### Abstract

*m*≥ 1 has been proposed. However, as of yet, no Lyapunov functions for this algorithm exist. This paper discusses the construction of Lyapunov functions by means of the sum-of-squares technique for

*m*= 1. Sign definiteness of both Lyapunov function and its time derivative is shown in spite of numerically obtained-and hence possibly inexact-sum-of-squares decompositions. By choosing the Lyapunov function to be a positive semidefinite, the finite time attractivity of the system's multiple equilibria is shown. A simple modification of this semidefinite function yields a positive definite Lyapunov function as well. Based on this approach, a set of the algorithm's tuning parameters ensuring finite-time convergence and stability in the presence of bounded uncertainties is proposed. Finally, a generalization of the approach for

*m*> 1 is outlined.

Originalsprache | englisch |
---|---|

Seiten (von - bis) | 3426-3433 |

Fachzeitschrift | IEEE Transactions on Automatic Control |

Jahrgang | 63 |

Ausgabenummer | 10 |

DOIs | |

Publikationsstatus | Veröffentlicht - 2018 |

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### Schlagwörter

### Fields of Expertise

- Information, Communication & Computing

### Dies zitieren

**A Lyapunov Function for an Extended Super-Twisting Algorithm.** / Seeber, Richard; Reichhartinger, Markus; Horn, Martin.

Publikation: Beitrag in einer Fachzeitschrift › Artikel › Forschung › Begutachtung

}

TY - JOUR

T1 - A Lyapunov Function for an Extended Super-Twisting Algorithm

AU - Seeber, Richard

AU - Reichhartinger, Markus

AU - Horn, Martin

PY - 2018

Y1 - 2018

N2 - Recently, an extension of the super-twisting algorithm for relative degrees m ≥ 1 has been proposed. However, as of yet, no Lyapunov functions for this algorithm exist. This paper discusses the construction of Lyapunov functions by means of the sum-of-squares technique for m = 1. Sign definiteness of both Lyapunov function and its time derivative is shown in spite of numerically obtained-and hence possibly inexact-sum-of-squares decompositions. By choosing the Lyapunov function to be a positive semidefinite, the finite time attractivity of the system's multiple equilibria is shown. A simple modification of this semidefinite function yields a positive definite Lyapunov function as well. Based on this approach, a set of the algorithm's tuning parameters ensuring finite-time convergence and stability in the presence of bounded uncertainties is proposed. Finally, a generalization of the approach for m > 1 is outlined.

AB - Recently, an extension of the super-twisting algorithm for relative degrees m ≥ 1 has been proposed. However, as of yet, no Lyapunov functions for this algorithm exist. This paper discusses the construction of Lyapunov functions by means of the sum-of-squares technique for m = 1. Sign definiteness of both Lyapunov function and its time derivative is shown in spite of numerically obtained-and hence possibly inexact-sum-of-squares decompositions. By choosing the Lyapunov function to be a positive semidefinite, the finite time attractivity of the system's multiple equilibria is shown. A simple modification of this semidefinite function yields a positive definite Lyapunov function as well. Based on this approach, a set of the algorithm's tuning parameters ensuring finite-time convergence and stability in the presence of bounded uncertainties is proposed. Finally, a generalization of the approach for m > 1 is outlined.

KW - Convex programming

KW - Multiple equilibria

KW - Polynomial methods

KW - Positive semidefinite Lyapunov function

KW - Sliding mode control

U2 - 10.1109/TAC.2018.2794411

DO - 10.1109/TAC.2018.2794411

M3 - Article

VL - 63

SP - 3426

EP - 3433

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 10

ER -