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:

${\sum}_{1}^{acef}\text{\hspace{0.17em}}:\text{\hspace{0.17em}}\{\begin{array}{l}\dot{\widehat{x}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(\widehat{x}\right)+{g}_{1}\left(\widehat{x}\right){\widehat{w}}^{\star}+\widehat{L}\left(\widehat{x},y\right)\left(y-{h}_{2}\left(\widehat{x}\right)-{k}_{21}\left(\widehat{x}\right){w}^{\star}\right)\\ \widehat{z}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{h}_{2}\left(\widehat{x}\right)\\ \tilde{z}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}y-{h}_{2}\left(\widehat{x}\right)\end{array}$ |
(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 ...

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