Modellordnungsreduktion durch Krylow-Unterraum-Methoden

In model order reduction of linear (and recently nonlinear systems), algorithms based on Krylov subspace methods (also known as moment matching and rational interpolation) have gained a wide popularity due to their simplicity, low computational cost and flexibility, making them a predestined candidate in the reduction of truly large-scale models.

In this setting, the moments are defined as the coefficients of the Taylor series expansion of the transfer function around a certain complex frequency. If the reduction is performed using bases of particular Krylov subspaces, then the reduced order model is guaranteed to match specific moments of the full order model. The main advantage of this approach is that it involves only the solution of sparse linear systems of equations and hence results in low computational effort and small memory requirements. The main open problems and thus active fields of research within this family of methods are:

• Guaranteeing preservation of certain properties (such as stability, passivity, ...) of the reduced-order model,
• Obtaining rigorous estimations on the approximation error,
• Finding a suitable parametrization for the reduction procedure, e.g. location of the matching frequencies, reduced order...
• Extending the validity to other system classes (DAEs, bilinear, quadratic bilinear, nonlinear, ...)

In our group, we actively work on all these topics. In particular, our methods can automatically choose suitable reduction parameters (interpolation frequencies, reduced order, ...) to obtain reduced order models of high fidelity (often being optimal with respect to system norms such as H2 and H8) and providing error bounds and/or error indicators.

The algorithms developed at MORLab are included in our MATLAB toolboxes and are free for download.

Related References

A. Castagnotto, C. Beattie and S. Gugercin, Interpolatory methods for H-infinity model reduction of multi-input/multi-output systems, In: Model Reduction of Parametrized Systems, 2017, 349-365.

A. Castagnotto, H.K.F. Panzer and B. Lohmann, Fast H2-optimal model order reduction exploiting the local nature of Krylov-subspace methods, European Control Conference (ECC), 2016, 1958-1969.

A. Castagnotto, H.K.F. Panzer, T. Wolf and B. Lohmann, Advances on the Adaptive Selection of Both Shifts and Reduced Order in H2-Pseudo-Optimal Model Reduction, IFAC-PapersOnLine 48 (1), 2015, 679-680.

T. Wolf, ℌ 2 Pseudo-Optimal Model Order Reduction, Ph.D Thesis, Institute of Automatic Control, Technical University of Munich, 2014.

H.K.F. Panzer, Model order reduction by Krylov subspace methods with global error bounds and automatic choice of parameters Ph.D Thesis, Institute of Automatic Control, Technical University of Munich, 2014.

T. Wolf, H.K.F. Panzer and B. Lohmann, ℌ 2 pseudo-optimality in model order reduction by Krylov subspace methods, European Control Conference (ECC), 2013, 3427-3432.

H.K.F. Panzer, T. Wolf and B. Lohmann, H2 and H∞ error bounds for model order reduction of second order systems by Krylov subspace methods, European Control Conference (ECC), 2013, 4484-4489.

H.K.F. Panzer, S. Jaensch, T. Wolf and B. Lohmann, A greedy rational Krylov method for ℋ 2-pseudooptimal model order reduction with preservation of stability, American Control Conference (ACC), 2013, 5512-5517.

R. Castane-Selga, B. Lohmann and R. Eid, Stability preservation in projection-based model order reduction of large scale systems, European Journal of Control, 2012.

T. Wolf, H. Panzer and B. Lohmann, Sylvester equations and a factorization of the error system in Krylov-based model reduction, Vienna Conference on Mathematical Modelling (MATHMOD), 2012, 25-35.

T. Wolf, H.K.F. Panzer and B. Lohmann, Gramian-based error bound in model reduction by Krylov subspace methods, IFAC Proceedings Volumes 44 (1), 2011, 3587-3592.

T. Wolf, B. Lohmann, R. Eid and P. Kotyczka, Passivity and structure preserving order reduction of linear port-Hamiltonian systems using Krylov subspaces European Journal of Control, 2010, 401-406.

R. Castane-Selga, R. Eid and B. Lohmann, On the stability of Krylov-based order reduction using invariance properties of the controllability subspace, In Roppencker, G. und Lohmann, B. (Hrsg.): Methoden und Anwendungen der Regelungstechnik. Erlangen-Münchener Workshops 2007 und 2008. Shaker Verlag, Aachen, 2009.

R. Eid, Time Domain Model Reduction by Moment Matching, Ph.D Thesis, Institute of Automatic Control, Technische Universität München, 2009.

B. Salimbahrami, R. Eid, and B. Lohmann, On the choice of an optimal interpolation point in Krylov-based order reduction, Proc. of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 2008.

R. Eid and B. Lohmann, Moment Matching Model Order Reduction in Time Domain via Laguerre Series, Proceedings of the IFAC world Congress, South Korea, July 2008.

Eid, R., Salimbahrami, B. and Lohmann, B., Equivalence of Laguerre-based Order Reduction and Moment Matching, IEEE Trans. on Automatic Control, Vol. 52, No. 6, June 2007, pp. 1104-1108.

Salimbahrami, B. and Lohmann, B., Stable Reduced Order Modelling of Large Scale Systems using Prescribed Poles, IFAC World Congress, Prag, Czech Rep., 2005.

Ji, L. Y., Salimbahrami, B. and Lohmann, B., Real Interpolation Points in Model Reduction: Justification, Two Schemes and Error Bound, IFAC World Congress, Prag, Czech Rep., 2005.

Salimbahrami, B., and Lohmann, B., Stopping Criterion in Order Reduction of Large Scale Systems Using Krylov Subspace Methods, Appl. Math. Mech. (PAMM ), Vol. 4, Issue 1, Dec. 2004, pp. 682-683. and presented in GAMM 75th Annual Meeting, TU-Dresden, March 2004.

Lohmann, B., Salimbahrami, B., Ordnungsreduktion mittels Krylov-Unterraummethoden, Automatisierungstechnik (at), Vol. 52, No. 1, pp. 30-38.

Lohmann, B., Salimbahrami, B., Introduction to Krylov-Subspace Methods in Model Order Reduction, In: Lohmann, B. und Gräser, A. (Hrsg.): Methoden und Anwendungen der Automatisierungstechnik. Ausgewählte Beiträge der internat. automatisierungstechnischen Kolloquien Salzhausen 2001 und 2002. Shaker Verlag, Aachen, 2003.

Bechtold, T., Salimbahrami, B., Rudnyi, E. B., Lohmann, B. and Korvink, J. G., Krylov-Subspace-Based Order Reduction Methods Applied to Generate Compact Thermo-Electric Models for MEMS, Proc. of 6th International Conference on Modeling and Simulation of Microsystems (MSM), San Francisco 2003.

Salimbahrami, B. and Lohmann, B., Modified Lanczos Algorithm in Model Order Reduction of MIMO Linear Systems, Internal Report, July 2002.

Salimbahrami, B., Lohmann, B., Bechtold, T. and Korvink, J.G., A Two-Sided Arnoldi-Algorithm with Stopping Criterion and MIMO selection procedure, Mathematical and Computer Modelling of Dynamical Systems, V. 11, No. 1, March 2005, pp. 79-93.

Salimbahrami, B., Lohmann, B. and Bechtold, T., Two-Sided Arnoldi in Order Reduction of Large Scale MIMO Systems, Scientific Report, University of Bremen, December 2002.

Salimbahrami, B., Lohmann, B., Bechtold, T., Korvink, J.G., Two-sided Arnoldi Algorithm and Its Application in Order Reduction of MEMS, Proceedings of the 4th  MATHMOD, Vienna 2003, pp. 1021-1028.

Salimbahrami, B., Lohmann, B., Krylov Subspace Methods in Linear Model Order Reduction: Introduction and Invariance Properties, Scientific Report, Univ. of Bremen, 2002.