Backstepping-sliding-mode-based prescribed-time control for a class of nonlinear systems with disturbances

Litong Zhou; Lichao Feng; Nan Ji; Ruicheng Zhang; Litong Zhou; Lichao Feng; Nan Ji; Ruicheng Zhang

doi:10.3934/math.20251013

AIMS Mathematics

2025, Volume 10, Issue 10: 22791-22816. doi: 10.3934/math.20251013

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Backstepping-sliding-mode-based prescribed-time control for a class of nonlinear systems with disturbances

1.
College of Electrical Engineering, North China University of Science and Technology, Tangshan 063210, China
2.
College of Science, North China University of Science and Technology, Tangshan 063210, China
3.
Hebei Key Laboratory of Data Science and Application, North China University of Science and Technology, Tangshan 063210, China

Received: 19 July 2025 Revised: 17 September 2025 Accepted: 24 September 2025 Published: 09 October 2025
MSC : 93C10, 93D05

This article addresses the prescribed-time stability (PTS) problem for a class of nonlinear strict-feedback systems subject to unknown disturbances. Herein, a novel backstepping-sliding-mode (BSM) composite control structure is proposed: a dynamic time-varying sliding-mode surface is integrated into the final step of the backstepping design, effectively suppressing matched disturbances and simplifying the implementation complexity. Specifically speaking, 1) for systems with known nonlinear terms, time-varying gains are directly embedded within the backstepping virtual control laws to achieve PTS; 2) for the case of unknown nonlinear terms, by combining adaptive laws with the BSM framework, the controlled system is guaranteed to converge within the prescribed-time and maintain globally stability thereafter. Moreover, the prescribed time $ {T}_{p} $ is designer-prespecified and does not depend on initial conditions or controller gains. Finally, two numerical simulation examples validate the convergence performance and robustness of the proposed approaches.
- nonlinear strict-feedback systems,
- sliding-mode control,
- backstepping method,
- adaptive control,
- prescribed-time control
Citation: Litong Zhou, Lichao Feng, Nan Ji, Ruicheng Zhang. Backstepping-sliding-mode-based prescribed-time control for a class of nonlinear systems with disturbances[J]. AIMS Mathematics, 2025, 10(10): 22791-22816. doi: 10.3934/math.20251013

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Abstract

This article addresses the prescribed-time stability (PTS) problem for a class of nonlinear strict-feedback systems subject to unknown disturbances. Herein, a novel backstepping-sliding-mode (BSM) composite control structure is proposed: a dynamic time-varying sliding-mode surface is integrated into the final step of the backstepping design, effectively suppressing matched disturbances and simplifying the implementation complexity. Specifically speaking, 1) for systems with known nonlinear terms, time-varying gains are directly embedded within the backstepping virtual control laws to achieve PTS; 2) for the case of unknown nonlinear terms, by combining adaptive laws with the BSM framework, the controlled system is guaranteed to converge within the prescribed-time and maintain globally stability thereafter. Moreover, the prescribed time $ {T}_{p} $ is designer-prespecified and does not depend on initial conditions or controller gains. Finally, two numerical simulation examples validate the convergence performance and robustness of the proposed approaches.

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