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

12.2.1  Solution to the Finite-Horizon Discrete-Time Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering Problem

We similarly consider the following class of estimators:

${\sum }^{daf}:\{\begin{array}{l}{\widehat{x}}_{k+1}=f\left(x_k\right)+L\left({\widehat{x}}_k,k\right)\left[y_k-h_2\left(x_k\right)\right],\widehat{x}\left(k_0\right)={\widehat{x}}^0\\ \widehat{z}=h_1\left({\widehat{x}}_k\right)\end{array}$

(12.46)

where ${\widehat{x}}_k\in \mathcal{X}$ is the estimated state, $L(.,.)\in {ℳ}^{n\times m}(\mathcal{X}\times \text{Z})$ is the error-gain matrix which is smooth and has to be determined, and $\widehat{z}\in {\Re }^s$ is the estimated output of the filter. We can now define the estimation error or penalty variable, $\stackrel{⌣}{z}$, which has to be controlled as:

${\stackrel{⌣}{z}}_k:=z_k-\widehat{z}=h_1\left({\widehat{x}}_k\right)-h_1\left({\widehat{x}}_k\right).$

Then, we combine the plant (12.42) and estimator (12.46) dynamics to obtain the following augmented system:

$\begin{array}{l}{\stackrel{⌣}{x}}_{k+1}=\stackrel{⌣}{f}\left({\stackrel{⌣}{x}}_k\right)+\stackrel{⌣}{g}\left({\stackrel{⌣}{x}}_k\right)w_{k,}\stackrel{⌣}{x}\left(k_0\right)={\left(x^{0^T}{\widehat{x}}^{0^T}\right)}^T\\ {\stackrel{⌣}{z}}_k=h_1\left({\stackrel{⌣}{x}}_k\right)\end{array}\},$

(12.47)

where

$\begin{array}{l}{\stackrel{⌣}{x}}_k=\left(\begin{array}{l}{x}_k\\ x_k\end{array}\right),\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)=(\begin{array}{l}\end{array}\end{array}$