Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning-Based Robust Fixed-Time Terminal Sliding Mode Control

Chaimae El Mortajinea, Moussa Labbadib, Adnane Saoudc, Mostafa Bouzia

TL;DR

This paper addresses robust fixed-time stabilization for nonlinear systems with partially unknown dynamics by developing an integral fixed-time sliding mode controller (FxT-SMC) featuring a non-singular terminal sliding variable. It first proves FxT stability for known dynamics, then extends to learning-based dynamics using Gaussian Process (GP) regression to estimate drift terms with probabilistic error bounds, yielding a learning-based FxT controller. The approach delivers explicit settling-time guarantees with contributions including simplified parameter tuning and a clear separation between the reaching phase and the fixed-time convergence. A PMSM simulation demonstrates that increased online data improves model accuracy, reduces settling times, and yields smoother control—the results validating the practical viability of GP-assisted FxT-SMC for time-critical nonlinear systems.

Abstract

In this paper, we develop and analyze an integral fixed-time sliding mode control method for a scenario in which the system model is only partially known, utilizing Gaussian processes. We present two theorems on fixed-time convergence. The first theorem addresses the fully known system model, while the second considers situations where the system's drift is approximated utilizing Gaussian processes (GP) for approximating unknown dynamics. Both theorems establish the global fixed-time stability of the closed-loop system. The stability analysis is based on a straightforward quadratic Lyapunov function. Our proposed method outperforms an established adaptive fixed-time sliding mode control approach, especially when ample training data is available.

Learning-Based Robust Fixed-Time Terminal Sliding Mode Control

TL;DR

This paper addresses robust fixed-time stabilization for nonlinear systems with partially unknown dynamics by developing an integral fixed-time sliding mode controller (FxT-SMC) featuring a non-singular terminal sliding variable. It first proves FxT stability for known dynamics, then extends to learning-based dynamics using Gaussian Process (GP) regression to estimate drift terms with probabilistic error bounds, yielding a learning-based FxT controller. The approach delivers explicit settling-time guarantees with contributions including simplified parameter tuning and a clear separation between the reaching phase and the fixed-time convergence. A PMSM simulation demonstrates that increased online data improves model accuracy, reduces settling times, and yields smoother control—the results validating the practical viability of GP-assisted FxT-SMC for time-critical nonlinear systems.

Abstract

In this paper, we develop and analyze an integral fixed-time sliding mode control method for a scenario in which the system model is only partially known, utilizing Gaussian processes. We present two theorems on fixed-time convergence. The first theorem addresses the fully known system model, while the second considers situations where the system's drift is approximated utilizing Gaussian processes (GP) for approximating unknown dynamics. Both theorems establish the global fixed-time stability of the closed-loop system. The stability analysis is based on a straightforward quadratic Lyapunov function. Our proposed method outperforms an established adaptive fixed-time sliding mode control approach, especially when ample training data is available.

Paper Structure

This paper contains 10 sections, 5 theorems, 46 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fixed- and finite-time stability
  4. Problem statement
  5. Fixed-time tracking with know dynamics
  6. Learning-Based Fixed-Time Control with Gaussian Process Regression
  7. Gaussian Process Regression
  8. Fixed-time control based on GPs
  9. Simulations
  10. Conclusions

Key Result

Lemma 1

(see polyakov2011nonlinear): If there exists a continuously differentiable positive definite radially unbounded function $V: \mathbb{R}^n \to \mathbb{R}^+$ such that where $V$ is called the Lyapuov function and $x \in \mathbb{R}^n$, $c_i> 0$, and $0 < h_1 < 1 < h_2$, then system eq:fz is globally FxT stable with a settling time bounded by

Figures (6)

  • Figure 1: State variables with $N= 5$ data.
  • Figure 2: Sliding variables with $N= 5$ data.
  • Figure 3: Input control with $N= 5$ data.
  • Figure 4: The unknown function estimations with $N= 5$ data.
  • Figure 5: State variables with $N= 50$ data.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • proof
  • Remark III.1
  • Remark III.2
  • Lemma 4
  • proof
  • ...and 3 more