A new class of fast finite-time discontinuous controllers

Abstract

We introduce a new class of high-order sliding mode (HOSM) controllers with improved convergence rate for a single-input uncertain system. The controllers are designed by means of Control Lyapunov Functions (CLF's) and properties of homogeneous systems in the bi-limit. Therefore, the proposed controllers in feedback with the system make the close-loop sys- tem homogeneous in the bi-limit. Robustness and convergence rate are improved in comparison with the reported HOSM. I. INTRODUCTION Sliding mode control (SMC) is considered to be effective to deal with uncertainties and disturbances in nonlinear dynamical systems.The classical SMC design is based on finding a discontinuous control input to drive the state trajec- tories to reach the so called sliding surface in finite time (18). The sliding surface is a manifold previously designed to at- tain some dynamic performance requirements. Unfortunately, standard SMC has two drawbacks: it is restricted to systems having relative degree equal to one and it yields the well- known chattering effect, a phenomenon of high-frequency and finite-amplitude oscillations of the controlled system provided by the use of a first-order sliding mode (FOSM) which degrades the performance in practical applications. Higher-order sliding modes HOSM's (11) preserve the main robustness properties of standard SMC but additional advantages are obtained. For instance, HOSM's can alleviate the chattering effect, finite time convergence to the origin is ensured and better accuracy with respect to discrete sampling time is attained (12). In contrast to the classical SMC design, the HOSM controllers are constructed by means of homogeneity tools. However, HOSM can deal only with bounded perturbations and the convergence rate is slower when the trajectories are far away from the origin. Fast Terminal Sliding Mode based Control (FTSMC) can deal with these problems. However, it has a stronger restriction for applicability purposes as the Terminal Slid- ing Mode based Control (TSMC), (14), (19). Since the control signal becomes unbounded for some sets, the state trajectories need to start in a prescribed sector of the state space in order to avoid an unbounded control signal. To overcome this drawback, the Nonsingular Fast Terminal Sliding Mode based Control (NFTSMC) rewrites the well- known fast terminal sliding surface (FTSS) to obtain non singular control inputs,(13). The terminal controllers are based on the standard SMC design and it has been suitable *Financial support from CONACyT CVU 267513, PAPIIT, UNAM, grant IN113614, and Fondo de Colaboraci´ on del II-FI,UNAM, IISGBAS-109-