## 3 Dynamic Neural Network Identification
### Identifier Based on Dynamic Neural Networks  
Inspired by the dynamic identifier scheme reported in [42; 43], we propose the following novel parallel identifier structure for the second-order system in Equation (24):  
\[ \begin{split}\dot{\xi}_{1}=&\xi_{2}\\ \dot{\xi}_{2}=&\hat{f}_{\phi}\Big{(}\dot{\xi}|W(t) \Big{)}+\hat{b}(t)u(\xi)+\mathbf{K}_{id}^{\top}\tilde{\xi},\end{split}  \tag{25}\]where \(\hat{f}_{\varrho}\big{(}\hat{\xi}|W(t)\big{)}\) is the dynamic identification operator of the unknown lumped disturbances \(\hat{f}_{\varrho}(\zeta,t)\). According to the dynamic neural network approach presented in such works, this operator is selected as the following Hopfield-type neural network:  
\[\hat{f}_{\varrho}\big{(}\hat{\xi}|W(t)\big{)}=a_{1}\hat{\xi}_{1}+a_{2}\hat{\xi}_ {2}+W(t)^{\top}\boldsymbol{\
abla}\big{(}\hat{\xi}\big{)}, \tag{26}\]  
where \(a_{1}\), \(a_{2}\) are negative constants to be chosen, \(\boldsymbol{\
abla}(\xi):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) is the vector of activation functions, and \(\boldsymbol{W}(t)\) is the weight matrix describing the neural network connections. Notice that the time dependence in \(\boldsymbol{W}(t)\) is explicitly considered hereafter to highlight that the weights of the dynamic neural network are time-varying. In the structure proposed in Equation (25), we integrate an adaptive parametric estimation \(\hat{b}(t)\) of the unknown constant \(b\) in contrast to the identifiers based on DNNs where another neural network is used. Moreover, the suggested identifier scheme in Equation (25) has a double integrator form unlike the \(n\)-dimensional first-order vector differential equation used by the renamed identifier structures ([42,43]). Additionally, another difference in the structure presented in Equation (25) relies on the addition of the correction term \(\boldsymbol{K_{id}}^{\top}\boldsymbol{\tilde{\zeta}}\), which is used to improve the convergence performance of the identifier. This term is based on an adjustable constant gain \(\boldsymbol{K_{id}}\in\mathbb{R}^{2\times 1}\) and the identification error \(\boldsymbol{\tilde{\zeta}}\) is defined as follows:  
\[\boldsymbol{\tilde{\zeta}}=\boldsymbol{\zeta}-\boldsymbol{\hat{\xi}}= \begin{bmatrix} \xi_{1}\\ \xi_{2}\end{bmatrix} = \begin{bmatrix}\xi_{1}-\xi_{1}\\ \xi_{2}-\xi_{2}\end{bmatrix}  \tag{27}\]  
Proceeding on the analysis method, let us suppose that, for the given parameters \(a_{1}\) and \(a_{2}\), it is possible to select a suitable matrix \(W_{0}\) in Equation (26), so we can closely match the function \(f_{\varrho}(\zeta,t)\) in the structure of Equation (24). However, there exists a bounded difference between the neural network and the real dynamic. In this sense, the function \(f_{\varrho}(\zeta,t)\) can be represented by the following neural network model:  
\[f_{\varrho}(\zeta,t)=a_{1}\zeta_{1}+a_{2}\zeta_{2}+\boldsymbol{W_{0}^{\top} \boldsymbol{\
abla}(\zeta)}+\tilde{f}_{\varrho}(\zeta,t), \tag{28}\]  
where \(\boldsymbol{W_{0}}\in\mathbb{R}^{2}\) denotes the vector of unknown constant weights and \(\tilde{f}_{\varrho}(\zeta,t)=f_{\varrho}(\zeta,t)-[a_{1}\zeta_{1}+a_{2}\zeta_ {2}+\boldsymbol{W_{0}^{\top}\boldsymbol{\
abla}(\zeta)}]\) defines the approximation error. **Assumption 4**. : _The upper bound for initial weights \(\boldsymbol{W_{0}}\) is available:_  
\[\boldsymbol{W_{0}}^{\top}\Delta\boldsymbol{W_{0}}\leq\boldsymbol{\tilde{W}}. \tag{29}\]  
where \(\Delta\) and \(\boldsymbol{\tilde{W}}\) are positive definite matrices with appropriate dimensions. The problem to tune the approximator Equation (28) in order to find the initial weights \(\boldsymbol{W_{0}}\) can be solved separately by implementing an offline training before carrying out the online system identification process. By means of input-output relation data \((\boldsymbol{\zeta}(k),u(k))|k=1,...,N\) produced by several experimental tests, we can use a certain numerical algorithm, for example, least squares [51,52] in order to obtain these baseline values. **Assumption 5**.