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

8.3    Certainty-Equivalent Filters (CEFs)

We discuss in this section a class of estimators for the nonlinear system (8.1) which we refer to as “certainty-equivalent” worst-case estimators (see also [139]). The estimator is constructed on the assumption that the asymptotic value of $\widehat{x}$ equals x, and the gain matrices are designed so that they are not functions of the state vector x, but of $\widehat{x}$ and y only. We begin with the one degree-of-freedom (1-DOF) case, then we discuss the 2-DOF case. Accordingly, we propose the following class of estimators:

${\sum }_1^{acef}:\{\begin{array}{l}\dot{\widehat{x}}=f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right){\widehat{w}}^{\star }+\widehat{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)w^{\star }\right)\\ \widehat{z}=h_2\left(\widehat{x}\right)\\ \tilde{z}=y-h_2\left(\widehat{x}\right)\end{array}$

(8.32)

where $\widehat{z}=\widehat{y}\in {\Re }^m$ is the estimated output, $\tilde{z}\in {\Re }^m$ is the new penalty variable, ${\widehat{w}}^{\star }$ is the estimated ...