Adaptive Stabilization for a Class of Fractional-Order Systems with Nonlinear Uncertainty

The stabilization task for uncertain integer-order systems has been widely and extensively investigated in the literature. However, stabilizing uncertain fractional-order systems (despite the recent great interest given by researchers to this research axis) is still considered as a fertile area of research. In this paper, an original adaptive scheme to handle this particular problem, under the nonlinear uncertainty modeling, is suggested. The approach consists of estimating the upper bound of uncertainties and designing an adaptive output feedback controller, using the Lyapunov direct method. The concept of uniform practical Mittag–Leffler stability is used throughout the paper. The feasibility and effectiveness of the theoretical results are shown through simulations via two numerical examples.

[1]  Manuel A. Duarte-Mermoud,et al.  Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems , 2015, Commun. Nonlinear Sci. Numer. Simul..

[2]  Huijun Gao,et al.  Reconfigurable Tolerant Control of Uncertain Mechanical Systems With Actuator Faults: A Sliding Mode Observer-Based Approach , 2018, IEEE Transactions on Control Systems Technology.

[3]  Yi Liu,et al.  Simple Nonlinear Predictive Control Strategy for Chemical Processes Using Sparse Kernel Learning with Polynomial Form , 2010 .

[4]  I. Podlubny.  Fractional differential equations , 1998 .

[5]  Ping Li,et al.  Kernel learning adaptive one‐step‐ahead predictive control for nonlinear processes , 2008 .

[6]  Sara Dadras,et al.  A New Fractional Order Observer Design for Fractional Order Nonlinear Systems , 2011 .

[7]  Yan-Jun Liu,et al.  Adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems , 2007, Inf. Sci..

[8]  Nabil Derbel,et al.  Robust sensor fault estimation for fractional-order systems with monotone nonlinearities , 2017 .

[9]  Haibo He,et al.  Adaptive Dynamic Programming for Decentralized Stabilization of Uncertain Nonlinear Large-Scale Systems With Mismatched Interconnections , 2020, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[10]  Luigi Fortuna,et al.  Special issue on Complexity in Engineering , 2018 .

[11]  Omar Naifar,et al.  Comments on "Lyapunov stability theorem about fractional system without and with delay" , 2016, Commun. Nonlinear Sci. Numer. Simul..

[12]  Yusheng Liu,et al.  Robust adaptive observer for nonlinear systems with unmodeled dynamics , 2009, Autom..

[13]  Omar Naifar,et al.  Comments on "Mittag-Leffler stability of fractional order nonlinear dynamic systems [Automatica 45(8) (2009) 1965-1969]" , 2017, Autom..

[14]  N. Laskin.  Fractional market dynamics , 2000 .

[15]  Takanori Yazawa,et al.  Tool wear estimation method in milling process using air turbine spindle rotation-control system equipped with disturbance force observer , 2018 .

[16]  M. Elloumi,et al.  State estimation for nonlinear conformable fractional‐order systems: A healthy operating case and a faulty operating case , 2019, Asian Journal of Control.

[17]  Nabil Derbel,et al.  Observer-based model reference control for linear fractional-order systems , 2018 .

[18]  S. Choi,et al.  Stability for Caputo Fractional Differential Systems , 2014 .

[19]  Jiqing Qiu,et al.  Stabilization of fractional-order singular uncertain systems. , 2015, ISA transactions.

[20]  Nabil Derbel,et al.  Sensor fault estimation for fractional-order descriptor one-sided Lipschitz systems , 2018 .

[21]  N. Engheta.  On fractional calculus and fractional multipoles in electromagnetism , 1996 .

[22]  A. Alikhanov.  A priori estimates for solutions of boundary value problems for fractional-order equations , 2010, 1105.4592.

[23]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[24]  Junwei Lu,et al.  Design of fuzzy state feedback controller for robust stabilization of uncertain fractional-order chaotic systems , 2014, J. Frankl. Inst..

[25]  Igor Podlubny,et al.  Mittag-Leffler stability of fractional order nonlinear dynamic systems , 2009, Autom..

[26]  Yi Liu,et al.  Adaptive Control of Nonlinear Time-Varying Processes Using Selective Recursive Kernel Learning Method , 2011 .

[27]  Guanghong Yang,et al.  Backstepping adaptive fuzzy control of uncertain nonlinear systems against actuator faults , 2009 .

[28]  Xinglong Zhang,et al.  The control method of a time-delay system with nonlinear uncertainty , 2010, 2010 International Conference on Computer Application and System Modeling (ICCASM 2010).

[29]  Hamid Reza Karimi,et al.  Observer-based adaptive stabilization of a class of uncertain nonlinear systems , 2014 .

[30]  Guisheng Zhai,et al.  Adaptive practical stabilization of a class of uncertain nonlinear systems via sampled-data control , 2018 .

[31]  Yong-Hong Lan,et al.  Observer-based robust control of a (1⩽ a ≪ 2) fractional-order uncertain systems: a linear matrix inequality approach , 2012 .

[32]  Shaocheng Tong,et al.  Observer-based adaptive fuzzy tracking control for a class of uncertain nonlinear MIMO systems , 2011, Fuzzy Sets Syst..

[33]  Chaoxu Mu,et al.  Adaptive Critic Control with Robust Stabilization for Uncertain Nonlinear Systems , 2018 .

[34]  Ryan P. Jenkins,et al.  A lumped parameter vane pump model for system stability analysis , 2018 .