Dynamic stabilisation for an Euler–Bernoulli beam equation with boundary control and matched nonlinear disturbance

Abstract

In this paper, we are concerned with dynamic stabilisation for a one-dimensional Euler–Bernoulli beam equation with boundary moment control and matched nonlinear uncertain disturbance. In the case of no disturbance, we show that a boundary feedback control law exponentially stabilises the system and Riesz basis generation holds for the closed-loop system. The well-posedness of the system in the sense of Salamon-Weiss, which is essentially important for the design of observer, is verified. We design an infinite-dimensional disturbance estimator, which doesn't need slow variation or high gain or boundedness of the derivation of the disturbance, to estimate the total disturbance. Based on the disturbance estimator, we design an output feedback control law. The Riesz basis generation and exponential stability of a couple system including the original equation is proved. Moreover, the boundedness of the closed-loop system is verified. Some numerical simulations are presented to illustrate the results.

Keywords:

• Euler–Bernoulli beam equation
• well-posedness
• Riesz basis
• exponential stabilisation
• disturbance estimator

2010 Mathematics Subject Classifications:

• 37L15
• 93D15
• 93B51
• 93B52

Acknowledgements

The author would like to thank the anonymous referees for their very careful reading and constructive suggestions which improved substantially the manuscript. *

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Science Foundation of Shaanxi Province [grant numbers 2018JM1051, 2014JQ1017] and the National Natural Science Foundation of China [grant number 11301412].