A survey on High-Order Sliding-Mode control design using Lyapunov functions

A survey of (some) recent Lyapunov-based methods to analyze and design High-Order Sliding-Mode (HOSM) controllers and observers is presented. This overview includes also some novel algorithms, taking advantage of a discontinuous integral action, to attain a High-Order Sliding-Mode using a continuous control signal. This account starts presenting the design based on Lyapunov functions of classical sliding-mode controllers using a discontinuous state feedback control law to impose a high-order sliding-mode of arbitrary order. In order to estimate in finite-time the required derivatives of the measured signals for the implementation of the controllers, a Lyapunov-based design of the well-known robust and exact differentiator is shown. As a mechanism to reduce the effect of the undesirable chattering effect a recent method for higher-order sliding-mode controllers has been developed, using a continuous state feedback in conjunction with a discontinuous integral action to enforce a sliding-mode by means of a continuous control signal. The analysis and design tool for this integral controller is also an explicit, strict and particularly tailored Lyapunov function. In order to make this recount accessible to a wide audience the presentation is restricted to the main results, without giving proofs or design details, and leaving aside the rigorous mathematical machinery. For this the main references are provided. Copyright (c) 2023 The Authors. This is an open access article under the CC BY- NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)