Two-stage extended recursive gradient algorithm for locally linear RBF-based autoregressive models with colored noises

Two‐stage and three‐stage recursive gradient identification of Hammerstein nonlinear systems based on the key term separation

“…Referring to the method in References 46,68‐70, Equation (13) can be viewed as a discrete‐time system where the state is $$$ \hat{\boldsymbol{\vartheta}}(t) $$$ and the input is $$$ \mu (t)\boldsymbol{\xi} (t) $$$. To ensure the convergence of $$$ \hat{\boldsymbol{\vartheta}}(t) $$$, the eigenvalues of matrix $$$ \left[{\boldsymbol{I}}_n-\mu (t)\boldsymbol{R}(t)\right] $$$ must be in the unit circle, this is to say $$$ \mu (t) $$$ should satisfy $$$ -{\boldsymbol{I}}_n\leqslant {\boldsymbol{I}}_n-\mu (t)\boldsymbol{R}(t)\leqslant {\boldsymbol{I}}_n $$$, so a reasonable choice of $$$ \mu (t) $$$ is to satisfy $$$$ \mu (t)\leqslant \frac{2}{\lambda_{\mathrm{max}}\left[\boldsymbol{R}(t)\right]}=2{\lambda}_{\mathrm{max}}^{-1}\left[\boldsymbol{R}(t)\right].$$…”

“…From the above derivation, we can obtain the KT‐AM‐RG algorithm 68‐70 : $$$$ \hat{\boldsymbol{\vartheta}}(t)\kern0.5em =\hat{\boldsymbol{\vartheta}}\left(t-1\right)+\frac{1}{r(t)}\left[\boldsymbol{\xi} (t)-\boldsymbol{R}(t)\hat{\boldsymbol{\vartheta}}\left(t-1\right)\right], $$$$ $$$$ r(t)\kern0.5em =r\left(t-1\right)+{\left\Vert \hat{\boldsymbol{\varphi}}(t)\right\Vert}^2, $$$$ $$$$ \boldsymbol{\xi} (t)\kern0.5em =\boldsymbol{\xi} \left(t-1\right)+\hat{\boldsymbol{\varphi}}(t)y(t)\in {\mathbb{R}}^n, $$$$ …”

A novel filter-based multi-stage parameter estimation for a class of hybrid nonlinear models

“…For the ExpAR model, Xu et al proposed a filter-based three-stage multi-innovation extended stochastic gradient algorithm [19] to estimate the parameters. In order to estimate the parameters of the RBF-AR model, Zhou et al provided a two-stage extended recursive gradient algorithm [20]. Gan et al proposed a variable projection (VP) algorithm to estimate parameters for plenty of hybrid models like the ExpAR model [21], the RBF-AR model [22] and the GRBF-AR model [16].…”