Startseite Completeness on the stability criterion of fractional order LTI systems
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Completeness on the stability criterion of fractional order LTI systems

  • Yiheng Wei EMAIL logo , Yuquan Chen , Songsong Cheng und Yong Wang
Veröffentlicht/Copyright: 18. Februar 2017
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

The importance of the concept of stability in fractional order system and control has been recognized for some time now. Recently, it has become evident that many conclusions were drawn, but little consensus was reached. Consequently, there is an urgent need for a much deeper understanding of such a concept. With the definition of fractional order positive definite matrix, a set of equivalent and elegant stability criteria are developed via revisiting a stability criterion we proposed before. All the results are formed in terms of linear matrix inequalities. Afterwards, a series of interesting properties of these criteria are revealed profoundly, including completeness, singularity, conservatism, etc. Eventually, a simulation study is provided to validate the effectiveness of the obtained results.

Acknowledgements

The work described in this paper was fully supported by the National Natural Science Foundation of China (No. 61573332, 61601431), the Fundamental Research Funds for the Central Universities (No. WK2100100028), the Anhui Provincial Natural Science Foundation (No. 1708085QF141) and the General Financial Grant from the China Postdoctoral Science Foundation (No. 2016M602032).

References

[1] H.S. Ahn and Y.Q. Chen, Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica44, No 11 (2008), 2985–2988; 10.1016/j.automatica.2008.07.003.Suche in Google Scholar

[2] R. Caponetto, S. Graziani, V. Tomasello, and A. Pisano, Identification and fractional super-twisting robust control of IPMC actuators. Fract. Calc. Appl. Anal. 18, No 6 (2015), 1358–1378; 10.1515/fca-2015-0079;https://www.degruyter.com/view/j fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.Suche in Google Scholar

[3] L.P. Chen, Y.G. He, Y. Chai, and R.C. Wu, New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dynam. 75, No 4 (2014), 633–641; 10.1007/s11071-013-1091-5.Suche in Google Scholar

[4] M. Chilali, P. Gahinet, and P. Apkarian, Robust pole placement in LMI regions. IEEE Trans. Autom. Control44, No 12 (1999), 2257–2270; 10.1109/9.811208.Suche in Google Scholar

[5] M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear. Sci. 22, No 1 (2015), 650–659; 10.1016/j.cnsns.2014.10.008.Suche in Google Scholar

[6] M.O. Efe, Fractional order systems in industrial automation-a survey. IEEE Trans. Ind. Inform. 7, No 4 (2011), 582–591; 10.1109/TII.2011.2166775.Suche in Google Scholar

[7] C. Farges, M. Moze, and J. Sabatier, Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica46, No 10 (2010), 1730–1734; 10.1016/j.automatica.2010.06.038.Suche in Google Scholar

[8] T.J. Freeborn, B. Maundy, and A.S. Elwakil, Fractional-order models of supercapacitors, batteries and fuel cells: A survey. Renew. Sust. Energ. Rev. 4, No 3 (2015), 1–7; 10.1007/s40243-015-0052-y.Suche in Google Scholar

[9] C. Ionescu and C. Muresan, Sliding mode control for a class of subsystems with fractional order varying trajectory dynamics. Fract. Calc. Appl. Anal. 18, No 6 (2015), 1441–1451; 10.1515/fca-2015-0083;https://www.degruyter.com/viewZj/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.Suche in Google Scholar

[10] Y. Li, Y.Q. Chen, and I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica45, No 8 (2009), 1965–1969; 10.1016/j.automatica.2009.04.003.Suche in Google Scholar

[11] S. Liang, C. Peng, and Y. Wang, Improved linear matrix inequalities stability criteria for fractional order systems and robust stabilization synthesis: The 0 < α < 1 case. Contl. Theor. Appl. 30, No 4 (2013), 531–535; 10.7641/CTA.2013.20674.Suche in Google Scholar

[12] J.G. Lu and Y.Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order α: The 0 < α < 1 case. IEEE Trans. Autom. Control55, No 1 (2010), 152–158; 10.1109/TAC.2009.2033738.Suche in Google Scholar

[13] J.G. Lu and Y.Q. Chen, Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fract. Calc. Appl. Anal. 16, No 1 (2013), 142–157; 10.2478/s13540-013-0010-2;https://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.Suche in Google Scholar

[14] J.A.T. Machado, V. Kiryakova, and F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear. Sci. 16, No 3 (2011), 1140–1153; 10.1016/j.cnsns.2010.05.027.Suche in Google Scholar

[15] B.B. Mandelbrot, A class of long-tailed probability distributions and the empirical distribution of city sizes. In: F. Massarik, P. Ratoosh (Eds.), Mathematical Explanations in Behavioral Science, Homewood Editions, New York (1965), 322–332.Suche in Google Scholar

[16] D. Matignon, Stability results for fractional differential equations with applications to control processing. In: IMACS Multiconference: Computational Engineering in Systems Applications, Lille, France (1996), 963–968.Suche in Google Scholar

[17] M. Moze, J. Sabatier, and A. Oustaloup, LMI tools for stability analysis of fractional systems. In: 5th Internat. Conference on Multibody Systems, Nonlinear Dynamics, and Control, Long Beach, USA (2005), 1611–1619; 10.1115/DETC2005-85182.Suche in Google Scholar

[18] A. Oustaloup, B. Mathieu, and P. Lanusse, The CRONE control of resonant plants: application to a flexible transmission. Eur. J. Control1, No 2 (1995), 113–121; 10.1016/S0947-3580(95)70014-0.Suche in Google Scholar

[19] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Eqnations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999).Suche in Google Scholar

[20] I. Podlubny, Fractional-order systems and PIλDμ controllers. IEEE Trans. Autom. Control44, No 1 (1999), 208–214; 10.1109/9.739144.Suche in Google Scholar

[21] J. Sabatier, M. Moze, and C. Farges, On stability of fractional order systems. In: Third IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey (2008), hal-00322949.Suche in Google Scholar

[22] M.S. Tavazoei, Time response analysis of fractional-order control systems: A survey on recent results. Fract. Calc. Appl. Anal. 17, No 2 (2014), 440–461; 10.2478/s13540-014-0179-z; https://www.degruyter.com/viewZj/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.Suche in Google Scholar

[23] M.S. Tavazoei and M. Haeri, A note on the stability of fractional order systems. Math. Comput. Simulat. 79, No 5 (2009), 1566–1576; 10.1016/j.matcom.2008.07.003.Suche in Google Scholar

[24] J.C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91, No 3 (2011), 437–445; 10.1016/j.sigpro.2010.04.024.Suche in Google Scholar

[25] S. Victor and P. Melchior, Improvements on flat output characterization for fractional systems. Fract. Calc. Appl. Anal. 18, No 1 (2015), 238–260; 10.1515/fca-2015-0016; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.Suche in Google Scholar

[26] Y.H. Wei, W.P. Tse, Z. Yao, and Y. Wang, Adaptive backstepping output feedback control for a class of nonlinear fractional order systems. Nonlinear Dynam. 86, No 2 (2016), 1047–1056; 10.1007/s11071-016-2945-4.Suche in Google Scholar

[27] Y.H. Wei, W.P. Tse, B. Du, and Y. Wang, An innovative fixed-pole numerical approximation for fractional order systems. ISA Transactions62 (2016), 94–102; 10.1016/j.isatra.2016.01.010.Suche in Google Scholar PubMed

[28] C. Yin, Y.Q. Chen, and S.M. Zhong, Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica50, No 12 (2014), 3173–3181; 10.1016/j.automatica.2014.10.027.Suche in Google Scholar

[29] J.M. Yu, H. Hu, S.B. Zhou, and X.R. Lin, Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica49, No 6 (2013), 1798–1803; 10.1016/j.automatica.2013.02.041.Suche in Google Scholar

[30] R.X. Zhang, G. Tian, S.P. Yang, and H.F. Cao, Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2). ISA Transactions56, (2015), 102–110; 10.1016/j.isatra.2014.12.006.Suche in Google Scholar PubMed

[31] X.F. Zhang and Y.Q. Chen, D-stability based LMI criteria of stability and stabilization for fractional order systems. In: Internat. Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Boston, USA (2015), DETC2015-46692; 10.1115/DETC2015-46692.Suche in Google Scholar

Received: 2016-1-14
Revised: 2016-9-20
Published Online: 2017-2-18
Published in Print: 2017-2-1

© 2017 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 20–1–2017)
  4. Survey paper
  5. Ten equivalent definitions of the fractional laplace operator
  6. Research paper
  7. Consensus of fractional-order multi-agent systems with input time delay
  8. Research paper
  9. Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives
  10. Research paper
  11. A preconditioned fast finite difference method for space-time fractional partial differential equations
  12. Research paper
  13. On existence and uniqueness of solutions for semilinear fractional wave equations
  14. Research paper
  15. Computational solutions of the tempered fractional wave-diffusion equation
  16. Research paper
  17. Completeness on the stability criterion of fractional order LTI systems
  18. Research paper
  19. Wavelet convolution product involving fractional fourier transform
  20. Research paper
  21. Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method
  22. Research paper
  23. A foundational approach to the Lie theory for fractional order partial differential equations
  24. Research paper
  25. Null-controllability of a fractional order diffusion equation
  26. Research paper
  27. New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions
  28. Research paper
  29. The stretched exponential behavior and its underlying dynamics. The phenomenological approach
  30. Short Paper
  31. Lyapunov-type inequality for an anti-periodic fractional boundary value problem
Heruntergeladen am 29.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2017-0008/html
Button zum nach oben scrollen