## 3 Adaptive Control of PEMFC Power Supply
### Adaptive SMC Design
\right]^{T}\), \(\alpha_{i}+m_{i}\beta_{i}^{2}>0\), \(\alpha_{v}+m_{v}\beta_{v}^{2}>0\), \(M_{i}\) and \(M_{v}\) can be regarded as disturbance caused by \(\Delta L\), \(\Delta C\), and \(\Delta R\). To make \(\mathbf{\Lambda}\) and **P** positive define matrix, \(\mathbf{\alpha}\), \(m\) and \(\mathbf{\beta}\), are satisfying:  
\[\mathbf{\alpha}m-\frac{1}{4}+m\mathbf{\beta}>0 \tag{26}\]  
According to Assumption 5, \(\Delta L\), \(\Delta C\), and \(\Delta R\) are limited by \(L_{m}\), \(C_{m}\), and \(R_{m}\), respectively. Thus, the maximum values of \(M_{i}\) and \(M_{v}\) are bounded by \(\overline{M}_{i}\) and \(\overline{M}_{v}\). Then:  
\[\dot{V}_{2,i}=-Z_{i}^{T}\Lambda Z_{i}-m_{i}n_{i}|s_{i}|+M_{i}\leq-m_{i}n_{i}|s_ {i}|+\overline{M}_{i} \tag{27}\]  
\[\dot{V}_{2,v}=-Z_{v}^{T}\mathds{P}Z_{V}-m_{v}n_{v}|s_{v}|+M_{v}\leq-m_{v}n_{v} |s_{v}|+\overline{M}_{v} \tag{28}\]  
\(m\) and \(n\) are limited by:  
\[\dot{V}_{2}\leq-mn|s|+\overline{M}\leq 0 \tag{29}\]  
By integrating Equation (29), yields:  
\[\int\limits_{0}^{\infty}(mn|s|\,-\,\overline{M})dx\leq\int\limits_{0}^{\infty} -\dot{V}_{2}dx=V_{2}(0)-V_{2}(\infty)<\infty \tag{30}\]  
where \(\underset{x\to 0}{lim}\int\limits_{0}^{t}(mn|s|-\overline{M})dx\) is bounded, according to Lemma 1, \(mn|s|-\overline{M}\) converges to zero when \(t\to\infty\). In other words, it converges to zero asymptotically. Therefore, the system is asymptotically stable. Furthermore, the adaptive laws are proposed to estimate parametric variations. \(\hat{L}\), \(\hat{C}\), and \(\hat{R}\) are defined as the estimations of \(L\), \(C\), and \(R\), respectively. Then, the model denoted by Equation (5) can be estimated by:  
\[\left\{ \begin{array}{l}\dot{i}_{L}=\xi_{1}^{T}\hat{\theta}\\ \dot{v}_{0}=\xi_{2}^{T}\hat{\theta}\end{array} \right. \tag{31}\]  
where \(\xi_{1}=\left[ \begin{array}{cccc}v_{fc}u-v_{o}&0&0\end{array} \right]\), \(\xi_{2}=\left[ \begin{array}{cccc}0&i_{L}&-v_{o}\end{array} \right]^{T}\), \(\hat{\theta}=\left[ \begin{array}{cccc}\frac{1}{L}&\frac{1}{C}&\frac{1}{RC} \end{array} \right]\). The second backstepping variables \(e_{2}\) is defined as:  
\[e_{2,i}=z_{2,i}-\xi_{1}^{T}\widetilde{\theta} \tag{32}\]  
\[e_{2,v}=z_{2,v}-\xi_{2}^{T}\widetilde{\theta} \tag{33}\]  
where \(\widetilde{\theta}=\theta-\hat{\theta}\). The sliding manifold \(w\) is selected as:  
\[w_{i}=\gamma_{i}z_{1,i}+e_{2,i} \tag{34}\]  
\[w_{v}=\gamma_{v}z_{1,v}+e_{2,v} \tag{35}\]  
where \(\gamma_{i}\) and \(\gamma_{v}\) are positive constants. Define the second Lyapunov function \(A_{2}\):  
\[A_{2i}=V_{1i}+\frac{1}{2}e_{i}^{2} \tag{36}\]  
\[A_{2v}=V_{1v}+\frac{1}{2}e_{v}^{2} \tag{37}\]  
To estimate \(\hat{\theta}\), define the third Lyapunov function \(A_{3}\):  
\[A_{3,i}=A_{2,i}+\frac{1}{2}\widetilde{\theta}^{T}\Psi_{i}{}^{-1}\widetilde{\theta} \tag{38}\]  
\[A_{3,v}=A_{2,v}+\frac{1}{2}\widetilde{\theta}^{T}\Psi_{v}{}^{-1}\widetilde{\theta} \tag{39}\]  
where \(\Psi_{i}\) and \(\Psi_{v}\) are positive definite matrix. To eliminate the estimation error \(\widetilde{\theta}\) existing in Equations (38) and (39), the adaptive laws \(\dot{\theta}\) are formulated by:  
\[\dot{\hat{\theta}}_{i}=\Psi_{i}\big{[}\varsigma_{1,i}+w_{i}\big{(}\gamma_{i} \xi_{1}+\alpha_{i}\xi_{1}-\hat{\theta}_{1}\xi_{2}\big{)}\big{]} \tag{40}\]  
\[\dot{\hat{\theta}}_{v}=\Psi_{v}\big{[}\varsigma_{1,v}+w_{v}\big{(}\gamma_{v} \xi_{2}+\alpha_{v}\xi_{2}-\hat{\theta}_{3}\xi_{2}+\hat{\theta}_{2}\xi_{1}\big{)} \big{]} \tag{41}\]  
Then, the control law is calculated by:  
\[ \begin{array}{l}\dot{u}_{i}=-\frac{1}{v_{fc}\delta_{1}}\big{\{}\gamma_{i}(- \alpha_{i}z_{1,i}+e_{2,i})\,+u_{i}\hat{\wp}(u_{i}i_{L})\hat{\theta}_{1}-\hat{ \varphi}_{1}\xi_{2}^{T}\hat{\theta}_{1}\\ +\alpha_{i}\xi_{1}^{T}\hat{\theta}+\xi_{1}^{T}\hat{\theta}-\alpha_{i}\dot{i}_ {L,ref}-\ddot{i}_{L,ref}+a_{i}[w_{i}+b_{i}sign(w_{i})]\Big{\}}\end{array}  \tag{42}\]  
\[