## 5 Adaptive System Identification  
In order to design an ideal feedback controller, \(A_{n}\) and \(B_{n}\) are required; thus, system identification is involved, with traditional identification methods usually considering \(A_{n}\) and \(B_{n}\) separately. Due to problems in \(B_{n}\) such as zero crossing, it is easy to cause singularity problems like system divergence. To solve this difficulty, we modify the system as follows. We rewrite the last subsystem and obtain  
\[u=\frac{1}{B_{n}}\dot{z}_{n}-\frac{A_{n}}{B_{n}} \tag{12}\]  
Therefore, the system can be identified for the two unknowns \(\frac{1}{B_{n}}\) and \(\frac{A_{n}}{B_{n}}\) to avoid the singularity problem. According to the basic structure of MTN, we obtain  
\[\left\{ \begin{array}{l}\frac{1}{B_{n}}=\mathbf{w}_{1}^{*\mathsf{T}}\eta(\mathbf{z})+ \varepsilon_{1}\\ \frac{A_{n}}{B_{n}}=\mathbf{w}_{2}^{*\mathsf{T}}\eta(\mathbf{z})+\varepsilon_{2}\end{array} \right. \tag{13}\]  
where \(\eta(\mathbf{z})\) is the polynomial combination of MTN. Figure 1: Basic structure of MTN. Since \(\mathbf{z}\) is unknown, it can be replaced by \(\mathbf{\hat{z}}\) from the foregoing equation. Despite an error between them, it can be compensated through weight adjustment. \(\mathbf{w}_{1}^{*}\) and \(\mathbf{w}_{2}^{*}\) are ideal-weight MTN vectors. Unlike traditional neural network methods, our technique requires two sets of basis vectors in which each is calculated separately, imposing a significant computational burden. Moreover, there is a suitable polynomial combination compared with the MTN, i.e., the identification effect can be achieved only by changing the parameter value. By introducing the MTN, the system input value can be rewritten as  
\[u=\mathbf{w}_{1}^{*\mathrm{T}}\eta(\mathbf{z})\dot{z}_{n}-\mathbf{w}_{2}^{*\mathrm{T}} \eta(\mathbf{z})+\varepsilon \tag{14}\]  
where \(\varepsilon=\varepsilon_{1}\dot{z}_{n}-\varepsilon_{2}\) is the total error of the MTN. In the above formula, \(\dot{z}_{n}\) is unknown and can be replaced by \(\dot{\hat{z}}_{n}\), so we can obtain  
\[u=\mathbf{w}_{1}^{*\mathrm{T}}\eta(\mathbf{\hat{z}})\dot{\hat{z}}_{n}-\mathbf{w}_{2}^{ *\mathrm{T}}\eta(\mathbf{\hat{z}})+\xi \tag{15}\]  
where \(\xi=\varepsilon+\mathbf{w}_{1}^{*\mathrm{T}}\eta(\mathbf{\hat{z}})\dot{\hat{z}}_{n}\) is the total system identification error. Since the unknown network weights \(\mathbf{w}_{1}^{*}\) and \(\mathbf{w}_{2}^{*}\) have not been estimated, the control strategy described below is designed. In the above calculations, only \(\dot{z}_{n}\) is estimated, while the derivative of the system \(\dot{z}_{n}\), i.e., \(\dot{\hat{z}}_{n}\), is unknown. Thus, a low-pass filter \(\frac{1}{1+\delta}\) is introduced where \(\theta\) is the filter constant. Using the inverse Laplace transform and without considering the influence of the initial value, the following formula can be obtained:  
\[\left\{ \begin{array}{l}\dot{z}_{n\theta}=\frac{\dot{z}_{n}}{1+\delta\dot{z}} \\ \dot{\hat{z}}_{n\theta}=\frac{\dot{z}_{n}-\dot{z}_{n\theta}}{\theta}\end{array} \right. \tag{16}\]  
letting the initial value of \(\dot{z}_{n\theta}\) be 0, i.e., \(\dot{z}_{n\theta}(0)=0\). Correspondingly, the low-pass filter can be applied to other variables, and the initial value is set to 0. **Lemma 1**. : _Consider the continuous function \(G(x)=G_{1}(x)G_{2}(x)\), where \(G_{1}(x)\) and \(G_{2}(x)\) are both continuous mappings.