Robust $$H_{\infty }$$ Observer-Based Stabilization of Disturbed Uncertain Fractional-Order Systems Using a Two-Step Procedure

The main objective of this work is the problem of robust \(H_{\infty }\) observer-based stabilization for a class of linear Disturbed Uncertain Fractional-Order Systems (DU-FOS) by using \(H_{\infty }\)-norm optimization. Based on the \(H_{\infty }\)-norm analysis for FOS, a new design methodology is established to stabilize a linear DU-FOS by using robust \(H_{\infty }\) Observer-Based Control (OBC). The existence conditions are derived, and by using the \(H_{\infty }\)-optimization technique, the stability of the estimation error and stabilization of the original system are given in an inequality condition, where all the observer matrices gains and the control law can be computed by solving a single inequality condition in two step. Finally, a simulation example is given to illustrate the validity of the results.

[1]  D. Luenberger.  An introduction to observers , 1971 .

[2]  Enrico Scalas,et al.  Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit , 2001 .

[3]  J. Sabatier,et al.  On Fractional Systems H∞, -Norm Computation , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[4]  Vincent Giovangigli,et al.  A threshold phenomenon in the propagation of a point source initiated flame , 1998 .

[5]  V. Uchaikin.  Fractional Derivatives for Physicists and Engineers , 2013 .

[6]  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.

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

[8]  M. Darouach,et al.  State estimation for a class of singular systems , 1992 .

[9]  Enrico Scalas,et al.  Coupled continuous time random walks in finance , 2006 .

[10]  Reyad El-Khazali,et al.  Fractional-order dynamical models of love , 2007 .

[11]  Mohamed Darouach,et al.  Linear functional observers for systems with delays in state variables , 2001, IEEE Trans. Autom. Control..

[12]  James Lam,et al.  State feedback H ∞ control of commensurate fractional-order systems , 2014, Int. J. Syst. Sci..

[13]  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.

[14]  Mohamed Darouach,et al.  Static output feedback ℋ∞ control for a fractional-order glucose-insulin system , 2013 .

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

[16]  J. Hristov,et al.  THERMAL IMPEDANCE ESTIMATIONS BY SEMI-DERIVATIVES AND SEMI-INTEGRALS 1-D semi-infinite cases , 2013 .

[17]  Mathieu Moze,et al.  On bounded real lemma for fractional systems , 2008 .

[18]  A. D. Freed,et al.  Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad , 2006, Biomechanics and modeling in mechanobiology.

[19]  Robert M. Darling,et al.  On the Short‐Time Behavior of Porous Intercalation Electrodes , 1997 .

[20]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[21]  Mohamed Darouach,et al.  H∞ observer design for linear fractional-order systems in time and frequency domains , 2014, 2014 European Control Conference (ECC).

[22]  Christophe Farges,et al.  H∞ state feedback control of commensurate fractional order systems , 2013 .

[23]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[24]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

[25]  O. Agrawal,et al.  Advances in Fractional Calculus , 2007 .

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

[27]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[28]  Noemi Wolanski,et al.  Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames , 2004 .

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

[30]  Jianying Yang,et al.  Dynamical models of happiness with fractional order , 2010 .