Disturbance Attenuation

Abstract

In this Chapter we will study problems of global stabilization of systems that can be modeled as feedback interconnection of two subsystems, one of which is accurately known while the other one is uncertain but has a finite L ₂ gain, for which an upper bound is available. More precisely, we consider systems modeled by equations of the form

$$\begin{array}{*{20}{l}} {{{\dot x}_1}}& = &{{f_1}({x_1},{h_2}({x_2}),u)} \\ {{{\dot x}_2}}& = &{{f_2}({x_2},{h_1}({x_1})),} \end{array}$$

(13.1)

which describe the feedback interconnection of a system

$$\begin{array}{*{20}{l}} {{{\dot x}_1}}& = &{{f_1}({x_1},w,u)} \\ y& = &{{h_1}({x_1})} \end{array}$$

(13.2)

in which ${x_1} \in {\mathbb{R}^{{n_1}}}$, w ∈ ℝ, u ∈ ℝ, y ∈ ℝ and f ₁(0,0,0)=0, h ₁(0)=0, a system

$$\begin{array}{*{20}{l}} {{{\dot x}_2}}& = &{{f_2}({x_2},y)} \\ w& = &{{h_2}({x_2})} \end{array}$$

(13.3)

in which ${x_2} \in {\mathbb{R}^{{n_2}}}$ and f ₂(0,0)=0,h ₂(0)=0.