Abstract
Previous method for determination of state matrices of fractional-order dynamic system by use of digraphs had problems with characteristic polynomials that were of full size (i.e. had all the terms) due to the constraints on matrix size minimality. This article proposes the alternative approach that allows to obtain solutions that can be easily computed even for full polynomials, but creates digraphs (and state matrices) of larger size. Proposed method is illustrated with the symbolic example of full characteristic polynomial showing the difference between the two approaches.
Research has been financed with the funds of the Statutory Research of 2016.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)
Kaczorek, T., Sajewski, L.: The Realization Problem for Positive and Fractional Systems. Springer, Berlin (2014)
Hryniów, K., Markowski, K.A.: Digraphs minimal realisations of state matrices for fractional positive systems. In: R. Szewczyk, C. Zieliński, M. Kaliczyńska (eds.) Progress in Automation, Robotics and Measuring Techniques, Advances in Intelligent Systems and Computing, vol. 350, pp. 63–72. Springer International Publishing (2015)
Hryniów, K., Markowski, K.A.: Classes of digraph structures corresponding to characteristic polynomials. In: R. Szewczyk, C. Zieliński, M. Kaliczyńska (eds.) Challenges in Automation, Robotics and Measurement Techniques, Advances in Intelligent Systems and Computing, pp. 329–339. Springer International Publishing (2015)
Berman, A., Neumann, M., Stern, R.J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York (1989)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge Univ, Press (1991)
Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, London (2009)
Fornasini, E., Valcher, M.E.: Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs. Linear Algebra and Its Applications 263, 275–310 (1997)
Hryniów, K., Markowski, K.A.: Digraphs-building method for finding a set of minimal realisations of positive 2-D dynamic systems. Systems & Control Letters (Submitted to) (2016)
Hryniów, K., Markowski, K.A.: Parallel digraphs-building algorithm for polynomial realisations. In: Proceedings of 2014 15th International Carpathian Control Conference (ICCC), pp. 174–179 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Markowski, K.A., Hryniów, K. (2017). Expansion of Digraph Size of 1-D Fractional System with Delay. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds) Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-319-45474-0_41
Download citation
DOI: https://doi.org/10.1007/978-3-319-45474-0_41
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45473-3
Online ISBN: 978-3-319-45474-0
eBook Packages: EngineeringEngineering (R0)