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

A

Proof of Theorem 5.7.1

The proof uses a back-stepping procedure and an inductive argument. Thus, under the assumptions (i), (ii) and (iv), a global change of coordinates for the system exists such that the system is in the strict-feedback form [140, 153, 199, 212]. By augmenting the ρ linearly independent set

$z_1=h\left(x\right),z_2=L_{f}h\left(x\right),\mathrm{\dots },z_{\rho }=L_f^{\rho -1}h\left(x\right)$

with an arbitrary n−ρ linearly independent set z_ρ+1 = ψ_ρ+1(x),…, z_n = ψ_n with ψ_i(0) = 0, 〈dψ_i, G_ρ−1〉 = 0, ρ + 1 ≤ i ≤ n. Then the state feedback

$u=\frac{1}{L_{g_2}{L}_f^{\rho -1}h\left(x\right)}\left(\upsilon -L_f^{\rho }h\left(x\right)\right)$

globally transforms the system into the form:

$\begin{array}{l}{\dot{z}}_i=z_{i+1}+{\text{Ψ}}_i^T\left(z_1,\mathrm{\dots },z_i\right)w1\le i\le \rho -1,\\ {\dot{z}}_{\rho }=\upsilon +{\text{Ψ}}_{\rho }^T\left(z_1,\mathrm{\dots }{z}_{\rho }\right)w\\ {\dot{z}}_{\mu }=\psi \left(z\right)+{\text{Ξ}}^T\left(z\right)w,\\ y=z_1,\end{array}\}$

(A.1)

where z_μ = (z_ρ+1,…, z_n). Moreover, in the z-coordinates we have

${\mathcal{G}}_j=span\left\{\frac{\partial }{\partial z_{\rho -j}},\mathrm{\dots },\frac{\partial }{\partial z_{\rho }}\right\},0\le j\le \rho -1,$

so ...