Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proper Implicit Discretization of the Super-Twisting Controller -- without and with Actuator Saturation

Richard Seeber, Benedikt Andritsch

TL;DR

This work addresses the discrete-time implementation of the high order Super-Twisting Controller for plants with disturbances of bounded slope and actuator constraints. It introduces a proper implicit discretization based on a modified sliding variable, derives explicit update formulas, and extends the approach to a conditioned STC to handle saturation while preventing windup. The authors provide complete stability proofs with simple conditions and show that the proposed scheme achieves the best possible worst-case disturbance rejection bound $|x| \le L T^2$, supported by numerical simulations that outperform existing discretizations. The results offer a robust, windup-free, finite-time disturbance rejection framework for discrete-time higher order sliding mode control with practical actuator limits, and open paths for extensions to other higher order laws.

Abstract

The discrete-time implementation of the super-twisting sliding mode controller for a plant with disturbances with bounded slope, zero-order hold actuation, and actuator constraints is considered. Motivated by restrictions of existing implicit or semi-implicit discretization variants, a new proper implicit discretization for the super-twisting controller is proposed. This discretization is then extended to the conditioned super-twisting controller, which mitigates windup in presence of actuator constraints by means of the conditioning technique. It is proven that the proposed controllers achieve best possible worst-case performance subject to similarly simple stability conditions as their continuous-time counterparts. Numerical simulations and comparisons demonstrate and illustrate the results.

Proper Implicit Discretization of the Super-Twisting Controller -- without and with Actuator Saturation

TL;DR

This work addresses the discrete-time implementation of the high order Super-Twisting Controller for plants with disturbances of bounded slope and actuator constraints. It introduces a proper implicit discretization based on a modified sliding variable, derives explicit update formulas, and extends the approach to a conditioned STC to handle saturation while preventing windup. The authors provide complete stability proofs with simple conditions and show that the proposed scheme achieves the best possible worst-case disturbance rejection bound , supported by numerical simulations that outperform existing discretizations. The results offer a robust, windup-free, finite-time disturbance rejection framework for discrete-time higher order sliding mode control with practical actuator limits, and open paths for extensions to other higher order laws.

Abstract

The discrete-time implementation of the super-twisting sliding mode controller for a plant with disturbances with bounded slope, zero-order hold actuation, and actuator constraints is considered. Motivated by restrictions of existing implicit or semi-implicit discretization variants, a new proper implicit discretization for the super-twisting controller is proposed. This discretization is then extended to the conditioned super-twisting controller, which mitigates windup in presence of actuator constraints by means of the conditioning technique. It is proven that the proposed controllers achieve best possible worst-case performance subject to similarly simple stability conditions as their continuous-time counterparts. Numerical simulations and comparisons demonstrate and illustrate the results.
Paper Structure (23 sections, 55 equations, 4 figures)

This paper contains 23 sections, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation and Problem Statement
  3. Zero-Order Hold Sampled Sliding Mode Control
  4. State-of-the-Art Implicit Super-Twisting Control
  5. Semi-Implicit Conditioned Super-Twisting Control
  6. Implicit Super-Twisting Control without Actuator Saturation
  7. Implicit Conditioned Super-Twisting Control with Actuator Saturation
  8. Derivation of Explicit Control Laws
  9. Unsaturated Control Input
  10. Saturated Control Input
  11. Stability Analysis
  12. Unsaturated Control Input
  13. Saturated Control Input
  14. Simulation Results
  15. Conclusion
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Example for a disturbance signal $w(t)$ and corresponsing $w_k$ limiting the worst-case error as in Proposition \ref{['prop:best']}.
  • Figure 2: Disturbance signal $w$ and corresponding discrete-time disturbance $w_k$ with $L = 5$, $W = 0.25$ and sampling time $T = 0.01$ applied in both simulations.
  • Figure 3: Results of the discrete-time STC without actuator saturation. Parameters: $k_2=10$, $k_1=27$.
  • Figure 4: Results of the discrete-time STC in case of saturated actuation. Parameters: $k_2=10$, $k_1=16$, $U=1.5$. Top: plant state $x$, bottom: control signal $u$.

Theorems & Definitions (5)

  • proof : Proof of Theorem \ref{['th:explicit']}
  • proof : Proof of Theorem \ref{['th:explicit:conditioned']}
  • proof : Proof of Theorem \ref{['th:stability']}
  • proof : Proof of Theorem \ref{['th:stability:conditioned']}
  • proof