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July
2021, 26(7): 3579-3594.
doi: 10.3934/dcdsb.2020247
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 |
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.
Mathematics Subject Classification:
Primary: 11Y35; Secondary: 12E12.
Citation:
Assia Boubidi, Sihem Kechida, Hicham Tebbikh. Analytical study of resonance regions for second kind commensurate fractional systems. Discrete and Continuous Dynamical Systems - B,
2021, 26
(7)
: 3579-3594.
doi: 10.3934/dcdsb.2020247
![]()
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doi: 10.1007/978-1-84996-335-0. |
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A. Oustaloup, La Commande CRONE, Edition, Hermès, Paris, France, 1991. |
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I. Podlubny,
Fractional-order systems and and $PI^\lambda D^\mu$-controllers, IEEE Trans. Automat. Control, 44 (1999), 208-214.
doi: 10.1109/9.739144. |
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T. Poinot and J.-C. Trigeassou,
Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38 (2004), 133-154.
doi: 10.1007/s11071-004-3751-y. |
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A. G. Redwan, A. S. Elwakil and A. M. Soliman,
On the generalization of second-order filters to the the fractional-order domain, J. Circuit Syst. Comp., 18 (2009), 361-386.
doi: 10.1142/S0218126609005125. |
| [26] |
A. G. Redwan, A. M. Soliman and A. S. Elwakil,
First-order filters generalized to the fractional domain, J. Circuit Syst. Comp., 17 (2008), 55-66.
doi: 10.1142/S0218126608004162. |
| [27] |
J. Sabatier, O. Cois and A. Oustaloup, Modal placement control method for fractional systems: Application to a testing bench, in IEEE 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, USA, (2003), 633–639.
doi: 10.1115/DETC2003/VIB-48373. |
| [28] |
J. Sabatier, Ch. Farges and J.-C. Trigeassou,
A stability test for non-commensurate fractional order systems, Syst. Control. Lett., 62 (2013), 739-746.
doi: 10.1016/j.sysconle.2013.04.008. |
| [29] |
P. Shah and S. Agashe,
Review of fractional PID controller, Mechatronics, 38 (2016), 29-41.
doi: 10.1016/j.mechatronics.2016.06.005. |
| [30] |
J.-C. Trigeassou and N. Maamri, A new approach to the stability of linear fractional systems, in 2009 6th International Multi-Conference on Systems, Signals and Devices, Tunisia, (2009), 1–14.
doi: 10.1109/SSD.2009.4956821. |
| [31] |
T. Yinggan, L. Haifang, W. Weiwei, L. Qiusheng and G. Xinping,
Parameter identification of fractional order systems using block pulse functions, Signal Process., 107 (2015), 272-281.
doi: 10.1016/j.sigpro.2014.04.011. |
| [32] |
L. Zeng, P. Cheng and W. Yong, Subspace identification for commensurate fractional order systems using instrumental variables, in IEEE 2011 Control Conference, China, (2011), 1636–1640.
doi: 10.1007/s12555-012-0511-5. |
| [33] |
W. Zhong, J. Lu and Y. Miao, Fault detection observer design for fractional-order systems, in 29th Chinese Control And Decision Conference (CCDC), China, (2017), 2796–2801.
doi: 10.1109/CCDC.2017.7978988. |
| [34] |
S. Zhou, J. Cao and Y. Chen,
Genetic algorithm-based identification of fractional-order systems, Entropy, 15 (2013), 1624-1642.
doi: 10.3390/e15051624. |
show all references
References:
| [1] |
M. Aoun, Systèmes Linéaires non Entiers et Identification par Bases Orthogonales non Entières, Ph.D thesis, Université Bordeaux 1, Talence, France, 2005. |
| [2] |
M. Aoun, A. Aribi, S. Najar and M. N. Abdelkrim, On the fractional systems' fault detection: A comparison between fractional and rational residual sensitivity, in Eighth International Multi-Conference on Systems, Signals & Devices, Sousse, (2011), 1–6.
doi: 10.1109/SSD.2011.5767424. |
| [3] |
A. Aribi, C. Farges, M. Aoun, P. Melchior, S. Najar and M. N. Abdelkrim,
Fault detection based on fractional order models: Application to diagnosis of thermal systems, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 3679-3693.
doi: 10.1016/j.cnsns.2014.03.006. |
| [4] |
H. Atitallah, A. Aribi and M. Aoun, Diagnosis of time-delay fractional systems, in 17th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), Sousse, (2016), 284–292.
doi: 10.1109/STA.2016.7952042. |
| [5] |
A. Ben Hmed, M. Amairi and M. Aoun,
Stability and resonance conditions of the non-commensurate elementary fractional transfer functions of the second kind, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 842-865.
doi: 10.1016/j.cnsns.2014.07.014. |
| [6] |
J. Cao, J. Liang and B. Cao, Optimization of fractional order PID controllers based on genetic algorithms, in IEEE 2005 Machine Learning and Cybernetics, China, (2005), 5686–5689.
doi: 10.1109/ICMLC.2005.1527950. |
| [7] |
D. R. Curtiss,
Recent extensions of Descartes' rule of signs, Ann. of Math., 19 (1918), 251-278.
doi: 10.2307/1967494. |
| [8] |
W. Du, Q. Miao, L. Tong and Y. Tang,
Identification of fractional-order systems with unknown initial values and structure, Phys. Lett. A, 381 (2017), 1943-1949.
doi: 10.1016/j.physleta.2017.03.048. |
| [9] |
A. S. Elwakil,
Fractional-order circuits and systems: An emerging interdisciplinary research area, IEEE Circuits and Systems Magazine, 10 (2010), 40-50.
doi: 10.1109/MCAS.2010.938637. |
| [10] |
T. T. Hartley and C. F. Lorenzo, A solution to the fundamental linear fractional order differential equation, Technical Report, 1998. Available from : https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990041952.pdf |
| [11] |
M. Ikeda and S. Takahashi,
Generalization of Routh's algorithm and stability criterion for non-integer integral system, Electron. Comm. Japan, 60 (1977), 41-50.
doi: 10.1155/2008/419046. |
| [12] |
E. Ivanova, X. Moreau and R. Malti,
Stability and resonance conditions of second-order fractional systems, J. Vib. Control, 24 (2018), 659-672.
doi: 10.1177/1077546316654790. |
| [13] |
F. Khemane, Estimation Fréquentielle par Modèle Non-entier et Approche Ensembliste: Application à la Modélisation de la Dynamique du Conducteur, Ph.D thesis, Université Bordeaux 1, Talence, France, 2011. |
| [14] |
M. R. Kumar, V. Deepak and S. Ghosh,
Fractional-order controller design in frequency domain using an improved nonlinear adaptive seeker optimization algorithm, Comp. Sci., 25 (2017), 4299-4310.
doi: 10.3906/elk-1701-294. |
| [15] |
J. Lin, Modélisation et Identification des Systèmes d'ordre non Entier, Ph.D thesis, Université de Poitiers, Poitiers, France, 2001. |
| [16] |
R. Malti,
A note on ${L}_p$-norms of fractional systems, Automatica J. IFAC, 49 (2013), 2923-2927.
doi: 10.1016/j.automatica.2013.06.002. |
| [17] |
R. Malti, M. Aoun, F. Levron and A. Oustaloup,
Analytical computation of the $H_2$-norm of fractional commensurate transfer functions, Automatica J. IFAC, 47 (2011), 2425-2432.
doi: 10.1016/j.automatica.2011.08.021. |
| [18] |
R. Malti, X. Moreau, F. Khemane and A. Oustaloup,
Stability and resonance conditions of elementary fractional transfer functions, Automatica J. IFAC, 47 (2011), 2462-2467.
doi: 10.1016/j.automatica.2011.08.029. |
| [19] |
D. Matignon, Stability properties for generalized fractional differential systems, in ESAIM Proc., Soc. Math. Appl. Indust., Paris, (1998), 145–158.
doi: 10.1051/proc:1998004. |
| [20] |
F. Merrikh-Bayat, M. Afshar and M. Karimi-Ghartemani,
Extension of the root-locus method to a certain class of fractional-order systems, ISA. Trans., 48 (2009), 48-53.
doi: 10.1016/j.isatra.2008.08.001. |
| [21] |
C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-order Systems and control: Fundamentals and Applications, Advances in Industrial Control. Springer, London, 2010.
doi: 10.1007/978-1-84996-335-0. |
| [22] |
A. Oustaloup, La Commande CRONE, Edition, Hermès, Paris, France, 1991. |
| [23] |
I. Podlubny,
Fractional-order systems and and $PI^\lambda D^\mu$-controllers, IEEE Trans. Automat. Control, 44 (1999), 208-214.
doi: 10.1109/9.739144. |
| [24] |
T. Poinot and J.-C. Trigeassou,
Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38 (2004), 133-154.
doi: 10.1007/s11071-004-3751-y. |
| [25] |
A. G. Redwan, A. S. Elwakil and A. M. Soliman,
On the generalization of second-order filters to the the fractional-order domain, J. Circuit Syst. Comp., 18 (2009), 361-386.
doi: 10.1142/S0218126609005125. |
| [26] |
A. G. Redwan, A. M. Soliman and A. S. Elwakil,
First-order filters generalized to the fractional domain, J. Circuit Syst. Comp., 17 (2008), 55-66.
doi: 10.1142/S0218126608004162. |
| [27] |
J. Sabatier, O. Cois and A. Oustaloup, Modal placement control method for fractional systems: Application to a testing bench, in IEEE 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, USA, (2003), 633–639.
doi: 10.1115/DETC2003/VIB-48373. |
| [28] |
J. Sabatier, Ch. Farges and J.-C. Trigeassou,
A stability test for non-commensurate fractional order systems, Syst. Control. Lett., 62 (2013), 739-746.
doi: 10.1016/j.sysconle.2013.04.008. |
| [29] |
P. Shah and S. Agashe,
Review of fractional PID controller, Mechatronics, 38 (2016), 29-41.
doi: 10.1016/j.mechatronics.2016.06.005. |
| [30] |
J.-C. Trigeassou and N. Maamri, A new approach to the stability of linear fractional systems, in 2009 6th International Multi-Conference on Systems, Signals and Devices, Tunisia, (2009), 1–14.
doi: 10.1109/SSD.2009.4956821. |
| [31] |
T. Yinggan, L. Haifang, W. Weiwei, L. Qiusheng and G. Xinping,
Parameter identification of fractional order systems using block pulse functions, Signal Process., 107 (2015), 272-281.
doi: 10.1016/j.sigpro.2014.04.011. |
| [32] |
L. Zeng, P. Cheng and W. Yong, Subspace identification for commensurate fractional order systems using instrumental variables, in IEEE 2011 Control Conference, China, (2011), 1636–1640.
doi: 10.1007/s12555-012-0511-5. |
| [33] |
W. Zhong, J. Lu and Y. Miao, Fault detection observer design for fractional-order systems, in 29th Chinese Control And Decision Conference (CCDC), China, (2017), 2796–2801.
doi: 10.1109/CCDC.2017.7978988. |
| [34] |
S. Zhou, J. Cao and Y. Chen,
Genetic algorithm-based identification of fractional-order systems, Entropy, 15 (2013), 1624-1642.
doi: 10.3390/e15051624. |
Figure 5.
Division of the $ v $ $ \xi $ -plane according to $ D $ signs and coefficients signs of $ f(x) $ and $ f(-x) $
Figure 8.
Division of the $ v $ $ \xi $ -plane according to the number of strictly positive roots for stable systems
Figure 9.
New division of resonance regions of the second kind commensurate fractional systems in the $ v $ $ \xi $ -plane
Figure 10.
Magnitude Bode diagram for each region
Figure 11.
Sallen-Key FLPF circuit
Table 1.
Application of Descartes Rule of Signs according to Figure 5 division
| Regions | 1 | 2 | 31 | 32 | 41 | 42 | 5 | 6 |
| + | + | - | - | - | - | + | + | |
| 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | |
| + + + + | + - + - | + + + + | + + - + | + - + - | + - - - | + + - + | + - - - | |
| Sign changes Nbr | 0 | 3 | 0 | 2 | 3 | 1 | 2 | 1 |
| 0 | 3 or 1 | 0 | 2 or 0 | 3 or 1 | 1 | 2 or 0 | 1 | |
| - + - + | - - - - | - + - + | - + + + | - - - - | - - + - | - + + + | - - + - | |
| Sign changes Nbr | 3 | 0 | 3 | 1 | 0 | 2 | 1 | 2 |
| 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 |
| 3 | 0 | 1 | 1 | 0 | 0 | 1 | 2 |
| Regions | 1 | 2 | 31 | 32 | 41 | 42 | 5 | 6 |
| + | + | - | - | - | - | + | + | |
| 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | |
| + + + + | + - + - | + + + + | + + - + | + - + - | + - - - | + + - + | + - - - | |
| Sign changes Nbr | 0 | 3 | 0 | 2 | 3 | 1 | 2 | 1 |
| 0 | 3 or 1 | 0 | 2 or 0 | 3 or 1 | 1 | 2 or 0 | 1 | |
| - + - + | - - - - | - + - + | - + + + | - - - - | - - + - | - + + + | - - + - | |
| Sign changes Nbr | 3 | 0 | 3 | 1 | 0 | 2 | 1 | 2 |
| 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 |
| 3 | 0 | 1 | 1 | 0 | 0 | 1 | 2 |
Table 2.
Positive roots number of (8) and corresponding regions
| 0 | 1 | 2 | 3 | |
| Regions | 1 and 3 (31 and 32) | 4 (41 and 42) and 6 | 5 | 2 |
| 0 | 1 | 2 | 3 | |
| Regions | 1 and 3 (31 and 32) | 4 (41 and 42) and 6 | 5 | 2 |
Table 3.
Resonance regions of the second kind commensurate fractional systems
| Region and defined domain | Exemple |
| Gray region | (8) |
| No resonant frequency | |
| $\left\{\begin{array}{l}0<v \leq 0.5 {\;and\;} 0 \leq \xi \\ 0.5<v \leq 1 {\;and\;} \xi_{r l} \leq \xi\end{array}\right.$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=-0.87 \\ \Omega_{2}=-0.37+i 0.08 \\ \Omega_{3}=-0.37-i 0.08\end{array}\right.$ |
| Yellow region | |
| A single resonant frequency | (8) |
| $\left\{\begin{array}{l}0<v<1 {\;and\;} cos (v \pi / 2)<\xi<0 \\ 0.5<v<1 {\;and\;} \xi=0 \\ v=1 {\;and\;} 0<\xi<0.7071 \\ 1<v \leq 1.7837 {\;and\;}-cos (v \pi / 2)<\xi \\ 1.7837<v<2 {\;and\;}-cos (v \pi / 2)<\xi \leq \xi_{r l}\end{array}\right.$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.93 \\ \Omega_{2}=0.73+i 0.52 \\ \Omega_{3}=0.73-i 0.52\end{array}\right.$ |
| Brown region | (8) |
| A single resonant frequency with anti-resonant frequency | |
| $0.5<v<1$ and $0<\xi<\xi_{r l}$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.09 \\ \Omega_{2}=0.52 \\ \Omega_{3}=-1.00\end{array}\right.$ |
| $\Rightarrow\left\{\begin{array}{l}\omega_{{anti}-r e s}=0.09 {rd} / {sec} \\ \omega_{ {res}}=0.52 {rd} / {sec}\end{array}\right.$ | |
| Green region | (8) |
| Two resonant frequencies | |
| $1.7837<v<2$ and $\xi_{r l}<\xi$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.44 \\ \Omega_{2}=1.65 \\ \Omega_{3}=2.23\end{array}\right.$ |
| $\Rightarrow\left\{\begin{array}{l}\omega_{r e s 1}=0.44 r d / s e c \\ \omega_{\text {anti}-r e s}=1.65 r d / s e c \\ \omega_{\text {res} 2}=2.23 r d / \text {sec}\end{array}\right.$ |
| Region and defined domain | Exemple |
| Gray region | (8) |
| No resonant frequency | |
| $\left\{\begin{array}{l}0<v \leq 0.5 {\;and\;} 0 \leq \xi \\ 0.5<v \leq 1 {\;and\;} \xi_{r l} \leq \xi\end{array}\right.$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=-0.87 \\ \Omega_{2}=-0.37+i 0.08 \\ \Omega_{3}=-0.37-i 0.08\end{array}\right.$ |
| Yellow region | |
| A single resonant frequency | (8) |
| $\left\{\begin{array}{l}0<v<1 {\;and\;} cos (v \pi / 2)<\xi<0 \\ 0.5<v<1 {\;and\;} \xi=0 \\ v=1 {\;and\;} 0<\xi<0.7071 \\ 1<v \leq 1.7837 {\;and\;}-cos (v \pi / 2)<\xi \\ 1.7837<v<2 {\;and\;}-cos (v \pi / 2)<\xi \leq \xi_{r l}\end{array}\right.$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.93 \\ \Omega_{2}=0.73+i 0.52 \\ \Omega_{3}=0.73-i 0.52\end{array}\right.$ |
| Brown region | (8) |
| A single resonant frequency with anti-resonant frequency | |
| $0.5<v<1$ and $0<\xi<\xi_{r l}$ | $\Rightarrow\left\{\begin{array}{l}\Omega_{1}=0.09 \\ \Omega_{2}=0.52 \\ \Omega_{3}=-1.00\end{array}\right.$ |
| $\Rightarrow\left\{\begin{array}{l}\omega_{{anti}-r e s}=0.09 {rd} / {sec} \\ \omega_{ {res}}=0.52 {rd} / {sec}\end{array}\right.$ | |
| Green region | (8) |
| Two resonant frequencies | |
| $1.7837<v<2$ 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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