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Fractional Calculus and Applied Analysis

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Volume 22, Issue 6

Issues

Robust stability analysis of LTI systems with fractional degree generalized frequency variables

Cuihong Wang / Yan Guo / Shiqi Zheng / YangQuan Chen
Published Online: 2019-12-31 | DOI: https://doi.org/10.1515/fca-2019-0085

Abstract

A novel linear time-invariant (LTI) system model with fractional degree generalized frequency variables (FDGFVs) is proposed in this paper. This model can provide a unified form for many complex systems, including fractional-order systems, distributed-order systems, multi-agent systems and so on. This study mainly investigates the stability and robust stability problems of LTI systems with FDGFVs. By characterizing the relationship between generalized frequency variable and system matrix, a necessary and sufficient stability condition is firstly presented for such systems. Then for LTI systems with uncertain FDGFVs, we present a robust stability method in virtue of zero exclusion principle. Finally, the effectiveness of the method proposed in this paper is demonstrated by analyzing the robust stability of gene regulatory networks.

MSC 2010: Primary 26A33; Secondary 37C75; 37D20; 93D09; 93D15 93D20

Key Words and Phrases: fractional-order system; linear time-invariant systems; stability analysis; generalized frequency variables; gene regulatory networks

Dedicated to the memory of late Professor Wen Chen

References

  • [1]

    B.B. Alagoz, Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations. ISA Trans. 70 (2017), 7–15; .CrossrefPubMedWeb of ScienceGoogle Scholar

  • [2]

    C. Bonnet, J. Partington, Coprime factorizations and stability of fractional differential systems. Syst. Control Lett. 41, No 3 (2000), 167–174; .CrossrefGoogle Scholar

  • [3]

    C. Bonnet, J. Partington, Analysis of fractional delay systems of retarded and neutral type. Automatica 38, No 7 (2002), 1133–1138; .CrossrefGoogle Scholar

  • [4]

    L. Chen, K. Aihara, Stability of genetic regulatory networks with time delay. IEEE Trans. Circuits-I. 49, No 5 (2002), 602–608; .CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Z. Gao, Analytical criterion on stabilization of fractional-order plants with interval uncertainties using fractional-order PDμ controllers with a filter. ISA Trans. 83 (2018), 25–34; .CrossrefPubMedWeb of ScienceGoogle Scholar

  • [6]

    S. Hara, T. Hayakawa, and H. Sugata, Stability analysis of linear systems with generalized frequency variables and its applications to formation control. 46th IEEE Conf. on Decision and Control, Dec 12–14, (2007), New Orleans, LA, USA, 1459–1466; .CrossrefGoogle Scholar

  • [7]

    S. Hara, T. Hayakawa, and H. Sugata, LTI systems with generalized frequency variables: A unified framework for homogeneous multi-agent dynamical systems. SICE J. Control, Measurement, and System Integration 2, No 5 (2009), 299–306; .CrossrefGoogle Scholar

  • [8]

    S. Hara, T. Iwasaki, Y. Hori, Robust stability analysis for LTI systems with generalized frequency variables and its application to gene regulatory networks. Automatica 105 (2019), 96–106; .CrossrefWeb of ScienceGoogle Scholar

  • [9]

    S. Hara, M. Kanno, and H. Tanaka, Cooperative gain output feedback stabilization for multi-agent dynamical systems. 48th IEEE Conf. on Decision and Control held jointly with the 28th Chinese Control Conf., Dec 15-18, (2009), Shanghai, China, 877–882; .CrossrefGoogle Scholar

  • [10]

    S. Hara, H. Tanaka, and T. Iwasaki, Stability analysis of systems with generalized frequency variables. IEEE Trans. Automat. Contr. 59, No 2 (2014), 313–326; .CrossrefWeb of ScienceGoogle Scholar

  • [11]

    Y. Hori, H. Miyazako, S. Kumagai, and S. Hara, Coordinated spatial pattern formation in biomolecular communication networks. IEEE Trans. Molecular, Biological and Multi-Scale Communications 1, No 2 (2015), 111–121; .CrossrefGoogle Scholar

  • [12]

    C. Huang, J. Cao, M. Xiao, Hybrid control on bifurcation for a delayed fractional gene regulatory network. Chaos Soliton. Fract. 87 (2016), 19–29; .CrossrefWeb of ScienceGoogle Scholar

  • [13]

    R. Ji, L. Ding, X. Yan and M. Xin, Modelling gene regulatory network by fractional order differential equations. IEEE 5th Internat. Conf. on Bio-Inspired Computing: Theories and Applications (BIC-TA), Sept 23-26, (2010), Changsha, China, 431–434; .CrossrefGoogle Scholar

  • [14]

    Z. Jiao and Y.Q. Chen, Stability analysis of fractional-order systems with double noncommensurate orders for matrix case. Fract. Calc. Appl. Anal. 14, No 3 (2011), 436–453; ; https://www.degruyter.com/view/j/fca.2011.14.issue-3/issue-files/fca.2011.14.issue-3.xml.CrossrefWeb of Science

  • [15]

    Z. Jiao and Y.Q. Chen, Stability of fractional-order linear time-invariant systems with multiple noncommensurate orders. Comput. Math. Appl. 64, No 10 (2012), 3053–3058; .CrossrefWeb of ScienceGoogle Scholar

  • [16]

    Z. Jiao, Y. Chen, and I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives. Springer, London (2013).Google Scholar

  • [17]

    T.H. Kim, Y. Hori, and S. Hara, Robust stability analysis of gene-protein regulatory networks with cyclic activation-repression interconnections. Syst. Control Lett. 60, No 6 (2011), 373–382; .CrossrefWeb of ScienceGoogle Scholar

  • [18]

    C. Li, Q. Yi, Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation. Fract. Calc. Appl. Anal. 21, No 4 (2018), 1046–1472; ; https://www.degruyter.com/view/j/fca.2018.21.issue-4/issue-files/fca.2018.21.issue-4.xml.CrossrefWeb of Science

  • [19]

    S. Liang, S.G. Wang, Y. Wang, Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees. J. Franklin. Inst. 354 No 1 (2017), 83–104; .CrossrefWeb of ScienceGoogle Scholar

  • [20]

    G. Ling, Z. Guan, R. Liao, and X. Cheng, Stability and bifurcation analysis of cyclic genetic regulatory networks with mixed time delays. SIAM J. Appl. Dyn. Syst. 14, No 1 (2015), 202–220; .CrossrefWeb of ScienceGoogle Scholar

  • [21]

    D. Matignon, Stability properties for generalized fractional differential systems. In: ESAIM: Proceedings and Surveys 5 (1998), 145–158; .CrossrefGoogle Scholar

  • [22]

    R. Mohsenipour, J.M. Fathi, Robust stability analysis of fractional-order interval systems with multiple time delays. Int. J. Robust Nonlin. Control 29 No 6 (2019), 1823–1839; .CrossrefWeb of ScienceGoogle Scholar

  • [23]

    C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue and V. Feliu-Batlle, Fractional-order Systems and Controls: Fundamentals and Applications. Springer Science & Business Media, London (2010).Google Scholar

  • [24]

    I. Petras, Stability of fractional order systems with rational orders: A survey. Fract. Calc. Appl. Anal. 12, No 3 (2009), 269–298.Google Scholar

  • [25]

    I. Petras, L. Dorcak, The frequency method for stability investigation of fractional control systems. Journal of SACTA 2, No. 1–2 (1999), 75–85.Google Scholar

  • [26]

    I. Petras, Y. Q. Chen and B. M. Vinagre, Robust Stability Test for Interval Fractional Order Linear Systems, Unsolved Problems in Mathematics and Control Systems. Princeton University Press, USA, (2004).Google Scholar

  • [27]

    I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications. 198, Academic Press, San Diego (1999).Google Scholar

  • [28]

    F. Ren, F. Cao, J. Cao, Mittag-Leffler stability and generalized Mittag-Leffler stability of fractional-order gene regulatory networks. Neurocomputing 160 (2015), 185–90; .CrossrefWeb of ScienceGoogle Scholar

  • [29]

    J. Sabatier, C. Farges, J.C. Trigeassou, A stability test for non-commensurate fractional order systems. Syst. Control Lett. 62, No 9 (2013), 739–746; .CrossrefWeb of ScienceGoogle Scholar

  • [30]

    J. Sabatier, M. Moze, and C. Farges, LMI stability conditions for fractional order systems. Comput. Math. Appl. 59, No 5 (2010), 1594–1609; .CrossrefWeb of ScienceGoogle Scholar

  • [31]

    H. Taghavian and M.S. Tavazoei, Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach. Int. J. Robust Nonlin. Control 28, No 4 (2017), 1127–1144; .CrossrefWeb of ScienceGoogle Scholar

  • [32]

    C.D. Thron, The secant condition for instability in biochemical feedback control–I. The role of cooperativity and saturability. Bull. Math. Biol. 53, No 3 (1991), 383–401; .CrossrefGoogle Scholar

  • [33]

    M.E. Valcher and P. Misra, On the controllability and stabilizability of non-homogeneous multi-agent dynamical systems. Syst. Control Lett. 61, No 7 (2012), 780–787; .CrossrefWeb of ScienceGoogle Scholar

  • [34]

    M.E. Valcher and P. Misra, On the stabilizability and consensus of positive homogeneous multi-agent dynamical systems. IEEE Trans. Automat. Contr. 59, No 7 (2014), 1936–1941; .CrossrefWeb of ScienceGoogle Scholar

  • [35]

    Y. Wang, H. Fujimoto, and S. Hara, Torque distribution-based range extension control system for longitudinal motion of electric vehicles by LTI modeling with generalized frequency variable. IEEE/ASME Trans. Mechatron. 21, No 1 (2016), 443–452; .CrossrefWeb of ScienceGoogle Scholar

  • [36]

    Z. Wang, H. Gao, J. Cao and X. Liu, On delayed genetic regulatory networks with polytopic uncertainties: robust stability analysis. IEEE Trans. Nanobiosci. 7, No 2 (2008), 154–163; .CrossrefWeb of ScienceGoogle Scholar

  • [37]

    C. Wang, Y. Zhao and Y.Q. Chen, The controllability, observability, and stability analysis of a class of composite systems with fractional degree generalized frequency variables. IEEE/CAA J. Automat. Sin. 6, No 3 (2019), 859–864; CrossrefGoogle Scholar

  • [38]

    Y. Wei, Y.Q. Chen, S. Cheng and Y. Wang, A note on short memory principle of fractional calculus. Frac. Calc. Appl. Anal. 20, No 6 (2017), 1382–1404; ; https://www.degruyter.com/view/j/fca.2017.20.issue-6/issue-files/fca.2017.20.issue-6.xml.Crossref

  • [39]

    S. Zheng, Robust stability of fractional order system with general interval uncertainties. Syst. Control Lett. 99 (2017), 1–8, .CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2019-07-27

Published Online: 2019-12-31

Published in Print: 2019-12-18


Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 6, Pages 1655–1674, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2019-0085.

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