Abstract
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.
Highlights
Analogous to the adaptive control where the estimation/cancellation strategy is adopted, the active disturbance rejection control (ADRC) can handle much more general disturbance that is estimated in real time by an extended state observer (ESO) and is compensated in the closed-loop by an ESO-based feedback control, which makes the control energy significantly reduced in practice [38]
When the boundary control ux(1, t) = U (t) of system (1.1) is replaced by ux(1, t) = U (t) + d(t) with d(t) being the disturbance, we can ask the second question: Problem II: Can we propose an output feedback control law to system (1.1) by rejecting the disturbance d(t) and maintaining the stability of the closed-loop system simultaneously?
From properties of the Mittag-Leffler function, it is easy to conclude that both u(·, t) L2(0,1) → +∞ and ut(·, t) L2(0,1) → +∞ as t → ∞
Summary
We consider boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Neumann boundary control governed by. Analogous to the adaptive control where the estimation/cancellation strategy is adopted, the ADRC can handle much more general disturbance that is estimated in real time by an extended state observer (ESO) and is compensated in the closed-loop by an ESO-based feedback control, which makes the control energy significantly reduced in practice [38] Very recently, this emerging control technology has been shown to be highly efficient in stabilizing the one dimensional partial differential equations (PDEs) [7, 12], multi-dimensional PDEs [13, 39] and stochastic differential equations [14, 15], which are all subject to external disturbance, among many others.
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