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Modified Implicit Discretization of the Super-Twisting Controller

Benedikt Andritsch, Lars Watermann, Stefan Koch, Markus Reichhartinger, Johann Reger, Martin Horn

TL;DR

This work presents a novel implicit discretization of the STC that preserves key continuous-time STC properties in discrete time, including finite-time convergence and rejection of Lipschitz disturbances with bound $L$. The proposed controller uses discrete-time functions $ ext{"Psi}_{1, ext{proposed}}$ and $ ext{"Psi}_{2, ext{proposed}}$ and defines an invariant set $oldsymbol{ extcal M}=ig\\{ |x_{1,k}| h^2eta,ig|h x_{2,k}-x_{1,k}ig| h^2etaig oig\,}$ to establish forward invariance and a finite-time convergence to this set, with disturbance-free global finite-time stability. In the presence of a Lipschitz disturbance with bound $L$, the steady-state error satisfies $ limsup_{k o\infty}|x_{1,k}| h^2 L$, and the steady-state error is independent of the controller gains, while avoiding discretization-chattering. Simulation studies show the proposed discrete-time STC outperforms existing implicit, semi-implicit, and low-chattering discretizations in terms of disturbance rejection, convergence time, and steady-state accuracy, making it particularly suitable for discrete-time hardware implementations of sliding-mode control.

Abstract

In this paper a novel discrete-time realization of the super-twisting controller is proposed. The closed-loop system is proven to converge to an invariant set around the origin in finite time. Furthermore, the steady-state error is shown to be independent of the controller gains. It only depends on the sampling time and the unknown disturbance. The proposed discrete-time controller is evaluated comparative to previously published discrete-time super-twisting controllers by means of the controller structure and in extensive simulation studies. The continuous-time super-twisting controller is capable of rejecting any unknown Lipschitz-continuous perturbation and converges in finite time. Furthermore, the convergence time decreases, if any of the gains is increased. The simulations demonstrate that the closed-loop systems with each of the known controllers lose one of these properties, introduce discretization-chattering, or do not yield the same accuracy level as with the proposed controller. The proposed controller, in contrast, is beneficial in terms of the above described properties.

Modified Implicit Discretization of the Super-Twisting Controller

TL;DR

This work presents a novel implicit discretization of the STC that preserves key continuous-time STC properties in discrete time, including finite-time convergence and rejection of Lipschitz disturbances with bound . The proposed controller uses discrete-time functions and and defines an invariant set to establish forward invariance and a finite-time convergence to this set, with disturbance-free global finite-time stability. In the presence of a Lipschitz disturbance with bound , the steady-state error satisfies , and the steady-state error is independent of the controller gains, while avoiding discretization-chattering. Simulation studies show the proposed discrete-time STC outperforms existing implicit, semi-implicit, and low-chattering discretizations in terms of disturbance rejection, convergence time, and steady-state accuracy, making it particularly suitable for discrete-time hardware implementations of sliding-mode control.

Abstract

In this paper a novel discrete-time realization of the super-twisting controller is proposed. The closed-loop system is proven to converge to an invariant set around the origin in finite time. Furthermore, the steady-state error is shown to be independent of the controller gains. It only depends on the sampling time and the unknown disturbance. The proposed discrete-time controller is evaluated comparative to previously published discrete-time super-twisting controllers by means of the controller structure and in extensive simulation studies. The continuous-time super-twisting controller is capable of rejecting any unknown Lipschitz-continuous perturbation and converges in finite time. Furthermore, the convergence time decreases, if any of the gains is increased. The simulations demonstrate that the closed-loop systems with each of the known controllers lose one of these properties, introduce discretization-chattering, or do not yield the same accuracy level as with the proposed controller. The proposed controller, in contrast, is beneficial in terms of the above described properties.
Paper Structure (16 sections, 2 theorems, 35 equations, 7 figures)

This paper contains 16 sections, 2 theorems, 35 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Implicit Discretization
  4. Discretization Based on Matching Approach
  5. Semi-Implicit Discretization
  6. Low-Chattering Discretization
  7. Proposed Discrete-Time STC
  8. Evaluation in Simulation Studies
  9. Disturbed Case
  10. Undisturbed Case
  11. Convergence Time when Varying One Parameter
  12. Steady-State Accuracy when Varying the Parameters
  13. State Trajectories
  14. Conclusion
  15. Derivation of the Low-Chattering Disc.
  16. ...and 1 more sections

Key Result

Proposition 1

Consider the closed-loop system dt-closedloop-system with the Lipschitz constant $L$, i.e. $|\Delta_k| \leq L~\forall k$, and $\beta > L$. Then $\mathcal{M}$ is a forward invariant set and if $x_k \in \mathcal{M}$ is fulfilled for some $k = K$, the steady-state error is limited, i.e. $\limsup_{k\geq

Figures (7)

  • Figure 1: Compared controller functions.
  • Figure 2: Invariant set $\mathcal{M}$ of the closed-loop system.
  • Figure 3: Simulation in time-domain with a disturbance, $\alpha = \sqrt{10}$, $\beta = 10$, $h = 0.01$ and $x_1(0) = 1$.
  • Figure 4: Simulations in time-domain in the undisturbed case, $\beta = 10$, $h = 0.01$ and $x_1(0) = 1$.
  • Figure 5: Convergence time $t_C$ over varying parameter $\alpha$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Theorem 2