## 3 Controller Design
### Sliding-Mode Control  
For the third case, the power control was realized using a sliding-mode controller. In general, the sliding-mode control is split into two phases [19]. First, the trajectory approaches a switching manifold  
Figure 5: Block diagram of an IMC controller. in finite time. Afterwards, the trajectory slides on the switching manifold along the desired trajectory, which is the second phase. Finally, the system remains stable on this trajectory or eventually approaches the equilibrium position. The sliding-mode enables robustness against variations of the parameters in the plant and matched disturbances influencing the system dynamics via the same channel as the physical control signal. However, high frequency switching could potentially trigger chattering, which induces premature wear of mechanical actuators and may excite unmodeled dynamics [20; 21; 22]. For the electric power control with a sliding-mode controller, the plant of the SOFC was approximated by the third-order system  
\[G_{\text{S,PT3}}(s)=\frac{b}{s^{3}+a_{33}\times s^{2}+a_{32}s+a_{31}}=\frac{71. 28}{s^{3}+0.06s^{2}+5.43\times 10^{-3}s+2.15\times 10^{-6}}\;\;. \tag{11}\]  
The states  
\[\mathbf{x}= \begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix} = \begin{bmatrix}\hat{P}_{\text{el}}\\ \hat{P}_{\text{el}}\\ \hat{P}_{\text{el}}\end{bmatrix}  \tag{12}\]  
were defined by the electric power and its estimated time derivatives, see the observer design in Section 3.5. Thereby, a linear controllable canonical form  
\[ \begin{array}{rl}\dot{x}_{1}&=x_{2}\\ \dot{x}_{2}&=x_{3}\\ \dot{x}_{3}&=-a_{31}x_{1}-a_{32}x_{2}-a_{33}x_{3}+bu\end{array}  \tag{13}\]  
can be formulated with the input variable \(u=\dot{n}_{\text{H2,SM}}\). Here, the trajectory tracking errors are defined as  
\[ \begin{array}{rl}\dot{x}_{1}&=x_{1,\text{d}}-x_{1}\\ \dot{x}_{2}&=x_{2,\text{d}}-x_{2}\\ \dot{x}_{3}&=x_{3,\text{d}}-x_{3}\end{array}  \tag{14}\]  
whereby the reference signal for the complete state vector  
\[\mathbf{x}_{\text{d}}= \begin{bmatrix}x_{1,\text{d}}\\ x_{2,\text{d}}\\ x_{3,\text{d}}\end{bmatrix} = \begin{bmatrix}P_{\text{el,d}}\\ P_{\text{el,d}}\\ \hat{P}_{\text{el,d}}\end{bmatrix}  \tag{15}\]  
is given by the desired electric power parameterized in terms of sufficiently often continuously differentiable Bernstein polynomials [23]. With the new resulting system of differential equations  
\[\begin{array}{rl}\dot{x}_{1}&=\dot{x}_{1,\text{d}}-\dot{x}_{1}=\dot{x}_{1, \text{d}}-x_{2}\\ \dot{\dot{x}}_{2}&=\dot{x}_{2,\text{d}}-\dot{x}_{2}=\dot{x}_{2,\text{d}}-x_{3} \\ \dot{\dot{x}}_{3}&=\dot{x}_{  
Under the assumption that the sliding-mode is reached with a perfect trajectory tracking  
\[S(\tilde{\mathbf{x}})=0\quad\text{and}\quad\dot{S}(\hat{\mathbf{x}})=0\enspace, \tag{19}\]  
the coefficients \(\alpha_{i}>0\) with \(i\in\{0,1\}\) are determined by necessary and sufficient criteria for Hurwitz stability. Otherwise, while the trajectory is still approaching the switching manifold in the first phase characterized by  
\[S(\tilde{\mathbf{x}})\
eq 0\enspace, \tag{20}\]  
a quadratic Lyapunov function  
\[V=\frac{1}{2}S(\tilde{\mathbf{x}})^{2}>0 \tag{21}\]  
with  
\[\dot{V}=S(\tilde{\mathbf{x}})\cdot\dot{S}(\hat{\mathbf{x}})<0 \tag{22}\]  
is employed to guarantee the convergence of \(S(\tilde{\mathbf{x}})\) toward zero with the sliding condition  
\[\dot{V}=S(\tilde{\mathbf{x}})\cdot\dot{S}(\hat{\mathbf{x}})<-\eta\cdot S( \tilde{\mathbf{x}})\cdot\text{sign}(S(\tilde{\mathbf{x}}))\enspace. \tag{23}\]  
**Remark 4**.