## 3 Adaptive Control of PEMFC Power Supply
### Adaptive SMC Design  
In order to address the nonlinearity of PEMFC, the control plant backs up to subsystems \(z_{1}\) and \(z_{2}\), which is combing with SMC to formulate backstepping SMC. **Assumption 6**. : _The desired value \(i_{L,ref}\) and \(v_{o,ref}\) are bounded and have \(n\)-order derivative. Meanwhile, \(I_{L,ref}=\left[ \begin{array}{cccc}i_{L,ref}&\dot{i}_{L,ref}&\cdots&i_{L,ref} ^{(n)}\end{array} \right]^{T}\) and \(V_{o,ref}=\left[ \begin{array}{cccc}v_{o,ref}&\dot{v}_{o,ref}&\cdots&v_{o,ref} ^{(n)}\end{array} \right]^{T}\) are bounded._  
**Remark 1**. : _If there is an \(i\) under a certain parameter, it represents the parameter used in current regulating, and the same as \(v\), i.e., \(c_{i}\) is the parameter \(c\) used in current regulating and \(p\) is the parameter \(p_{v}\) used in voltage regulating._  
**Lemma 1**. : _If f(x) is a uniformly continuous function and \(\lim_{t\rightarrow\infty}\int_{o}^{t}f(x)dx\) exists, then f(x) converges to zero asymptotically._  
The tracking errors \(z_{1}\) are defined as below, where \(z_{1,i}\) and \(z_{1,v}\) denote current error and voltage error, respectively:  
\[z_{1,i}=i_{L}-i_{L,ref} \tag{9}\]  
\[z_{1,v}=v_{o}-v_{o,ref} \tag{10}\]  
To ensure the stability, define the first Lyapunov function \(V_{1:}\)  
\[V_{1,i}=\frac{1}{2}z_{1,i}^{2} \tag{11}\]  
\[V_{1,v}=\frac{1}{2}z_{1,v}^{2} \tag{12}\]  
The virtual control values \(p\) is represented by:  
\[p_{i}=\alpha_{i}z_{1,i} \tag{13}\]  
\[p_{v}=\alpha_{v}z_{1,v} \tag{14}\]  
where \(\alpha_{i}\) and \(\alpha_{v}\) are positive constants. Via the virtual control values, the second backstepping variables \(z_{2}\) can be obtained by:  
\[z_{2,i}=\dot{i}_{L}-\dot{i}_{L,ref}+p_{i} \tag{15}\]  
\[z_{2,v}=\dot{v}_{o}-\dot{v}_{o,ref}+p_{v} \tag{16}\]  
The derivative of \(V_{1}\) is expressed as:  
\[\dot{V}_{1,i}=-\alpha_{i}{z_{1,i}}^{2}+{z_{1,i}}{z_{2,i}} \tag{17}\]  
\[\dot{V}_{1,v}=-\alpha_{v}{z_{1,v}}^{2}+{z_{1,v}}{z_{2,v}} \tag{18}\]  
Then, the sliding manifolds \(s\) are selected as:  
\[z_{1,v}=v_{o}-v_{o,ref} \tag{10}\]0  
\[z_{1,v}=v_{o}-v_{o,ref} \tag{10}\]1  
where \(\beta_{i}\), \(\beta_{v}\) are positive constants. To guarantee the stability and to maintain \(i_{L}\) and \(v_{o}\) on the sliding manifolds, the second Lyapunov function \(V_{2}\) are defined as:  
\[z_{1,v}=v_{o}-v_{o,ref} \tag{10}\]2  
\[z_{1,v}=v_{o}-v_{o,ref} \tag{10}\]3  
The control law \(u\) is formulated by:  
\[ \begin{array}{l}\dot{u}_{i}=-\frac{L_{0}}{\wp_{fc}}\{\beta_{i}(-\alpha_{i}z_ {1,i}+z_{2,i})\,-\,\frac{1}{L_{0}\zeta_{0}}+\frac{1}{L_{0}\zeta_{0}R_{0}}\\ \,+\frac{u_{i}V_{i}\zeta_{0}-\alpha_{i}}{L_{0}}-\alpha_{i}\dot{i}_{L,ref}-\ddot {i}_{L,ref}+m_{i}[s_{i}+n_{i}sign(s_{i})]\}\end{array}  \tag{23}\]  
\[ \begin{array}{l}u_{v}=-\frac{L_{0}\zeta_{0}}{\wp_{fc}}\{\wp_{v}(-\alpha_{v} z_{1,v}+z_{2,v})\,-\,\frac{1}{L_{0}\zeta_{0}}-\frac{1}{R_{0}\zeta_{0}^{2}}+\frac{1}{R_ {0}^{2}\zeta_{0}^{2}}\\ \,+\frac{\alpha_{i}\dot{i}_{L}}{\zeta_{0}}-\frac{1}{R_{0}\zeta_{0}}-\alpha_{v }\dot{v}_{o,ref}-\ddot{v}_{o,ref}+m_{v}[s_{v}+n_{i}sign(s_{v})]\}\end{array}  \tag{24}\]  
where \(m\) and \(n\) are positive constants. The derivative of \(V_{2}\) can be obtained by:  
\[ \begin{array}{l}\dot{V}_{2,i}=-\alpha_{i}z_{1,i}^{2}+z_{1,i}z_{2,i}-m_{i}s_{ i}^{2}-m_{i}n_{i}|s_{i}|+M_{i}\\ =-Z_{i}^{T}\left[\begin{array}{cc}\alpha_{i}+m_{i}\beta_{i}^{2}&-\frac{1}{2} +m_{i}\dot{\beta}_{i}\\ -\frac{1}{2}+m_{i}\dot{\beta}_{i}&m_{i}\end{array}\right]Z_{i}-m_{i}n_{i}|s_{i} |+M_{i}\\ =-Z_{i}^{T}\Lambda Z_{i}-m_{i}n_{i}|s_{i}|+M_{i}\end{array}  \tag{25}\]  
Similarly, \(\dot{V}_{2,v}=-Z_{v}^{T}\mathds{P}Z_{V}-m_{v}n_{v}|s_{v}|+M_{v}\),  
where \(\textbf{P}=\left[ \begin{array}{cc}\alpha_{v}+m_{v}\beta_{v}^{2}&-\frac{1}{ 2}+m_{v}\beta_{v}\\ -\frac{1}{2}+m_{v}\beta_{v}&m_{v}\end{array} \right]\), \(Z_{i}=\left[ \begin{array}{cc}z_{1,i}&z_{2,i}\end{array} \right]^{T}\), \(Z_{v}=\left[ \begin{array}{cc}z_{1,v}&z_{2,v}\end{array}