Approximation of large-scale dynamical systems for Bench-mark Collection

Authors:

Santosh Kumar Suman,Awadhesh Kumar,

DOI NO:

https://doi.org/10.26782/jmcms.2019.06.00016

Keywords:

Benchmarks Example,Order reduction,Error estimation,Krylov,Balanced Truncation,Modal method,

Abstract

In this contribution,We present a benchmark collection Inclusive of some needful real-world examples, which can be used to assessment and compare numerical methods for model reduction. In this paper the reduction method is explored for getting structure preserving reduced order model of a large-scale dynamical system, we have considered model order reduction of higher orderLTIsystems) with SISO and MIMO [XXXII] that aims at finding Error estimation using Approximation of both system. This enables a new evaluation of the error system Provided that the Observability Gramian of the original system has once been considered, an H∞and H2 error bound can be computed with negligible numerical attempt for any reduced model attributable to The reduced order model (ROM) of a large-scale dynamical system is necessary to effortlessness the analysis of the system using approximation Algorithms. The response evaluation is considered in terms ofresponse constraints and graphical assessments.the application of Approximation methodsis offered for arisingROMof the large-scaleLTI systems which includes benchmark problems. It is reported that the reduced order model using compare numerical methods is almost alike in performance to that of with original systems.all simulation resultshave been obtained via MATLAB based software (sssMOR toolbox).

Refference:

I.Antoulas, A. C. (2004). Approximation of large-scale dynamical systems: An overview. IFAC
Proceedings Volumes (IFAC-PapersOnline).https://doi.org/10.1016/S1474-6670(17)31584-7
II.Antoulas, A. C. (2005). An overview of approximation methods for large-scale dynamical systems.
Annual Reviews in Control.https://doi.org/10.1016/j.arcontrol.2005.08.002
III.Antoulas, A. C., Benner, P., & Feng, L. (2018). Model reduction by iterative error system approximation.
Mathematical and Computer Modelling of Dynamical Systems. https://doi.org/10.1080/13873954.2018.1427116
IV.Antoulas, A. C., Sorensen, D. C., & Gugercin, S. (2012).A surveyof model reduction methods for
large-scale systems. https://doi.org/10.1090/conm/280/04630
V.Antoulas, Athanasios C. (2011a). 8. Hankel-Norm Approximation. En Approximation of Large-Scale
Dynamical Systems.https://doi.org/10.1137/1.9780898718713.ch8
VI.Antoulas, Athanasios C. (2011b). Approximation of Large-Scale Dynamical Systems. En Approximation of Large-Scale Dynamical Systems.https://doi.org/10.1137/1.9780898718713
VII.Antoulas, Athanasios C., Beattie, C. A., & Gugercin, S. (2010). Interpolatory model reduction of large-
scale dynamical systems. En Efficient Modeling and Control of Large-Scale Systems. https://doi.org/10.1007/978-1-4419-5757-3_1
VIII.Antoulas, Athanasios C., & Sorensen, D. C. (2001). Approximation of large-scale dynamical systems: An overview.Int.J. Appl. Math. Comput. Sci.
IX.Beattie, C. A., & Gugercin, S. (2011). Weighted model reduction via interpolation.IFAC Proceedings
Volumes(IFAC-PapersOnline).https://doi.org/10.3182/20110828-6-IT-1002.03419
X.Benner, P. (2007). A MATLAB repository for model reduction based on spectral projection.Proceedings of the 2006 IEEE Conference on Computer Aided Control Systems Design, CACSD.https://doi.org/10.1109/CACSD.2006.285438
XI.Benner, P., & Faßbender, H. (2011). On the numerical solution of large-scale sparse discrete-time Riccati equations.Advances in Computational Mathematics. https://doi.org/10.1007/s10444-011-9174-7
XII.Benner, P., Gugercin, S., & Willcox, K. (2015). A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems.SIAM Review. https://doi.org/10.1137/130932715
XIII.Castagnotto, A., Cruz Varona, M., Jeschek, L., & Lohmann, B. (2017). Sss &sssMOR: Analysis and reduction of large-scale dynamic systems inMATLAB.At-Automatisierungstechnik. https://doi.org/10.1515/auto-
2016-0137
XIV.Castagnotto, A., Hu, S., & Lohmann, B. (2018). An Approach for GlobalizedH2-Optima lModel Reduction.
IFAC-PapersOnLine.https://doi.org/10.1016/j.ifacol.2018.03.034
XV.Castagnotto, A., Panzer, H. K. F., & Lohmann, B. (2017). Fast H 2-optimalmodel order reduction exploiting the local nature of Krylov-subspace methods.2016 European Control Conference,ECC 2016.
https://doi.org/10.1109/ECC.2016.7810578
XVI.Chahlaoui, Younès. (2011). Two efficient SVD/Krylov algorithms for model order reduction of large scale systems.Electronic Transactions on Numerical Analysis.
XVII.Chahlaoui, Younes, & Dooren, P. Van. (2002). A collection of Benchmark examples for model reduction of linear time invariant dynamical systems.SLICOT Working Notes. https://doi.org/10.1007/3-540-27909-1_24
XVIII.Chahlaoui, Younes, & Van Dooren, P. (2005).Benchmark Examples for Model Reduction of Linear Time
-Invariant Dynamical Systems.https://doi.org/10.1007/3-540-27909-1_24
XIX.Chidambara, M. R. (1967). Further Remarks on Simplifying Linear Dynamic Systems.IEEET ransactions
on Automatic Control.https://doi.org/10.1109/TAC.1967.1098557
XX.Davison, E. J. (1966). A method for simplifying linear dynamic systems.IEEE Transactionson AutomaticControl.https://doi.org/10.1109/TAC.1966.1098264
XXI.Dax, A. (2013). From Eigenvalues to Singular Values: A Review.Advances in Pure Mathematics
. https://doi.org/10.4236/apm.2013.39a2002
XXII.Ferranti, F., Deschrijver, D., Knockaert, L., & Dhaene, T. (2011). Data-driven parameterized model order reduction using z-domain multivariate orthonormal vector fitting technique.Lecture Notes in Electrical
Engineering. https://doi.org/10.1007/978-94-007-0089-5_7
XXIII.Grimme, E. (1997). Krylov projection mezhods for model reduction.Vasa.
XXIV.Gugercin, S., Antoulas, A. C., & Beattie, C. (2008). $\mathcal{H}_2$Model Reduction for Large-
Scale Linear Dynamical Systems.SIAM Journal on Matrix Analysis and Applications. https://doi.org/10.1137/060666123
XXV.Gugercin, Serkan, & Antoulas, A. C. (2006). Model reduction of large-scalesystems by least squares.
Linear Algebra and Its Applications.https://doi.org/10.1016/j.laa.2004.12.022
XXVI.Korvink, J. G., & Rudnyi, E. B. (2005). Oberwolfach Benchmark Collection.En Dimension Reduction of Large-Scale Systems. https://doi.org/10.1007/3-540-27909-1_11
XXVII.Litz, L. (1979). Ordnungsreduktion linearer zustandsraummodelle durch beibehaltung der dominanten eigenbewegungen.At-Automatisierungstechnik.https://doi.org/10.1524/auto.1979.27.112.80
XXVIII.Model Order Reduction: Theory, Research Aspects and Applications. (2008).https://doi.org/10.1007/978
-3-540-78841-6
XXIX.Mohamed, K. S. (2018). Machine learning for model order reduction. En Machine Learning for Model Order Reduction. https://doi.org/10.1007/978-3-319-75714-8
XXX.Moore, B. C. (1981). Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction.IEEE Transactions on Automatic Control. https://doi.org/10.1109/TAC.1981.1102568
XXXI.Pinnau, R. (2008). Model Reduction via Proper Orthogonal Decomposition.https://doi.org/10.1007/978
-3-540-78841-6_5
XXXII.Samba riya, D. K., & Sharma, O. (2016). Routh Approximation: An Approach of Model Order Reduction in SISO and MIMO Systems.Indonesian Journal of Electrical Engineering and Computer Science.
https://doi.org/10.11591/ijeecs.v2.i3.pp486-500
XXXIII.Schilders, W. (2008). Introduction to Model Order Reduction.https://doi.org/10.1007/978
-3-540-78841-6_1
XXXIV.Segalman, D. J. (2007). Model Reduction of Systems With Localized Nonlinearities.Journal of Computational and Nonlinear Dynamics.https://doi.org/10.1115/1.2727495
XXXV.Varga, A.(1995). Enhanced modal approach for model reduction.Mathematical Modelling of
Systems.https://doi.org/10.1080/13873959508837010
XXXVI.Varga, Andras. (2011). Model Reduction Software in the SLICOT Library. En Applied and Computational Control, Signals, and Circuits.https://doi.org/10.1007/978-1-4615-1471-8_7
XXXVII.Verbeek, M. E. (2004). Partial element equivalent circuit (PEEC) models for on-chip passives and interconnects.International Journal of Numerical Modelling:Electronic Networks,Devices and Fields.https://doi.org/10.1002/jnm.524
XXXVIII.Willcox, K. E., & Peraire, J. (2002). Balanced Model Reduction via the Proper Introduction.
AIAA Journal. https://doi.org/10.2514/2.1570
XXXIX.Yogarathinam, A., Kaur, J., & Chaudhuri, N. R. (2019). A New H-IRKA Approach for Model Reduct
ion with Explicit Modal Preservation:Application on Grids with Renewable Penetration.IEEE Transactions on
Control Systems Technology. https://doi.org/10.1109/TCST.2017.2779104
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