In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.
Keywords: Diffusion-wave equation, disturbance rejection, feedback stabilization, boundary control, backstepping method
@article{COCV_2022__28_1_A7_0,
author = {Zhou, Hua-Cheng and Wu, Ze-Hao and Guo, Bao-Zhu and Chen, Yangquan},
title = {Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022003},
mrnumber = {4368390},
zbl = {1482.35263},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022003/}
}
TY - JOUR AU - Zhou, Hua-Cheng AU - Wu, Ze-Hao AU - Guo, Bao-Zhu AU - Chen, Yangquan TI - Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022003/ DO - 10.1051/cocv/2022003 LA - en ID - COCV_2022__28_1_A7_0 ER -
%0 Journal Article %A Zhou, Hua-Cheng %A Wu, Ze-Hao %A Guo, Bao-Zhu %A Chen, Yangquan %T Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022003/ %R 10.1051/cocv/2022003 %G en %F COCV_2022__28_1_A7_0
Zhou, Hua-Cheng; Wu, Ze-Hao; Guo, Bao-Zhu; Chen, Yangquan. Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 7. doi: 10.1051/cocv/2022003
[1] , and , Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 2951–2957. | MR | Zbl | DOI
[2] and , Stabilization and Control of Fractional Order Systems: a Sliding Mode Approach. Vol. 317 of Lecture Notes in Electrical Engineering. Springer, Cham (2015). | MR | Zbl
[3] , , and , Period input estimation for linear periodic systems: Automotive engine applications. Automatica 43 (2007) 971–980. | MR | Zbl | DOI
[4] , and , Observer-based output feedback control for a boundary controlled fractional reaction diffusion system with spatially-varying diffusivity. IET Control Theory Appl. 12 (2018) 1561–1572. | MR | Zbl | DOI
[5] and , UDE-based robust boundary control for an unstable parabolic PDE with unknown input disturbance. Automatica 93 (2018) 363–368. | MR | Zbl | DOI
[6] , , and , Adaptive Mittag-Leffler stabilization of commensurate fractional-order nonlinear systems. 53rd IEEE CDC, Los Angeles, California, USA (2014) 15–17.
[7] and , A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance. IEEE Trans. Autom. Control 62 (2017) 3774–3787. | MR | Zbl | DOI
[8] , Active disturbance rejection control for nonlinear fractional-order systems. Internat. J. Robust Nonlinear Control 26 (2016) 876–892. | MR | Zbl | DOI
[9] , and , Mittag-Leffler convergent backstepping observers for coupled semilinear subdiffusion systems with spatially varying parameters. Systems Control Lett. 122 (2018) 86–92. | MR | Zbl | DOI
[10] and , Regional output feedback stabilization of semilinear time-fractional diffusion systems in a parallelepipedon with control constraints. Internat. J. Robust Nonlinear Control 30 (2020) 3639–3652. | MR | Zbl | DOI
[11] , , and , Mittag-Leffler Functions Related Topics and Applications. Springer, Heidelberg (2014). | MR | Zbl | DOI
[12] and , Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input. IEEE Trans. Autom. Control 58 (2013) 1269–1274. | MR | Zbl | DOI
[13] and , The active disturbance rejection control to stabilization for multi-dimensional wave equation with boundary control matched disturbance, IEEE Trans. Autom. Control 60 (2015) 143–157. | MR | Zbl | DOI
[14] , and , Active disturbance rejection control approach to output-feedback stabilization of a class of uncertain nonlinear systems subject to stochastic disturbance. IEEE Trans. Automat. Control 61 (2016) 1613–1618. | MR | Zbl | DOI
[15] and , Active disturbance rejection control approach to output-feedback stabilization of lower triangular nonlinearsystems with stochastic uncertainty. Internat. J. Robust Nonlinear Control 27 (2017) 2773–2797. | MR | Zbl | DOI
[16] and , Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance. SIAM J. Control Optim. 51 (2013) 1679–1706. | MR | Zbl | DOI
[17] , From PID to active disturbance rejection control. IEEE Trans. Indust. Electron. 56 (2009) 900–906. | DOI
[18] , , and , Asymptotical stability of fractional order systems with time delay via an integral inequality. IET Control Theory Appl. 12 (2018) 1748–1754. | MR | Zbl | DOI
[19] , and , Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). | MR | Zbl
[20] , , and , Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45 (2009) 1965–1969. | MR | Zbl | DOI
[21] , and , Boundary stabilization and disturbance rejection for time fractional order diffusion-wave equations. Nonlinear Dynam. 38 (2004) 339–354. | MR | Zbl | DOI
[22] , , and , Fractional order boundary stabilization of a time-fractional wave equation, in Fractional Derivatives an Their Applications, edited by , , and . UBooks Augsburg, Germany (2005).
[23] , , , and , Fractional-order boundary control of fractional wave equation with delayed boundary measurement using smith predictor. 43rd IEEE CDC, Atlantis, Bahamas (2004).
[24] , Analytical and Numerical Methods for Volterra Equations. SIAM Studies in Applied Mathematics, Philadelphia, PA (1985). | MR | Zbl
[25] and , Stabilization of one-dimensional wave equation with nonlinear boundary condition subject to boundary control matched disturbance. IEEE Trans. Automat. Control 64 (2019) 3068–3073. | MR | Zbl | DOI
[26] and , On the lack of controllability of fractional in time ODE and PDE. Math. Control Signals Systems 28 (2016) Art. 10 21. | MR | Zbl
[27] and , Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38 (2016) 2699–2724. | MR | Zbl | DOI
[28] , Fractional diffusive waves in viscoelastic solids, in Nonlinear Waves in Solids, edited by , . ASME/AMR, Fairfield (1995) 93–7.
[29] , Stability results for fractional differential equations with applications to control processing, IMACS/IEEE-SMC Multiconference, Symposium on Control, Optimization and Supervision (CESA) (1996) 963–968.
[30] and , Boundary fractional derivative control of the wave equation. IEEE Trans. Automat. Control 40 (1995) 378–382. | MR | Zbl | DOI
[31] and , The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Mathematics in Science and Engineering, vol. III. Academic Press, New York and London (1974). | MR | Zbl
[32] , Fractional Differential Equations. Academic Press, San Diego (1999). | MR
[33] and , Fractional diffusion and wave equations. J. Math. Phys. 30 (1989) 134–144. | MR | Zbl | DOI
[34] , Comparing Fréchet and positive stable laws. Electr. J. Probab. 19 (2014) 1–25. | MR | Zbl
[35] and , Adaptive Control of Parabolic PDEs. Princeton University Press, Princeton, NJ (2010). | MR | Zbl | DOI
[36] and , Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54 (2005) 613–625. | MR | Zbl | DOI
[37] , , and , Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance. Automatica 52 (2015) 23–34. | MR | Zbl | DOI
[38] and , An energy saving, factory-validated disturbance decoupling control design for extrusion processes, The 10th World Congress on Intelligent Control and Automation. IEEE, Piscataway, NJ (2012) 2891–2896. | DOI
[39] and , Unknown input observer design and output feedback stabilization for multi-dimensional wave equation with boundary control matched uncertainty. J. Differ. Equ. 263 (2017) 2213–2246. | MR | Zbl | DOI
[40] , , and , Mittag-Leffler stabilization for an unstable time fractional anomalous diffusion equation with boundary control matched disturbance. Internat. J. Robust Nonlinear Control 29 (2019) 4384–4401. | MR | Zbl | DOI
[41] and , Boundary feedback stabilization for an unstable time fractional reaction diffusion equations. SIAM J. Control Optim. 56 (2018) 75–101. | MR | Zbl | DOI
Cité par Sources :
