Observer‐based output feedback control for a boundary controlled fractional reaction diffusion system with spatially‐varying diffusivity

Chen, Juan; Cui, Baotong; Chen, YangQuan

doi:10.1049/iet-cta.2017.1352

Cited by 30 publications

(28 citation statements)

References 36 publications

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“…The results obtained here could provide some insights into the qualitative analysis of the design of fractional PDEs with boundary disturbance. The approach is potentially usable for treating other fractional PDEs with boundary control matched disturbance, such as the following model

({, \begin{matrix} array_{0}^{C} D_{t}^{α} w (x, t) = w_{x x} (x, t) + λ (x) w (x, t), x \in (0, 1), t \geq 0, \\ array w (0, t) = 0, w (1, t) = u (t) + d (t), t \geq 0, \\ array w (x, 0) = w_{0} (x), 0 \leq x \leq 1, \end{matrix})

where w ( x , t ) is the state, u ( t ) is the control input, and d ( t ) is the unknown external disturbance, as well as the spatially‐varying diffusion coefficient for fractional reaction diffusion model in the work of Chen et al, which are of great interest and will be considered in our future works. Finally, the feedback in this paper is about the full state feedback, and a more interesting future work is on output‐feedback stabilization for fractional PDEs, based certainly on the state‐feedback stabilization results developed in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…where w(x, t) is the state, u(t) is the control input, and d(t) is the unknown external disturbance, as well as the spatially-varying diffusion coefficient for fractional reaction diffusion model in the work of Chen et al, 7 which are of great interest and will be considered in our future works. Finally, the feedback in this paper is about the full state feedback, and a more interesting future work is on output-feedback stabilization for fractional PDEs, based certainly on the state-feedback stabilization results developed in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…Actually, when = 1, we can write system (5) as . v(·, t) = Av(·, t) + Bd(t), where the operator A is given by (7) and B is given by B = 1 , where a is the Dirac pulse at x = a. It is well known that A generates an exponentially stable C 0 -semigroup e At and B is admissible for e At .…”

Section: Disturbance Estimator Designmentioning

confidence: 99%

“…The time‐fractional anomalous diffusion equations can better characterize a subdiffusion process that describes the continuous time random walks phenomenon such as the particles to jump at fixed‐time intervals with an incorporating waiting times . When there is no disturbance, the stabilization of time‐fractional anomalous diffusion equations has been investigated in the works of Zhou and Guo and Chen et al To the best of authors' knowledge, stabilization of time‐fractional anomalous diffusion equations with boundary disturbance has not been addressed so far. In this paper, we adopt the ADRC and the backstepping approach to achieve the Mittag‐Leffler stability for a class of factional PDEs with disturbance.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control matched disturbance

Zhou

Guo

et al. 2019

Intl J Robust & Nonlinear

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Summary This paper addresses the Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control subject to the control matched disturbance. The active disturbance rejection control (ADRC) approach is adopted for developing the control law. A state‐feedback scheme is designed to estimate the disturbance by constructing two auxiliary systems: One is to separate the disturbance from the original system to a Mittag‐Leffler stable system and the other is to estimate the disturbance finally. The proposed control law compensates the disturbance using its estimation and stabilizes system asymptotically. The closed‐loop system is shown to be Mittag‐Leffler stable and the constructed auxiliary systems in the closed loop are proved to be bounded. This is the first time for ADRC to be applied to a system described by the fractional partial differential system without using the high gain.

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({, \begin{matrix} array_{0}^{C} D_{t}^{α} w (x, t) = w_{x x} (x, t) + λ (x) w (x, t), x \in (0, 1), t \geq 0, \\ array w (0, t) = 0, w (1, t) = u (t) + d (t), t \geq 0, \\ array w (x, 0) = w_{0} (x), 0 \leq x \leq 1, \end{matrix})

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Disturbance Estimator Designmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control matched disturbance

Zhou

Guo

et al. 2019

Intl J Robust & Nonlinear

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“…Due to the great role of the backstepping method [1], [2] in controlling/stability of integer-order PDEs [3], researchers concern this method for time fractional PDEs. There are some representative results [4]- [7] on Mittag-Leffler stability/stabilization of time fractional PDEs. Some other work related to control of more general fractional differential equations has also emerged in [8]- [10].…”

Section: Introduction a Previous Work And Motivationmentioning

confidence: 99%

Stabilization and Stability Robustness of Coupled Non-Constant Parameter Time Fractional PDEs

et al. 2019

Self Cite

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This paper considers observer-based output feedback stabilization and stability robustness against small diffusivity perturbations of coupled time fractional partial differential equations (PDEs) with space-dependent (non-constant) parameters. Herein, the plant is equipped with the only available measurement at x = 0 and actuation at x = 1. By backstepping transformation, the well-posedness of the kernel matrix PDE and the observer gains are obtained. Then an output feedback controller is introduced and the Mittag-Leffler stability of the closed-loop system is proved by the fractional Lyapunov method. Robustness analysis of diffusion coefficients uncertainty (with a small perturbation in the diffusion coefficient) is also provided. The output feedback stabilization of the closed-loop system is tested by a numerical example. INDEX TERMS Coupled time fractional PDEs, output feedback boundary stabilization, uncertainty analysis, space-dependent coefficients.

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Boundary output feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems

Cai

Zhou

Fan

et al. 2021

Asian Journal of Control

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This paper studies the boundary feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems with non‐collocated or collocated outputs in the case that the system state is unmeasurable. By employing the backstepping method, we introduce a target system and analyze its Mittag–Leffler stability. For each pair of sides of the region boundary, assuming that system state is hinged on one side, while the controller is designed on the opposite side. These boundary controllers work together to achieve the state feedback Mittag–Leffler stabilization of the considered system. In addition, for both kinds of outputs, feedback controllers are also established to achieve the asymptotical stability of the corresponding closed‐loop system. Two numerical experiments are carried out to illustrate our results.

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Observer‐based output feedback control for a boundary controlled fractional reaction diffusion system with spatially‐varying diffusivity

Cited by 30 publications

References 36 publications

Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control matched disturbance

Mittag‐Leffler stabilization for an unstable time‐fractional anomalous diffusion equation with boundary control matched disturbance

Stabilization and Stability Robustness of Coupled Non-Constant Parameter Time Fractional PDEs

Boundary output feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems

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