0
Research Papers

Initial Conditions and Initialization of Fractional Systems

[+] Author and Article Information
Massinissa Tari

LIAS,
University of Poitiers,
Poitiers Cedex 8600, France
e-mail: massinissa.tari@univ-poitiers.fr

Nezha Maamri

LIAS,
University of Poitiers,
Poitiers Cedex 8600, France
e-mail: nezha.maamri@univ-poitiers.fr

Jean-Claude Trigeassou

IMS-LAPS,
University of Bordeaux,
Talence Cedex 33405, France
e-mail: jean-claude.trigeassou@ims-bordeaux.fr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 15, 2015; final manuscript received January 28, 2016; published online March 16, 2016. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(4), 041014 (Mar 16, 2016) (7 pages) Paper No: CND-15-1098; doi: 10.1115/1.4032695 History: Received April 15, 2015; Revised January 28, 2016

Abstract

In this paper, the initialization of fractional order systems is analyzed. The objective is to prove that the usual pseudostate variable $x(t)$ is unable to predict the future behavior of the system, whereas the infinite dimensional variable $z(ω, t)$ fulfills the requirements of a true state variable. Two fractional systems, a fractional integrator and a one-derivative fractional system, are analyzed with the help of elementary tests and numerical simulations. It is proved that the dynamic behaviors of these two fractional systems differ completely from that of their integer order counterparts. More specifically, initialization of these systems requires knowledge of $z(ω,t0)$ initial condition.

<>

References

Kalman, R. E. , 1960, “ On the General Theory of Control System,” First IFAC Congress Automatic Control, Moscow, USSR, Vol. 1, pp. 481–492.
Kailath, T. , 1980, Linear System, Prentice-Hall, Inc./Englewood Cliffs, NJ.
Zadeh, L. A. , and Desoer, C. A. , 2008, Linear System Theory: The State Space Approach, Dover Publications, New York.
Heymans, N. , and Podlubny, I. , 2005, “ Physical Interpretation of Initial Condition for Fractional Differential Equation With Rieamman–Liouville Fractional Derivatives,” Rheol. Acta, 29(5), pp. 765–771.
Machado, J. T. , 2014, “ Numerical Analysis of the Initial Conditions of Fractional Systems,” Commun. Nonlinear Sci. Numer. Simul., 19(9), pp. 2935–2941.
Lorenzo, C. F. , and Hartley, T. T. , 2001, “ Initialization in Fractional Order Systems,” European Control Conference, Porto, Portugal, Sept. 4–7, pp. 1471–1476.
Lorenzo, C. F. , and Hartley, T. T. , 2008, “ Initialization of Fractional Differential Equation,” ASME J. Comput. Nonlinear Dyn., 3(2), p. 021101.
Hartley, T. T. , and Lorenzo, C. F. , 2009, “ The Initialization Response of Linear Fractional-Order Systems With Constant History Function,” ASME Paper No. DETC2009-87631.
Sabatier, J. , Merveillaut, M. , Malti, R. , and Oustaloup, A. , 2010, “ How to Impose Physically Coherent Initial Conditions to a Fractional System?,” Commun. Nonlinear Sci. Numer. Simul., 15(5), pp. 1318–1326.
Trigeassou, J. C. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2012, “ State Variables and Transients of Fractional Order Differential Systems,” Comput. Math. Appl., 64(10), pp. 3117–3140.
Du, M. , and Wang, Z. , 2011, “ Initialized Fractional Differential Equations With Rieman-Liouville Fractional Order Derivative,” Conference ENOC 2011, Rome, Italy, pp. 49–60.
Du, M. , and Wang, Z. , 2015, “ Correcting the Initialization of Models With Fractional Derivatives Via History Dependent Conditions,” Acta Mech. Sin., 31, pp. 1–6.
Fukunaga, M. , and Shimizu, N. , 2004, “ Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations,” Nonlinear Dyn. Fractional, 38(1), pp. 207–220.
Trigeassou, J. C. , and Maamri, N. , 2011, “ Initial Conditions and Initialization of Linear Fractional Differential Equations,” Signal Process., 91(3), pp. 427–436.
Trigeassou, J. C. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2012, “ Transients of Fractional Order Integrator and Derivatives,” Signal, Image Video Process., 6(3), pp. 359–372.
Hartley, T. T. , Lorenzo, C. F. , Trigeassou, J. C. , and Maamri, N. , 2013, “ Equivalence of History Function Based and Infinite Dimensional State Initializations for Fractional Order Operators,” ASME J. Comput. Nonlinear Dyn., 8(4), p. 041014.
Oldham, K. B. , and Spanier, J. , 1974, The Fractional Calculus, Academic Press, New York.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2013, “ The Infinite State Approach: Origin and Necessity,” Comput. Math. Appl., 6(5), pp. 892–907.
Heleschewitz, D. , and Matignon, D. , 1998, “ Diffusive Realizations of Fractional Integro-Differential Operators: Structural Analysis Under Approximation,” System, Structure and Control, Conference IFAC, Nantes, France, Vol. 2, pp. 243–248.
Helechewitz, D. , 2000, “ Analyse et simulation de systèmes différentiels fractionnaires et pseudo-différentiels sous representation diffusive,” Ph.D. thesis, ENST, Paris, France.
Montseny, G. , 1998, “ Diffusive Representation of Pseudo Differential Time Operators,” ESSAIM, Vol. 5, pp. 159–175.
Ortigueira, M. D. , Machado, J. T. , Rivero, M. , and Trujillo, J. J. , 2015, “ Integer/Fractional Decomposition of the Impulse Response of Fractional Linear Systems,” Signal Process., 114, pp. 85–88.
Sabatier, J. , Farges, C. , and Fadiga, L. , “ Approximation of a Fractional Order Model by an Integer Order Model: A New Approach Taking Into Account Approximation Error as an Uncertainty,” J. Vib. Control (Published online).
Trigeassou, J.-C. , Maamri, N. , and Oustaloup, A. , 2014, “ Lyapunov Stability of Fractional Order Systems: The Two Derivatives Case,” ICFDA’14, Catania, Italy.
Trigeassou, J. C. , and Maamri, N. , 2010, “ The Initial Conditions of Riemann–Liouville and Caputo Derivatives: An Integrator Interpretation,” FDA’10, Badajoz, Spain.
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2013, “ The Caputo Derivative and the Infinite State Approach,” IFAC Joint Conference FDA 13, Grenoble, France, pp. 587–592.
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2015, “ Analysis of the Caputo Derivative and Pseudo State Representation With the Infinite State Approach,” Fractional Calculus Theory, R. A. Z. Daou , and X. Moreau , eds., Nova Science Publishers, New York.
Sabatier, J. , Farges, C. , and Trigeassou, J. C. , 2014, “ Fractional Systems State Space Description: Some Wrong Ideas and Proposed Solutions,” J. Vib. Control, 20(7), pp. 1076–1084.
Maamri, N. , Massinissa, T. , and Trigeassou, J.-C. , 2014, “ Physical Interpretation and Initialization of the Fractional Integrator,” ICFDA’14, Catania, Italy, pp. June 24–25.
Sabatier, J. , Farges, C. , Merveillaut, M. , and Feneteau, L. , 2012, “ On Observability and Pseudo State Estimation of Fractional Order Systems,” Eur. J. Control, 18(3), pp. 1–12.
Gambone, T. , Hartley, T. T. , Lorenzo, C. F. , Adams, J. L. , and Veillette, R. J. , 2011, “ An Experimental Validation of the Time-Varying Initialization Response in Fractional-Order Systems,” ASME Paper No. DETC2011-47250.

Figures

Fig. 1

Pseudo impulse excitation u(t)

Fig. 4

Pseudo impulse excitation u¯(t)

Fig. 3

zj(t) : components of z(ω,t) at T = 5 s

Fig. 2

Comparison between the theoretical response x(t) and the numerical approximation xnum(t)

Fig. 5

Comparison between x¯(t) and x(t)

Fig. 6

Comparison between the components zj(t) and z¯j(t) at T = 5

Fig. 8

x¯(t) and x(t): responses in the integer case

Fig. 9

Comparison between the theoretical response x(t) and the numerical approximation xnum(t)

Fig. 10

Comparison between x¯(t) and x(t)

Fig. 11

Comparison between the component zj(t) and z¯j(t) at T = 5 s

Fig. 7

x¯(t) and x(t): responses in the integer order case

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections