Abstract
In this paper, the problems of robust stability and stabilization, for the first time, are studied for delayed fractional-order linear systems with convex polytopic uncertainties. The authors derive some sufficient conditions for the problems based on linear matrix inequality technique combined with fractional Razumikhin stability theorem. All the results are obtained in terms of linear matrix inequalities that are numerically tractable. The proposed results are quite general and improve those given in the literature since many factors, such as discrete and distributed delays, convex polytopic uncertainties, global stability and stabilizability, are considered. Numerical examples and simulation results are given to illustrate the effectiveness of the effectiveness of our results.
Similar content being viewed by others
References
Hollkamp J P, Sen M, and Semperlotti F, Vibration analysis of discrete parameter systems using fractional-order models, SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, Portland, Oregon, United States, 2017.
Tenreiro Machado J A and Mata M E, Pseudo phase plane and fractional calculus modeling of western global economic downturn, Communications in Nonlinear Science and Numerical Simulation, 2015, 22(1–3): 396–406.
Abrashov S, Malti R, Moze M, et al., Simple and robust experiment design for system identification using fractional models, IEEE Transactions on Automatic Control, 2017, 62(6): 2648–2658.
Bagley R L and Calico R A, Fractional order state equations for the control of viscoelastically damped structures, Journal of Guidance, Control, and Dynamics, 1991, 14(2): 304–311.
Chen W C, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, Solitons & Fractals, 2008, 36(5): 1305–1314.
Heaviside O, Electromagnetic Theory, Chelsea, New York, 1971.
Debnath L, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 2003, 54: 3413–3442.
Hilfer R, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
Ahn H S, Chen Y Q, and Podlubny I, Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, Applied Mathematics and Computation, 2007, 187(1): 27–34.
Odibat Z M, Analytic study on linear systems of fractional differential equations, Computers & Mathematics with Applications, 2010, 59(3): 1171–1183.
Li Y, Chen Y Q, and Podlubny I, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computers & Mathematics with Applications, 2010, 59(5): 1810–1821.
Chen L, He Y, Chai Y, et al., New results on stability and stabilization of a class of nonlinear fractional-order systems, Nonlinear Dynamics, 2014, 75(4): 633–641.
Zhang S, Yu Y, and Wang H, Mittag-leffler stability of fractional-order Hopfield neural networks, Nonlinear Analysis: Hybrid Systems, 2015, 16: 104–121.
Zhang S, Yu Y, and Yu J, LMI conditions for global stability of fractional-order neural networks, IEEE Transactions on Neural Networks and Learning Systems, 2017, 28(10): 2423–2433.
Yin C, Zhong S, Huang X, et al., Robust stability analysis of fractional-order uncertain singular nonlinear system with external disturbance, Applied Mathematics and Computation, 2015, 269: 351–362.
Liu S, Zhou X F, Li X, et al., Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time-varying delays, Applied Mathematics Letters, 2017, 65: 32–39.
Zhang J, Zhao X, and Chen Y, Finite-time stability and stabilization of fractional order positive switched systems, Circuits, Systems, and Signal Processing, 2016, 35(7): 2450–2470.
Shen J and Lam J, Stability and performance analysis for positive fractional-order systems with time-varying delays, IEEE Transactions on Automatic Control, 2016, 61(9): 2676–2681.
Benzaouia A and El Hajjaji A, Stabilization of continuous-time fractional positive T-S fuzzy systems by using a Lyapunov function, Circuits, Systems, and Signal Processing, 2017, 36(10): 3944–3957.
Liu L, Cao X Y, Fu Z, et al., Guaranteed cost finite-time control of fractional-order nonlinear positive switched systems with D-perturbations via MDAD, Journal of Systems Science and Complexity, 2019, 32(3): 857–874.
Liang J, Wu B W, Wang Y E, et al., Input-output finite-time stability of fractional-order positive switched systems, Circuits, Systems, and Signal Processing, 2019, 38(4): 1619–1638.
Zhang R and Yang S, Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller, Nonlinear Dynamics, 2012, 68(1–2): 45–51.
Faieghi M, Kuntanapreeda S, Delavari H, et al., LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynamics, 2013, 72(1–2): 301–309.
Ji Y and Qiu J, Stabilization of fractional-order singular uncertain systems, ISA Transactions, 2015, 56: 53–64.
Lenka B K and Banerjee S, Asymptotic stability and stabilization of a class of nonautonomous fractional order systems, Nonlinear Dynamics, 2016, 85(1): 167–177.
Wei Y, Tse P W, Yao Z, et al., The output feedback control synthesis for a class of singular fractional order systems, ISA Transactions, 2017, 69: 1–9.
Ji Y, Du M, and Guo Y, Stabilization of non-linear fractional-order uncertain systems, Asian Journal of Control, 2018, 20(2): 669–677.
Thuan M V and Huong D C, New results on stabilization of fractional-order nonlinear systems via an LMI approach, Asian Journal of Control, 2018, 20(4): 1541–1550.
Zhang J E, Stabilization of uncertain fractional-order complex switched networks via impulsive control and its application to blind source separation, IEEE Access, 2018, 6: 32780–32789.
Chen Y Q, Ahn H S, and Podlubny I, Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal Processing, 2006, 86: 2611–2618.
Ahn H S and Chen Y Q, Necessary and sufficient stability condition of fractional-order interval linear systems, Automatica, 2008, 44: 2985–2988.
Lu J G and Chen G, Robust stability and stabilization of fractional-order interval systems: An LMI approach, IEEE Transactions on Automatic Control, 2009, 54(6): 1294–1299.
Lu J G and Chen Y Q, Robust stability and stabilization of fractional-order interval systems with the fractional order α: The 0 < α < 1 case, IEEE Transactions on Automatic Control, 2010, 55(1): 152–158.
Thuan M V, Trinh H, and Huong D C, Reachable sets bounding for switched systems with timevarying delay and bounded disturbances, International Journal of Systems Science, 2017, 48(3): 494–504.
Adelipour S, Abooee A, and Haeri M, LMI-based sufficient conditions for robust stability and stabilization of LTI-fractional-order systems subjected to interval and polytopic uncertainties, Transactions of the Institute of Measurement and Control, 2015, 37(10): 1207–1216.
Li P, Chen L, Wu R, et al., Robust asymptotic stability of interval fractional-order nonlinear systems with time-delay, Journal of the Franklin Institute, 2018, 355(15): 7749–7763.
Farges C, Sabatier J, and Moze M, Fractional order polytopic systems: Robust stability and stabilisation, Advances in Difference Equations, 2011, 35: 1–10.
Jiao Z and Zhong Y, Robust stability for fractional-order systems with structured and unstructured uncertainties, Computers and Mathematics with Applications, 2012, 64(10): 3258–3266.
Lu J G and Chen Y Q, Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties, Fractional Calculus and Applied Analysis, 2013, 16(1): 142–157.
Chen L, Wu R, He Y, et al., Robust stability and stabilization of fractional-order linear systems with polytopic uncertainties, Applied Mathematics and Computation, 2015, 257: 274–284.
Li S, Robust stability and stabilization of LTI fractional-order systems with poly-topic and two-norm bounded uncertainties, Advances in Difference Equations, 2018, 88: 1–13.
Mori T and Kokame H, A parameter-dependent Lyapunov function for a polytope of matrices, IEEE Transactions on Automatic Control, 2000, 45(8): 1516–1519.
Phat V N, Ha Q P, and Trinh H, Parameter-dependent H8 control for time-varying delay polytopic systems, Journal of Optimization Theory and Applications, 2010, 147: 58–70.
Duarte-Mermoud M A, Aguila-Camacho N, Gallegos J A, et al., Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simulat., 2015, 22(1–3): 650–659.
Kilbas A, Srivastava H, and Trujillo J, Theory and Application of Fractional Differential Equations, Elsevier, New York, 2006.
Gu K, An integral inequality in the stability problem of time-delay systems, Proc. IEEE Conf. Dec. Contr., Sydney, Australia, 2000, 2805–2810.
Boyd S, Ghaoui L E, Feron E, et al., Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.
Wen Y, Zhou X F, Zhang Z, et al., Lyapunov method for nonlinear fractional differential systems with delay, Nonlinear Dynamics, 2015, 82: 1015–1025.
Deng W, Li C, and Lü J, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynamics, 2007, 48: 409–416.
Thanh N T, Trinh H, and Phat V N, Stability analysis of fractional differential time-delay equations, IET Control Theory Appl., 2017, 11(7): 1006–1015.
Liu S, Yang R, Zhou X F, et al., Stability analysis of fractional delayed equations and its applications on consensus of multi-agent systems, Communications in Nonlinear Science and Numerical Simulation, 2019, 73: 351–362.
Author information
Authors and Affiliations
Corresponding authors
Additional information
The research of Mai Viet Thuan is was supported by Ministry of Education and Training of Vietnam (B2020-TNA).
This paper was recommended for publication by Editor SUN Jian.
Rights and permissions
About this article
Cite this article
Dinh, C.H., Mai, V.T. & Duong, T.H. New Results on Stability and Stabilization of Delayed Caputo Fractional Order Systems with Convex Polytopic Uncertainties. J Syst Sci Complex 33, 563–583 (2020). https://doi.org/10.1007/s11424-020-8338-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-020-8338-2