SCIENCE CHINA Information Sciences, Volume 63 , Issue 5 : 152202(2020) https://doi.org/10.1007/s11432-019-1521-5

Necessary and sufficient conditions for normalization and sliding mode control of singular fractional-order systems with uncertainties

More info
  • ReceivedJun 1, 2019
  • AcceptedJul 22, 2019
  • PublishedMar 27, 2020



This work was supported by National Natural Science Foundation of China (Grant No. 61573008), Natural Science Foundation of Shandong Province (Grant No. ZR2016FM16), and Post-Doctoral Applied Research Projects of Qingdao (Grant No. 2015122). The authors would like to thank the anonymous reviewers for their valuable suggestions.


[1] Podlubny I. Fractional Differential Equations. NewYork: Academic Press, 1999. Google Scholar

[2] Petrá? I. Modeling and numerical analysis of fractional-order Bloch equations. Comput Math Appl, 2011, 61: 341-356 CrossRef Google Scholar

[3] Sabatier J, Farges C, Trigeassou J C. Fractional systems state space description: some wrong ideas and proposed solutions. J Vib Control, 2014, 20: 1076-1084 CrossRef Google Scholar

[4] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006. Google Scholar

[5] Wang Z. A Numerical Method for Delayed Fractional-Order Differential Equations. J Appl Math, 2013, 2013(2): 1-7 CrossRef Google Scholar

[6] Wang Z, Huang X, Zhou J. A Numerical Method for Delayed Fractional-Order Differential Equations: Based on G-L Definition. Appl Math Inf Sci, 2013, 7: 525-529 CrossRef Google Scholar

[7] Zhang B, Xu S, Ma Q. Output-feedback stabilization of singular LPV systems subject to inexact scheduling parameters. Automatica, 2019, 104: 1-7 CrossRef Google Scholar

[8] Feng Y, Yagoubi M. Robust Control of Linear Descriptor Systems. Berlin: Springer, 2017. Google Scholar

[9] Xu S Y, Lam J. Robust Control and Filtering of Singular Systems. Berlin: Springer, 2006. Google Scholar

[10] Dai L Y. Singular Control Systems. Berlin: Springer, 1989. Google Scholar

[11] N'Doye I, Darouach M, Zasadzinski M. Robust stabilization of uncertain descriptor fractional-order systems. Automatica, 2013, 49: 1907-1913 CrossRef Google Scholar

[12] Kaczorek T. Singular fractional linear systems and electrical circuits. Int J Appl Math Comput Sci, 2011, 21: 379-384 CrossRef Google Scholar

[13] Yu Y, Jiao Z, Sun C Y. Sufficient and Necessary Condition of Admissibility for Fractional-order Singular System. Acta Automatica Sin, 2013, 39: 2160-2164 CrossRef Google Scholar

[14] Zhang X, Chen Y Q. Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order $\alpha$: The $0<\alpha<1$ case. ISA Trans, 2018, 82: 42-50 CrossRef PubMed Google Scholar

[15] Lin C, Chen B, Shi P. Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems. Syst Control Lett, 2018, 112: 31-35 CrossRef Google Scholar

[16] Shiri B, Baleanu D. System of fractional differential algebraic equations with applications. Chaos Solitons Fractals, 2019, 120: 203-212 CrossRef ADS Google Scholar

[17] Ji Y, Qiu J. Stabilization of fractional-order singular uncertain systems.. ISA Trans, 2015, 56: 53-64 CrossRef PubMed Google Scholar

[18] Wei Y, Tse P W, Yao Z. The output feedback control synthesis for a class of singular fractional order systems.. ISA Trans, 2017, 69: 1-9 CrossRef PubMed Google Scholar

[19] Liu S, Zhou X F, Li X. Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time-varying delays. Appl Math Lett, 2017, 65: 32-39 CrossRef Google Scholar

[20] Dassios I K, Baleanu D I. Caputo and related fractional derivatives in singular systems. Appl Math Computation, 2018, 337: 591-606 CrossRef Google Scholar

[21] Wei Y, Wang J, Liu T. Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state. J Franklin Institute, 2019, 356: 1975-1990 CrossRef Google Scholar

[22] Meng B, Wang X, Wang Z. Synthesis of Sliding Mode Control for a Class of Uncertain Singular Fractional-Order Systems-Based Restricted Equivalent. IEEE Access, 2019, 7: 96191-96197 CrossRef Google Scholar

[23] Utkin V I, Poznyak A S. Adaptive sliding mode control. In: Advances in Sliding Mode Control. Berlin: Springer, 2013. Google Scholar

[24] Dong S L, Chen C L P, Fang M, et al. Dissipativity-based asynchronous fuzzy sliding mode control for T-S fuzzy hidden Markov jump systems. IEEE Trans Cybern, 2019. doi: 10.1109/TCYB.2019.2919299. Google Scholar

[25] Meng B, Wang X. Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control. Math Problems Eng, 2018, 2018(2): 1-8 CrossRef Google Scholar

[26] Meng B, Wang Z, Wang Z. Adaptive sliding mode control for a class of uncertain nonlinear fractional-order Hopfield neural networks. AIP Adv, 2019, 9: 065301 CrossRef ADS Google Scholar

[27] Pisano A, Tanelli M, Ferrara A. Switched/time-based adaptation for second-order sliding mode control. Automatica, 2016, 64: 126-132 CrossRef Google Scholar

[28] Edwards C, Shtessel Y B. Adaptive continuous higher order sliding mode control. Automatica, 2016, 65: 183-190 CrossRef Google Scholar

[29] Pérez-Ventura U, Fridman L. Design of super-twisting control gains: A describing function based methodology. Automatica, 2019, 99: 175-180 CrossRef Google Scholar

[30] Casta?os F, Hernández D, Fridman L M. Integral sliding-mode control for linear time-invariant implicit systems. Automatica, 2014, 50: 971-975 CrossRef Google Scholar

[31] Obeid H, Fridman L M, Laghrouche S. Barrier function-based adaptive sliding mode control. Automatica, 2018, 93: 540-544 CrossRef Google Scholar

[32] Guo Y, Lin C, Chen B. Necessary and sufficient conditions for the dynamic output feedback stabilization of fractional-order systems with order 0 < α < 1. Sci China Inf Sci, 2019, 62: 199201 CrossRef Google Scholar

[33] Wei Y, Chen Y, Cheng S. Completeness on the stability criterion of fractional order LTI systems. Fractional Calculus Appl Anal, 2017, 20 CrossRef Google Scholar

[34] Cobb D. Controllability, observability, and duality in singular systems. IEEE Trans Automat Contr, 1984, 29: 1076-1082 CrossRef Google Scholar

[35] Chen Y Q, Ahn H S, Xue D. Robust controllability of interval fractional order linear time invariant systems. Signal Processing, 2006, 86: 2794-2802 CrossRef Google Scholar

[36] Aguila-Camacho N, Duarte-Mermoud M A, Gallegos J A. Lyapunov functions for fractional order systems. Commun NOnlinear Sci Numer Simul, 2014, 19: 2951-2957 CrossRef ADS Google Scholar

[37] Wu Z G, Dong S L, Shi P, et al. Reliable filter design of Takagi-Sugeno fuzzy switched systems with imprecise modes. IEEE Trans Cybern, 2019. doi: 10.1109/TCYB.2018.2885505. Google Scholar

[38] Dong S L, Fang M, Shi P, et al. Dissipativity-based control for fuzzy systems with asynchronous modes and intermittent measurements. IEEE Trans Cybern, 2019. doi: 10.1109/TCYB.2018.2887060. Google Scholar

[39] Hu X, Xia J, Wei Y. Passivity-based state synchronization for semi-Markov jump coupled chaotic neural networks with randomly occurring time delays. Appl Math Computation, 2019, 361: 32-41 CrossRef Google Scholar