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Euler implicit time-discretization of multivariable sliding-mode controllers.

Mohammad Rasool Mojallizadeh1, Félicien Bonnefoy2, Franck Plestan3

  • 1Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France; Nantes Université, École Centrale Nantes, CNRS, LHEEA, UMR 6598, F-44000 Nantes, France.

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Summary

This study introduces an implicit Euler time-discretization for multivariable sliding-mode controllers to eliminate numerical chattering. The developed algorithm effectively suppresses chattering and ensures finite-time convergence in complex control systems.

Keywords:
Backward discretizationEuler implicit discretizationMultivariable generalized equationMultivariable sliding-mode controlMultivariable systems

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Area of Science:

  • Control Systems Engineering
  • Numerical Analysis
  • Applied Mathematics

Background:

  • Sliding-mode control (SMC) is effective but suffers from numerical chattering.
  • Conventional explicit time-discretization methods can exacerbate chattering in multivariable systems.
  • Existing methods often require modifying the continuous-time control law to mitigate chattering.

Purpose of the Study:

  • To develop an Euler implicit time-discretization for multivariable sliding-mode controllers.
  • To address the numerical chattering problem without altering the continuous-time control law.
  • To propose a novel algorithm for solving the implicit discretization equations.

Main Methods:

  • Developed an Euler implicit time-discretization for multivariable sliding-mode controllers.
  • Proposed a novel algorithm to solve the resulting multivariable generalized equations.
  • Analyzed properties such as finite-time convergence and chattering suppression.
  • Implemented the controller on a digital processor for a six-input, six-output system.

Main Results:

  • The implicit discretization leads to a multivariable generalized equation solvable by the proposed algorithm.
  • Finite-time convergence, gain insensitivity, and chattering suppression were analytically demonstrated.
  • Numerical simulations confirmed the effectiveness of the implicit method over explicit discretization.
  • Successful implementation on a six-component thrust generator demonstrated practical applicability.

Conclusions:

  • The Euler implicit time-discretization offers a robust solution to chattering in multivariable sliding-mode control.
  • The developed algorithm provides a practical means for synthesizing implicit sliding-mode controllers.
  • This approach maintains the advantages of sliding-mode control while enhancing numerical stability.