ABSTRACT

We assume knowledge of two bounding functions ρ1(x1) and ρ2(x) so that all admissible uncertainties are characterized by the inequalities,

| Δ1(x, t) | ≤ ρ1(x1) (49.25) | Δ2(x, t) | ≤ ρ2(x) (49.26)

for all x ∈ R2 and all t ∈ R. Note that the bound ρ1 on the uncertainty Δ1 is allowed to depend only on the state x1; this is the structural condition suggested in the previous section and will be characterized more completely below. We will take a recursive approach to the design of a robust controller for this system. This approach is based on the integrator backstepping technique developed by [11] for the adaptive control of nonlinear systems. The first step in this approach is to consider the scalar system,

x˙1 = u¯+Δ1(x1, u¯, t), (49.27) which we obtain by treating the state variable x2 in Equation 49.23 as a control variable u¯. This new system (Equation 49.27) is only conceptual; its relationship to the actual system (Equations 49.23-49.24) will be explored later. Let us next design a robust controller u¯ = μ(x1) for this conceptual system. By construction, this new system satisfies the matching condition, and so we may use the Lyapunov redesign method to construct the feedback u¯ = μ(x1). The nominal system is simply x˙1 = u¯which can be stabilized by a nominal feedback u¯nom = −x1. A suitable Lyapunov function for the nominal closed-loop system x˙1 = −x1 would be V1(x1) = x21 . We then choose u¯ = u¯nom + u¯rob, where u¯rob is given, for example, by Equation 49.9 with ρ¯ = ρ1. The resulting feedback function for u¯ is

μ(x1) = −x1 − ρ1(x1)sgn(x1). (49.28) If we now apply the feedback u¯ = μ(x1) to the conceptual system (Equation 49.27), we achieve V˙1 ≤ −2x21 and thus guarantee stability for every admissible uncertainty Δ1. Let us assume for now that this function μ is (sufficiently) smooth; we will return to the question of smoothness in Section 49.5.