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Eigenvalue Mapping-based Discretization of the Generalized Super-Twisting Algorithm

Ningning Ding

TL;DR

This work tackles the challenge of discretizing the continuous-time $GSTA$ without inducing discretization chattering and with high steady-state accuracy. It extends eigenvalue mapping to the complex domain and introduces four new eigenvalue mapping functions (including Gudermannian, ReLU, and Tanh) plus a hybrid EMF, providing explicit recursions for a robust, chattering-free discrete-time $GSTA$ implementation. The proposed scheme broadens the parameter space and reduces sensitivity to gain overestimation, with simulations showing superior performance over conventional discretization methods under both disturbance-free and disturbed conditions. Overall, the approach offers a practical, high-accuracy framework for digital sliding-mode control with improved robustness and tunability.

Abstract

In this paper, an eigenvalue mapping-based discretization method is applied to discretize the generalized super-twisting algorithm. The existing eigenvalue mapping is extended to the complex domain which greatly enlarges the range of parameter selection. Furthermore, we present the clue to find new eigenvalue mapping functions (EMFs). One new hybrid EMF and three brand-new EMFs are proposed in this paper. In contrast to the conventional methods, the proposed discretization method totally avoids the discretization chattering and the control precision is enhanced in terms of the steady-state error. Besides, the control precision is insensitive to the overestimation of the control gains, which benefits the gain tuning of the controller in practice. Simulation examples verify the effectiveness and superiority of the proposed discretization methodology.

Eigenvalue Mapping-based Discretization of the Generalized Super-Twisting Algorithm

TL;DR

This work tackles the challenge of discretizing the continuous-time without inducing discretization chattering and with high steady-state accuracy. It extends eigenvalue mapping to the complex domain and introduces four new eigenvalue mapping functions (including Gudermannian, ReLU, and Tanh) plus a hybrid EMF, providing explicit recursions for a robust, chattering-free discrete-time implementation. The proposed scheme broadens the parameter space and reduces sensitivity to gain overestimation, with simulations showing superior performance over conventional discretization methods under both disturbance-free and disturbed conditions. Overall, the approach offers a practical, high-accuracy framework for digital sliding-mode control with improved robustness and tunability.

Abstract

In this paper, an eigenvalue mapping-based discretization method is applied to discretize the generalized super-twisting algorithm. The existing eigenvalue mapping is extended to the complex domain which greatly enlarges the range of parameter selection. Furthermore, we present the clue to find new eigenvalue mapping functions (EMFs). One new hybrid EMF and three brand-new EMFs are proposed in this paper. In contrast to the conventional methods, the proposed discretization method totally avoids the discretization chattering and the control precision is enhanced in terms of the steady-state error. Besides, the control precision is insensitive to the overestimation of the control gains, which benefits the gain tuning of the controller in practice. Simulation examples verify the effectiveness and superiority of the proposed discretization methodology.
Paper Structure (12 sections, 39 equations, 6 figures)

This paper contains 12 sections, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. PROPOSED DISCRETIZATION SCHEME
  4. SIMULATION EXAMPLES
  5. Simulations without Disturbance in Time-Domain
  6. Simulations with Disturbance in Time-Domain
  7. Steady-State Accuracy when Varying the Sampling Time
  8. Steady-State Accuracy when Varying the Controller Parameters
  9. CONCLUSIONS
  10. Limitation of the matching-explicit EMF when the sampling time approximates zero
  11. Limitation of the tanh EMF when the sampling time approximates zero
  12. Limitation of the Gudermannian EMF when the sampling time approximates zero

Figures (6)

  • Figure 1: Curve of eigenvalue functions. The sampling time $h$ is 0.05s
  • Figure 2: Comparison of the convergence rate of the state variable $\left| {{x}_{1}} \right|$. The parameters are ${{k}_{1}}=1.5\sqrt{\Lambda}$, ${{k}_{2}}=2.2\Lambda$, ${{\mu }_{1}}=1$, ${{\mu}_{2}}=1$ and $h=0.05$.
  • Figure 3: The system states $x_{1,k}$, $x_{2,k}$, the control input $u_{k}$ and the variable $\nu_{k}$ of the system (4) in the time domain in the unbounded disturbance case.
  • Figure 4: The system states $x_{1,k}$, $x_{2,k}$, the control input $u_{k}$ and the variable $\nu_{k}$ of the system (4) in the time domain in the bounded disturbance case.
  • Figure 5: Precision w.r.t. an increase of sampling time.
  • ...and 1 more figures