Higher Order Super-Twisting for Perturbed Chains of Integrators

In this paper, we present a generalization of the super-twisting algorithm for perturbed chains of integrators of arbitrary order. This higher order super-twisting (HOST) controller is homogeneous with respect to a family of dilations and is continuous. It is built as a dynamic controller (with respect to the state variable of the chain of integrators) and the convergence analysis is performed by the use of a homogeneous strict Lyapunov function which is explicitly constructed. The effectiveness of the controller is finally illustrated with simulations for a chain of integrators of order four, first pure then perturbed, where we compare the performances of two HOST controllers.

I. Introduction

Many higher order sliding mode control (HOSMC) algorithms exist in contemporary literature for control of nonlinear systems with bounded uncertainty. These algorithms are robust, they preserve the insensitivity of classical sliding mode, and maintain the performance characteristics of the closed loop system. Levant for example, has presented a method of designing arbitrary order sliding mode controllers for single input single output (SISO) systems in [1]. Laghrouche et al. [2] have proposed a two part integral sliding mode-based control to deal with the finite time stabilization problem and uncertainty rejection problem separately. Dinuzzo et al. have proposed another method in [3], where the problem of HOSMC has been treated as Robust Fuller's problem. Defoort et al. [4] have developed a robust multiinput multioutput HOSMC controller, using a constructive algorithm with weighted homogeneity-based finite time stabilization of an integrator chain. Sliding mode with homogeneity approach was also used in [5] and [6], to demonstrate finite time stabilization of the arbitrary order sliding mode controllers for SISO systems [1] . A Lyapunov-based approach for arbitrary HOSMC controller design was presented in [7] and [8]. In this paper, it was shown that a class of homogeneous controllers that satisfies certain conditions, could be used to stabilize perturbed integrator chains.