Lyapunov-Based Design of Homogeneous High-Order Sliding Modes

Abstract

We provide a Lyapunov-based design of homogeneous High-Order Sliding Mode (HOSM) Control and Observation (Differentiation) algorithms of arbitrary order for a class of Single-Input-Single-Output uncertain nonlinear systems. First, we recall the standard problem of HOSM control, which corresponds to the design of a state feedback control and an observer for a particular Differential Inclusion (DI), which represents a family of dynamic systems, including bounded matched perturbations/uncertainties. Next we provide a large family of zero-degree homogeneous discontinuous controllers solving the state feedback problem based on a family of explicit and smooth homogeneous Lyapunov functions. We show the formal relationship between the control laws and the Lyapunov functions. This also gives a method for the calculation of controller gains ensuring the robust and finite time stability of the sliding set. The required unmeasured states can be estimated robustly and in finite time by means of an observer or differentiator, originally proposed by A. Levant. We give explicit and smooth Lyapunov functions for the design of gains ensuring the convergence of the estimated states to the actual ones in finite time, despite the non vanishing bounded perturbations or uncertainties acting on the system. Finally, it is shown that a kind of separation principle is valid for the interconnection of the HOSM controller and observer, and we illustrate the results by means of a simulation on an electro-mechanical system.