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doi: 10.3934/dcdsb.2021274
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Stability and applications of multi-order fractional systems

Javier Gallegos

Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile

Received  September 2020 Revised  August 2021 Early access November 2021

Fund Project: The author thanks the anonymous reviewers for their comments. This research was supported by CONICYTPCHA/National PhD scholarship program, 2018

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.

Keywords: Fractional ODE, caputo derivative, multi-order, robust control, population dynamics.
Mathematics Subject Classification: 34A08, 93D09.
Citation: Javier Gallegos. Stability and applications of multi-order fractional systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021274
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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7] C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[16]

W. LePage, Complex Variables and the Laplace Transform for Engineers, Dover Publications, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7] C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[16]

W. LePage, Complex Variables and the Laplace Transform for Engineers, Dover Publications, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Robust performance
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Figure 2.  Population dynamics depending on the derivation order
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