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September
2022, 27(9): 5283-5296.
doi: 10.3934/dcdsb.2021274
Stability and applications of multi-order fractional systems
Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile |
This paper establishes conditions for global/local robust asymptotic stability for a class of multi-order nonlinear fractional systems consisting of a linear part plus a global/local Lipschitz nonlinear term. The derivation order can be different in each coordinate and take values in $ (0, 2) $. As a consequence, a linearized stability theorem for multi-order systems is also obtained. The stability conditions are order-dependent, reducing the conservatism of order-independent ones. Detailed examples in robust control and population dynamics show the applicability of our results. Simulations are attached, showing the distinctive features that justify multi-order modelling.
Mathematics Subject Classification:
34A08, 93D09.
Citation:
Javier Gallegos. Stability and applications of multi-order fractional systems. Discrete and Continuous Dynamical Systems - B,
2022, 27
(9)
: 5283-5296.
doi: 10.3934/dcdsb.2021274
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References:
| [1] |
D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, 2 edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
|
| [2] |
C. Bonnet and J. Partington,
Coprime factorizations and stability of fractional differential systems, Syst. Control. Lett., 41 (2000), 167-174.
doi: 10.1016/S0167-6911(00)00050-5. |
| [3] |
O. Brandibur and E. Kaslik,
Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model, Math. Methods. Appl. Sci., 41 (2018), 7182-7194.
doi: 10.1002/mma.4768. |
| [4] |
J. Chen, K. Lundberg, D. Davison and D. Bernstein,
The Final Value Theorem Revisited - Infinite Limits and Irrational Functions, IEEE. Control. Syst. Mag., 27 (2007), 97-99.
|
| [5] |
N. Cong, T. Doan, S. Siegmund and H. Tuan,
Linearized asymptotic stability for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 39 (2016), 1-13.
doi: 10.14232/ejqtde.2016.1.39. |
| [6] |
W. Deng, C. Li and J. Lü,
Stability analysis of linear fractional differential system with multiple delays, Nonlinear Dynam., 48 (2007), 409-416.
doi: 10.1007/s11071-006-9094-0. |
| [7] |
C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic, New York, 1975.
|
| [8] |
K. Diethelm, S. Siegmund and H. T. Tuan,
Asymptotic behavior of solutions of linear multi-order fractional differential equation system, Fract. Calc. Appl. Anal., 20 (2017), 1165-1195.
doi: 10.1515/fca-2017-0062. |
| [9] |
J. A. Gallegos, N. Aguila-Camacho and M. A. Duarte-Mermoud,
Smooth solutions to mixed-order fractional differential systems with applications to stability analysis, J. Integral Equations Appl., 31 (2019), 59-84.
doi: 10.1216/jie-2019-31-1-59. |
| [10] |
J. A. Gallegos, N. Aguila-Camacho and M. A. Duarte-Mermoud,
Vector Lyapunov-like functions for multi-order fractional systems with multiple time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 105089.
doi: 10.1016/j.cnsns.2019.105089. |
| [11] |
J. A. Gallegos and M. A. Duarte-Mermoud,
Robustness and convergence of fractional systems and their applications to adaptive systems, Fract. Calc. Appl. Anal., 20 (2017), 895-913.
doi: 10.1515/fca-2017-0047. |
| [12] |
J. A. Gallegos and M. A. Duarte-Mermoud,
Converse theorems in Lyapunov's second method and applications for fractional order systems, Turkish J. Math., 43 (2019), 1626-1639.
doi: 10.3906/mat-1808-75. |
| [13] |
A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.
|
| [14] |
V. Lakshmikantham, V. M. Matrosov and S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Kluwer Academic Publishers, 1991.
doi: 10.1007/978-94-015-7939-1. |
| [15] |
B. Lenka,
Fractional comparison method and asymptotic stability of multivariable fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 398-415.
doi: 10.1016/j.cnsns.2018.09.016. |
| [16] |
W. LePage, Complex Variables and the Laplace Transform for Engineers, Dover Publications, 1980. |
| [17] |
C. M. A. Pinto, A. Mendes Lopes and J. A. T. Machado,
A review of power laws in real life phenomena, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3558-3578.
doi: 10.1016/j.cnsns.2012.01.013. |
| [18] |
H. Taghavian and M. Tavazoei,
Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach, Internat. J. Robust Nonlinear Control, 28 (2017), 1127-1144.
doi: 10.1002/rnc.3919. |
| [19] |
M. Tavazoei and M. Asemani,
Fractional-order-dependent global stability analysis and observer-based synthesis for a class of nonlinear fractional-order systems, Internat. J. Robust Nonlinear Control, 28 (2018), 4549-4564.
doi: 10.1002/rnc.4250. |
| [20] |
H. Tuan and H. Trinh,
Stability of fractional-order nonlinear systems by Lyapunov direct method, IET Control Theory Appl., 12 (2018), 2417-2422.
doi: 10.1049/iet-cta.2018.5233. |
| [21] |
Z. Wang, D. Yang and H. Zhang,
Stability analysis on a class of nonlinear fractional-order system, Nonlinear Dynam., 86 (2016), 1023-1033.
doi: 10.1007/s11071-016-2943-6. |
show all references
References:
| [1] |
D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, 2 edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
|
| [2] |
C. Bonnet and J. Partington,
Coprime factorizations and stability of fractional differential systems, Syst. Control. Lett., 41 (2000), 167-174.
doi: 10.1016/S0167-6911(00)00050-5. |
| [3] |
O. Brandibur and E. Kaslik,
Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model, Math. Methods. Appl. Sci., 41 (2018), 7182-7194.
doi: 10.1002/mma.4768. |
| [4] |
J. Chen, K. Lundberg, D. Davison and D. Bernstein,
The Final Value Theorem Revisited - Infinite Limits and Irrational Functions, IEEE. Control. Syst. Mag., 27 (2007), 97-99.
|
| [5] |
N. Cong, T. Doan, S. Siegmund and H. Tuan,
Linearized asymptotic stability for fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 39 (2016), 1-13.
doi: 10.14232/ejqtde.2016.1.39. |
| [6] |
W. Deng, C. Li and J. Lü,
Stability analysis of linear fractional differential system with multiple delays, Nonlinear Dynam., 48 (2007), 409-416.
doi: 10.1007/s11071-006-9094-0. |
| [7] |
C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic, New York, 1975.
|
| [8] |
K. Diethelm, S. Siegmund and H. T. Tuan,
Asymptotic behavior of solutions of linear multi-order fractional differential equation system, Fract. Calc. Appl. Anal., 20 (2017), 1165-1195.
doi: 10.1515/fca-2017-0062. |
| [9] |
J. A. Gallegos, N. Aguila-Camacho and M. A. Duarte-Mermoud,
Smooth solutions to mixed-order fractional differential systems with applications to stability analysis, J. Integral Equations Appl., 31 (2019), 59-84.
doi: 10.1216/jie-2019-31-1-59. |
| [10] |
J. A. Gallegos, N. Aguila-Camacho and M. A. Duarte-Mermoud,
Vector Lyapunov-like functions for multi-order fractional systems with multiple time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 105089.
doi: 10.1016/j.cnsns.2019.105089. |
| [11] |
J. A. Gallegos and M. A. Duarte-Mermoud,
Robustness and convergence of fractional systems and their applications to adaptive systems, Fract. Calc. Appl. Anal., 20 (2017), 895-913.
doi: 10.1515/fca-2017-0047. |
| [12] |
J. A. Gallegos and M. A. Duarte-Mermoud,
Converse theorems in Lyapunov's second method and applications for fractional order systems, Turkish J. Math., 43 (2019), 1626-1639.
doi: 10.3906/mat-1808-75. |
| [13] |
A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.
|
| [14] |
V. Lakshmikantham, V. M. Matrosov and S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Kluwer Academic Publishers, 1991.
doi: 10.1007/978-94-015-7939-1. |
| [15] |
B. Lenka,
Fractional comparison method and asymptotic stability of multivariable fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 398-415.
doi: 10.1016/j.cnsns.2018.09.016. |
| [16] |
W. LePage, Complex Variables and the Laplace Transform for Engineers, Dover Publications, 1980. |
| [17] |
C. M. A. Pinto, A. Mendes Lopes and J. A. T. Machado,
A review of power laws in real life phenomena, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3558-3578.
doi: 10.1016/j.cnsns.2012.01.013. |
| [18] |
H. Taghavian and M. Tavazoei,
Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach, Internat. J. Robust Nonlinear Control, 28 (2017), 1127-1144.
doi: 10.1002/rnc.3919. |
| [19] |
M. Tavazoei and M. Asemani,
Fractional-order-dependent global stability analysis and observer-based synthesis for a class of nonlinear fractional-order systems, Internat. J. Robust Nonlinear Control, 28 (2018), 4549-4564.
doi: 10.1002/rnc.4250. |
| [20] |
H. Tuan and H. Trinh,
Stability of fractional-order nonlinear systems by Lyapunov direct method, IET Control Theory Appl., 12 (2018), 2417-2422.
doi: 10.1049/iet-cta.2018.5233. |
| [21] |
Z. Wang, D. Yang and H. Zhang,
Stability analysis on a class of nonlinear fractional-order system, Nonlinear Dynam., 86 (2016), 1023-1033.
doi: 10.1007/s11071-016-2943-6. |
Figure 2.
Population dynamics depending on the derivation order
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