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Mittag-Leffler convergent backstepping observers for coupled semilinear subdiffusion systems with spatially varying parameters. (English) Zbl 1408.93097

Summary: The purpose of this paper is to investigate the observer-based boundary output feedback control for subdiffusion processes governed by coupled semilinear Time Fractional Diffusion Systems (TFDSs) with spatially varying parameters. For this, backstepping technique is used to Mittag-Leffler stabilize the coupled semilinear observer error dynamic systems. We then design an observer-based output feedback controller at the right boundary to realize the Mittag-Leffler stability of the closed-loop systems at hand. A numerical example is finally included to test our methods.

MSC:

93D15 Stabilization of systems by feedback
93B52 Feedback control
93C10 Nonlinear systems in control theory
93C20 Control/observation systems governed by partial differential equations
26A33 Fractional derivatives and integrals
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[1] Li, Y.; Chen, Y.; Podlubny, I., Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 8, 1965-1969 (2009) · Zbl 1185.93062
[2] Li, C.; Zhang, F., A survey on the stability of fractional differential equations, Eur. Phys. J. Special Top., 193, 1, 27-47 (2011)
[3] Monje, C. A.; Chen, Y.; Vinagre, B. M.; Xue, D.; Feliu-Batlle, V., Fractional-Order Systems and Controls: Fundamentals and Applications (2010), Springer Science & Business Media
[4] Podlubny, I., Fractional-order systems and \(PI^\lambda D^\mu \)-controllers, IEEE Trans. Automat. Control, 44, 1, 208-214 (1999) · Zbl 1056.93542
[5] Sabatier, J.; Moze, M.; Farges, C., LMI stability conditions for fractional order systems, Comput. Math. Appl., 59, 5, 1594-1609 (2010) · Zbl 1189.34020
[6] Liang, J.; Chen, Y.; Fullmer, R., Boundary stabilization and disturbance rejection for time fractional order diffusion-wave equations, Nonlinear Dynam., 38, 1, 339-354 (2004) · Zbl 1094.74042
[7] Ge, F.; Chen, Y.; Kou, C., Boundary feedback stabilisation for the time fractional-order anomalous diffusion system, IET Control Theory Appl., 10, 11, 1250-1257 (2016)
[8] Zhou, H.; Guo, B., Boundary feedback stabilization for an unstable time fractional reaction diffusion equation, SIAM J. Control Optim., 56, 1, 75-101 (2018) · Zbl 1386.35464
[9] Ge, F.; Chen, Y., Extended Luenberger-type observer for a class of semilinear time fractional diffusion systems, Chaos Solitons Fractals, 102, 229-235 (2017) · Zbl 1374.93170
[10] Meurer, T., On the extended Luenberger-type observer for semilinear distributed-parameter systems, IEEE Trans. Automat. Control, 58, 7, 1732-1743 (2013) · Zbl 1369.93105
[11] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier Science Limited · Zbl 1092.45003
[12] Henry, B. I.; Wearne, S. L., Fractional reaction-diffusion, Physica A, 276, 3, 448-455 (2000)
[13] Uchaikin, V. V.; Sibatov, R. T., Fractional theory for transport in disordered semiconductors, Commun. Nonlinear Sci. Numer. Simul., 13, 4, 715-727 (2008) · Zbl 1221.82141
[14] Ge, F.; Chen, Y.; Kou, C., Regional Analysis of Time-Fractional Diffusion Processes (2018), Springer · Zbl 1382.93002
[15] Tang, S.; Xie, C., State and output feedback boundary control for a coupled PDE-ODE system, Systems Control Lett., 60, 8, 540-545 (2011) · Zbl 1236.93076
[16] Baccoli, A.; Pisano, A.; Orlov, Y., Boundary control of coupled reaction-diffusion processes with constant parameters, Automatica, 54, 80-90 (2015) · Zbl 1318.93072
[17] Liu, B.; Boutat, D.; Liu, D., Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions, Systems Control Lett., 97, 61-69 (2016) · Zbl 1350.93041
[18] Vazquez, R.; Krstić, M., Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients, IEEE Trans. Automat. Control, 62, 4, 2026-2033 (2017) · Zbl 1366.93533
[19] Deutscher, J.; Kerschbaum, S., Backstepping control of coupled linear parabolic PIDEs with spatially-varying coefficients, IEEE Trans. Automat. Control (2018), In Press · Zbl 1423.93155
[20] Meurer, T., Flatness-based trajectory planning for diffusion-reaction systems in a parallelepipedona spectral approach, Automatica, 47, 5, 935-949 (2011) · Zbl 1233.93025
[21] Krstić, M.; Smyshlyaev, A., Boundary Control of PDEs: A Course on Backstepping Designs, Vol. 16 (2008), SIAM · Zbl 1149.93004
[22] Smyshlyaev, A.; Krstić, M., Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Trans. Automat. Control, 49, 12, 2185-2202 (2004) · Zbl 1365.93193
[23] Aguila-Camacho, N.; Duarte-Mermoud, M. A.; Gallegos, J. A., Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19, 9, 2951-2957 (2014) · Zbl 1510.34111
[24] Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S. V., Mittag-Leffler Functions, Related Topics and Applications (2014), Springer · Zbl 1309.33001
[25] Meurer, T.; Kugi, A., Tracking control for boundary controlled parabolic PDEs with varying parameters: Combining backstepping and differential flatness, Automatica, 45, 5, 1182-1194 (2009) · Zbl 1162.93016
[26] Meurer, T.; Kugi, A., Trajectory planning for boundary controlled parabolic PDEs with varying parameters on higher-dimensional spatial domains, IEEE Trans. Automat. Control, 54, 8, 1854-1868 (2009) · Zbl 1367.93278
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