Boundary stabilisation via event-triggered output-feedback for a class of 2 × 2 hyperbolic systems

Abstract

This paper is devoted to the boundary stabilisation via event-triggered output-feedback for a class of $2\times 2$ hyperbolic systems. Different from the closely related works, the system parameters are spatially varying and only one boundary measurement is available for feedback, which brings essential difficulties to the analysis and synthesis of event-triggered control. To solve the problem, by using the time-varying strategy and the infinite-dimensional backstepping technique, a novel event-triggered output-feedback controller is designed for the systems. Particularly, a time-varying event-triggering mechanism, with an extra decaying signal introduced into the threshold, is determined. It is rigorously shown that the designed event-triggered controller can guarantee that all the states of the resulting closed-loop system exponentially converge to zero in the sense of $L^2$ norm while no Zeno phenomenon occurs. Two simulation examples are given to illustrate the effectiveness of the established boundary control strategy.

Keywords:

• Hyperbolic systems
• event-triggered control
• time-varying strategy
• output-feedback stabilisation

Disclosure statement

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

Notes

1 To reduce the notational burden, the arguments x, t and ζ are omitted in this paper if there is no confusion.

2 For any function $u(x)$ defined on $[0,1],$ the $L^2$ norm of $u(x)$ is simply denoted by $\Vert u(\cdot ){\Vert }_{L^2}=\sqrt{{\int }_0^1{u}^2(x)\mathrm{d}x}$.

Additional information

Funding

This work was partly supported by National Natural Science Foundation of China (Grant Nos. 61821004, 61873146, 61973186, 61703237 and 61773332), by the Key and Development Plan of Shandong Province (Grant No. 2019JZZY010433), by the Taishan Scholars Climbing Program of Shandong Province, and by the Fundamental Research Funds of Shandong University.

Notes on contributors

Xia Li

Xia Li received the B.S. degree in School of Mathematical Sciences, Qufu Normal University, in 2013. She is currently a doctoral student in School of Control Science and Engineering, Shandong University. Her current research interests include adaptive control and event-triggered control.

Yungang Liu

Yungang Liu received the Ph.D. degree in Control Theory and Control Engineering from Shanghai Jiao Tong University, Shanghai, China, in 2000.

He is now a Changjiang Scholar Chair Professor with the School of Control Science and Engineering, Shandong University, Jinan, China. He is also the Director of the Institute of Artificial Intelligence and Systems & Control at the same university. His current research interests include stochastic control, nonlinear control design and system analysis, distributed parameter system control, adaptive control and applications in power systems, and big data & artificial intelligence.

Dr. Liu was a recipient of the Guan Zhaozhi Award in the Chinese Control Conference in 2004, the National Outstanding Youth Science Foundation of China in 2013, and the Second Prize of the National Natural Science Award of China in 2015. He currently serves as Associate Editor for seven scientific journals.

Fengzhong Li

Fengzhong Li received the Ph.D. degree in control theory and control engineering from Shandong University, Jinan, China, in 2016. He is currently an Associate Professor with Shandong University. His current research interests include stochastic control, nonlinear control, prescribed-performance control, and event-triggered control.

Dr. Li was a recipient of the Excellent Doctoral Dissertation Award of the Chinese Association of Automation in 2018.

Jian Li

Jian Li received the Ph.D. degree in control theory and control engineering from Shandong University, Jinan, China, in 2013. He is currently an associate professor in Yantai University. His current research interests include controls of distributed parameter systems and mechanical systems.