4.10 Discontinuous Robust Controllers

In this section we consider the problem of constructing stabilizing state-feedback controllers for nonlinear dynamical systems whose mathematical models contain uncertainties. The design of the controllers is based on the Lyapunov theory presented in the previous sections. The class of dynamical systems considered here is modeled by

where , and represent the lumped uncertainties. We assume that the uncertain element ? is bounded by a known continuous nonnegative function ?, that is,

We also assume that the equilibrium state x = 0 of the uncontrolled nominal system

is globally uniformly asymptotically stable and that we were able to construct a Lyapunov function corresponding to this equilibrium state. The Lyapunov function W is assumed to satisfy the following conditions: There exist three continuous and strictly increasing functions ? _i, i = 1, 2, 3, such that

for all t and all x. We will show that if we apply the control strategy

where ?( t, x) = G ^T ( t, x) ? _x W( t, x) and ? = {( t, x): ? = 0}, then the equilibrium state x = 0 is a globally uniformly asymptotically stable solution of the closed-loop system. For this, consider the Lyapunov function W for the uncontrolled nominal system. The time derivative of W evaluated on the solutions of the closed-loop...