Stability Preservation Analysis for Frequency-Based Methods in Numerical Simulation of Fractional Order Systems

In this paper, the frequency domain-based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation. First, the stability boundary curve is exactly determined for these methods. Then, this boundary is analyzed and compared with an accurate (ideal) boundary in different frequency ranges. Also, the critical regions in which the stability does not preserve are determined. Finally, the analytical achievements are confirmed via some numerical illustrations.

[1]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[2]  P. Lino,et al.  New tuning rules for fractional PIα controllers , 2007 .

[3]  Victor George Jenson,et al.  Mathematical Methods in Chemical Engineering , 1978 .

[4]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[5]  Rumen L. Mishkov,et al.  Generalization of the formula of Faa di Bruno for a composite function with a vector argument , 2000 .

[6]  Guoqing Chen,et al.  An RLC interconnect model based on fourier analysis , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[7]  Manuel Duarte Ortigueira,et al.  Introduction to fractional linear systems. Part 1. Continuous-time case , 2000 .

[8]  Marina V. Shitikova,et al.  Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems , 1997 .

[9]  N. Engheta.  On fractional calculus and fractional multipoles in electromagnetism , 1996 .

[10]  Alain Oustaloup,et al.  From fractal robustness to the CRONE control , 1999 .

[11]  Leang-San Shieh,et al.  State-Space Self-Tuning Control for Stochastic Fractional-Order Chaotic Systems , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[12]  D. Matignon.  Stability properties for generalized fractional differential systems , 1998 .

[13]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[14]  R. Bagley,et al.  Fractional order state equations for the control of viscoelasticallydamped structures , 1991 .

[15]  O. Agrawal,et al.  A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives , 2002 .

[16]  Julien Clinton Sprott,et al.  Chaos in fractional-order autonomous nonlinear systems , 2003 .

[17]  N. Laskin.  Fractional market dynamics , 2000 .

[18]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[19]  K. Cole.  ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMS , 1933 .

[20]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[21]  Kai Diethelm,et al.  Multi-order fractional differential equations and their numerical solution , 2004, Appl. Math. Comput..

[22]  Guanrong Chen,et al.  A note on the fractional-order Chen system , 2006 .

[23]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[24]  A. El-Sayed,et al.  Fractional-order diffusion-wave equation , 1996 .

[25]  Guang-ren Duan,et al.  Parametric Control Systems Design - Theory and Applications , 2006, 2006 SICE-ICASE International Joint Conference.

[26]  Pankaj Kumar,et al.  An approximate method for numerical solution of fractional differential equations , 2006, Signal Process..

[27]  C. F. Lorenzo,et al.  Chaos in a fractional order Chua's system , 1995 .

[28]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[29]  Junguo Lu.  Chaotic dynamics of the fractional-order Lü system and its synchronization , 2006 .

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

[31]  Mohammad Saleh Tavazoei,et al.  Chaos control via a simple fractional-order controller , 2008 .

[32]  Vicente Feliu-Batlle,et al.  Fractional robust control of main irrigation canals with variable dynamic parameters , 2007 .

[33]  A. Oustaloup,et al.  Numerical Simulations of Fractional Systems: An Overview of Existing Methods and Improvements , 2004 .

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

[35]  Anissa Zergaïnoh-Mokraoui,et al.  State-space representation for fractional order controllers , 2000, Autom..

[36]  C. Halijak,et al.  Approximation of Fractional Capacitors (1/s)^(1/n) by a Regular Newton Process , 1964 .

[37]  Carl F. Lorenzo,et al.  Chaos in a Chua system with order less than two , 2007 .

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

[39]  M. Haeri,et al.  Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems , 2007 .

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