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

11.2  Discrete-Time Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Control

In this section, we discuss the state-feedback mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-control problem for discrete-time nonlinear systems. We begin with an affine discrete-time state-space model defined on $\mathcal{X}\subset {\Re }^n$ in coordinates (x₁,…,x_n) :

${\sum }^{da}:\{\begin{array}{l}{\dot{x}}_{k+1}=f(x_k)+g_1(x_k)w_k+g_2(x_k)u_k;x(k_0)=x^0\\ z_k=h_1(x_k)+k_{12}(x_k)u_k\\ y_k=x_k,\end{array}$

(11.29)

where all the variables and system matrices have their previous meanings and dimensions respectively. We assume similarly that the system has a unique equilibrium at x = 0, and is such that f(0) = 0 and h₁0) = 0. For simplicity, we similarly also assume the following hold for the system matrices.

Assumption 11.2.1 The system matrices are such that

$\begin{array}{l}{h}_1^T\end{array}$