{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T15:51:58Z","timestamp":1758124318354,"version":"3.37.3"},"reference-count":68,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,9,8]],"date-time":"2018-09-08T00:00:00Z","timestamp":1536364800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004281","name":"Narodowe Centrum Nauki","doi-asserted-by":"crossref","award":["DEC-2017\/01\/X\/ST7\/00885"],"award-info":[{"award-number":["DEC-2017\/01\/X\/ST7\/00885"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s10444-018-9633-5","type":"journal-article","created":{"date-parts":[[2018,9,8]],"date-time":"2018-09-08T14:17:29Z","timestamp":1536416249000},"page":"631-653","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian"],"prefix":"10.1007","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7753-0558","authenticated-orcid":false,"given":"Marek","family":"Rydel","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,9,8]]},"reference":[{"issue":"1","key":"9633_CR1","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s11043-005-3442-1","volume":"9","author":"K Adolfsson","year":"2005","unstructured":"Adolfsson, K., Enelund, M., Olsson, P.: On the fractional order model of viscoelasticity. Mec. Time-Dependent Mater. 9(1), 15\u201334 (2005)","journal-title":"Mec. Time-Dependent Mater."},{"issue":"6","key":"9633_CR2","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1007\/s00202-007-0092-0","volume":"90","author":"MA Al-Alaoui","year":"2008","unstructured":"Al-Alaoui, M.A.: Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems. Electr. Eng. 90(6), 455\u2013467 (2008)","journal-title":"Electr. Eng."},{"issue":"3","key":"9633_CR3","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1080\/00207179108953642","volume":"53","author":"RW Aldhaheri","year":"1991","unstructured":"Aldhaheri, R.W.: Model order reduction via real Schur-form decomposition. Int. J. Control. 53(3), 709\u2013716 (1991)","journal-title":"Int. J. Control."},{"key":"9633_CR4","unstructured":"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)"},{"issue":"6","key":"9633_CR5","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1016\/j.aeue.2005.05.016","volume":"60","author":"RW Aldhaheri","year":"2006","unstructured":"Aldhaheri, R.W.: Frequency-domain model reduction approach to design IIR digital filters using orthonormal bases. AEU - Int. J. Electron. Commun. 60(6), 413\u2013420 (2006)","journal-title":"AEU - Int. J. Electron. Commun."},{"key":"9633_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718713","volume-title":"Approximation of Large-Scale Dynamical System","author":"A Antoulas","year":"2005","unstructured":"Antoulas, A.: Approximation of Large-Scale Dynamical System. SIAM, Philadelphia (2005)"},{"issue":"6","key":"9633_CR7","doi-asserted-by":"publisher","first-page":"2083","DOI":"10.1007\/s00034-016-0269-8","volume":"35","author":"J Baranowski","year":"2016","unstructured":"Baranowski, J., Bauer, W., Zag\u00f3rowska, M., Piatek, P.: On digital realizations of non-integer order filters. Circ. Syst. Signal Process. 35(6), 2083\u20132107 (2016)","journal-title":"Circ. Syst. Signal Process."},{"key":"9633_CR8","doi-asserted-by":"crossref","unstructured":"Baur, U.: Low rank solution of data-sparse Sylvester equations. Numer. Linear Algebra Appl., 15(9) (2008)","DOI":"10.1002\/nla.605"},{"key":"9633_CR9","first-page":"256","volume":"31","author":"U Baur","year":"2008","unstructured":"Baur, U., Benner, P.: Cross-Gramian based model reduction for data-sparse systems. Electron. Trans. Numer. Anal. 31, 256\u2013270 (2008)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"1","key":"9633_CR10","doi-asserted-by":"publisher","first-page":"A471","DOI":"10.1137\/15M1030911","volume":"38","author":"P Benner","year":"2016","unstructured":"Benner, P., K\u00fcrschner, P., Saak, J.: Frequency-limited balanced truncation with low-rank approximations. SIAM J. Sci. Comput. 38(1), A471\u2013A499 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"9633_CR11","doi-asserted-by":"crossref","unstructured":"Biswas, K., Bohannan, G., Caponetto, R., Mendes Lopes, A., Tenreiro Machado, J.A.: Fractional-Order Devices. SpringerBriefs in Nonlinear Circuits Springer International Publishing (2017)","DOI":"10.1007\/978-3-319-54460-1"},{"issue":"3","key":"9633_CR12","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1109\/81.989172","volume":"49","author":"YQ Chen","year":"2002","unstructured":"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\u2013365 (2002)","journal-title":"IEEE Trans. Circuts Systems I: Fund. Theory Appl."},{"issue":"5","key":"9633_CR13","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/s00332-004-0617-5","volume":"14","author":"M Condon","year":"2004","unstructured":"Condon, M., Ivanov, R.: Empirical balanced truncation of nonlinear systems. J. Nonlinear Sci. 14(5), 405\u2013414 (2004)","journal-title":"J. Nonlinear Sci."},{"key":"9633_CR14","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/0024-3795(88)90200-5","volume":"109","author":"K Datta","year":"1988","unstructured":"Datta, K.: The matrix equation XA \u2013 BX = R and its applications. Linear Algebra Appl. 109, 91\u2013105 (1988)","journal-title":"Linear Algebra Appl."},{"issue":"24","key":"9633_CR15","doi-asserted-by":"publisher","first-page":"8097","DOI":"10.1016\/j.jpowsour.2010.07.053","volume":"195","author":"Z Deng","year":"2010","unstructured":"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\u20138103 (2010)","journal-title":"J. Power. Sources"},{"key":"9633_CR16","doi-asserted-by":"crossref","unstructured":"Enns, D.: Model reduction with balanced realizations: an error bound and frequency weighted generalization. In: 23rd IEEE Conf. on Decision and Control, pp. 127\u2013132 (1984)","DOI":"10.1109\/CDC.1984.272286"},{"issue":"2","key":"9633_CR17","doi-asserted-by":"publisher","first-page":"228","DOI":"10.1109\/TAC.1983.1103195","volume":"28","author":"K Fernando","year":"1983","unstructured":"Fernando, K., Nicholson, H.: On the structure of balanced and other principal representations of SISO systems. IEEE Trans. Autom. Control 28(2), 228\u2013231 (1983)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"5","key":"9633_CR18","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1109\/TCS.1985.1085737","volume":"32","author":"K Fernando","year":"1985","unstructured":"Fernando, K., Nicholson, H.: On the cross-Gramian for symmetric MIMO systems. IEEE Trans. Circ. Syst. 32(5), 487\u2013489 (1985)","journal-title":"IEEE Trans. Circ. Syst."},{"issue":"3","key":"9633_CR19","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1016\/j.sigpro.2010.02.005","volume":"91","author":"J Gabano","year":"2011","unstructured":"Gabano, J., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Process. 91(3), 531\u2013541 (2011)","journal-title":"Signal Process."},{"key":"9633_CR20","doi-asserted-by":"publisher","first-page":"164","DOI":"10.1016\/j.cnsns.2016.11.016","volume":"47","author":"JD Gabano","year":"2017","unstructured":"Gabano, J.D., Poinot, T., Huard, B.: Bounded diffusion impedance characterization of battery electrodes using fractional modeling. Commun. Nonlinear Sci. Numer. Simul. 47, 164\u2013177 (2017)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"1","key":"9633_CR21","doi-asserted-by":"publisher","first-page":"143","DOI":"10.3182\/20130204-3-FR-4032.00138","volume":"46","author":"R Garrappa","year":"2013","unstructured":"Garrappa, R., Maione, G.: Model order reduction on Krylov subspaces for fractional linear systems. IFAC Proc. 46(1), 143\u2013148 (2013). 6th IFAC Workshop on Fractional Differentiation and Its Applications","journal-title":"IFAC Proc."},{"issue":"2","key":"9633_CR22","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1080\/00207729008910366","volume":"21","author":"W Gawronski","year":"1990","unstructured":"Gawronski, W., Juang, J.: Model reduction in limited time and frequency intervals. Int. J. Syst. Sci. 21(2), 349\u2013376 (1990)","journal-title":"Int. J. Syst. Sci."},{"key":"9633_CR23","doi-asserted-by":"publisher","unstructured":"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)","DOI":"10.1007\/s10444-018-9610-z"},{"issue":"10","key":"9633_CR24","doi-asserted-by":"publisher","first-page":"1379","DOI":"10.1016\/S0098-1354(02)00120-5","volume":"26","author":"J Hahn","year":"2002","unstructured":"Hahn, J., Edgar, T.F.: An improved method for nonlinear model reduction using balancing of empirical Gramians. Comput. Chem. Eng. 26(10), 1379\u20131397 (2002)","journal-title":"Comput. Chem. Eng."},{"issue":"3","key":"9633_CR25","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1002\/asjc.1444","volume":"19","author":"KS Haider","year":"2017","unstructured":"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\u20131227 (2017)","journal-title":"Asian J. Control"},{"issue":"1","key":"9633_CR26","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1080\/21642583.2016.1215273","volume":"4","author":"C Himpe","year":"2016","unstructured":"Himpe, C., Ohlberger, M.: A note on the cross Gramian for non-symmetric systems. Syst. Sci. Control Eng. 4(1), 199\u2013208 (2016)","journal-title":"Syst. Sci. Control Eng."},{"key":"9633_CR27","doi-asserted-by":"crossref","unstructured":"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\u2013283. Springer International Publishing (2017)","DOI":"10.1007\/978-3-319-58786-8_17"},{"issue":"1","key":"9633_CR28","first-page":"86","volume":"65","author":"M Imran","year":"2018","unstructured":"Imran, M., Ghafoor, A., Imran, M.: Frequency limited model reduction techniques with error bounds. IEEE Trans. Circ. Syst. II: Express Briefs 65(1), 86\u201390 (2018)","journal-title":"IEEE Trans. Circ. Syst. II: Express Briefs"},{"issue":"12","key":"9633_CR29","doi-asserted-by":"publisher","first-page":"3304","DOI":"10.1016\/j.automatica.2014.10.062","volume":"50","author":"M Imran","year":"2014","unstructured":"Imran, M., Ghafoor, A., Sreeram, V.: A frequency weighted model order reduction technique and error bounds. Automatica 50(12), 3304\u20133309 (2014)","journal-title":"Automatica"},{"key":"9633_CR30","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.cnsns.2017.04.001","volume":"51","author":"C Ionescu","year":"2017","unstructured":"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\u2013159 (2017)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"1","key":"9633_CR31","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1002\/asjc.1330","volume":"19","author":"A Jazlan","year":"2017","unstructured":"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\u201334 (2017)","journal-title":"Asian J. Control"},{"key":"9633_CR32","unstructured":"Jazlan, A., Sreeram, V., Togneri, R.: Cross Gramian based time interval model reduction. In: Australian Control Conference, pp. 139\u2013141 (2015)"},{"issue":"1","key":"9633_CR33","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.amc.2005.11.014","volume":"177","author":"K Jbilou","year":"2006","unstructured":"Jbilou, K.: Low rank approximate solutions to large Sylvester matrix equations. Appl. Math. Comput. 177(1), 365\u2013376 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"9633_CR34","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/s11071-007-9322-2","volume":"54","author":"IS Jesus","year":"2008","unstructured":"Jesus, I.S., Tenreiro Machado, J.A.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263\u2013282 (2008)","journal-title":"Nonlinear Dyn."},{"issue":"8","key":"9633_CR35","doi-asserted-by":"publisher","first-page":"3210","DOI":"10.1016\/j.jfranklin.2017.02.019","volume":"354","author":"Y Jiang","year":"2017","unstructured":"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\u20133224 (2017)","journal-title":"J. Franklin Inst."},{"issue":"1","key":"9633_CR36","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1016\/j.jfranklin.2013.09.005","volume":"351","author":"W Krajewski","year":"2014","unstructured":"Krajewski, W., Viaro, U.: A method for the integer-order approximation of fractional-order systems. J. Franklin Inst. 351(1), 555\u2013564 (2014)","journal-title":"J. Franklin Inst."},{"key":"9633_CR37","doi-asserted-by":"publisher","unstructured":"K\u00fcrschner, 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)","DOI":"10.1007\/s10444-018-9608-6"},{"issue":"2","key":"9633_CR38","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1109\/TAC.1987.1104549","volume":"32","author":"A Laub","year":"1987","unstructured":"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\u2013122 (1987)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"7","key":"9633_CR39","doi-asserted-by":"publisher","first-page":"904","DOI":"10.1109\/PROC.1983.12688","volume":"71","author":"AJ Laub","year":"1983","unstructured":"Laub, A.J., Silverman, L.M., Verma, M.: A note on cross-Grammians for symmetric realizations. Proc. IEEE 71(7), 904\u2013905 (1983)","journal-title":"Proc. IEEE"},{"key":"9633_CR40","first-page":"341","volume":"8","author":"C Lin","year":"1992","unstructured":"Lin, C., Chiu, T.: Model reduction via frequency weighted balanced realization. Theory Adv. Technol. 8, 341\u2013451 (1992)","journal-title":"Theory Adv. Technol."},{"issue":"1","key":"9633_CR41","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1063\/1.2952963","volume":"1019","author":"R Mansouri","year":"2008","unstructured":"Mansouri, R., Bettayeb, M., Djennoune, S.: Optimal reduced-order approximation of fractional dynamical systems. AIP Conf. Proc. 1019(1), 127\u2013132 (2008)","journal-title":"AIP Conf. Proc."},{"key":"9633_CR42","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1016\/j.jmps.2017.10.013","volume":"111","author":"S Mashayekhi","year":"2018","unstructured":"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\u2013156 (2018)","journal-title":"J. Mech. Phys. Solids"},{"issue":"4","key":"9633_CR43","doi-asserted-by":"publisher","first-page":"608","DOI":"10.3923\/jas.2008.608.614","volume":"8","author":"B Moaveni","year":"2008","unstructured":"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\u2013614 (2008)","journal-title":"J. Appl. Sci."},{"key":"9633_CR44","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84996-335-0","volume-title":"Fractional-Order Systems and Controls: Fundamentals and Applications. Series on Advances in Industrial Control","author":"C Monje","year":"2010","unstructured":"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)"},{"issue":"1","key":"9633_CR45","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1109\/TAC.1981.1102568","volume":"AC\u201326","author":"B Moore","year":"1981","unstructured":"Moore, B.: Principal component analysis in linear systems: Controllability, observability and model reduction. IEEE Trans. Autom. Control AC\u201326(1), 17\u201332 (1981)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"1","key":"9633_CR46","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1109\/81.817385","volume":"47","author":"A Oustaloup","year":"2000","unstructured":"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\u201339 (2000)","journal-title":"IEEE Trans. Circ. Syst. I: Fund. Theory Appl."},{"key":"9633_CR47","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, Orlando (1999)"},{"issue":"1","key":"9633_CR48","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1109\/9.739144","volume":"44","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional-order systems and P I \u03bb D \u03bc controllers. IEEE Trans. Autom. Control 44(1), 208\u2013214 (1999)","journal-title":"IEEE Trans. Autom. Control"},{"key":"9633_CR49","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.automatica.2017.11.016","volume":"88","author":"M Rydel","year":"2018","unstructured":"Rydel, M., Stanis\u0142awski, R.: A new frequency weighted Fourier-based method for model order reduction. Automatica 88, 107\u2013112 (2018)","journal-title":"Automatica"},{"key":"9633_CR50","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.matcom.2017.03.005","volume":"140","author":"M Rydel","year":"2017","unstructured":"Rydel, M., Stanis\u0142awski, W.: Selection of reduction parameters for complex plant MIMO LTI models using the evolutionary algorithm. Math. Comput. Simul. 140, 94\u2013106 (2017)","journal-title":"Math. Comput. Simul."},{"issue":"10","key":"9633_CR51","doi-asserted-by":"publisher","first-page":"2645","DOI":"10.1016\/j.sigpro.2006.02.030","volume":"86","author":"J Sabatier","year":"2006","unstructured":"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\u20132657 (2006)","journal-title":"Signal Process."},{"issue":"7","key":"9633_CR52","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1109\/9.29399","volume":"34","author":"MG Safonov","year":"1989","unstructured":"Safonov, M.G., Chiang, R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control 34(7), 729\u2013733 (1989)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"1","key":"9633_CR53","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1080\/00207179.2014.938250","volume":"88","author":"HR Shaker","year":"2015","unstructured":"Shaker, H.R., Tahavori, M.: Control configuration selection for bilinear systems via generalised Hankel interaction index array. Int. J. Control. 88(1), 30\u201337 (2015)","journal-title":"Int. J. Control."},{"issue":"2","key":"9633_CR54","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1002\/asjc.694","volume":"16","author":"J Shen","year":"2014","unstructured":"Shen, J., Lam, J.: H \u221e $H_{\\infty }$ model reduction for positive fractional order systems. Asian J. Control 16(2), 441\u2013450 (2014)","journal-title":"Asian J. Control"},{"key":"9633_CR55","doi-asserted-by":"crossref","unstructured":"Sierociuk, D., Dzieli\u0144ski, 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)","DOI":"10.1098\/rsta.2012.0146"},{"key":"9633_CR56","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/j.automatica.2016.09.014","volume":"75","author":"P Sopasakis","year":"2017","unstructured":"Sopasakis, P., Sarimveis, H.: Stabilising model predictive control for discrete-time fractional-order systems. Automatica 75, 24\u201331 (2017)","journal-title":"Automatica"},{"key":"9633_CR57","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1016\/S0024-3795(02)00283-5","volume":"351-352","author":"DC Sorensen","year":"2002","unstructured":"Sorensen, D.C., Antoulas, A.C.: The Sylvester equation and approximate balanced reduction. Linear Algebra Appl. 351-352, 671\u2013700 (2002)","journal-title":"Linear Algebra Appl."},{"key":"9633_CR58","doi-asserted-by":"crossref","unstructured":"Stanis\u0142awski, R., Latawiec, K.J.: Fractional-order discrete-time Laguerre filters \u2013 a new tool for modeling and stability analysis of fractional-order LTI SISO systems. Discret. Dyn. Nat. Soc., 1\u20139: Article ID: 9590687 (2016)","DOI":"10.1155\/2016\/9590687"},{"key":"9633_CR59","doi-asserted-by":"crossref","unstructured":"Stanis\u0142awski, R., Latawiec, K.J., Ga\u0142ek, M., \u0141ukaniszyn, 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\u2013168. Miedzyzdroje (2014)","DOI":"10.1109\/MMAR.2014.6957343"},{"issue":"7","key":"9633_CR60","doi-asserted-by":"publisher","first-page":"3008","DOI":"10.1016\/j.jfranklin.2017.02.003","volume":"354","author":"R Stanis\u0142awski","year":"2017","unstructured":"Stanis\u0142awski, 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\u20133020 (2017)","journal-title":"J. Franklin Inst."},{"issue":"3","key":"9633_CR61","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1080\/00207721.2011.604741","volume":"44","author":"M Tahavori","year":"2013","unstructured":"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\u2013501 (2013)","journal-title":"Int. J. Syst. Sci."},{"issue":"4","key":"9633_CR62","first-page":"493","volume":"223","author":"M Tavakoli-Kakhki","year":"2009","unstructured":"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\u2013505 (2009)","journal-title":"Proc. Institut. Mech. Eng. Part I: J. Syst. Control Eng."},{"issue":"5","key":"9633_CR63","doi-asserted-by":"publisher","first-page":"919","DOI":"10.1016\/S0005-1098(03)00030-X","volume":"39","author":"A Varga","year":"2003","unstructured":"Varga, A., Anderson, B.: Accuracy-enhancing methods for balancing-related frequency-weighted model and controller reduction. Automatica 39(5), 919\u2013927 (2003)","journal-title":"Automatica"},{"issue":"1","key":"9633_CR64","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1023\/A:1016504620249","volume":"29","author":"BM Vinagre","year":"2002","unstructured":"Vinagre, B.M., Petr\u00e1\u0161, 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\u2013279 (2002)","journal-title":"Nonlin. Dyn."},{"issue":"3","key":"9633_CR65","first-page":"231","volume":"3","author":"BM Vinagre","year":"2000","unstructured":"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\u2013248 (2000)","journal-title":"Fract. Calcul. Appl. Anal."},{"issue":"9","key":"9633_CR66","doi-asserted-by":"publisher","first-page":"1734","DOI":"10.1109\/9.788542","volume":"44","author":"G Wang","year":"1999","unstructured":"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\u20131737 (1999)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"9633_CR67","doi-asserted-by":"publisher","first-page":"944","DOI":"10.1137\/S1064827502418768","volume":"26","author":"K Willcox","year":"2005","unstructured":"Willcox, K., Megretski, A.: Fourier series for accurate, stable, reduced-order models in large-scale linear applications. SIAM J. Sci. Comput. 26(3), 944\u2013962 (2005)","journal-title":"SIAM J. Sci. Comput."},{"issue":"16","key":"9633_CR68","doi-asserted-by":"publisher","first-page":"1376","DOI":"10.1049\/el.2016.1777","volume":"52","author":"U Zulfiqar","year":"2016","unstructured":"Zulfiqar, U., Imran, M., Ghafoor, A.: Cross-Gramian based frequency-weighted model order reduction technique. Electron. Lett. 52(16), 1376\u20131377 (2016)","journal-title":"Electron. Lett."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9633-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-018-9633-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9633-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T23:52:43Z","timestamp":1661989963000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-018-9633-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,8]]},"references-count":68,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["9633"],"URL":"https:\/\/doi.org\/10.1007\/s10444-018-9633-5","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2018,9,8]]},"assertion":[{"value":"15 January 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}