## 3 Design of Fixed-Time Consensus Protocol
### Design of Consistency Protocol  
The fixed-time consensus protocol is designed as follows:  
\[ \begin{array}{ll}u_{i}&=\frac{k_{1}}{\hat{\theta}_{i}}\left[\sum\limits_{j=1}^{ N}a_{ij}(\omega_{r,j}-\omega_{r,i})\right]^{\frac{p}{q}}+\frac{k_{2}}{\hat{ \theta}_{i}}\left[\sum\limits_{j=1}^{N}a_{ij}(\omega_{r,j}-\omega_{r,i}) \right]^{\frac{r}{s}}\\ &+\frac{k_{3}}{\hat{\theta}_{i}}\left[\sum\limits_{j=1}^{N}a_{ij}(\omega_{r,j} -\omega_{r,i})-m_{i}(\omega_{r,i}-\omega_{r,0})\right]-\frac{\hat{f}_{i}}{ \hat{\theta}_{i}}\end{array}  \tag{9}\]  
\(k_{1},k_{2},k_{3}>0\) represent the control gain and \(A=\left[a_{ij}\right]\in R^{N\times N}\) represents the adjacency matrix of the information transmission network among the MASs. \(p,q,r,s\) are all controller gains, they are all positive odd numbers, and they satisfy \(p<q,r>s\). **Theorem 1**. : _In an undirected connected multi-agent network, under the influence of consensus protocol (9), for any solution satisfying conditions on_  
\[\dot{V}\leq-aV^{\xi}-bV^{\eta}-cV\ \ \ \ a,b,c>0,0<\xi<1,\eta>1\]  
_it enables the multi-PMSM system (1) to achieve speed synchronization control within a fixed time, and the fixed convergence time is:_  
\[T_{\max}\triangleq\frac{1}{c(1-\xi)}\ln(1+\frac{c}{a})+\frac{1}{c(\eta-1)}\ln (1+\frac{c}{b})\]  
Proof. : Define the system error equation as:  
\[ \begin{array}{ll}\dot{e}_{r,i}&=\hat{\theta}_{i}u_{i}+f_{i}\\ &=k_{1}\left[\sum\limits_{j=1}^{N}a_{ij}(\omega_{r,j}-\omega_{r,j})\right]^{ \frac{p}{q}}+k_{2}\left[\sum\limits_{j=1}^{N}a_{ij}(\omega_{r,j}-\omega_{r,i} )\right]^{\frac{r}{s}}\\ &+k_{3}\sum\limits_{j=1}^{N}a_{ij}(\omega_{r,j}-\omega_{r,i})-k_{3}m_{i}( \omega_{r,i}-\omega_{r,0})\end{array}  \tag{10}\]  
Define the semi-positive definite function as:  
\[V=\frac{1}{2}e^{T}L(A)e,A=\left[a_{ij}\right]_{N\times N} \tag{11}\]  
Figure 2: Schematic diagram of multi-PMSM control method. Based on the symmetry of \(A\), it can be determined that:  
\[\frac{\partial V}{\partial e_{r,i}}=-\sum\limits_{j=1}^{N}a_{ij}(e_{r,j}-e_{r,i}) \tag{12}\]  
The derivative of \(V\) versus time is:  
\[ \begin{array}{ll}\dot{V}&=\sum\limits_{i=1}^{N}\frac{\partial V}{\partial e_{ r,i}}\dot{e}_{r,i}\\ &=-k_{1}\sum\limits_{i=1}^{N}\left[\sum\limits_{j=1}^{N}a_{ij}(e_{r,j}-e_{r,i })\right]^{\frac{q+p}{q}}-k_{2}\sum\limits_{i=1}^{N}\left[\sum\limits_{j=1}^{ N}a_{ij}(e_{r,j}-e_{r,i})\right]^{\frac{s+r}{s}}\\ &-k_{3}\sum\limits_{i=1}^{N}\left[\sum\limits_{j=1}^{N}a_{ij}(e_{r,j}-e_{r,i })\right]^{2}-k_{3}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}a_{ij}(e_{r,j}-e_ {r,i})m_{i}e_{r,i}\\ \end{array}  \tag{13}\]  
\(k_{3}\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}(e_{r,j}-e_{r,i})m_{i}e_{r,i}=-k_{3}e^ {T}L(M)e\), \(M=diag\{m_{1},m_{2},\cdots m_{n}\}\), and \(e^{T}L(M)e/e^{T}e\geq\lambda_{\min}L(M)\). According to \(\lambda_{\min}L(A)\leq e^{T}L(A)e/e^{T}e\leq\lambda_{\max}L(A)\), one has:  
\[ \begin{array}{ll}-k_{3}e^{T}L(M)e&\leq-k_{3}\lambda_{\min}L(M)e^{T}e\\ &=-k_{3}\lambda_{\min}L(M)\frac{e^{T}L(A)e}{\lambda_{\max}L(A)}\\ &=-k_{3}\frac{\lambda_{\min}L(M)}{\lambda_{\max}L(A)}2V\leq 0\end{array}  \tag{14}\]  
According to Lemma 1, if \(p<q,r>s\geq 0\), then \(0<(q+p)/2q<1,\,(r+s)/2s>1\). The following inequalities can be obtained:  
\[ \begin{array}{ll}\dot{V}\leq-k_{1}\left(\sum\limits_{i=1}^{N}\left(\sum \limits_{j=1}^{N}a_{ij}(e_{r,j}-e_{r,i})\right)^{2}\right)^{\frac{q+p}{2q}}\\ -k_{2}N\frac{s-r}{2s}\left(\sum\limits_{i=1}^{N}\left(\sum\limits_{j=1}^{N}a_ {ij}(e_{r,j}-e_{r,i})\right)^{2}\right)^{\frac{s+r}{2s}}-k_{3}\sum\limits_{i= 1}^{N}\left(\sum\limits_{j=1}^{N}a_{ij}(e_{r,j}-e_{r,i})\right)^{2}\end{array}  \tag{15}\]  
There exists a semi-positive matrix \(Q\in R^{N\times N}\) such that \(L(A)=Q^{T}Q\). Using Lemma 2, one has  
\[ \begin{array}{ll}\frac{\sum\limits_{i=1}^{N}\left(\sum\limits_{j=1}^{N}a_ {ij}(e_{r,j}-e_{r,i})\right)^{2}}{V}=\frac{e^{T}L(A)^{T}L(A)e}{\frac{1}{2}e^{ T}L(A)e}\\ \\ =\frac{2e^{T}Q^{T}QQ^{T}Qe}{e^{T}Qe}\geq 2\lambda_{2}L(A)\end{array}