Robust $H_{\infty }$ Observer-Based Control of Fractional-Order Systems With Gain Parametrization

This paper investigates the robust <inline-formula><tex-math notation="LaTeX">$H_{\infty }$</tex-math> </inline-formula> observer-based control (OBC) for linear time-invariant disturbed uncertain fractional-order systems (DU-FOS). First, the existence conditions for robust <inline-formula><tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> OBC are given. Then, based on the <inline-formula><tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula>-norm analysis using the generalized Kalman–Yakubovich–Popov lemma for FOS, and following the fractional derivative order <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math> </inline-formula>, new sufficient linear matrix inequalities (LMIs) conditions are obtained to ensure the stability of the estimation errors and the stabilization of the DU-FOS simultaneously. All observer matrices gains and control laws can be computed by solving a unique LMI condition in one step. Numerical simulation is given to illustrate the validity of the proposed method.

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

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

[3]  M. Darouach,et al.  Design of full and reduced orders observers for linear fractional-order systems in the time and frequency domains , 2013, 3rd International Conference on Systems and Control.

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

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

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

[7]  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.

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

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

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

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

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

[13]  Christophe Farges,et al.  H∞ analysis and control of commensurate fractional order systems , 2013 .

[14]  Christophe Farges,et al.  Fractional systems state space description: some wrong ideas and proposed solutions , 2014 .

[15]  A. Oustaloup,et al.  On Fractional Systems H-Norm Computation , 2005 .

[16]  Patrick Lanusse,et al.  CRONE control system design toolbox for the control engineering community: tutorial and case study , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  Alain Oustaloup,et al.  Analytical computation of the H2-norm of fractional commensurate transfer functions , 2011, Autom..

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

[19]  Mathieu Moze,et al.  On computation of H∞ norm for commensurate fractional order systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

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

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

[22]  Mohamed Darouach,et al.  Robust $$H_{\infty }$$ Observer-Based Stabilization of Disturbed Uncertain Fractional-Order Systems Using a Two-Step Procedure , 2015, RRNR.

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

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

[25]  O. Agrawal,et al.  Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , 2007 .

[26]  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.

[27]  Carl F. Lorenzo,et al.  THE INITIALIZATION RESPONSE OF LINEAR FRACTIONAL-ORDER SYSTEMS WITH CONSTANT HISTORY FUNCTION , 2009 .

[28]  Shu Liang,et al.  Bounded real lemmas for fractional order systems , 2015, Int. J. Autom. Comput..

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

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

[31]  Christophe Farges,et al.  Analysis of fractional models physical consistency , 2017 .

[32]  Mohamed Darouach,et al.  Static output feedback ℋ∞ control for a fractional-order glucose-insulin system , 2013, International Journal of Control, Automation and Systems.

[33]  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.

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

[35]  Alain Oustaloup,et al.  Object-oriented CRONE toolbox for fractional differential signal processing , 2012, Signal Image Video Process..

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

[37]  Carl F. Lorenzo,et al.  Initialization in fractional order systems , 2001, 2001 European Control Conference (ECC).

[38]  Minyue Fu,et al.  A linear matrix inequality approach to robust H∞ filtering , 1997, IEEE Trans. Signal Process..

[39]  Christophe Farges,et al.  H∞ output feedback control of commensurate fractional order systems , 2013, 2013 European Control Conference (ECC).

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

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

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

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

[44]  Christophe Farges,et al.  Long memory models: a first solution to the infinite energy storage ability of linear time-invariant fractional models , 2014 .

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