Globally Stable Implicit Euler Time-Discretization of a Nonlinear Single-Input Sliding-Mode Control System

Bernard Brogliato (1) , Andrey Polyakov (2, 3)

1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
2 NON-A - Non-Asymptotic estimation for online systems
3 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189

In this note we study the effect of an implicit Euler time-discretization method on the stability of the discretization of a globally fixed-time stable, scalar differential inclusion representing a simple nonlinear system with a set-valued signum controller. The controller nonlinearity is a cubic term and it is shown that the fully-implicit method preserves the global Lyapunov stability property of the continuous-time system, contrarily the explicit discretization which does not. It allows to obtain finite-time convergence to the origin when the plant is undisturbed, while the cubic term provides the hyper-exponential convergence rate.

Mots clés: hyper-exponential convergence; sliding mode; Implicit Euler discretization; fixed-time stability

Domaines: Automatique

Bernard Brogliato, Andrey Polyakov. Globally Stable Implicit Euler Time-Discretization of a Nonlinear Single-Input Sliding-Mode Control System. 54th Conference on Decision and Control 2015, Dec 2015, Osaka, Japan. pp.5426-5431, ⟨10.1109/CDC.2015.7403069⟩. ⟨hal-01212601⟩