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

7.3    Extensions to a General Class of Discrete-Time Nonlinear Systems

In this section, we extend the results of the previous sections to a more general class of discrete-time nonlinear systems that may not necessarily be affine in u and w. We consider the general class of systems described by the state-space equations on $\mathcal{X}\subset {\Re }^n$ containing the origin {0}:

${\sum }^d:\{\begin{array}{l}{x}_{k+1}=F(x_k,w_k,u_k);x(k_0)=x^0\\ z_k=Z(x_k,w_k,u_k)\\ y_k=Y(x_k,w_k)\end{array}$

(7.64)

where all the variables have their usual meanings, and $F:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to \mathcal{X},F:\left(0,0,0\right)=0,Z:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to {\Re }^s,Z\left(0,0,0\right)=0,Y:\mathcal{X}\times \mathcal{W}\to {\Re }^m$. We begin with the full-information and state-feedback problems.

7.3.1    Full-Information ${\mathcal{H}}_{\infty }$-Control for a General Class of Discrete-Time Nonlinear Systems ...