## 3 Decentralized Neural Block Control (DNBC-UKF)  
This section shows the proposed SI optimization approach for UKF learning of decentralized neural block control (DNBC-UKF) [16] applied to a 2-DOF robot manipulator. For this purpose, we take the system to the following form  
\[\begin{split}\boldsymbol{\mathcal{X}}_{\mathrm{i},k+1}^{1}& =f_{\mathrm{i}}^{1}\Big{(}\boldsymbol{\mathcal{X}}_{\mathrm{i}}^{1} \Big{)}+B_{\mathrm{i}}^{1}\Big{(}\boldsymbol{\mathcal{X}}_{\mathrm{i}}^{1} \Big{)}\boldsymbol{\mathcal{X}}_{\mathrm{i}}^{2}+\Gamma_{i\ell}^{1},\\ &\vdots\\ \boldsymbol{\mathcal{X}}_{\mathrm{i},k+1}^{r}&=f_{ \mathrm{i}}^{r}\Big{(}\boldsymbol{\mathcal{X}}_{\mathrm{i}}^{1},\cdots, \boldsymbol  
The following RHONN structure is used in order to identify the behavior of system (1)  
\[ \begin{split}\mathbf{x}_{i,k+1}^{1}&=\mathbf{w}_{i,k}^{1} \mathcal{S}\Big{(}\mathbf{\mathcal{X}}_{i,k}^{1}\Big{)}+\mathbf{w}_{i}^{{}^{\prime}1} \mathbf{\mathcal{X}}_{i,k}^{2},\\ &\vdots\\ \mathbf{x}_{i,k+1}^{r}&=\mathbf{w}_{i,k}^{r}\mathcal{S}\Big{(}\mathbf{ \mathcal{X}}_{i,k}^{1},\cdots,\mathbf{\mathcal{X}}_{i,k}^{r}\Big{)}+\mathbf{w}_{i}^{ {}^{\prime}r}\mathbf{u}_{i,k}\end{split}  \tag{2}\]  
where \(\mathbf{x}_{i,k+1}^{j}= \begin{bmatrix}x_{i}^{1}&x_{i}^{2}&\cdots&x_{i}^{r}\end{bmatrix} ^{\top}\) is the \(j\)-th block neuron state with \(i=1,\ldots,N\) and \(j=1,\ldots,r-1\); \(\mathbf{w}_{i,k}^{{}^{\prime}j}\) are fixed parameters with \(rank(\mathbf{w}_{i}^{{}^{\prime}j})=m_{ij}\). \(S(\bullet)\) is the activation function and \(\mathbf{u}_{i,k}\) represents the input vector. The NN training task consists of finding values of \(w_{i}^{j,k}\) that minimize the identification error. For this reason, we propose to use a learning method using only the identification error information, such as the UKF described in Figure 2.  
The UKF is a powerful estimation and control tool: with wide applications in control theory; for the identification of nonlinear systems; and for the training of neural networks. Its versatility lies in its ability to handle nonlinear dynamics, non-Gaussian distributions, and uncertainties associated with real-world systems. In the context of control theory, the UKF serves as an efficient means for state estimation in nonlinear systems. It allows for the real-time estimation of the system's internal states, which are often unobservable or difficult to measure directly [17]. By incorporating nonlinear models and the measured system outputs, the UKF provides accurate and reliable estimates of the system's states, enabling effective control strategies to be devised. The estimated states obtained from the UKF can then be utilized for feedback control, trajectory tracking, and system stabilization in a wide range of dynamic systems [18; 19]. The UKF also plays a significant role in the identification of nonlinear systems. Identification refers to the process of determining the mathematical models or parameters that represent the underlying dynamics of a system based on observed input-output data. Non  
Figure 2: UKF framework. linear systems pose significant challenges in identification due to their complex dynamics. The UKF addresses these challenges by iteratively updating the system model parameters, enabling an accurate estimation of the nonlinear system's behavior. By leveraging the filtering and estimation capabilities of the UKF, researchers and engineers can effectively identify the dynamics, parameters, and structure of complex nonlinear systems, leading to an improved understanding and control of such systems [20]. Additionally, the UKF is employed in the training of neural networks, specifically in the context of Recurrent Neural Networks (RNNs). RNNs are powerful architectures for modeling sequential data and time series. It is possible to use the UKF in the training process to optimize the internal states, weights, and biases of the network in order to improve its learning capability and prediction accuracy [21].