A. S. Bruns, Bilinear H₂-optimal Model Order Reduction with Applications to Thermal Parametric Systems, PhD thesis, Otto-von-Guericke Universität Magdeburg, 2015.

[3a] M. M. Uddin, Computational Methods for Model Reduction of Large-Scale Sparse Structured Descriptor Systems, PhD thesis, Otto-von-Guericke Universität Magdeburg, 2015.

Formar Project Members

Dr. Tobias Breiten
Dr. Monir M. Uddin
Dr. Angelika S. Bruns