Use of Homogeneity in Sliding Mode Control Design: Tutorial

This tutorial is devoted to homogeneous sliding mode control (SMC) algorithms. Many popular SMC algorithms are shown to be homogeneous in the standard or in a generalized sense. The issues of modeling, analysis, and design of SMC systems are considered. In particular, Filippov and Utkin approaches to regularization of discontinuous differential equations with SMCs are surveyed. Stability analysis using smooth and non‐smooth Lyapunov functions is discussed. An introduction to the homogeneity theory is given. Various well‐known conventional and modern SMC algorithms for finite‐time output regulation of affine‐in‐control nonlinear systems are interpreted, designed, and analyzed using homogeneity theory. In particular, homogeneous Lyapunov functions allowing simple analysis for systems with SMCs are presented. It is shown that the SMC parameters can be tuned using Linear Matrix Inequalities. Homogeneity‐based robustness analysis with respect to mismatched perturbations is provided. An approach to chattering attenuation using the homogeneous Lyapunov function is proposed. The theoretical results are supported by numerical simulations.