Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

6.4    Output Measurement-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Nonlinear Systems

In this section, we look at the output measurement-feedback problem for a more general class of nonlinear systems. For this purpose, we consider the following class of nonlinear systems defined on a manifold $\mathcal{X}\subseteq {\Re }^n$ containing the origin in coordinates x = (x₁, …, x_n):

${\mathrm{\Sigma }}^g:\{\begin{array}{l}\dot{x}=F\left(x,w,u\right);x\left(t_0\right)=x_0\\ z=Z\left(x,u\right)\\ y=Y\left(x,w\right)\end{array}$

(6.52)

where all the variables have their previous meanings, while $F:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to \mathcal{X}$ is the state dynamics function, $Z:\mathcal{X}\times \mathcal{U}\to {\Re }^s$ is the controlled output function and $Y:\mathcal{X}\times \mathcal{W}\to {\Re }^m$ is the measurement output function. Moreover, the functions F (., ., .), Z(., .) and $\mathcal{Y}$ (., .) are smooth C^r, r ≥ 1 functions of their arguments, and the point x = 0 ...