Time-Varying Sliding Modes for the Third Order Systems

Abstract

In this chapter we further develop the sliding mode design method for possibly time-varying and nonlinear continuous time plants. We still consider the linear sliding surfaces, however now we focus our attention on more complex, i.e. third order systems, described by the following equations

$\dot{x}_{1}=x_{2}$

$\dot{x}_{2}=x_{3}$

$\dot{x}_{3}=f(x,t)+\Delta f(x,t)+ b(x,t)u+d(t)$ (3.1)

where x₁, x₂, x₃ are the state variables of the system and $x(t) = [{x}_{1}(t) {x}_{2}(t){x}_{3}(t)]^T$ is the state vector. Similarly as in chapter 2, also in this part of the book t denotes time, u is the input signal, b, f - are a priori known, bounded functions of time and the system state, Δf and d are functions representing the system uncertainty and external disturbances, respectively.