Abstract
This paper presents a new approach to modeling of linear time-invariant discrete-time non-commensurate fractional-order single-input single-output state space systems by means of the Balanced Truncation and Frequency Weighted model order reduction methods based on the cross Gramian. These reduction methods are applied to the specific rational (integer-order) FIR-based approximation to the fractional-order system, which enables to introduce simple, analytical formulae for determination of the cross Gramian of the system. This leads to significant decrease of computational burden in the reduction algorithm. As a result, a rational and relatively low-order state space approximator for the fractional-order system is obtained. A simulation experiment illustrates an efficiency of the introduced methodology in terms of high approximation accuracy and low time complexity of the proposed method.
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Adolfsson, K., Enelund, M., Olsson, P.: On the fractional order model of viscoelasticity. Mec. Time-Dependent Mater. 9(1), 15–34 (2005)
Al-Alaoui, M.A.: Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems. Electr. Eng. 90(6), 455–467 (2008)
Aldhaheri, R.W.: Model order reduction via real Schur-form decomposition. Int. J. Control. 53(3), 709–716 (1991)
Aldhaheri, R.W.: Design of low order IIR filters with an arbitrary frequency range via frequency domain cross Gramian. In: Proc. of the 1st International Conference on Modeling, Simulation and Applied Optimization (2005)
Aldhaheri, R.W.: Frequency-domain model reduction approach to design IIR digital filters using orthonormal bases. AEU - Int. J. Electron. Commun. 60(6), 413–420 (2006)
Antoulas, A.: Approximation of Large-Scale Dynamical System. SIAM, Philadelphia (2005)
Baranowski, J., Bauer, W., Zagórowska, M., Piatek, P.: On digital realizations of non-integer order filters. Circ. Syst. Signal Process. 35(6), 2083–2107 (2016)
Baur, U.: Low rank solution of data-sparse Sylvester equations. Numer. Linear Algebra Appl., 15(9) (2008)
Baur, U., Benner, P.: Cross-Gramian based model reduction for data-sparse systems. Electron. Trans. Numer. Anal. 31, 256–270 (2008)
Benner, P., Kürschner, P., Saak, J.: Frequency-limited balanced truncation with low-rank approximations. SIAM J. Sci. Comput. 38(1), A471–A499 (2016)
Biswas, K., Bohannan, G., Caponetto, R., Mendes Lopes, A., Tenreiro Machado, J.A.: Fractional-Order Devices. SpringerBriefs in Nonlinear Circuits Springer International Publishing (2017)
Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuts Systems I: Fund. Theory Appl. 49(3), 363–365 (2002)
Condon, M., Ivanov, R.: Empirical balanced truncation of nonlinear systems. J. Nonlinear Sci. 14(5), 405–414 (2004)
Datta, K.: The matrix equation XA – BX = R and its applications. Linear Algebra Appl. 109, 91–105 (1988)
Deng, Z., Cao, H., Li, X., Jiang, J., Yang, J., Qin, Y.: Generalized predictive control for fractional order dynamic model of solid oxide fuel cell output power. J. Power. Sources 195(24), 8097–8103 (2010)
Enns, D.: Model reduction with balanced realizations: an error bound and frequency weighted generalization. In: 23rd IEEE Conf. on Decision and Control, pp. 127–132 (1984)
Fernando, K., Nicholson, H.: On the structure of balanced and other principal representations of SISO systems. IEEE Trans. Autom. Control 28(2), 228–231 (1983)
Fernando, K., Nicholson, H.: On the cross-Gramian for symmetric MIMO systems. IEEE Trans. Circ. Syst. 32(5), 487–489 (1985)
Gabano, J., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Process. 91(3), 531–541 (2011)
Gabano, J.D., Poinot, T., Huard, B.: Bounded diffusion impedance characterization of battery electrodes using fractional modeling. Commun. Nonlinear Sci. Numer. Simul. 47, 164–177 (2017)
Garrappa, R., Maione, G.: Model order reduction on Krylov subspaces for fractional linear systems. IFAC Proc. 46(1), 143–148 (2013). 6th IFAC Workshop on Fractional Differentiation and Its Applications
Gawronski, W., Juang, J.: Model reduction in limited time and frequency intervals. Int. J. Syst. Sci. 21(2), 349–376 (1990)
Gosea, I.V., Petreczky, M., Antoulas, A.C., Fiter, C.: Balanced truncation for linear switched systems. Advances in Computational Mathematics. https://doi.org/10.1007/s10444-018-9610-z (2018)
Hahn, J., Edgar, T.F.: An improved method for nonlinear model reduction using balancing of empirical Gramians. Comput. Chem. Eng. 26(10), 1379–1397 (2002)
Haider, K.S., Ghafoor, A., Imran, M., Malik, F.M.: Model reduction of large scale descriptor systems using time limited Gramians. Asian J. Control 19(3), 1217–1227 (2017)
Himpe, C., Ohlberger, M.: A note on the cross Gramian for non-symmetric systems. Syst. Sci. Control Eng. 4(1), 199–208 (2016)
Himpe, C., Ohlberger, M.: Cross-Gramian-based model reduction: a comparison. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, pp 271–283. Springer International Publishing (2017)
Imran, M., Ghafoor, A., Imran, M.: Frequency limited model reduction techniques with error bounds. IEEE Trans. Circ. Syst. II: Express Briefs 65(1), 86–90 (2018)
Imran, M., Ghafoor, A., Sreeram, V.: A frequency weighted model order reduction technique and error bounds. Automatica 50(12), 3304–3309 (2014)
Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T., Bates, J.H.T.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017)
Jazlan, A., Sreeram, V., Shaker, H.R., Togneri, R., Minh, H.B.: Frequency interval cross Gramians for linear and bilinear systems. Asian J. Control 19(1), 22–34 (2017)
Jazlan, A., Sreeram, V., Togneri, R.: Cross Gramian based time interval model reduction. In: Australian Control Conference, pp. 139–141 (2015)
Jbilou, K.: Low rank approximate solutions to large Sylvester matrix equations. Appl. Math. Comput. 177(1), 365–376 (2006)
Jesus, I.S., Tenreiro Machado, J.A.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)
Jiang, Y., Xu, K.: H2 optimal reduced models of general MIMO LTI systems via the cross Gramian on the Stiefel manifold. J. Franklin Inst. 354(8), 3210–3224 (2017)
Krajewski, W., Viaro, U.: A method for the integer-order approximation of fractional-order systems. J. Franklin Inst. 351(1), 555–564 (2014)
Kürschner, P.: Balanced truncation model order reduction in limited time intervals for large systems. Advances in Computational Mathematics. https://doi.org/10.1007/s10444-018-9608-6 (2018)
Laub, A., Heath, M., Paige, C., Ward, R.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Autom. Control 32(2), 115–122 (1987)
Laub, A.J., Silverman, L.M., Verma, M.: A note on cross-Grammians for symmetric realizations. Proc. IEEE 71(7), 904–905 (1983)
Lin, C., Chiu, T.: Model reduction via frequency weighted balanced realization. Theory Adv. Technol. 8, 341–451 (1992)
Mansouri, R., Bettayeb, M., Djennoune, S.: Optimal reduced-order approximation of fractional dynamical systems. AIP Conf. Proc. 1019(1), 127–132 (2008)
Mashayekhi, S., Miles, P., Hussaini, M.Y., Oates, W.S.: Fractional viscoelasticity in fractal and non-fractal media: Theory, experimental validation, and uncertainty analysis. J. Mech. Phys. Solids 111, 134–156 (2018)
Moaveni, B., Khaki-Sedigh, A.: A new approach to compute the cross-Gramian matrix and its application in input-output pairing of linear multivariable plants. J. Appl. Sci. 8(4), 608–614 (2008)
Monje, C., Chen, Y., Vinagre, B., Xue, D., Feliu-Batlle, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Series on Advances in Industrial Control. Springer, London (2010)
Moore, B.: Principal component analysis in linear systems: Controllability, observability and model reduction. IEEE Trans. Autom. Control AC–26(1), 17–32 (1981)
Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Trans. Circ. Syst. I: Fund. Theory Appl. 47(1), 25–39 (2000)
Podlubny, I.: Fractional Differential Equations. Academic Press, Orlando (1999)
Podlubny, I.: Fractional-order systems and P I λ D μ controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)
Rydel, M., Stanisławski, R.: A new frequency weighted Fourier-based method for model order reduction. Automatica 88, 107–112 (2018)
Rydel, M., Stanisławski, W.: Selection of reduction parameters for complex plant MIMO LTI models using the evolutionary algorithm. Math. Comput. Simul. 140, 94–106 (2017)
Sabatier, J., Aoun, M., Oustaloup, A., Grgoire, G., Ragot, F., Roy, P.: Fractional system identification for lead acid battery state of charge estimation. Signal Process. 86(10), 2645–2657 (2006)
Safonov, M.G., Chiang, R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control 34(7), 729–733 (1989)
Shaker, H.R., Tahavori, M.: Control configuration selection for bilinear systems via generalised Hankel interaction index array. Int. J. Control. 88(1), 30–37 (2015)
Shen, J., Lam, J.: \(H_{\infty }\) model reduction for positive fractional order systems. Asian J. Control 16(2), 441–450 (2014)
Sierociuk, D., Dzieliński, A., Sarwas, G., Petras, I., Podlubny, I., Skovranek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. London A: Math. Phys. Eng. Sci., 371(1990) (2013)
Sopasakis, P., Sarimveis, H.: Stabilising model predictive control for discrete-time fractional-order systems. Automatica 75, 24–31 (2017)
Sorensen, D.C., Antoulas, A.C.: The Sylvester equation and approximate balanced reduction. Linear Algebra Appl. 351-352, 671–700 (2002)
Stanisławski, R., Latawiec, K.J.: Fractional-order discrete-time Laguerre filters – a new tool for modeling and stability analysis of fractional-order LTI SISO systems. Discret. Dyn. Nat. Soc., 1–9: Article ID: 9590687 (2016)
Stanisławski, R., Latawiec, K.J., Gałek, M., Łukaniszyn, M.: Modeling and identification of a fractional-order discrete-time SISO Laguerre-Wiener system. In: Proceedings of the 19th International Conference on Methods and Models in Automation and Robotics, pp. 165–168. Miedzyzdroje (2014)
Stanisławski, R., Rydel, M., Latawiec, K.J.: Modeling of discrete-time fractional-order state space systems using the balanced truncation method. J. Franklin Inst. 354(7), 3008–3020 (2017)
Tahavori, M., Shaker, H.R.: Model reduction via time-interval balanced stochastic truncation for linear time invariant systems. Int. J. Syst. Sci. 44(3), 493–501 (2013)
Tavakoli-Kakhki, M., Haeri, M.: Model reduction in commensurate fractional-order linear systems. Proc. Institut. Mech. Eng. Part I: J. Syst. Control Eng. 223(4), 493–505 (2009)
Varga, A., Anderson, B.: Accuracy-enhancing methods for balancing-related frequency-weighted model and controller reduction. Automatica 39(5), 919–927 (2003)
Vinagre, B.M., Petráš, I., Podlubny, I., Chen, Y.Q.: Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlin. Dyn. 29(1), 269–279 (2002)
Vinagre, B.M., Podlubny, I., Hernandez, A., Feliu, V.: Some approximations of fractional order operators used in control theory and applications. Fract. Calcul. Appl. Anal. 3(3), 231–248 (2000)
Wang, G., Sreeram, V., Liu, W.Q.: A new frequency-weighted balanced truncation method and an error bound. IEEE Trans. Autom. Control 44(9), 1734–1737 (1999)
Willcox, K., Megretski, A.: Fourier series for accurate, stable, reduced-order models in large-scale linear applications. SIAM J. Sci. Comput. 26(3), 944–962 (2005)
Zulfiqar, U., Imran, M., Ghafoor, A.: Cross-Gramian based frequency-weighted model order reduction technique. Electron. Lett. 52(16), 1376–1377 (2016)
Acknowledgements
The author is indebted to Prof. Krzysztof J. Latawiec and Prof. Rafał Stanisławski for their stimulating discussions and to anonymous reviewers for their instructive comments.
Funding
This work was supported by the Polish National Science Centre on the base of decision no. DEC-2017/01/X/ST7/00885.
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Communicated by: Peter Benner
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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Rydel, M. New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian. Adv Comput Math 45, 631–653 (2019). https://doi.org/10.1007/s10444-018-9633-5
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DOI: https://doi.org/10.1007/s10444-018-9633-5
Keywords
- Model order reduction
- Cross Gramian
- Frequency weighted
- Non-commensurate fractional order system
- FIR-based approximation