Analytical study of resonance regions for second kind commensurate fractional systems

Assia Boubidi; Sihem Kechida; Hicham Tebbikh

doi:10.3934/dcdsb.2020247

Article Contents

2021, Volume 26, Issue 7: 3579-3594. Doi: 10.3934/dcdsb.2020247

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Analytical study of resonance regions for second kind commensurate fractional systems

Laboratoire d'Automatique et Informatique de Guelma (LAIG), 8 Mai 1945-University, BP 401, 24000 Guelma, Algeria

^* Corresponding author: Sihem Kechida
^* Corresponding author: Sihem Kechida

Received: May 2019

Revised: May 2020

Early access: August 2020

Published: July 2021

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Abstract

The aim of this paper is to determine analytically the resonance limits for second kind commensurate fractional systems in terms of the pseudo damping factor $ \xi $ and the commensurate order $ v $ and in addition specify the different resonance regions. In the literature, these limits and regions have never been discussed mathematically, they are determined numerically. Second kind commensurate fractional systems are resonant if the equation : $ \Omega^{3v}+3\xi cos(v \pi/2)\Omega^{2v}+(2\xi^{2}+cos(v\pi))\Omega{^v}+\xi cos(v\pi/2) = 0 $, obtained by setting the first derivative of the amplitude-frequency response equal to zero, has at last one strictly positive root. As in the conventional case, resonance limits correspond to zero discriminant of the last equation. This discriminant is a cubic equation in $ \xi{^2} $ whose coefficients change depending on $ v $. To resolve this equation, the tangent trigonometric solving method is used and the relationship between $ \xi $ and $ v $ is established, which represents the resonance limits expression. To search resonance regions, a mathematical study is conducted on the first equation to find the positive roots number for each ($ v $, $ \xi $) combination. Compared to works already achieved, a new region appeared in the region of single resonant frequency with an anti-resonant one. The results are tested through numerical examples and applied to a fractional filter.

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Mathematics Subject Classification: Primary: 11Y35; Secondary: 12E12.

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Figure 1. Stability and Resonance regions of the second kind commensurate fractional systems in the $ v $$ \xi $-plane [18]

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Figure 2. Solution in $ y $ of equation $ D = 0 $

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Figure 3. Division of $ v $$ \xi $-plane according to discriminant signs

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Figure 4. Coefficients signs of $ f(x) $ and $ f(-x) $ in the $ v $$ \xi $-plane

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Figure 5. Division of the $ v $$ \xi $-plane according to $ D $ signs and coefficients signs of $ f(x) $ and $ f(-x) $

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Figure 6. Roots of (8) for different regions of the $ v $$ \xi $-plane according to Figure 3 division

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Figure 7. Division of the $ v $$ \xi $-plane according to the number of strictly positive roots

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Figure 8. Division of the $ v $$ \xi $-plane according to the number of strictly positive roots for stable systems

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Figure 9. New division of resonance regions of the second kind commensurate fractional systems in the $ v $$ \xi $-plane

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Figure 10. Magnitude Bode diagram for each region

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Figure 11. Sallen-Key FLPF circuit

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Table 1. Application of Descartes Rule of Signs according to Figure 5 division

Regions	1	2	31	32	41	42	5	6
$ D $ sign	+	+	-	-	-	-	+	+
$ R $ Nbr	3	3	1	1	1	1	3	3
$ f(x) $ coefs signs	+ + + +	+ - + -	+ + + +	+ + - +	+ - + -	+ - - -	+ + - +	+ - - -
Sign changes Nbr	0	3	0	2	3	1	2	1
$ R^{+} $ Nbr Possibility	0	3 or 1	0	2 or 0	3 or 1	1	2 or 0	1
$ f(-x) $ coefs signs	- + - +	- - - -	- + - +	- + + +	- - - -	- - + -	- + + +	- - + -
Sign changes Nbr	3	0	3	1	0	2	1	2
$ R^{-} $ Nbr Possibility	3 or 1	0	3 or 1	1	0	2 or 0	1	2 or 0
R⁺ Nbr	0	3	0	0	1	1	2	1
$ R^{-} $ Nbr	3	0	1	1	0	0	1	2

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Table 2. Positive roots number of (8) and corresponding regions

$ R^{+} $ Nbr	0	1	2	3
Regions	1 and 3 (31 and 32)	4 (41 and 42) and 6	5	2

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Table 3. Resonance regions of the second kind commensurate fractional systems

Region and defined domain	Exemple $ (\omega_{n}=1) $
	$ v=0.5 \;and\;\xi $=0.5
Gray region	(8)
No resonant frequency	$ \Downarrow $
	$ \Omega^{1.5}+1.06\Omega^{1}+0.5\Omega^{0.5}+0.35=0 $
$\left\{\begin{array}{l}0<v \leq 0.5 {\;and\;} 0 \leq \xi \\ 0.5<v \leq 1 {\;and\;} \xi_{r l} \leq \xi\end{array}\right.$	$\Rightarrow\left\{\begin{array}{l}\Omega_{1}=-0.87 \\ \Omega_{2}=-0.37+i 0.08 \\ \Omega_{3}=-0.37-i 0.08\end{array}\right.$
Yellow region	$ v $=1.5 and $ \xi $=0.9
A single resonant frequency	(8)
	$ \Downarrow $
	$ \Omega^{4.5}-1.91\Omega^{3}+1.62\Omega^{1.5}-0.64=0 $
$\left\{\begin{array}{l}0<v<1 {\;and\;} cos (v \pi / 2)<\xi<0 \\ 0.5<v<1 {\;and\;} \xi=0 \\ v=1 {\;and\;} 0<\xi<0.7071 \\ 1<v \leq 1.7837 {\;and\;}-cos (v \pi / 2)<\xi \\ 1.7837<v<2 {\;and\;}-cos (v \pi / 2)<\xi \leq \xi_{r l}\end{array}\right.$	$\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.93 \\ \Omega_{2}=0.73+i 0.52 \\ \Omega_{3}=0.73-i 0.52\end{array}\right.$
	$ \Rightarrow \omega_{res} =0.93 rd/sec $
	$ v=0.8 \;and \;\xi=0.28 $
Brown region	(8)
A single resonant frequency with anti-resonant frequency	$ \Downarrow $
A single resonant frequency with anti-resonant frequency	$ \Omega^{2.4}+0.26\Omega^{1.6}-0.65\Omega^{0.8}+0.09=0 $
$0.5<v<1$ and $0<\xi<\xi_{r l}$	$\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.09 \\ \Omega_{2}=0.52 \\ \Omega_{3}=-1.00\end{array}\right.$
	$\Rightarrow\left\{\begin{array}{l}\omega_{{anti}-r e s}=0.09 {rd} / {sec} \\ \omega_{ {res}}=0.52 {rd} / {sec}\end{array}\right.$
	$ v=1.9 \;and \;\xi=2.5 $
Green region	(8)
Two resonant frequencies	$ \Downarrow $
	$ \Omega^{5.7}-7.41\Omega^{3.8}+13.45\Omega^{1.9}-2.47=0 $
$1.7837<v<2$ and $\xi_{r l}<\xi$	$\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.44 \\ \Omega_{2}=1.65 \\ \Omega_{3}=2.23\end{array}\right.$
	$\Rightarrow\left\{\begin{array}{l}\omega_{r e s 1}=0.44 r d / s e c \\ \omega_{\text {anti}-r e s}=1.65 r d / s e c \\ \omega_{\text {res} 2}=2.23 r d / \text {sec}\end{array}\right.$

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