Backstepping control of ODE/wave PDE cascaded systems with dynamic boundary conditions

Abstract

We consider backstepping control for ODE/wave PDE cascaded systems with dynamic boundary conditions. The ODE state is driven by the uncontrolled boundary of the wave PDE. A backstepping compensator is designed such that the closed-loop system is globally asymptotically stable. The controller design and stability analysis are based on coordinate and backstepping transformations. Changes of coordinate recast the original system into a coupled $2\times 2$ first-order hyperbolic system with spatially-varying coefficients cascading into an ODE. Backstepping transformations are exploited to design a compensator for the resulting coupled system. The validity of the proposed design is illustrated by controlling a bit of the oil well drilling system to operate at a constant uniform rotary speed.

Keywords:

• Cascaded system
• wave dynamics
• predictor control
• backstepping transformations

Acknowledgements

The authors thank the anonymous reviewers for their many helpful suggestions. This work was supported by the National Natural Science Foundation of China (grant nos. 61773350, 62072164), the Natural Science Foundation of Zhejiang Province (grant nos. LY17F030001), and Science Fund for Distinguished Young Scholars of Hubei Province of China, Grant/Award Number: 2017CFA034.

Disclosure statement

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

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 61773350, 62072164], the NaturalScience Foundation of Zhejiang Province [grant number LY17F030001], and Outstanding young talents project of Natural Science Foundation of Hubei Province, China [grant/award number 2017CFA034].