Adaptive Sliding Mode Control Based on the Extended Equivalent Control Concept for Disturbances with Unknown Bounds

Abstract

In this chapter, we propose an adaptive sliding mode approach based on the extended equivalent control concept to cope with disturbances of unknown bounds in nonlinear systems. Some advantages with respect to previous proposed methods are its simplicity and capability of rejecting non smooth disturbances. Unlike other adaptive approaches, overestimation of the controller gain and the loss of the sliding motion can be avoided. The developed method guarantees ideal sliding modes and alleviates the chattering phenomena. Theoretically, the sliding variable becomes identically null after some finite time in spite the disturbances. Global stability is proved. Simulation results illustrate the potential and limitations of this novel adaptation strategy.

Appendix

To define the concept of input-to-state stability/stable (ISS) [11, Sect. 4.9], consider the system

$$\begin{aligned} \dot{x} = f(x,u,t)\,, \end{aligned}$$

(6.32)

where $f:{\mathbb {R}}^n\times {\mathbb {R}}^m\times {\mathbb {R}}_+\rightarrow {\mathbb {R}}^n$ is piecewise continuous in t and locally Lipschitz in x and u. The input signal u(t) is piecewise continuous and uniformly bounded. Assume that this system has a globally uniformly asymptotically stable equilibrium point at $x=0$ when $u(t)\equiv 0$.

Definition 6.1

The system (6.32) is said to be input-to-state stable (ISS) if there exist a class ${\mathscr {KL}}$ function $\beta$ and a class ${\mathscr {K}}$ function $\gamma$ such that for any initial state $x(t_0)$ and any bounded input signal u(t), the solution x(t) exists $\forall t\ge t_0\ge 0$ and satisfies

$$\begin{aligned} \Vert x(t)\Vert \le \beta \left( \Vert x(t_0)\Vert ,t-t_0\right) +\gamma \left( \sup _{t_0\le \tau \le t}\Vert u(\tau )\Vert \right) \,. \end{aligned}$$

(6.33)