Stability analysis of distributed-order nonlinear dynamic systems

References

• Abate, J., & Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS Journal on Computing, 18(4), 408–421.
   Web of Science ®Google Scholar
• Agarwal, R., Hristova, S., & O'Regan, D. (2015). Lyapunov functions and strict stability of Caputo fractional differential equations. Advances in Difference Equations, 2015(1), 346.
   Web of Science ®Google Scholar
• Aguila-Camacho, N., Duarte-Mermoud, M. A., & Gallegos, J. A. (2014). Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9), 2951–2957.
   Web of Science ®Google Scholar
• Atanackovic, T. M. (2003). On a distributed derivative model of a viscoelastic body. Comptes Rendus Mecanique, 331(10), 687–692.
   Web of Science ®Google Scholar
• Atanacković, T. M., Dolićanin, D., Konjik, S., & Pilipović, S. (2011). Dissipativity and stability for a nonlinear differential equation with distributed order symmetrized fractional derivative. Applied Mathematics Letters, 24(6), 1020–1025.
   Web of Science ®Google Scholar
• Atanacković, T. M., Oparnica, L., & Pilipović, S. (2007). On a nonlinear distributed order fractional differential equation. Journal of Mathematical Analysis and Applications, 328(1), 590–608.
   Web of Science ®Google Scholar
• Caputo, M. (1969, Elasticitá e dissipazione [Elasticity and elastic dissipation]. Rigorous time domain responses of polarizable media, 40(2), 423–434, Bologna: Zanichelli Publisher.
   Google Scholar
• Caputo, M. (1997). Elasticitá e dissipazione [Elasticity and elastic dissipation]. Rigorous time domain responses of polarizable media, 40(2), 423–434.
   Google Scholar
• Chen, J., Li, C., Huang, T., & Yang, X. (2017). Global stabilization of memristor-based fractional-order neural networks with delay via output-feedback control. Modern Physics Letters B, 31(05), 1750031.
   Web of Science ®Google Scholar
• Chen, L., Chai, Y., Wu, R., & Yang, J. (2012). Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative. IEEE Transactions on Circuits and Systems II: Express Briefs, 59(9), 602–606.
   Web of Science ®Google Scholar
• Chen, L., He, Y., Chai, Y., & Wu, R. (2014). New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dynamics, 75(4), 633–641.
   Web of Science ®Google Scholar
• Cho, Y., Kim, I., & Sheen, D. (2015). A fractional-order model for MINMOD Millennium. Mathematical Biosciences, 262, 36–45.
   PubMed Web of Science ®Google Scholar
• Du, B., Lam, J., Shu, Z., & Chen, Y. (2016). On reachable sets for positive linear systems under constrained exogenous inputs. Automatica, 74, 230–237.
   Web of Science ®Google Scholar
• Duarte-Mermoud, M. A., Aguila-Camacho, N., Gallegos, J. A., & Castro-Linares, R. (2015). Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 22(1), 650–659.
   Web of Science ®Google Scholar
• Duong, P. L. T., Kwok, E., & Lee, M. (2015). Optimal design of stochastic distributed order linear SISO systems using hybrid spectral method. Mathematical Problems in Engineering, 2015, 989542.
   Web of Science ®Google Scholar
• Duong, P. L. T., Kwok, E., & Lee, M. (2016). Deterministic analysis of distributed order systems using operational matrix. Applied Mathematical Modelling, 40(3), 1929–1940.
   Web of Science ®Google Scholar
• Fernández-Anaya, G., Nava-Antonio, G., Jamous-Galante, J., Muñoz-Vega, R., & Hernández-Martínez, E. G. (2017). Asymptotic stability of distributed order nonlinear dynamical systems. Communications in Nonlinear Science and Numerical Simulation, 48, 541–549.
   Web of Science ®Google Scholar
• Freeborn, T. J., Maundy, B., & Elwakil, A. S. (2015). Fractional-order models of supercapacitors, batteries and fuel cells: A survey. Materials for Renewable and Sustainable Energy, 4(3), 1–7.
   Web of Science ®Google Scholar
• Gao, G. H., Sun, H. W., & Sun, Z. Z. (2015). Some high-order difference schemes for the distributed-order differential equations. Journal of Computational Physics, 298, 337–359.
   Web of Science ®Google Scholar
• Hasan, M. M., & Drapaca, C. (2015). A fractional order model for local electric fields in tissues. Mechanics of Biological Systems and Materials, 7, 75–79.
   Google Scholar
• Hu, J. B., Lu, G. P., Zhang, S. B., & Zhao, L. D. (2015). Lyapunov stability theorem about fractional system without and with delay. Communications in Nonlinear Science and Numerical Simulation, 20(3), 905–913.
   Web of Science ®Google Scholar
• Ibrir, S., & Bettayeb, M. (2015). New sufficient conditions for observer-based control of fractional-order uncertain systems. Automatica, 59, 216–223.
   Web of Science ®Google Scholar
• Jakovljević, B. B., Rapaić, M. R., Jelicić, Z. D., & Sekara, T. B. (2014, June). Optimization of distributed order fractional PID controller under constraints on robustness and sensitivity to measurement noise. In 2014 International Conference on Fractional Differentiation and Its Applications (ICFDA) (pp. 1–6). Catania, Italy: IEEE.
   Google Scholar
• Jiao, Z., Chen, Y. Q., & Podlubny, I. (2012). Distributed-order dynamic systems: Stability, simulation, applications and perspectives. Springer briefs in electrical and computer engineering/Springer briefs in control, automation and robotics. Springer.
   Google Scholar
• Kamal, S., Raman, A., & Bandyopadhyay, B. (2013). Finite-time stabilization of fractional order uncertain chain of integrator: An integral sliding mode approach. IEEE Transactions on Automatic Control, 58(6), 1597–1602.
   Web of Science ®Google Scholar
• Khalil, H. K., & Grizzle, J. W. (1996). Nonlinear systems (Vol. 3). London, UK: Prentice Hall.
   Google Scholar
• Kochubei, A. N. (2009). Distributed order derivatives and relaxation patterns . Journal of Physics A: Mathematical and Theoretical, 42, 315203. arXiv preprint arXiv:0905.0616.
   Web of Science ®Google Scholar
• Krstic, M., & Smyshlyaev, A. Lecture notes of control of PDE systems. Retrieved from http://www.lehigh.edu/∼eus204/Teaching/ME450_CDPS/ lectures/lecture03.pdf
   Google Scholar
• Lazović, G., Vosika, Z., Lazarević, M., Simić-Krstić, J., & Koruga, Đ. (2014). Modeling of bioimpedance for human skin based on fractional distributed-order modified Cole model. FME Transactions, 42(1), 74–81.
   Google Scholar
• Li, D., Cao, J., Zhou, S., & Chen, Y. (2015, August). Fractional order model of broadband piezoelectric energy harvesters. In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, V009T07A019–V009T07A019, Boston, Massachusetts, USA: American Society of Mechanical Engineers.
   Google Scholar
• Li, X. Y., & Wu, B. Y. (2016). A numerical method for solving distributed order diffusion equations. Applied Mathematics Letters, 53, 92–99.
   Web of Science ®Google Scholar
• Li, Y., Chen, Y., & Podlubny, I. (2009). Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45(8), 1965–1969.
   Web of Science ®Google Scholar
• Li, Y., Chen, Y., & Podlubny, I. (2010). Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Computers & Mathematics with Applications, 59(5), 1810–1821.
   Web of Science ®Google Scholar
• McClure, T. (2012). Numerical inverse Laplace transform. Retrieved fromhttps://www.mathworks.com/ matlabcentral/fileexchange/39035-numerical-inverse-laplace-transform?requestedDomain= www.mathworks.com
   Google Scholar
• Meerschaert, M. M., & Tadjeran, C. (2004). Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1), 65–77.
   Web of Science ®Google Scholar
• Noroozi, H., & Ansari, A. (2013). Extremal positive solutions for the distributed order fractional hybrid differential equations. Computational Methods for Differential Equations, 1(2), 120–134.
   Google Scholar
• Noroozi, H., & Ansari, A. (2014). Basic results on distributed order fractional hybrid differential equations with linear perturbations. Journal of Mathematical Modeling, 2(1), 55–73.
   Google Scholar
• Odabasi, M., & Misirli, E. (2015). On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences. doi: 10.1002/mma.3533
   PubMed Web of Science ®Google Scholar
• Petras, I. (2011). Fractional-order nonlinear systems: Modeling, analysis and simulation. Berlin, Germany: Springer Science & Business Media.
   Google Scholar
• Pillonetto, G., Chen, T., Chiuso, A., De Nicolao, G., & Ljung, L. (2016). Regularized linear system identification using atomic, nuclear and kernel-based norms: The role of the stability constraint. Automatica, 69, 137–149.
   Web of Science ®Google Scholar
• Podlubny, I. (1998). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). San Diego, USA: Academic Press.
   Google Scholar
• Sandev, T., Chechkin, A. V., Korabel, N., Kantz, H., Sokolov, I. M., & Metzler, R. (2015). Distributed-order diffusion equations and multifractality: Models and solutions. Physical Review E, 92(4), 042117.
   Web of Science ®Google Scholar
• Shen, J., & Lam, J. (2014). Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica, 50(2), 547–551.
   Web of Science ®Google Scholar
• Shen, J., & Lam, J. (2016). Stability and performance analysis for positive fractional-order systems with time-varying delays. IEEE Transactions on Automatic Control, 61(9), 2676–2681.
   Web of Science ®Google Scholar
• Stanisławski, R., & Latawiec, K. J. (2013). Stability analysis for discrete-time fractional-order LTI state-space systems. Part I: New necessary and sufficient conditions for the asymptotic stability. Bulletin of the Polish Academy of Sciences: Technical Sciences, 61(2), 353–361.
   Web of Science ®Google Scholar
• Su, N., Nelson, P. N., & Connor, S. (2015). The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests. Journal of Hydrology, 529, 1262–1273.
   Web of Science ®Google Scholar
• Süli, E., & Mayers, D. F. (2003). An introduction to numerical analysis. Cambridge UK: Cambridge University Press.
   Google Scholar
• Tavazoei, M. S. (2012). From traditional to fractional PI control: A key for generalization. IEEE Industrial Electronics Magazine, 6, 41–51.
   Web of Science ®Google Scholar
• Tavazoei, M. S. (2013). On type number concept in fractional-order systems. Automatica, 49(1), 301–304.
   Web of Science ®Google Scholar
• Tavazoei, M. S. (2014). Toward searching possible oscillatory region in order space for nonlinear fractional-order systems. ASME Journal of Computational and Nonlinear Dynamics, 9, 021011.
   Web of Science ®Google Scholar
• Vargas-De-León, C. (2015). Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation, 24(1), 75–85.
   Web of Science ®Google Scholar
• Yang, H., & Jiang, B. (2016). Stability of fractional-order switched non-linear systems. IET Control Theory & Applications, 10(8), 965–970.
   Web of Science ®Google Scholar
• Yang, X., Li, C., Huang, T., Song, Q., & Chen, X. (2017). Quasi-uniform synchronization of fractional-order memristor-based neural networks with delay. Neurocomputing, 234, 205–215.
   Web of Science ®Google Scholar
• Yang, X., Li, C., Song, Q., Huang, T., & Chen, X. (2016). Mittag–Leffler stability analysis on variable-time impulsive fractional-order neural networks. Neurocomputing, 207, 276–286.
   Web of Science ®Google Scholar
• Yang, Y., & Chen, G. (2015). Finite‐time stability of fractional order impulsive switched systems. International Journal of Robust and Nonlinear Control, 25(13), 2207–2222.
   Web of Science ®Google Scholar
• Yu, J., Hu, H., Zhou, S., & Lin, X. (2013). Generalized Mittag–Leffler stability of multi-variables fractional order nonlinear systems. Automatica, 49(6), 1798–1803.
   Web of Science ®Google Scholar
• Zhang, R., Tian, G., Yang, S., & Cao, H. (2015). Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2). ISA Transactions, 56, 102–110.
   PubMed Web of Science ®Google Scholar
• Zhang, X. M., & Han, Q. L. (2015). Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems. Automatica, 57, 199–202.
   Web of Science ®Google Scholar
• Zhao, L. D., Hu, J. B., Fang, J. A., & Zhang, W. B. (2012). Studying on the stability of fractional-order nonlinear system. Nonlinear Dynamics, 70(1), 475–479.
   Web of Science ®Google Scholar
• Zhao, X., Liu, H., Zhang, J., & Li, H. (2015). Multiple-mode observer design for a class of switched linear systems. IEEE Transactions on Automation Science and Engineering, 12(1), 272–280.
   Web of Science ®Google Scholar
• Zheng, G., Bejarano, F. J., Perruquetti, W., & Richard, J. P. (2015). Unknown input observer for linear time-delay systems. Automatica, 61, 35–43.
   Web of Science ®Google Scholar
• Zhou, F., Zhao, Y., Li, Y., & Chen, Y. (2013). Design, implementation and application of distributed order PI control. ISA Transactions, 52(3), 429–437.
   PubMed Web of Science ®Google Scholar