A novel algorithm on adaptive backstepping control of fractional order systems

This article presents a new method of designing the adaptive backstepping controller for triangular fractional order systems with non-commensurate orders. The fractional order indirect Lyapunov method is available for the 0 < α < 1 case, by introducing the appropriate transformations of frequency distributed model. Meanwhile, based on the semigroup property of fractional order derivative and the fractional order tracking differentiator, we extend the result to the 1 < α < 2 case. The case of controlling fractional order systems with unknown control input coefficient is also investigated. Simulation examples are given to demonstrate the effectiveness of the proposed controllers.

[1]  Jianbin Qiu,et al.  Fuzzy-Model-Based Piecewise ${\mathscr H}_{\infty }$ Static-Output-Feedback Controller Design for Networked Nonlinear Systems , 2010, IEEE Transactions on Fuzzy Systems.

[2]  Alain Oustaloup,et al.  A Lyapunov approach to the stability of fractional differential equations , 2009, Signal Process..

[3]  Mehmet Önder Efe,et al.  Fractional Order Systems in Industrial Automation—A Survey , 2011, IEEE Transactions on Industrial Informatics.

[4]  Samir Ladaci,et al.  On Fractional Adaptive Control , 2006 .

[5]  Jianbin Qiu,et al.  Observer-Based Piecewise Affine Output Feedback Controller Synthesis of Continuous-Time T–S Fuzzy Affine Dynamic Systems Using Quantized Measurements , 2012, IEEE Transactions on Fuzzy Systems.

[6]  Alain Oustaloup,et al.  Lyapunov Stability of Linear Fractional Systems: Part 1 — Definition of Fractional Energy , 2013 .

[7]  Guoping Lu,et al.  Lyapunov stability theorem about fractional system without and with delay , 2015, Commun. Nonlinear Sci. Numer. Simul..

[8]  Jianbin Qiu,et al.  Static-Output-Feedback ${\mathscr H}_{\bm \infty }$ Control of Continuous-Time T - S Fuzzy Affine Systems Via Piecewise Lyapunov Functions , 2013, IEEE Transactions on Fuzzy Systems.

[9]  Li Xu,et al.  Adaptive Mittag-Leffler stabilization of commensurate fractional-order nonlinear systems , 2014, 53rd IEEE Conference on Decision and Control.

[10]  Lan Song,et al.  Tracking Differentiator Based Fractional Order Model Reference Adaptive Control: The 1 < α < 2 Case , 2014, 53rd IEEE Conference on Decision and Control.

[11]  Y. Chen,et al.  Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications , 2011 .

[12]  Xinzhi Liu,et al.  Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks , 2014, Commun. Nonlinear Sci. Numer. Simul..

[13]  Dongli Zhang,et al.  Fractional order sliding mode controller design for antilock braking systems , 2013, Neurocomputing.

[14]  Yan Zhou,et al.  An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks , 2014, Neurocomputing.

[15]  I. Podlubny.  Fractional-order systems and PIλDμ-controllers , 1999, IEEE Trans. Autom. Control..

[16]  Quan-Sen Sun,et al.  Fractional-order embedding multiset canonical correlations with applications to multi-feature fusion and recognition , 2013, Neurocomputing.

[17]  Yong Wang,et al.  A rational approximate method to fractional order systems , 2014 .

[18]  Alain Oustaloup,et al.  How to impose physically coherent initial conditions to a fractional system , 2010 .

[19]  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..

[20]  Zhen Wang,et al.  Control of an uncertain fractional order economic system via adaptive sliding mode , 2012, Neurocomputing.

[21]  Lin Shi,et al.  Analysis and control of a hyperchaotic system with only one nonlinear term , 2012 .

[22]  Alain Oustaloup,et al.  The infinite state approach: Origin and necessity , 2013, Comput. Math. Appl..

[23]  Jianbin Qiu,et al.  A New Design of Delay-Dependent Robust ${\cal H}_{\bm \infty}$ Filtering for Discrete-Time T--S Fuzzy Systems With Time-Varying Delay , 2009, IEEE Transactions on Fuzzy Systems.

[24]  Tadeusz Kaczorek,et al.  Selected Problems of Fractional Systems Theory , 2011 .

[25]  Alain Oustaloup,et al.  CRONE Control: Principles and Extension to Time-Variant Plants with Asymptotically Constant Coefficients , 2002 .

[26]  Saptarshi Das,et al.  Intelligent Fractional Order Systems and Control - An Introduction , 2012, Studies in Computational Intelligence.

[27]  Koksal Erenturk,et al.  Fractional-Order $\hbox{PI}^{\lambda}\hbox{D}^{\mu}$ and Active Disturbance Rejection Control of Nonlinear Two-Mass Drive System , 2013, IEEE Transactions on Industrial Electronics.

[28]  Alain Oustaloup,et al.  Lyapunov Stability of Linear Fractional Systems: Part 2 — Derivation of a Stability Condition , 2013 .

[29]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[30]  A. Luo,et al.  Fractional Dynamics and Control , 2011 .

[31]  Manuel A. Duarte-Mermoud,et al.  Lyapunov functions for fractional order systems , 2014, Commun. Nonlinear Sci. Numer. Simul..