Abstract
This example-oriented contribution deals with the value set-based numerical analysis of robust stability for the family of fractional-order retarded quasi-polynomials with both uncertain parameters and uncertain fractional orders. The specific investigated feedback control system consists of the fractional-order PID controller and the controlled plant, represented by a heat transfer process described by the linear time-invariant fractional-order time-delay model with parametric uncertainty (with three uncertain parameters, namely, gain, fractional time constant, and fractional time-delay term, and furthermore two fractional orders). The graphical robust stability analysis is based on the numerical calculation of the value sets and the application of the zero exclusion principle.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bhattacharyya, S.P.: Robust control under parametric uncertainty: an overview and recent results. Annu. Rev. Control. 44, 45–77 (2017)
Barmish, B.R.: New Tools for Robustness of Linear Systems. Macmillan, New York (1994)
Machado, J.A.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)
Chen, Y.Q., Petráš, I., Xue, D.: Fractional order control – a tutorial. In: Proceedings of the 2009 American Control Conference, St. Louis, MO, USA (2009)
Xue, D.: Fractional-Order Control Systems: Fundamentals and Numerical Implementations. De Gruyter, Berlin, Germany (2017)
Tan, N., Özgüven, Ö.F., Özyetkin, M.M.: Robust stability analysis of fractional order interval polynomials. ISA Trans. 48(2), 166–172 (2009)
Jiao, Z., Zhong, Y.: Robust stability for fractional-order systems with structured and unstructured uncertainties. Comput. Math. Appl. 64(10), 3258–3266 (2012)
Matušů, R., Şenol, B., Pekař, L.: Robust stability of fractional order polynomials with complicated uncertainty structure. PLOS ONE 12(6), e0180274 (2017)
Busłowicz, M.: Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4), 319–324 (2008)
Petráš, I., Chen, Y.Q., Vinagre, B.M., Podlubný, I.: Stability of linear time invariant systems with interval fractional orders and interval coefficients. In: Second IEEE International Conference on Computational Cybernetics, Vienna, Austria (2004)
Gómez-Aguilar, J.F., Razo-Hernández, R., Granados-Lieberman, D.: A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response. Rev. Mex. Fís. 60(1), 32–38 (2014)
Macias, M., Sierociuk, D.: Fractional order calculus for modeling and fractional PID control of the heating process. In: Proceedings of the 13th International Carpathian Control Conference, High Tatras, Podbanské, Slovakia (2012)
Sierociuk, D., Dzieliński, A., Sarwas, G., Petráš, I., Podlubný, I., Škovránek, T.: Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A, 371(1990), 20120146 (2013)
Matušů, R., Pekař, L.: Robust stability of thermal control systems with uncertain parameters: the graphical analysis examples. Appl. Therm. Eng. 125, 1157–1163 (2017)
Yeroğlu, C., Özyetkin, M.M., Tan, N.: Frequency response computation of fractional order interval transfer functions. Int. J. Control Autom. Syst. 8(5), 1009–1017 (2010)
Şenol, B., Yeroğlu, C.: Computation of the value set of fractional order uncertain polynomials: a 2q convex parpolygonal approach. In: Proceedings of the 2012 IEEE International Conference on Control Applications, Dubrovnik, Croatia (2012)
Yeroğlu, C., Şenol, B.: Investigation of robust stability of fractional order multilinear affine systems: 2q-convex parpolygon approach. Syst. Control Lett. 62(10), 845–855 (2013)
Labora, D.C., Nieto, J.J., Rodríguez-López, R.: Is it possible to construct a fractional derivative such that the index law holds? Prog. Fract. Differ. Appl. 4(1), 1–3 (2018)
Bhalekar, S., Patil, M.: Can we split fractional derivative while analyzing fractional differential equations? Commun. Nonlinear Sci. Numer. Simul. 76, 12–24 (2019)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Matušů, R., Senol, B., Alagoz, B.B., Ates, A. (2021). Value Set-Based Numerical Analysis of Robust Stability for Fractional-Order Retarded Quasi-Polynomials with Uncertain Parameters and Uncertain Fractional Orders. In: Silhavy, R., Silhavy, P., Prokopova, Z. (eds) Data Science and Intelligent Systems. CoMeSySo 2021. Lecture Notes in Networks and Systems, vol 231. Springer, Cham. https://doi.org/10.1007/978-3-030-90321-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-90321-3_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-90320-6
Online ISBN: 978-3-030-90321-3
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)