## 2 Problem Formulation For SOFC
Therefore, one of the main control objectives is to eliminate the negative impact due to changes of the current load to obtain a stable output voltage. 2. Based on the stationary voltage-current characteristics of the SOFC (see Figure 3), it is shown that the behavior of the SOFC reflects the typical nonlinear characteristic over a wide operating regime. In addition, the nonlinear characteristic of system responses in the SOFC system at low- and high-current loads is obvious. Even under the condition of an overloaded current, the operating stack voltage may deteriorate rapidly, which can also reflect the strong nonlinear characteristic of SOFCs. Thus, the nonlinear control strategy must be better than the linear method for SOFCs. In the design of the nonlinear controller, \(q_{f}\), \(V_{dc}\) and \(I\) need to be defined as the control input \(u\), control output \(y\) and measurable disturbance \(d\), respectively. This input-output relationship can be redefined as an unknown discrete-time nonlinear autoregressive exogenous model (NARX): \[ \begin{split} y(k)=& f\left(y(k-1),\cdots,y(k-n_{a} ),u(k),u(k-1),\cdots,\right.\\ &\left.u(k-n_{b}),d(k-1),\cdots,d(k-n_{c})\right)\end{split} \] (7) where \(n_{a}\), \(n_{b}\) and \(n_{c}\) denote the orders of dynamics. Moreover, \(n_{a}\geq n_{b}\), \(n_{a}\geq n_{c}\). 3. In practice, the control input \(q_{f}=u\) does not change rapidly in a short time with inertia existing in the SOFC control system. For this SOFC system, the control inputs must be constrained by the amplitude and rate conditions, which can be written as: \[q_{f\min}\leq u(t)\leq q_{f\max},\ \ \dot{q}_{f\min}\leq\dot{u}(t)\leq\dot{q}_{f \max}\] (8) In addition to the constraints of the control input, fuel utilization is another important operating condition that impacts the performance of an SOFC. Thus, the fuel utilization needs to be maintained within a limited scope, which can be described as: \[\rho=\frac{q_{\text{H}_{2}}^{in}-q_{\text{H}_{2}}^{o}}{q_{\text{H}_{2}}^{in}}= \frac{q_{\text{H}_{2}}^{r}}{q_{\text{H}_{2}}^{in}}=\frac{2K_{r}I}{q_{\text{H} _{2}}^{in}}\] (9) The expected utilization must be from \(\rho_{\min}=0.7\) to \(\rho_{\max}=0.9\). The hydrogen flow should be within \(\left[q_{\text{H}_{2}\min}^{in},q_{\text{H}_{2}\max}^{in}\right]\) from Equation (9) to maintain safe utilization. Then, the result \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(T_{0}\) & 1273 & \(\tau_{\text{H}_{2}}\) & 26.1 \\ \(F_{0}\) & 96,485 & \(\tau_{\text{H}_{\text{O}}}\) & 78.3 \\ \(R_{0}\) & 8.314 & \(\tau_{\text{O}_{2}}\) & 2.91 \\ \(E_{0}\) & 1.18 & \(\tau_{\text{H}-\text{O}}\) & 1.145 \\ \(N_{0}\) & 384 & \(r\) & 0.126 \\ \(K_{r}\) & \(0.996\times 10^{-3}\) & \(\tau_{f}\) & 5 \\ \(\text{K}_{\text{H}_{2}}\) & \(8.32\times 10^{-6}\) & \(\dot{\beta}\) & 0.05 \\ \(\text{K}_{\text{H}_{2}\text{O}}\) & \(2.77\times 10^{-6}\) & \(\beta\) & 0.11 \\ \(\text{K}_{\text{O}_{2}}\) & \(2.49\times 10^{-5}\) & \(I_{L}\) & 800 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the SOFC system model. \(\left[\hat{q}_{\text{H}_{2}\min}^{in},q_{\text{H}_{2}\max}^{in}\right]\) can be employed to compute the constrained condition \(\left[\hat{q}_{f\min},\hat{q}_{f\max}\right]\) of the fuel flow. Finally, the ultimate dynamic amplitude constraint \(\left[\hat{q}_{f\min},\hat{q}_{f\max}\right]\) can be obtained as:  
\[ \begin{split}\bar{q}_{f\min}&=\max\left(q_{f\min}, \hat{q}_{f\min}\right)\\ \bar{q}_{f\max}&=\min\left(q_{f\max},\hat{q}_{f\max} \right)\end{split}  \tag{10}\]  
Therefore, the following relationship can be achieved as:  
\[\bar{q}_{f\min}\leq u(k)\leq\bar{q}_{f\max} \tag{11}\]  
Due to the final dynamic magnitude constraint, the change rate of \(u(t)\) can be approximated via the first-order Euler approximation, which is written as:  
\[\dot{u}(t)=\frac{u(k)-u(k-1)}{T_{s}} \tag{12}\]  
where \(T_{s}\) denotes the sampling time.