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Minimal positive realizations of linear continuous-time fractional descriptor systems: Two cases of an input-output digraph structure

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International Journal of Applied Mathematics and Computer Science
Issues in Parameter Identification and Control (special section, pp. 9-122), Abdel Aitouche (Ed.)

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Bang-Jensen, J. and Gutin, G. (2009). Digraphs: Theory, Algorithms and Applications, Springer-Verlag, London.10.1007/978-1-84800-998-1Search in Google Scholar

Benvenuti, L. and Farina, L. (2004). A tutorial on the positive realization problem, IEEE Transactions on Automatic Control 49(5): 651-664.10.1109/TAC.2004.826715Open DOISearch in Google Scholar

Berman, A. and Plemmons, R.J. (1979). Nonnegative Matrices in the Mathematical Sciences, SIAM, London.10.1016/B978-0-12-092250-5.50009-6Search in Google Scholar

Caputo, M. (1967). Linear models of dissipation whose q is almost frequency independent-II, Geophysical Journal International 13(5): 529, DOI: 10.1111/ j.1365-246X.1967.tb02303.x.10.1111/j.1365-246X.1967.tb02303.xOpen DOISearch in Google Scholar

Dai, L. (Ed.) (1989). System Analysis via Transfer Matrix, Springer, Berlin/Heidelberg, DOI: 10.1007/BFb0002482.10.1007/BFb0002482Open DOISearch in Google Scholar

Das, S. (2011). Functional Fractional Calculus, Springer, Berlin/Heidelberg, DOI: 10.1007/978-3-642-20545-3.10.1007/978-3-642-20545-3Open DOISearch in Google Scholar

Dodig, M. and Stoi, M. (2009). Singular systems, state feedback problem, Linear Algebra and Its Applications 431(8): 1267-1292, DOI:10.1016/j.laa.2009.04.024.10.1016/j.laa.2009.04.024Search in Google Scholar

Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, Wiley-Interscience, New York, NY.10.1002/9781118033029Search in Google Scholar

Fornasini, E. and Valcher, M.E. (1997). Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs, Linear Algebra and Its Applications 263: 275-310.10.1016/S0024-3795(96)00540-XSearch in Google Scholar

Fornasini, E. and Valcher, M.E. (2005). Controllability and reachability of 2D positive systems: A graph theoretic approach, IEEE Transactions on Circuits and Systems I 52(3): 576-585.10.1109/TCSI.2004.842872Open DOISearch in Google Scholar

Godsil, C. and Royle, G. (2001). Algebraic Graph Theory, Springer Verlag, New York, NY.10.1007/978-1-4613-0163-9Search in Google Scholar

Guang-Ren, D. (2010). Analysis and Design of Descriptor Linear Systems, Springer, New York, NY, DOI: 10.1007/978-1-4419-6397-0.10.1007/978-1-4419-6397-0Open DOISearch in Google Scholar

Horn, R.A. and Johnson, C.R. (1991). Topics in Matrix Analysis, Cambridge University Press, Cambridge. 10.1017/CBO9780511840371Open DOISearch in Google Scholar

Hryniów, K. and Markowski, K.A. (2014). Parallel digraphs-building algorithm for polynomial realisations, Proceedings of 15th International Carpathian Control Conference (ICCC), Velke Karlovice, Czech Republic, pp. 174-179, DOI: 10.1109/CarpathianCC.2014.6843592.10.1109/CarpathianCC.2014.6843592Open DOISearch in Google Scholar

Hryniów, K. and Markowski, K.A. (2015). Optimisation of digraphs creation for parallel algorithm for finding a complete set of solutions of characteristic polynomial, Proceedings of the 20th International Conference on Methods and Models in Automation and Robotics, MMAR 2015, Miedzyzdroje, Poland, pp. 1139-1144, DOI: 10.1109/MMAR.2015.7284039.10.1109/MMAR.2015.7284039Open DOISearch in Google Scholar

Hryniów, K. and Markowski, K.A. (2016a). Classes of digraph structures corresponding to characteristic polynomials, in R. Szewczyk et al. (Eds.), Challenges in Automation, Robotics and Measurement Techniques: Proceedings of Automation 2016, Warsaw, Poland, Springer International Publishing, Cham, pp. 329-339, DOI: 10.1007/978-3-319-29357-8 30.10.1007/978-3-319-29357-830Open DOISearch in Google Scholar

Hryniów, K. and Markowski, K.A. (2016b). Parallel digraphs-building computer algorithm for finding a set of characteristic polynomial realisations of dynamic system, Journal of Automation, Mobile Robotics and Intelligent Systems 10(03): 38-51, DOI: 10.14313/JAMRIS 3-2016/23.10.14313/JAMRIS3-2016/23Open DOISearch in Google Scholar

Ionescu, C.M., Kosinski, W. and De Keyser, R. (2010). Viscoelasticity and fractal structure in a model of human lungs, Archives of Mechanics 62(1): 21-48.Search in Google Scholar

Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer Verlag, London.10.1007/978-1-4471-0221-2Search in Google Scholar

Kaczorek, T. (2007). Polynomial and Rational Matrices, Springer Verlag, London.10.1007/978-1-84628-605-6Search in Google Scholar

Kaczorek, T. (2011). Singular fractional linear systems and electrical circuits, International Journal of Applied Mathematics and Computer Science 21(2): 379-384, DOI: 10.2478/v10006-011-0028-8.10.2478/v10006-011-0028-8Open DOISearch in Google Scholar

Kaczorek, T. and Sajewski, L. (2014). The Realization Problem for Positive and Fractional Systems, Springer International Publishing, Berlin, DOI: 10.1007/978-3-319-04834-5.10.1007/978-3-319-04834-5Open DOISearch in Google Scholar

Kublanovskaya, V.N. (1983). Analysis of singular matrix pencils, Journal of Soviet Mathematics 23(1): 1939-1950, DOI: 10.1007/BF01093276.10.1007/BF01093276Open DOISearch in Google Scholar

Lewis, F. (1984). Descriptor systems: Decomposition into forward and backward subsystems, IEEE Transactions on Automatic Control 29(2): 167-170, DOI: 10.1109/TAC.1984.1103467.10.1109/TAC.1984.1103467Open DOISearch in Google Scholar

Lewis, F.L. (1986). A survey of linear singular systems, Circuits, Systems and Signal Processing 5(1): 3-36, DOI: 10.1007/BF01600184.10.1007/BF01600184Open DOISearch in Google Scholar

Luenberger, D.G. (1979). Introduction to Dynamic Systems: Theory, Models, and Applications, Wiley, New York, NY.Search in Google Scholar

Machado, J. and Lopes, A.M. (2015). Fractional state space analysis of temperature time series, Fractional Calculus and Applied Analysis 18(6): 1518-1536.10.1515/fca-2015-0088Search in Google Scholar

Machado, J., Mata, M.E. and Lopes, A.M. (2015). Fractional state space analysis of economic systems, Entropy 17(8): 5402-5421.10.3390/e17085402Search in Google Scholar

Magin, R., Ortigueira, M.D., Podlubny, I. and Trujillo, J. (2011). On the fractional signals and systems, Signal Processing 91(3): 350-371.10.1016/j.sigpro.2010.08.003Search in Google Scholar

Markowski, K.A. (2016). Digraphs structures corresponding to minimal realisation of fractional continuous-time linear systems with all-pole and all-zero transfer function, 2016 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), Cluj-Napoca, Romania, pp. 1-6, DOI: 10.1109/AQTR.2016.7501367.10.1109/AQTR.2016.7501367Open DOISearch in Google Scholar

Markowski, K.A. (2017a). Determination of minimal realisation of one-dimensional continuous-time fractional linear system, International Journal of Dynamics and Control 5(1): 40-50, DOI: 10.1007/s40435-016-0232-3.10.1007/s40435-016-0232-3Open DOISearch in Google Scholar

Markowski, K.A. (2017b). Realisation of continuous-time (fractional) descriptor linear systems, in R. Szewczyk et al. (Eds.), Automation 2017, Springer International Publishing, Cham, pp. 204-214, DOI: 10.1007/978-3-319-54042-9 19.10.1007/978-3-319-54042-919Open DOISearch in Google Scholar

Markowski, K.A. (2017c). Realisation of linear continuous-time fractional singular systems using digraph-based method: First approach, Journal of Physics: Conference Series 783(1): 012052, DOI: 10.1088/1742-6596/783/1/012052.10.1088/1742-6596/783/1/012052Open DOISearch in Google Scholar

Markowski, K.A. (2018). Classes of digraphs structures with weights corresponding to 1D fractional systems, International Conference on Automation, Quality and Testing, Robotics, AQTR 2018, Cluj-Napoca, Romania, (submitted).Search in Google Scholar

Markowski, K.A. and Hryniów, K. (2017a). Expansion of digraph size of 1-D fractional system with delay, in A. Babiarz et al. (Eds.), Theory and Applications of Non-integer Order Systems, Springer International Publishing, Cham, pp. 467-476, DOI: 10.1007/978-3-319-45474-0 41.10.1007/978-3-319-45474-041Open DOISearch in Google Scholar

Markowski, K.A. and Hryniów, K. (2017b). Finding a set of (A, B, C, D) realisations for fractional one-dimensional systems with digraph-based algorithm, in A. Babiarz et al. (Eds.), Theory and Applications of Non-integer Order Systems, Springer International Publishing, Cham, pp. 357-368, DOI: 10.1007/978-3-319-45474-0 32.10.1007/978-3-319-45474-032Open DOISearch in Google Scholar

Miller, K. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY.Search in Google Scholar

Mitkowski, W. (2008). Dynamical properties of Metzler systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 309-312.Search in Google Scholar

Muresan, C.I., Dulf, E.H. and Prodan, O. (2016a). A fractional order controller for seismic mitigation of tructures equipped with viscoelastic mass dampers, Journal of Vibration and Control 22(8): 1980-1992, DOI: 10.1177/1077546314557553.10.1177/1077546314557553Search in Google Scholar

Muresan, C.I., Dutta, A., Dulf, E.H., Pinar, Z., Maxim, A. and Ionescu, C.M. (2016b). Tuning algorithms for fractional order internal model controllers for time delay processes, International Journal of Control 89(3): 579-593, DOI: 10.1080/00207179.2015.1086027.10.1080/00207179.2015.1086027Open DOISearch in Google Scholar

Nishimoto, K. (1984). Fractional Calculus, Decartess Press, Koriama. Search in Google Scholar

Ortigueira, M.D. (2011). Fractional Calculus for Scientists and Engineers, Academic Press, Springer, Dordrecht, DOI: 10.1007/978-94-007-0747-4.10.1007/978-94-007-0747-4Open DOISearch in Google Scholar

Petras, I., Sierociuk, D. and Podlubny, I. (2012). Identification of parameters of a half-order system, IEEE Transactions on Signal Processing 60(10): 5561-5566.10.1109/TSP.2012.2205920Search in Google Scholar

Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA. Search in Google Scholar

Podlubny, I., Skovranek, T. and Datsko, B. (2014). Recent advances in numerical methods for partial fractional differential equations, 2014 15th International Carpathian Control Conference (ICCC), Velke Karlovice, Czech Republic, pp. 454-457.10.1109/CarpathianCC.2014.6843647Search in Google Scholar

Sajewski, L. (2012). Positive realization of fractional continuous-time linear systems with delays, Pomiary Automatyka Robotyka 2: 413-417.Search in Google Scholar

Sikora, B. (2016). Controllability criteria for time-delay fractional systems with a retarded state, International Journal of Applied Mathematics and Computer Science 26(3): 521-531, DOI: 10.1515/amcs-2016-0036.10.1515/amcs-2016-0036Open DOISearch in Google Scholar

Vandoorn, T.L., Ionescu, C.M., De Kooning, J.D.M., De Keyser, R. and Vandevelde, L. (2013). Theoretical analysis and experimental validation of single-phase direct versus cascade voltage control in islanded microgrids, IEEE Transactions on Industrial Electronics 60(2): 789-798.10.1109/TIE.2012.2205362Open DOISearch in Google Scholar

eISSN:
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Language:
English
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Journal Subjects:
Mathematics, Applied Mathematics