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

5.3    Robust Nonlinear ${\mathcal{H}}_{\infty }$ State-Feedback Control

In this section, we consider the state-feedback ${\mathcal{H}}_{\infty }$-control problem for the affine nonlinear system Σ^a in the presence of unmodelled dynamics and/or parameter variations. This problem has been considered in many references [6, 7, 209, 223, 245, 261, 265, 284]. The approach presented here is based on [6, 7] and is known as guaranteed-cost control. It is an extension of quadratic-stabilization, and was first developed by Chang [78] and later popularized by Petersen [226, 227, 235]. For this purpose, the system is represented by the model:

${\mathrm{\Sigma }}_{\text{Δ}}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+\text{Δ}f\left(x,\theta ,t\right)+g_1\left(x\right)w+\left[g_2\left(x\right)+\text{Δ}{g}_2\left(x,\theta ,t\right)\right]u;\\ x\left(t_0\right)=x_0\\ y=x\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ u\end{array}\right]\end{array}$

(5.41)

where all the variables ...