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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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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