## 4 Design of Controller
\begin{array}{l}\dot{V}\leq-\mathbf{r}^{\rm T}\mathbf{K}_{\rm DLr}\mathbf{r}+\mathbf{r}^{\rm T}( \mathbf{\Xi}_{\chi}+\mathbf{W}_{\rm S}^{\rm T}\mathbf{\tilde{\psi}})-\mathbf{r}^{\rm T}\mathbf{\tau}_ {\rm H}+(\mathbf{W}_{\rm S}^{\rm T}\mathbf{\hat{\psi}})^{\rm T}\mathbf{\Gamma}\mathbf{\hat{W}}_{ \rm C}^{\rm T}\mathbf{\hat{\psi}}-(\mathbf{\hat{W}}_{\rm S}^{\rm T}\mathbf{\hat{\psi}})^{ \rm T}\mathbf{\Gamma}\mathbf{\
u}_{1}\\ \leq-\mathbf{r}^{\rm T}\mathbf{K}_{\rm DLr}\mathbf{r}+\mathbf{r}^{\rm T}\mathbf{v}-\mathbf{r}^{\rm T}\bm {\tau}_{\rm H}\end{array}  \tag{40}\]  
where \(\mathbf{v}=\mathbf{\Xi}_{\chi}+\mathbf{W}_{\rm S}^{\rm T}\mathbf{\tilde{\psi}}+\text{diag}(\bm {W}_{\rm S}^{\rm T}\mathbf{\hat{\psi}})\mathbf{\hat{W}}_{\rm C}^{\rm T}\mathbf{\hat{\psi}}- \text{diag}(\mathbf{\hat{W}}_{\rm S}^{\rm T}\mathbf{\hat{\psi}})\mathbf{\
u}_{1}\). The Equation (40) can be rewrite:  
\[ \begin{array}{l}\dot{V}\leq-\frac{1}{2}\mathbf{r}^{\rm T}\big{[}2\mathbf{K}_{\rm DLr }+(\frac{2}{K_{\rm d}}-\frac{1}{\gamma^{2}})\mathbf{E}\big{]}\mathbf{r}-\frac{1}{2}( \frac{1}{\gamma}\mathbf{r}-\mathbf{\gamma}\mathbf{v})^{\rm T}(\frac{1}{\gamma}\mathbf{r}-\mathbf{ \gamma}\mathbf{v})+\frac{1}{2}\gamma^{2}\mathbf{v}^{\rm T}\mathbf{v}\\ \leq-\frac{1}{2}\mathbf{r}^{\rm T}\mathbf{Q}\mathbf{r}+\frac{1}{2}\gamma^{2}\mathbf{v}^{\rm T} \mathbf{v}\end{array}  \tag{41}\]  
where \(\mathbf{Q}=2\mathbf{K}_{\rm DLr}+(2/k_{\rm d}-1/\gamma^{2})\mathbf{E}\) is a positive definite diagonal matrix, \(\mathbf{E}\) is a unit matrix. By integrating Equation (41) to obtain:  
\[V(T)-V(0)\leq\frac{1}{2}\int_{0}^{T}\mathbf{r}^{\rm T}\mathbf{Q}\mathbf{r}dt+\frac{1}{2} \gamma^{2}\int_{0}^{T}\mathbf{v}^{\rm T}\mathbf{v}dt \tag{42}\]  
Then  
\[ \begin{array}{l}\int_{0}^{T}\mathbf{r}^{\rm T}\mathbf{Q}\mathbf{r}dt\leq\mathbf{r}(0)^{\rm T }\mathbf{M}_{\rm c}\mathbf{r}(0)+\sum\limits_{k=1}^{m}\frac{1}{k_{\rm max}}\mathbf{\tilde {w}}_{Sk}^{\rm T}(0)\mathbf{\tilde{w}}_{Sk}(0)+\sum\limits_{k=1}^{m}\frac{1}{k_{ \rm max}}\mathbf{\tilde{w}}_{Ck}^{\rm T}(0)\mathbf{\tilde{w}}_{Ck}(0)+\\ \frac{1}{k_{\rm c}}\mathbf{\tilde{\omega}}^{\rm T}(0)\mathbf{\tilde{\omega}}(0)+\frac {1}{k_{\rm f}^{\rm T}}\mathbf{\tilde{I}}^{\rm T}(0)\mathbf{\tilde{J}}(0)+\gamma^{2} \int_{0}^{T}\mathbf{v}^{\rm T}\mathbf{v}dt\end{array}  \tag{43}\]  
Equation (43) can be written in the following form through references [37,38]. \[\int_{0}^{T}\lVert\mathbf{r}\rVert^{2}Qdt\leq\gamma^{2}\int_{0}^{T}\lVert\mathbf{v} \rVert^{2}dt+\beta_{\rm H} \tag{44}\]  
where \(\beta_{\rm H}=\mathbf{r}(0)^{\rm T}\mathbf{M}_{\rm c}\mathbf{r}(0)+\sum\limits_{k=1}^{m} \frac{1}{k_{\rm max}}\mathbf{\tilde{w}}_{Sk}^{\rm T}(0)\mathbf{\tilde{w}}_{Sk}(0)+ \sum\limits_{k=1}^{m}\frac{1}{k_{\rm max}}\mathbf{\tilde{w}}_{Ck}^{\rm T}(0)\mathbf{ \tilde{w}}_{Ck}(0)+\frac{1}{k_{\rm w}}\mathbf{\tilde{\omega}}^{\rm T}(0)\mathbf{ \tilde{\omega}}(0)+\frac{1}{k_{\rm w}}\mathbf{\tilde{\omega}}^{\rm T}(0)\mathbf{ \tilde{\omega}}(0)+\frac{1}{k_{\rm w}}\mathbf{\tilde{\omega}}^{\rm T}(0)\mathbf{ \tilde{\omega}}(0)+\frac{1}{k_{\rm w}}\mathbf{\tilde{\omega}}^{\rm T}(0)\mathbf{ \tilde{\omega}}(0)\). Through references [37,38], It can be known that \(\mathbf{v}\) is bounded by Assumption 1 and Assumption 2, further \(\sup\limits_{v\in L_{2}[0,T]}\left\lVert Q^{1/2}r\right\rVert_{L2T}/ \left\lVert v\right\rVert_{L2T}\leq\gamma\) when the initial states of the hybrid system are all zero. According to the definition of \(\mathbf{Q}\) in Equation (41), it can be explained that the system is stable according to the \(H\infty\) control theory when \(k_{\rm d}=2\gamma^{2}\), and it can be seen that with the decrease of \(\gamma\), \(\lVert\mathbf{r}\rVert\) will also decrease. \(\square\)