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

3.5    Dissipativity and Passive Properties of Discrete-Time Nonlinear Systems

In this section, we discuss the discrete-time counterparts of the previous sections. For this purpose, we again consider a discrete-time nonlinear state-space system defined on $\mathcal{X}$ ⊆ ℜⁿ containing the origin x = {0} in coordinates (x = x₁,…, x_n):

${\sum }^d:\{\begin{array}{l}{x}_{k+1}=f(x_k,u_k),x(k_0)=x^0\\ y_k=h(x_k,u_k)\end{array}$

(3.59)

where x_k ∈ $\mathcal{X}$ is the state vector, u_k ∈ $\mathcal{U}$ ⊆ ℜ^m is the input function belonging to an input space $\mathcal{U}$, y_k ∈ $\mathcal{Y}$ ⊆^m is the output function which belongs to the output space $\mathcal{Y}$ (i.e., Σ^d is square). The functions f : $\mathcal{X}$ × $\mathcal{U}$ → $\mathcal{X}$ and h : $\mathcal{X}$ × $\mathcal{U}$ → $\mathcal{Y}$ are real C^r functions of their arguments such that there exists a unique solution x(k, k₀, x₀, u_k) to the system ...