## 3 Constrained Adaptive Tracking Control
### Adaptive Wiener Model Identification  
The scheme of the training Wiener model can easily identify the parameters \(a_{i}\), \(b_{j}\), \(c_{l}\) and \(\sigma_{m}\). The Wiener model output \(\hat{y}(k)\) is defined in the following form:  
\[\hat{y}(k)=\sum_{m=1}^{p}\hat{\sigma}_{m}\hat{x}^{m}(k) \tag{17}\]  
In the Wiener model, the function of the hidden layer output is described as:  
\[\hat{x}(k)= -\hat{a}_{1}\hat{x}(k-1)-\hat{a}_{2}\hat{x}(k-2)-\cdots-\hat{a}_{n _{a}}\hat{x}(k-n_{a})\] \[+\hat{b}_{0}u(k)+\hat{b}_{1}u(k-1)+\cdots+\hat{b}_{n_{b}}u(k-n_{b})\] \[+\hat{c}_{1}d(k-1)+\hat{c}_{2}d(k-2)+\cdots+\hat{c}_{n_{c}}d(k-n_{ c}) \tag{18}\] \[= -\sum_{i=1}^{n_{a}}\hat{a}_{i}\hat{x}(k-i)+\sum_{j=0}^{n_{b}}\hat {b}_{j}u(k-j)+\sum_{l=1}^{n_{c}}\hat{c}_{l}d(k-l)\]  
where the weights \(\hat{a}_{i}\), \(\hat{b}_{j}\), \(\hat{c}_{l}\) and \(\hat{\sigma}_{m}\) are related to the parameters \(a_{i}\), \(b_{j}\), \(c_{l}\) and \(\sigma_{m}\) in Equation (15) and (16), respectively.  
Figure 4: The structure of the Wiener model.  
\[\begin{split}\hat{\sigma}_{m}(k+1)&=\hat{\sigma}_{m}(k)+ \eta_{w1}\hat{e}(k)\frac{\partial\hat{y}(k)}{\partial\hat{\sigma}_{m}}\\ \hat{a}_{i}(k+1)&=\hat{a}_{i}(k)+\eta_{w1}\hat{e}(k )\frac{\partial\hat{y}(k)}{\partial\hat{a}_{i}}\\ \hat{b}_{j}(k+1)&=\hat{b}_{j}(k)+\eta_{w1}\hat{e}( k)\frac{\partial\hat{y}(k)}{\partial\hat{b}_{j}}\\ \hat{c}_{l}(k+1)&=\hat{c}_{l}(k)+\eta_{w1}\hat{e}( k)\frac{\partial\hat{y}(k)}{\partial\hat{c}_{l}}\end{split} \tag{27}\]  
According to these update rules, the output \(\hat{y}(k)\) is determined. At the same time, the parameters \(\hat{a}_{i}\), \(\hat{b}_{j}\), \(\hat{c}_{l}\) and \(\hat{\sigma}_{m}\) in the Wiener model can be tuned. Based on the above research, the convergence criterion is proposed to select an adequate learning rate.  
**Theorem 1**.: _If \(\eta_{w1}\) is chosen as:_  
\[0<\eta_{w1}<\frac{2}{\bar{\Psi}_{1}^{2}} \tag{28}\]  
_where \(\bar{\Psi}_{1}=\max_{k}\left\|\frac{\partial\hat{y}(k)}{\partial\bar{W}_{1}}\right\|\), the modeling error \(\hat{e}(k)\) converges asymptotically to zero under the update law (27)._  
Proof of Theorem 1.: Choosing a Lyapunov function as \(V_{1}(k)=\frac{1}{2}\hat{e}^{2}(k)\), from (21), we get:  
\[\begin{split}\Delta V_{1}(k)&=V_{1}(k+1)-V_{1}(k)\\ &=\Delta\hat{e}(k)\left(\hat{e}(k)+\frac{\Delta\hat{e}(k)}{2} \right)\\ &=\left(\frac{\partial\hat{e}(k)}{\partial W_{1}}\right)^{T} \Delta W_{1}\left[\hat{e}(k)+\frac{1}{2}\left(\frac{\partial\hat{e}(k)}{ \partial W_{1}}\right)^{T}\Delta W_{1}\right]\\ &=-\left(\frac{\partial\hat{y}(k)}{\partial W_{1}}\right)^{T} \Delta W_{1}\left[\hat{e}(k)-\frac{1}{2}\left(\frac{\partial\hat{y}(k)}{ \partial W_{1}}\right)^{T}\Delta W_{1}\right]\\ &=-\eta_{w1}\hat{e}^{2}(k)\left\|\frac{\partial\hat{y}(k)}{ \partial W_{1}}\right\|^{2}+\frac{1}{2}\eta_{w1}^{2}\hat{e}^{2}(k)\left\| \frac{\partial\hat{y}(k)}{\partial W_{1}}\right\|^{4}\\ &\leq-\frac{1}{2}\eta_{w1}\left(2-\eta_{w1}\bar{\Psi}_{1}^{2} \right)\hat{e}^{2}(k)\left\|\frac{\partial\hat{y}(k)}{\partial W_{1}}\right\|^ {2}\end{split}\]  
Notice that \(\Delta V_{1}(k)\) can be negative definite when \(\eta_{w1}\) satisfies Equation (28). Since \(V_{1}(k)\) is reducing and non-negative, this function is converging to the constant \(V_{1}^{\infty}\geq 0\), as \(k\rightarrow\infty\); therefore, \(\Delta V_{1}(k)\to 0\). It can be inferred that \(\lim_{k\rightarrow\infty}\hat{e}(k)=0\).  
Based on Equations (18) and (22), the Jacobian information algorithm can be derived as follows:  
\[\frac{\partial y(k)}{\partial u(k)}\cong\frac{\partial\hat{y}(k)}{\partial u(k )}=\frac{\partial\hat{y}(k)}{\partial\hat{x}(k)}\cdot\frac{\partial\hat{x}(k )}{\partial u(k)}=\hat{b}_{0}\sum_{m=1}^{p}m\hat{\sigma}_{m}\hat{x}^{m-1}(k) \tag{29}\]  
In this part, the presented modified MFACapproach can be used in the SOFC to obtain safe fuel utilization and meet operating constraint conditions under the condition that its current \(I\) and voltage \(V_{dc}\) can be available.