A Lyapunov approach to second-order sliding mode controllers and observers

Abstract

In this paper a strong Lyapunov function is obtained, for the first time, for the supertwisting algorithm, an important class of second order sliding modes (SOSM). This algorithm is widely used in the sliding modes literature to design controllers, observers and exact differentiators. The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm. These modification allows the system to deal with linearly growing perturbations, that are not endured by the basic supertwisting algorithm. Moreover, the introduction of Lyapunov functions opens many new analysis and design tools to the Higher Order Sliding Modes research area.

A multivariable super-twisting sliding mode approach

“…Remark 5: These conditions are not identical to the ones in [6], perhaps because of the different approximations used to obtain the expressions in (17).…”

“…Using similar arguments to [6], the Lyapunov function in (12) can be written as V = X T P X for an appropriate symmetric positive definite matrix P ∈ R 3m×3m and V ≤ λ max (P ) X 2 from Rayleigh's inequality. Therefore from (27)V…”

“…Until very recently stability, robustness and convergence rates in higher order sliding mode methods have been analyzed in terms of homogeneity or geometric arguments [5]. However in a succession of papers [6,16,14], Lyapunov methods were employed successfully for the first time to analyze the properties of the super-twisting algorithm for uncertain systems. This has opened the door for the integration of these ideas with other nonlinear tools including gain adaptation [13,10,7].…”

Adaptive continuous higher order sliding mode control

“…A recent overview of this field is given in Utkin & Poznyak (2013a). The creation of new Lyapunov functions for higher order sliding mode control structures -particularly twisting and super-twisting controllers -has also rejuvenated interest in this area (Moreno & Osorio (2008)) and the literature expanding these Lyapunov ideas into the realm of adaptive sliding mode control is developing rapidly (Plestan et al (2010); Alwi & Edwards (2013); Shtessel et al (2012); Bartolini et al (2103); ). Whilst it is intuitively clear that when sliding begins to deteriorate the controller gains must be increased, devising an effective way of lowering unnecessarily large gains once sliding is achieved, has proved more elusive.…”

On autonomous and teleoperated aerial service robots

“…Inspired by [76], the designed observer is given by, From (4.22) and (4.23), the estimation error dynamics is …”