Feedback Design for Robust Semiglobal Stability

Abstract

In section 9.3 we have introduced the concept of semiglobal stabilizability, and we have shown (Theorem 9.3.1) how, using a linear feedback, it is possible to stabilize in a semiglobal sense (i.e. imposing that the domain of attraction of the equilibrium contains a prescribed compact set) a system of the form (9.23), under the hypothesis that the equilibrium z = 0 of its zero dynamics is globally asymptotically stable. In this section, in preparation to the subsequent study of the problem of robust semiglobal stabilization using output feedback, we extend the result of Theorem 9.3.1 to the case of a system modeled by equations of the form

$$\begin{array}{*{20}{l}} {\dot z}& = &{{f_0}(z,\xi )} \\ {{{\dot \xi }_1}}& = &{{\xi _2}} \\ {{{\dot \xi }_2}}& = &{{\xi _3}} \\ {}&{}& \cdots \\ {{{\dot \xi }_r}}& = &{q(z,{\xi _1}, \ldots ,{\xi _r},\mu ) + b(z,{\xi _1}, \ldots ,{\xi _r},\mu )u,} \end{array}$$

(1)

in which z ∈ ℝⁿ, ξ _i ∈ ℝ for i=1,…,r, u ∈ ℝ and $\mu \in \mathcal{P} \subset {\mathbb{R}^p}$ is a vector of unknown parameters, ranging over a compact set $\mathcal{P}$.