Adaptive stabilisation via passive identifier for an ODE system cascaded by coupled parabolic equations with serious uncertainties

Abstract

Stabilisation via adaptive feedback is investigated for an ODE system cascaded by coupled parabolic equations. The remarkable characteristic of the system is the presence of serious uncertainties since no known or bounded intervals are needed for the unknown parameters. Besides, strong coupling exists in the PDE subsystem which brings essential difference compared with those of the related literature. For this, a passive identifier is skillfully constructed firstly, and in turn adaptive laws are given to compensate the serious parametric unknowns. Then, by constructing a backstepping transformation and its inverse one, the identifier joint with the original system is transformed into a new system. From the new system, an adaptive state-feedback controller is designed which ensures that all the states of the closed-loop system are bounded and particularly those of the original system converge to zero. Finally, a numerical example is provided to validate the proposed algorithm.

Keywords:

• Cascaded PDE–ODE system
• passive identifier
• stabilisation
• adaptive control

Disclosure statement

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

Notes

1 Throughout the paper, we suppress the time or space dependence for some functions where no confusion will arise so as to reduce the notation burden.

Additional information

Funding

This work was supported by the National Natural Science Foundations of China [grant numbers 61773332, 61673332, 61873146].