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September  2022, 12(3): 481-493. doi: 10.3934/naco.2021016

Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications

1. 

School of Mechatronic Engineering and Automation and, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai - 200444, China

2. 

Department of Mathematics and Physics, North South University, Dhaka - 1229, Bangladesh

3. 

Department of Mathematics, Chittagong University, Chittagong - 4331, Bangladesh

4. 

Department of Electrical and Computer Engineering, North South University, Dhaka - 1229, Bangladesh

Received  January 2021 Revised  April 2021 Published  September 2022 Early access  May 2021

This paper discusses model order reduction of large sparse second-order index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA). In general, such DAEs arise in constraint mechanics, multibody dynamics, mechatronics and many other branches of sciences and technologies. By deflecting the algebraic equations the second-order index-3 system can be altered into an equivalent standard second-order system. This can be done by projecting the system onto the null space of the constraint matrix. However, creating the projector is computationally expensive and it yields huge bottleneck during the implementation. This paper shows how to find a reduce order model without projecting the system onto the null space of the constraint matrix explicitly. To show the efficiency of the theoretical works we apply them to several data of second-order index-3 models and experimental resultants are discussed in the paper.

Citation: Xin Du, M. Monir Uddin, A. Mostakim Fony, Md. Tanzim Hossain, Md. Nazmul Islam Shuzan. Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 481-493. doi: 10.3934/naco.2021016
References:
[1]

M. I. Ahmad and P. Benner, Interpolatory model reduction techniques for linear second-order descriptor systems, in Proc. European Control Conf. ECC 2014, Strasbourg, IEEE, (2014), 1075–1079.

[2]

A. Antoulas, Approximation of Large-Scale Dynamical Systems, Ser. Advances in Design and Control. Philadelphia, PA: SIAM Publications, 6 (2005). doi: 10.1137/1.9780898718713.

[3]

F. Bennini, Ordnungsreduktion von elektrostatisch-mechanischen Finite Elemente Modellen auf der Basis der modalen Zerlegung, Ph. D. Thesis, Technische Universität Chemnitz, Chemnitz, 2005.

[4]

P. BennerJ. Saak and M. M. Uddin, Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control, Numerical Algebra, Control and Optimization, 6 (2016), 1-20.  doi: 10.3934/naco.2016.6.1.

[5]

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics, Ser. European Consortium for Mathematics in Industry, Teubner, 1998. doi: 10.1007/978-3-663-09828-7.

[6]

S. GugercinA. C. Antoulas and C. A. Beattie, $\mathcal{H}_2$ model reduction for large-scale dynamical systems, SIAM J. Matrix Anal. Appl., 30 (2008), 609-638.  doi: 10.1137/060666123.

[7]

S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput., 35 (2013), B1010–B1033. doi: 10.1137/130906635.

[8]

M. HeinkenschlossD. 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), 1038-1063.  doi: 10.1137/070681910.

[9]

V. Mehrmann and T. Stykel, Balanced truncation model reduction for large-scale systems in descriptor form, Chapter 20 of [3], (2005), 357–361. doi: 10.1007/3-540-27909-1_3.

[10]

B. C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Autom. Control, AC–26 (1981), 17-32.  doi: 10.1109/TAC.1981.1102568.

[11]

M. M. Rahman, M. M. Uddin, L. S. Andallah and M. Uddin, Tangential interpolatory projections for a class of second-order index-1 descriptor systems and application to mechatronics, Production Engineering, (2020), 1–11.

[12]

R. Riaza, Differential-Algebraic Systems. Analytical Aspects and Circuit Applications, World Scientific Publishing Co. Pte. Ltd., Singapore, 2008. doi: 10.1142/6746.

[13]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235-286.  doi: 10.1137/S0036144500381988.

[14]

N. Truhar and K. Veselić, Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix, Syst. Cont. Lett., 56 (2007), 493-503.  doi: 10.1016/j.sysconle.2007.02.003.

[15]

M. M. Uddin, Computational Methods for Model Reduction of Large-Scale Sparse Structured Descriptor Systems, Ph. D. Thesis, Otto-von-Guericke-Universität, Magdeburg, Germany, 2015.

[16]

M. M. Uddin, Gramian-based model-order reduction of constrained structural dynamic systems, IET Control Theory & Applications, 12 (2018), 2337-2346.  doi: 10.1049/iet-cta.2018.5580.

[17]

M. M. Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, Chapman and Hall/CRC, New York, USA, 2019. doi: 10.1201/9781351028622.

[18]

M. M. Uddin, Structure preserving model order reduction of a class of second-order descriptor systems via balanced truncation, Applied Numerical Mathematics, 152 (2020), 185-198.  doi: 10.1016/j.apnum.2019.12.010.

[19]

M. M. Uddin, Computational Techniques for Structure Preserving Model Reduction, in Proceedings of International Joint Conference on Computational Intelligence: IJCCI, Springer Nature, 2019.

[20]

S. Wyatt, Issues in Interpolatory Model Reduction: Inexact Solves, Second Order Systems and Daes, Ph. D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 2012.

show all references

References:
[1]

M. I. Ahmad and P. Benner, Interpolatory model reduction techniques for linear second-order descriptor systems, in Proc. European Control Conf. ECC 2014, Strasbourg, IEEE, (2014), 1075–1079.

[2]

A. Antoulas, Approximation of Large-Scale Dynamical Systems, Ser. Advances in Design and Control. Philadelphia, PA: SIAM Publications, 6 (2005). doi: 10.1137/1.9780898718713.

[3]

F. Bennini, Ordnungsreduktion von elektrostatisch-mechanischen Finite Elemente Modellen auf der Basis der modalen Zerlegung, Ph. D. Thesis, Technische Universität Chemnitz, Chemnitz, 2005.

[4]

P. BennerJ. Saak and M. M. Uddin, Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control, Numerical Algebra, Control and Optimization, 6 (2016), 1-20.  doi: 10.3934/naco.2016.6.1.

[5]

E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics, Ser. European Consortium for Mathematics in Industry, Teubner, 1998. doi: 10.1007/978-3-663-09828-7.

[6]

S. GugercinA. C. Antoulas and C. A. Beattie, $\mathcal{H}_2$ model reduction for large-scale dynamical systems, SIAM J. Matrix Anal. Appl., 30 (2008), 609-638.  doi: 10.1137/060666123.

[7]

S. Gugercin, T. Stykel and S. Wyatt, Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput., 35 (2013), B1010–B1033. doi: 10.1137/130906635.

[8]

M. HeinkenschlossD. 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), 1038-1063.  doi: 10.1137/070681910.

[9]

V. Mehrmann and T. Stykel, Balanced truncation model reduction for large-scale systems in descriptor form, Chapter 20 of [3], (2005), 357–361. doi: 10.1007/3-540-27909-1_3.

[10]

B. C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Autom. Control, AC–26 (1981), 17-32.  doi: 10.1109/TAC.1981.1102568.

[11]

M. M. Rahman, M. M. Uddin, L. S. Andallah and M. Uddin, Tangential interpolatory projections for a class of second-order index-1 descriptor systems and application to mechatronics, Production Engineering, (2020), 1–11.

[12]

R. Riaza, Differential-Algebraic Systems. Analytical Aspects and Circuit Applications, World Scientific Publishing Co. Pte. Ltd., Singapore, 2008. doi: 10.1142/6746.

[13]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235-286.  doi: 10.1137/S0036144500381988.

[14]

N. Truhar and K. Veselić, Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix, Syst. Cont. Lett., 56 (2007), 493-503.  doi: 10.1016/j.sysconle.2007.02.003.

[15]

M. M. Uddin, Computational Methods for Model Reduction of Large-Scale Sparse Structured Descriptor Systems, Ph. D. Thesis, Otto-von-Guericke-Universität, Magdeburg, Germany, 2015.

[16]

M. M. Uddin, Gramian-based model-order reduction of constrained structural dynamic systems, IET Control Theory & Applications, 12 (2018), 2337-2346.  doi: 10.1049/iet-cta.2018.5580.

[17]

M. M. Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, Chapman and Hall/CRC, New York, USA, 2019. doi: 10.1201/9781351028622.

[18]

M. M. Uddin, Structure preserving model order reduction of a class of second-order descriptor systems via balanced truncation, Applied Numerical Mathematics, 152 (2020), 185-198.  doi: 10.1016/j.apnum.2019.12.010.

[19]

M. M. Uddin, Computational Techniques for Structure Preserving Model Reduction, in Proceedings of International Joint Conference on Computational Intelligence: IJCCI, Springer Nature, 2019.

[20]

S. Wyatt, Issues in Interpolatory Model Reduction: Inexact Solves, Second Order Systems and Daes, Ph. D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, 2012.

Figure 1.  Comparison of original and the 30 dimensional reduced models for the DSMS
Figure 2.  Comparison of the original and 30 dimensional reduced models for the TCOM
Figure 3.  Comparison of the original and 30 dimensional reduced models computed by IRKA and balanced truncation for the TCOM
Figure 4.  Time comparisons of both balanced truncation and IRKA for the TCOM
Table 1.  The dimension of the tested models including number of differential and algebraic variables, inputs and outputs
models dimension n1 and n2 inputs/outputs
DSMS 2200 2000 and 200 1/3
TCOM 11001 6001 and 5000 1/1
models dimension n1 and n2 inputs/outputs
DSMS 2200 2000 and 200 1/3
TCOM 11001 6001 and 5000 1/1
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