Robustness to In-Domain Viscous Damping of a Collocated Boundary Adaptive Feedback Law for an Antidamped Boundary Wave PDE

Adaptive output‐feedback control for damped wave equations with unknown damping coefficient and corrupted boundary measurement

“…Therefore, the convergence of w ( x , t ) and cannot be derived. Moreover, it can be verified that when the parameters , 's, and ϑ i 's are negative, the control method presented in the article still holds, although when , the coefficient of w t ( L , t ) in the third equation in system (1) becomes a positive constant, called the antidamping boundary coefficient 4 …”

“…Over the past decades, output‐feedback control has been intensively investigated for the systems described by second‐order hyperbolic partial differential equations (PDEs) (see, eg, References 1‐9), which can describe many practical processes, such as the propagation of waves in an elastic string, 1,5 the vibration of a drillstring, 7 and an axially moving accelerated string 9 . However, the corresponding output‐feedback theory is far from mature, particularly for the systems with uncertainties.…”

“…In recent years, much effort has been devoted to the output‐feedback control for the second‐order hyperbolic PDE systems with disturbances or unknown parameters (see, eg, References 2‐4,8‐11), specifically, by adaptive technique. References 3 and 4 investigated the stabilization of the systems with unknown constant parameter and disturbances, respectively. In References 8, 10, and 11, sliding mode control and active disturbance rejection control are, respectively, utilized to solve the stabilization for the systems with input disturbances.…”

“…Based on this, in recent years, infinite‐dimensional disturbance estimator method is proposed to handle the disturbances 2,9 . It is necessary to point out that, although disturbances and unknown constant parameter are allowed in References 2,3,8‐11, and 4, respectively, all parameters are required to be exactly known in References 2,3,8‐11, and no disturbance is allowed in Reference 4. To the authors' knowledge, no report has been found on the output‐feedback control for a second‐order hyperbolic system simultaneously with a disturbance and an unknown damping coefficient.…”

“…Remark that, in References 9,19‐21, all states along the space domain are required to be available for feedback, which is usually difficult or even impossible in practice since an infinite number of sensors are needed. Although the output‐feedback control is considered in References 1‐9, the proposed static/observer‐based control schemes do not allow any output disturbance.…”