On the robustness of linear and non-linear fractional-order systems with non-linear uncertain parameters

This paper presents the robust stabilization problem of linear and non-linear fractional-order systems with non-linear uncertain parameters. The uncertainty in the model appears in the form of the combination of ‘additive perturbation’ and ‘multiplicative perturbation’. Sufficient conditions for the robust asymptotical stabilization of linear fractional-order systems are presented in terms of linear matrix inequalities (LMIs) with the fractional-order 0 <α < 1. Sufficient conditions for the robust asymptotical stabilization of non-linear fractional-order systems are then derived using a generalization of the Gronwall–Bellman approach. Finally, a numerical example is given to illustrate the effectiveness of the proposed results.

[1]  Yangquan Chen,et al.  Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality , 2007, Appl. Math. Comput..

[2]  P. Khargonekar,et al.  Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory , 1990 .

[3]  Mohamed Darouach,et al.  Observer-based control for fractional-order continuous-time systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[4]  Bor-Sen Chen,et al.  Robust linear controller design: Time domain approach , 1987 .

[5]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[6]  YangQuan Chen,et al.  Stability of linear time invariant systems with interval fractional orders and interval coefficients , 2004, Second IEEE International Conference on Computational Cybernetics, 2004. ICCC 2004..

[7]  Jun-Guo Lu,et al.  Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach , 2009 .

[8]  Yangquan Chen,et al.  Robust stability check of fractional order linear time invariant systems with interval uncertainties , 2005, IEEE International Conference Mechatronics and Automation, 2005.

[9]  Mohamed Darouach,et al.  On the robustness of linear systems with nonlinear uncertain parameters , 1998, Autom..

[10]  Valentina Orsini,et al.  Relaxed sufficient conditions for the stability of continuous and discrete-time linear time-varying systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[11]  S. H. Zak,et al.  On the stabilization and observation of nonlinear/uncertain dynamic systems , 1990, IEEE 1989 International Conference on Systems Engineering.

[12]  L. Ghaoui,et al.  History of linear matrix inequalities in control theory , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[13]  Jean-Claude Trigeassou,et al.  Initial conditions and initialization of linear fractional differential equations , 2011, Signal Process..

[14]  Mohamed Darouach,et al.  Stabilization of a Class of Nonlinear Affine Fractional-Order Systems , 2009 .

[15]  T. Hartley,et al.  Initialization of Fractional-Order Operators and Fractional Differential Equations , 2008 .

[16]  Yan Li,et al.  H∞ and Sliding Mode Observers for Linear Time-Invariant Fractional-Order Dynamic Systems With Initial Memory Effect , 2014 .

[17]  Kiyotaka Shimizu,et al.  Nonlinear state observers by gradient descent method , 2000, Proceedings of the 2000. IEEE International Conference on Control Applications. Conference Proceedings (Cat. No.00CH37162).

[18]  Mathieu Moze,et al.  Pseudo state feedback stabilization of commensurate fractional order systems , 2009, 2009 European Control Conference (ECC).

[19]  Carl F. Lorenzo,et al.  Time-Varying Initialization and Laplace Transform of the Caputo Derivative: With Order Between Zero and One , 2011 .

[20]  Stanislaw H. Zak,et al.  On the stabilization and observation of nonlinear/uncertain dynamic systems , 1990 .

[21]  Mathieu Moze,et al.  LMI stability conditions for fractional order systems , 2010, Comput. Math. Appl..

[22]  Margarita Rivero,et al.  Stability of Fractional Order Systems , 2013 .

[23]  Carl F. Lorenzo,et al.  Initialization Issues of the Caputo Fractional Derivative , 2005 .

[24]  Alain Oustaloup,et al.  State variables and transients of fractional order differential systems , 2012, Comput. Math. Appl..

[25]  Jun-Guo Lu,et al.  Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order $\alpha$: The $0≪\alpha≪1$ Case , 2010, IEEE Transactions on Automatic Control.

[26]  Alexandr A. Zevin,et al.  Exponential stability and solution bounds for systems with bounded nonlinearities , 2003, IEEE Trans. Autom. Control..

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

[28]  M. Darouach,et al.  Exponential stabilization of a class of nonlinear systems: A generalized GronwallBellman lemma appr , 2011 .

[29]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

[30]  B. G. Pachpatte.  A note on Gronwall-Bellman inequality , 1973 .

[31]  Christophe Farges,et al.  A stability test for non-commensurate fractional order systems , 2010, Syst. Control. Lett..

[32]  Christophe Farges,et al.  On Observability and Pseudo State Estimation of Fractional Order Systems , 2012, Eur. J. Control.

[33]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[34]  Michel Zasadzinski,et al.  Robust stabilization of linear and nonlinear fractional-order systems with nonlinear uncertain parameters , 2010, 49th IEEE Conference on Decision and Control (CDC).

[35]  Alain Oustaloup,et al.  Automatic initialization of the Caputo fractional derivative , 2011, IEEE Conference on Decision and Control and European Control Conference.

[36]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[37]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

[38]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

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

[40]  Jun-Guo Lu,et al.  Stability Analysis of a Class of Nonlinear Fractional-Order Systems , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[41]  C. A. Desoer,et al.  IV – LINEAR SYSTEMS , 1975 .

[42]  M. P. Aghababa.  Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller , 2012 .

[43]  Carl F. Lorenzo,et al.  Equivalence of History-Function Based and Infinite-Dimensional-State Initializations for Fractional-Order Operators , 2013 .

[44]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .