Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems

Mohammad Saleh Tavazoei

doi:10.1115/1.4025477

Journal of Computational and Nonlinear Dynamics | Volume 9 | Issue 2 | research-article

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Research Papers

Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems

Mohammad Saleh Tavazoei

[+] Author and Article Information

Mohammad Saleh Tavazoei

Electrical Engineering Department,
Sharif University of Technology,
Tehran 11155-4363, Iran
e-mail: tavazoei@sharif.edu

The power functions appearing in the function $P_{X^{*}}^{(α_{1}, α_{2}, . . ., α_{n})} (s)$ are generally multivalued functions. In this paper, in order to calculate the value of the function $P_{X^{*}}^{(α_{1}, α_{2}, . . ., α_{n})} (s)$ in a particular point, the principal values [36] of its involved power functions at such a point are considered. By considering this point, function $P_{X^{*}}^{(α_{1}, α_{2}, . . ., α_{n})} (s)$ will be a single-valued function.

It is worth mentioning that Eqs. (14-1) and (14-2) are similar to conditions (5-1) and (5-2) in Ref. [50]. The aforementioned conditions have been presented in Ref. [50] as the necessary and sufficient conditions which should be satisfied by LTI 2-inner dimension fractional-order systems to generate sinusoidal oscillations with frequency $ω$ , whereas in our problem (14-1) and (14-2) are not necessary conditions for the existence of oscillations in a nonlinear system (12). These equations are only used to specify the boundary of the possible oscillatory region of the fractional-order Van der Pol system (12). As a consequence, $ω$ appearing in (14-1) and (14-2) does not have any relation to the frequency of oscillations in the fractional-order Van der Pol system (12).

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received May 18, 2013; final manuscript received July 19, 2013; published online October 30, 2013. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 9(2), 021011 (Oct 30, 2013) (7 pages) Paper No: CND-13-1104; doi: 10.1115/1.4025477 History: Received May 18, 2013; Revised July 19, 2013

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Abstract

Finding the oscillatory region in the order space is one of the most challenging problems in nonlinear fractional-order systems. This paper proposes a method to find the possible oscillatory region in the order space for a nonlinear fractional-order system. The effectiveness of the proposed method in finding the oscillatory region and special order sets placed in its boundary is confirmed by presenting some examples.

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References

Guyomar, D., Ducharne, B., and Sebald, G., 2010, “The Use of Fractional Derivation in Modeling Ferroelectric Dynamic Hysteresis Behavior Over Large Frequency Bandwidth,” J. Appl. Phys., 107(11), p. 114108. [CrossRef]

Ionescu, C. M., Machado, J. A. T., and De Keyser, R., 2011, “Modeling of the Lung Impedance Using a Fractional-Order Ladder Network With Constant Phase Elements,” IEEE Trans. Biomed. Circuits Syst., 5(1), pp. 83–89. [CrossRef] [PubMed]

Gabano, J. D., Poinot, T., and Kanoun, H., 2011, “Identification of a Thermal System Using Continuous Linear Parameter-Varying Fractional Modelling,” Control Theory Appl., 5(7), pp. 889–899. [CrossRef]

Coussot, C., Kalyanam, S., Yapp, R., and Insana, M., 2009, “Fractional Derivative Models for Ultrasonic Characterization of Polymer and Breast Tissue Viscoelasticity,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 56(4), pp. 715–726. [CrossRef] [PubMed]

Narang, A., Shah, S. L., and Chen, T., 2011, “Continuous-Time Model Identification of Fractional-Order Models With Time Delays,” Control Theory Appl., 5(7), pp. 900–912. [CrossRef]

Efe, M., 2011, “Fractional-Order Systems in Industrial Automation—A Survey,” IEEE Trans. Ind. Inf., 7, pp. 582–591. [CrossRef]

Tavazoei, M. S., Haeri, M., Jafari, S., Bolouki, S., and Siami, M., 2008, “Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations,” IEEE Trans. Ind. Electron., 55(11), pp. 4094–4101. [CrossRef]

Tavazoei, M. S., 2012, “From Traditional to Fractional PI Control: A Key for Generalization,” IEEE Ind. Electron. Mag., 6(3), pp. 41–51. [CrossRef]

Tavakoli-Kakhki, M., Haeri, M., and Tavazoei, M. S., 2010, “Simple Fractional-Order Model Structures and Their Applications in Control System Design,” Eur.J. Control, 6, pp. 680–694. [CrossRef]

Monje, C. A., Vinagre, B. M., Feliu, V., and Chen, Y. Q., 2008, “Tuning and Auto-Tuning of Fractional-Order Controllers for Industry Applications,” Control Eng. Pract., 16(7), pp. 798–812. [CrossRef]

Ortigueira, M. D., 2010, “On the Fractional Linear Scale Invariant Systems,” IEEE Trans. Signal Process., 58(12), pp. 6406–6410. [CrossRef]

Tavazoei, M. S., 2011, “On Monotonic and Nonmonotonic Step Responses in Fractional-Order Systems,” IEEE Trans. Circuits Syst., II: Express Briefs, 58(7), pp. 447–451. [CrossRef]

Kaczorek, T., 2011, “Positive Linear Systems Consisting of n Subsystems With Different Fractional-Orders,” IEEE Trans. Circuits Syst., I: Regul. Pap., 58(6), pp. 1203–1210. [CrossRef]

Tavakoli-Kakhki, M., Haeri, M., and Tavazoei, M. S., 2010, “Over and Under Convergent Step Responses in Fractional-Order Transfer Functions,” Trans. Inst. Meas. Control (London), 32, pp. 376–394. [CrossRef]

Tavazoei, M. S., 2010, “Notes on Integral Performance Indices in Fractional-Order Control Systems,” J. Process Control, 20, pp. 285–291. [CrossRef]

Ahmad, W., El-Khazali, R., and El-Wakil, A., 2001, “Fractional-Order Wien-Bridge Oscillator,” Electron. Lett., 37, pp. 1110–1112. [CrossRef]

Gafiychuk, V., Datsko, B., and Meleshko, V., 2008, “Analysis of Fractional-Order Bonhoeffer–Van der Pol Oscillator,” Physica A, 387, pp. 418–424. [CrossRef]

Gafiychuk, V., and Datsko, B., 2008, “Stability Analysis and Limit Cycle in Fractional System With Brusselator Nonlinearities,” Phys. Lett. A, 372, pp. 4902–4904. [CrossRef]

Tavazoei, M. S., and Haeri, M., 2008, “Regular Oscillations or Chaos in a Fractional-Order System With any Effective Dimension,” Nonlinear Dyn., 54, pp. 213–222. [CrossRef]

Hartley, T. T., Lorenzo, C. F., and Qammer, H. K., 1995, “Chaos in a Fractional-Order Chua's System,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl., 42, pp. 485–490. [CrossRef]

Tavazoei, M. S., and Haeri, M., 2007, “A Necessary Condition for Double Scroll Attractor Existence in Fractional-Order Systems,” Phys. Lett. A, 367(1–2), pp. 102–113. [CrossRef]

Datsko, B. Y., and Gafiychuk, V. V., 2011, “Chaotic Dynamics in Bonhoffer–Van der Pol Fractional Reaction–Diffusion System,” Signal Process., 91, pp. 452–460. [CrossRef]

Tavazoei, M. S., and Haeri, M., 2009, “A Proof for Nonexistence of Periodic Solutions in Time Invariant Fractional-Order Systems,” Automatica, 45, pp 1886–1890. [CrossRef]

Tavazoei, M. S., 2010, “A Note on Fractional-Order Derivatives of Periodic Functions,” Automatica, 46, pp. 945–948. [CrossRef]

Tavazoei, M. S., Haeri, M., Siami, M., and Bolouki, S., 2010, “Maximum Number of Frequencies in Oscillations Generated by Fractional-Order LTI Systems,” IEEE Trans. Signal Process., 58, pp. 4003–4012. [CrossRef]

Tavazoei, M. S., HaeriM., and Nazari, N., 2008, “Analysis of Undamped Oscillations Generated by Marginally Stable Fractional-Order Systems,” Signal Process., 88(12), pp. 2971–2978. [CrossRef]

Deng, W., and Li, C., 2008, “The Evolution of Chaotic Dynamics for Fractional Unified System,” Phys. Lett. A, 372, 401–407. [CrossRef]

Chen, W. C., 2008, “Nonlinear Dynamics and Chaos in a Fractional-Order Financial System,” Chaos, Solitons Fractals, 36(5), pp. 1305–1314. [CrossRef]

Sheu, L. J., Chen, H. K., Chen, J. H., Tam, L. M., Chen, W. C., Lin, K. T., and KangY., 2008, “Chaos in the Newton-Leipnik System With Fractional-Order,” Chaos, Solitons Fractals, 36(1), pp. 98–103. [CrossRef]

Chen, J. H., and Chen, W. C., 2008, “Chaotic Dynamics of the Fractionally Damped Van der Pol Equation,” Chaos, Solitons Fractals, 35(1), pp. 188–198. [CrossRef]

Tavazoei, M. S., and Haeri, M., 2008, “Chaotic Attractors in Incommensurate Fractional-Order Systems,” Physica D, 237, pp. 2628–2637. [CrossRef]

Podlubny, I., 1999, Fractional Differential Equations, Academic, San Diego.

Wang, Y., and Li, C., 2007, “Does the Fractional Brusselator With Efficient Dimension Less Than 1 Have a Limit Cycle?,” Phys. Lett. A, 363, pp. 414–419. [CrossRef]

Daftardar-Gejji, V., and Jafari, H., 2007, “Analysis of a System of Nonautonomous Fractional Differential Equations Involving Caputo Derivatives,” J. Math.Anal. Appl., 328, pp. 1026–1033. [CrossRef]

Linz, S. J., 2000, “No-Chaos Criteria for Certain Jerky Dynamics,” Phys. Lett. A, 275, pp. 204–210. [CrossRef]

Kreyszig, E., 2006, Advanced Engineering Mathematics, 9th ed., John Wiley and Sons, New York.

Matignon, D., 1996, “Stability Results for Fractional Differential Equations With Applications to Control Processing,” Computational Engineering in Systems Applications, IMACS, IEEE-SMC Proceedings, Lille, France, July, Vol. 2, pp. 963–968.

Bonnet, C., and Partington, J. R., 2000, “Coprime Factorizations and Stability of Fractional Differential Systems,” Syst. Control Lett., 41, pp. 167–174. [CrossRef]

Ahlfors, L. V., 1966, Complex Analysis, 2nd ed., McGraw-Hill, New York.

Wiggins, S., 2003, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York.

Li, C., and Chen, G., 2004, “Chaos and Hyperchaos in the Fractional-Order Rössler Equations,”Physica A, 341, pp. 55–61. [CrossRef]

Tavakoli-Kakhki, M., Haeri, M., and Tavazoei, M. S., 2011, “Notes on the State Space Realizations of Rational Order Transfer Functions,” IEEE Trans. Circuits Syst., I: Regul. Pap., 58, pp. 1099–1108. [CrossRef]

Petras, I., 2010, “Fractional-Order Memristor-Based Chua's Circuit,” IEEE Trans. Circuits Syst., II:Express Briefs, 57, pp. 975–979. [CrossRef]

Diethelm, K., Ford, N. J., and Freed, A. D., 2002, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dyn., 29, pp. 3–22. [CrossRef]

Tavazoei, M. S., and Haeri, M., 2007, “Unreliability of Frequency-Domain Approximation in Recognizing Chaos in Fractional-Order Systems,” IET Signal Process., 1(4), pp. 171–181. [CrossRef]

Spasic, A. M., and Lazarevic, M. P., 2005, “Electroviscoelasticity of Liquid/Liquid Interfaces: Fractional-Order Model,” J. Colloid Interface Sci., 282, pp. 223–230. [CrossRef] [PubMed]

Spasic, A. M., and Lazarevic, M. P., 2007, “A New Approach to the Phenomena at the Interfaces of Finely Dispersed Systems,” J. Colloid Interface Sci., 316, pp. 984–995. [CrossRef] [PubMed]

Barbosa, R. S., Machado, J. A. T., Vingare, B. M., and Calderon, A. J., 2007, “Analysis of the Van der Pol Oscillator Containing Derivatives of Fractional-Order,” J. Vib. Control, 13(9–10), pp. 1291–1301. [CrossRef]

Tavazoei, M. S., Haeri, M., Attari, M., Bolouki, S., and Siami, M., 2009, “More Details on Analysis of Fractional-Order Van der Pol Oscillator,” J. Vib. Control, 15(6), pp. 803–819. [CrossRef]

Radwan, A. G., El-Wakil, A. S., and Soliman, A. M., 2008, “Fractional-Order Sinusoidal Oscillators: Design Procedure and Practical Examples,” IEEE Trans. Circuits Syst., I: Regul. Pap., 55(7), pp. 2051–2063. [CrossRef]

Chen, G., and Ueta, T., 1999, “Yet Another Attractor,” Int. J. Bifurcation Chaos, 9, pp. 1465–1466. [CrossRef]

Tavazoei, M. S., Haeri, M., and Jafari, S., 2008, “Fractional Controller to Stabilize Fixed Points of Uncertain Chaotic Systems: Theoretical and Experimental Study,” Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng., 222(3), pp. 175–184. [CrossRef]

Biswas, K., Sen, S., and Dutta, P. K., 2006, “Realization of a Constant Phase Element and its Performance Study in a Differentiator Circuit,” IEEE Trans. Circuits Syst.,II: Express Briefs.53, pp. 802–806. [CrossRef]

Jesus, I. S., and Machado, J. A. T., 2009, “Development of Fractional-Order Capacitors Based on Electrolyte Processes,” Nonlinear Dyn., 56, pp. 45–55. [CrossRef]

Mondal, D., and Biswas, K., 2011, “Performance Study of Fractional-Order Integrator Using Single-Component Fractional-Order Element,” Circuits Devices Syst., 5(4), pp. 334–342. [CrossRef]

Figures

Fig. 1

Region ΔR in the s-plane

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Fig. 2

Region Δ∧R in the s-plane

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Fig. 3

Regions (a) PX1*(0.9,0.8,0.7)(Δ∧3.1), and (b) PX1*(0.7,0.8,0.7)(Δ∧3.1) in example 1

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Fig. 4

The gray region specifies the possible oscillatory region in the order space R(0,1]2 for the fractional-order system (12) with parameters a = 1 (above the figure) and a = 1.5 (below the figure)

$The gray region specifies the possible oscillatory region in the order space R(0,1]2 for the fractional-order system (12) with parameters a = 1 (above the figure) and a = 1.5 (below the figure)$

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$The gray region specifies the possible oscillatory region in the order space R(0,1]2 for the fractional-order system (12) with parameters a = 1 (above the figure) and a = 1.5 (below the figure)$

Fig. 5

Numerical simulation results of system (12) where a = 1.5 and (α1,α2) = (0.5,0.45)

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Fig. 6

Numerical simulation results of system (12) where a = 1.5 and (α1,α2) = (0.5,0.4)

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Fig. 7

Numerical simulation results of system (18) where α1 = 0.57

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Fig. 8

Numerical simulation results of system (18) where α1 = 0.55

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Tavazoei M. Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems. ASME. J. Comput. Nonlinear Dynam. 2013;9(2):021011-021011-7. doi:10.1115/1.4025477.

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Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems

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