Minimal realization and approximation of commensurate linear fractional-order systems via Loewner matrix method

Lihong Meng; Xu Yang; Umair Zulfiqar; Xin Du; Lihong Meng; Xu Yang; Umair Zulfiqar; Xin Du

doi:10.3934/mbe.2021058

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1063-1076. doi: 10.3934/mbe.2021058

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Minimal realization and approximation of commensurate linear fractional-order systems via Loewner matrix method

1.
School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China
2.
Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, Beijing 100083, China
3.
School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China
4.
School of Electrical, Electronics and Computer Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
5.
Key Laboratory of Modern Power System Simulation and Control & Renewable Energy Technology, Ministry of Education (Northeast Electric Power University), Jilin 132012, China
6.
Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai 200444, China

Received: 31 August 2020 Accepted: 16 December 2020 Published: 08 January 2021

In this paper we propose a data driven realization and model order reduction (MOR) for linear fractional-order system (FoS) by applying the Loewner-matrix method. Given the interpolation data which obtained by sampling the transfer function of a FoS, the minimal fractional-order state space descriptor model that matching the interpolation data is constructed with low computational cost. Based on the framework, the commensurate order $ \alpha $ of the fractional-order system is estimated by solving a least squares optimization in terms of sample data in case of unknown order-$ \alpha $. In addition, we present an integer-order approximation model using the interpolation method in the Loewner framework for FoS with delay. Finally, several numerical examples demonstrate the validity of our approach.
- data driven model reduction,
- fractional-order system,
- Loewner framework,
- interpolation,
- time delay,
- commensurate order
Citation: Lihong Meng, Xu Yang, Umair Zulfiqar, Xin Du. Minimal realization and approximation of commensurate linear fractional-order systems via Loewner matrix method[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1063-1076. doi: 10.3934/mbe.2021058

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Abstract

In this paper we propose a data driven realization and model order reduction (MOR) for linear fractional-order system (FoS) by applying the Loewner-matrix method. Given the interpolation data which obtained by sampling the transfer function of a FoS, the minimal fractional-order state space descriptor model that matching the interpolation data is constructed with low computational cost. Based on the framework, the commensurate order $ \alpha $ of the fractional-order system is estimated by solving a least squares optimization in terms of sample data in case of unknown order-$ \alpha $. In addition, we present an integer-order approximation model using the interpolation method in the Loewner framework for FoS with delay. Finally, several numerical examples demonstrate the validity of our approach.

References

[1]	A. J. Mayo, A. C. Antoulas, A framework for the solution of the generalized realization problem, Linear Algebra Its Appl., 425 (2007), 634–662.
[2]	A. C. Antoulas, The Loewner framework and transfer functions of singular/rectangular systems, Appl. Math. Letters, 54 (2016), 36–47.
[3]	A. C. Antoulas, S. Lefteriu, A. C. Ionita, P. Benner, A. Cohen, A tutorial introduction to the Loewner framework for model reduction, Model Reduction Approxim.: Theory Algorithms, 15 (2017), 335.
[4]	A. C. Antoulas, C. A. Beattie, S. G. gercin, Interpolatory methods for model reduction, SIAM Comput. Sci. Eng., 2020.
[5]	B. Peherstorfer, S. Gugercin, K. Willcox, Data-driven reduced model construction with time-domain Loewner models, SIAM J. Sci. Comput., 39 (2017), 2152–2178.
[6]	M. Sahouli, S. Wahid, A. Dounavis, Iterative Loewner matrix macromodeling approach for noisy frequency responses, IEEE Trans. Microwave Theory Tech., 67 (2018), 634–641.
[7]	S. Lefteriu, A. C. Ionita, A. C. Antoulas, Modeling systems based on noisy frequency and time domain measurements, Perspect. Math. Syst. Theory, Control, Signal Process., 2010,365–378.
[8]	P. Schulze, B. Unger, C. Beattie, S. Gugercin, Data-driven structured realization, Linear Algebra Appl., 537 (2018), 250–286. doi: 10.1016/j.laa.2017.09.030
[9]	E. H. Dulf, D. C. Vodnar, A. Danku, C. I. Muresan, Fractional-order models for biochemical processes, Fractal Fract., 4 (2020), 12. doi: 10.3390/fractalfract4020012
[10]	R. Toledo-Hernandez, V. Rico-Ramirez, G. A. Iglesias-Silva, U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems Part I: Fractional models for biological reactions, Chem. Eng. Sci., 117 (2014), 217–228.
[11]	R. Toledo-Hernandez, V. Rico-Ramirez, G. A. Iglesias-Silva, U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems Part II: Numerical solution of fractional optimal control problems, Chem. Eng. Sci., 117 (2014), 239–247. doi: 10.1016/j.ces.2014.06.033
[12]	A. M. Khan, L.Mistri, Stability analysis and numerical solution for the fractional order biochemical reaction model, Nonlinear Anal. Diff. Equations HIKARI, (2016), 521–530.
[13]	L. Wang, P. Cheng, Y. Wang, Frequency domain subspace identification of commensurate fractional order input time delay systems, Int. J. Control, Autom. Syst., 9 (2011), 310–316. doi: 10.1007/s12555-011-0213-4
[14]	Z. Gao, Stable model order reduction method for fractional-order systems based on unsymmetric Lanczos algorithm, IEEE/CAA J. Autom. Sin., 6 (2019), 485–492. doi: 10.1109/JAS.2019.1911399
[15]	Y. Li, S. Yu, Identification of non-integer order systems in frequency domain, Acta Autom. Sin., 23 (2007), 882.
[16]	Z. Liao, C. Peng, Y. Wang, Subspace identification in time-domain for fractional order systems based on short memory principle, J. Appl. Sci., 29 (2011), 209–215.
[17]	N. Gehring, J. Rudolph, An algebraic approach to the identification of linear systems with fractional derivatives, IFAC-PapersOnLine, 50 (2017), 6214–6219. doi: 10.1016/j.ifacol.2017.08.1018
[18]	D. Casagrande, W. Krajewski, U. Viaro, The integer-order approximation of fractional-order systems in the Loewner framework, IFAC-PapersOnLine, 52 (2019), 43–48.
[19]	M. Bettayeb, S. Djennoune, A note on the controllability and the observability of fractional dynamical systems, IFAC Proceed. Vol., 39 (2006), 493–498.
[20]	C. M. Qin, N. M. Qi, K. Zhu, State space modeling and stability theory of variable fractional order system, Control Decis., 26 (2011), 1757–1760.
[21]	I. Podlubny, Fractional-order systems and fractional-order controllers, Inst. Exp. Phys., Slovak Acad. Sci., Kose, 12 (1994), 1–18.
[22]	M. Tavakoli-Kakhki, M. Haeri, The minimal state space realization for a class of fractional order transfer functions, SIAM J. Control Opt., 48 (2010), 4317–4326.
[23]	I. V. Gosea, M. Petreczky, A. C. Antoulas, Data-driven model order reduction of linear switched systems in the Loewner framework, SIAM J. Sci. Comput., 40 (2018), 572–610.
[24]	P. Schulze, B. Unger, Data-driven interpolation of dynamical systems with delay, Sys. Control Letters, 97 (2016), 125–131. doi: 10.1016/j.sysconle.2016.09.007
[25]	B. D. O. Anderson, A. C. Antoulas, Rational interpolation and state-variable realizations, Linear Algebra Its Appl., 137 (1990), 479–509.
[26]	S. Lefteriu, A. C. Antoulas, A new approach to modeling multiport systems from frequency-domain data, IEEE Trans. Comput.-Aided Design Int. Circuits.Syst., 29 (2009), 14–27.
[27]	A. C. Antoulas, Approximation of large-scale dynamical systems, Soc. Ind. Appl. Math., 2005.
[28]	A. C. Antoulas, C. A. Beattie, S. Gugercin, Interpolatory model reduction of large-scale dynamical systems, Effic. Model. Control Large-Scale Systems, 2010, 5–38.

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