## 3 Design of Fixed-Time Consensus Protocol
### Disturbance Observer Design  
This paper designs a TISMO to estimate the comprehensive disturbances in the system. The TISMO is established as follows:  
\[\left\{\begin{array}{l}\dot{\omega}_{r,i}=-\frac{B_{i}}{\dot{I}_{i}}\hat{\omega }_{r,i}+\frac{1.5n_{p}{}^{2}\hat{\varphi}_{r,i}}{I_{i}}u_{i}-\frac{n_{p}}{\dot{ I}_{i}}\hat{T}_{L,i}-\frac{n_{p}}{\dot{I}_{i}}h\\ \hat{T}_{L,i}=lh\end{array}\right. \tag{20}\]  
where \(u_{i}\) represents the system control input; \(T_{L,i}\) represents the load torque; \(\hat{T}_{L,i}\) is the estimated value of \(T_{L,i}\); \(l\) is a constant, and \(h\) is the yet-to-be-designed sliding mode control input term.  
Define the estimation error of the system as follows:  
\[\left\{\begin{array}{l}e_{\omega,i}=\omega_{r,i}-\hat{\omega}_{r,i}\\ e_{T_{L,i}}=T_{L,i}-\hat{T}_{L,i}\end{array}\right. \tag{21}\]  
where \(\hat{\omega}_{r,i}\) is the estimated value of \(\omega_{r,i}\); \(e_{\omega,i}\) and \(e_{T_{L,i}}\) are estimated errors.  
Obtained from (20) and (21)  
\[\dot{e}_{\omega,i}=-\frac{B_{i}}{\dot{I}_{i}}e_{\omega,i}-\frac{n_{p}}{\dot{I} _{i}}e_{T_{L,i}}+\frac{n_{p}}{\dot{I}_{i}}h \tag{22}\]  
Define the TISMO surface \(S_{0}\) as follows:  
\[S_{0}=e_{\omega,i}+\int_{0}^{t}\left(k_{4}e_{\omega,i}+k_{S}sign^{\gamma}(e_{ \omega,i})\right)d_{\tau} \tag{23}\]  
\(k_{4},k_{5}\) represent the given positive constant; \(\gamma\) is a positive fraction less than 1; \(sign(e_{\omega,i})\) represents the sign function, defined as:  
\[sign(e_{\omega,i})=\left\{\begin{array}{l}1,e_{\omega,i}>0\\ 0,\ e_{\omega,i}=0\\ -1,\ e_{\omega,i}<0\end{array}\right.\]  
The designed sliding mode reaching law is as follows:  
\[\dot{S}_{0}=-\omega_{1}S_{0}-\omega_{2}sign^{\sigma}(S_{0})-\omega_{3}sign^{ \delta}(S_{0}) \tag{24}\]  
\(\omega_{1}>0,\omega_{2}>0,\omega_{3}>0;0<\sigma<1;1<\delta\).  
Choose the Lyapunov function \(V_{1}\) as follows:  
\[V_{1}=\frac{1}{2}{S_{0}}^{2} \tag{25}\]  
The derivation \(V_{1}\) can be obtained as follows:  
\[\begin{array}{ll}\dot{V}_{1}&=S_{0}\dot{S}_{0}\\ &=S_{0}(-\omega_{1}S_{0}-\omega_{2}sign^{\sigma}(S_{0})-\omega_{3}sign^{\delta }(S_{0}))\\ &=-\omega_{1}{S_{0}}^{2}-\omega_{2}|S_{0}|^{\sigma+1}-\omega_{3}|S_{0}|^{ \delta+1}\leq 0\end{array} \tag{26}\]  
Obtained from (20) and (26)  
\[h=-\frac{J_{i}}{n_{p}}(-\frac{B_{i}}{\dot{I}_{i}}e_{\omega,i}+k_{4}e_{\omega,i }+k_{5}sign^{\gamma}(e_{\omega,i})+\omega_{1}S_{0}+\omega_{2}sign^{\sigma}(S_{ 0})+\omega_{3}sign^{\delta}(S_{0})) \tag{27}\]  
The design of the TISMO is as follows:\[\left\{\begin{array}{l}e_{\omega,j}=\omega_{r,j}-\hat{\omega}_{r,j}\\ S_{0}=e_{\omega,j}+\int_{0}^{t}(k_{4}e_{\omega,i}+k_{5}sign^{\gamma}(e_{\omega,i }))d_{\tau}\\ h=-\frac{j_{i}}{n_{p}}(-\frac{B_{i}}{j_{i}}i_{\omega,i}+k_{4}e_{\omega,i}+k_{5} sign^{\gamma}(e_{\omega,i})+\omega_{1}S_{0}+\omega_{2}sign^{\sigma}(S_{0})+ \omega_{3}sign^{\delta}(S_{0}))\\ \hat{\omega}_{r,i}=-\frac{B_{i}}{j_{i}}\hat{\omega}_{r,j}+\frac{1.5n_{p}^{2}\Psi _{f,i}}{J}u_{i}-\frac{n_{p}}{J}\hat{T}_{L,i}-\frac{n_{p}}{J}h_{\omega,i}\\ \hat{T}_{L,j}=lh\\ f_{i}=-\frac{n_{p}}{J_{i}}\hat{T}_{L,i}-\frac{B_{i}}{j_{i}}\hat{\omega}_{r,j} \end{array}\right. \tag{28}\]