A Lyapunov Function for an Extended Super-Twisting Algorithm

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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.
Original languageEnglish
Pages (from-to)3426-3433
JournalIEEE Transactions on Automatic Control
Issue number10
Publication statusPublished - 2018


  • Convex programming
  • Multiple equilibria
  • Polynomial methods
  • Positive semidefinite Lyapunov function
  • Sliding mode control

Fields of Expertise

  • Information, Communication & Computing


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