# Finite-Time Consensus Tracking Control for Speed Sensorless Multi-Motor Systems
## 4 Finite-Time Consensus Tracking Protocol Design
 \tag{21}\]  
Step 2: The following Lyapunov function is constructed:  
\[V=V_{0}+\sum_{i=1}^{n}V_{1i} \tag{22}\]  
where \(V_{1i}=\frac{1}{(2-1/p)2^{1-1/p}q_{2}^{1+p}}\int_{\omega_{i}^{*}}^{\varpi_{i }}(\tau^{p}-\omega_{i}^{*p})^{2-1/p}d\tau\), and \(V_{1i}\) is a positive definite function. Finding the derivation of \(V_{1i}\) yields:  
\[ \begin{split}\dot{V}_{1i}&=\frac{(\bar{k}_{i}\omega _{i}+\bar{b}_{i}u_{i}+d_{i})}{(2-1/p)2^{1-1/p}q_{2}^{1+p}}\rho_{i}^{2-1/p}-\\ &\frac{1}{2^{1-1/p}q_{2}^{1+p}}(\frac{d\omega_{i}^{*p}}{dt})\int_ {\omega_{i}^{*}}^{\varpi_{i}}(\tau^{p}-\omega_{i}^{*p})^{1-1/p}d\tau.\end{split}  \tag{23}\]\[ \begin{split}\text{Because}\ \frac{d(\omega_{i}^{*p})}{dt}=-q_{2}^{p}\sum \limits_{j\in N_{i}}a_{ij}(\omega_{i}-\omega_{j})+a_{i0}(\omega_{i}-\omega_{0}) \leq q_{2}^{p}(\alpha|\tilde{\omega}_{i}|+\gamma\sum\limits_{m=1}^{n}|\omega_{ m}|),\text{where}\\ \omega_{m}=\omega_{i}-\omega_{0},\ \alpha=\max\{\sum\limits_{j\in N_{i} }a_{ij}\},\ \gamma=\max\{a_{ij},\forall i,j\},\ \text{and}\ \int_{\omega_{i}^{*}}^{\tilde{\omega}_{i}}(\tau p- \omega_{i}^{*p})^{1-1/p}d\tau\leq\left|\tilde{\omega}_{i}-\omega_{i}^{*}|| \rho_{i}|^{1-1/p},\text{ it is substituted in Equation (23). Thus,}\\ \hat{V}_{1i}\leq\frac{(\bar{k}_{i}\omega_{i}+\bar{b}_{i}u_{i}+d _{i})}{(2-1/p)2^{1-1/p}q_{2}^{1+p}}\rho_{i}^{2-1/p}+\\ \frac{(\alpha|\tilde{\omega}_{i}|+\gamma\sum\limits_{m=1}^{n}| \omega_{m}|)}{2^{1-1/p}q_{2}}|\tilde{\omega}_{i}-\omega_{i}^{*}||\rho_{i}|^{1 -1/p}.\end{split}  \tag{24}\]  
According to Lemmas 2 and 3, the following inequalities are obtained:  
\[ \begin{split}|\tilde{\omega}_{i}||\tilde{\omega}_{i}-\omega_{i}^ {*}||\rho_{i}|^{1-1/p}&\leq 2^{1-1/p}|\tilde{\omega}_{i}||\rho_{i}|^{ 1/p}|\rho_{i}|^{1-1/p}\\ &=2^{1-1/p}|\tilde{\omega}_{i}||\rho_{i}|\\ &\leq 2^{1-1/p}(|\tilde{\omega}_{i}-\omega_{i}^{*}|+|\omega_{i}^{*}|) |\rho_{i}|\\ &\leq 2^{1-1/p}(2^{1-1/p}|\rho_{i}|^{1+1/p}+q_{2}|\rho_{i}||\sigma_{ i}|^{1/p})\\ &\leq 2^{1-1/p}(2^{1-1/p}|\rho_{i}|^{1+1/p}+\\ &\frac{q_{2}}{1+p}|\sigma_{i}|^{1+1/p}+\frac{q_{2}p}{1+p}|\rho_{ i}|^{1+1/p})\\ \end{split}  \tag{25}\]  
\[ \begin{split}|\omega_{m}||\tilde{\omega}_{i}-\omega_{i}^{*}|| \rho_{i}|^{1-1/p}&\leq 2^{1-1/p}|\omega_{m}||\rho_{i}|^{1/p}|\rho_{ i}|^{1-1/p}\\ &=2^{1-1/p}|\omega_{m}||\rho_{i}|\\ &\leq 2^{1-1/p}(|\omega_{m}-\omega_{m}^{*}|+|\omega_{m}^{*}|)|\rho_{ i}|\\ &\leq 2^{1-1/p}(2^{1-1/p}|\rho_{m}|^{1/p}+q_{2}|\sigma_{m}|^{1/p})| \rho_{i}|\\ &\leq 2^{1-1/p}((2^{1-1/p}+q_{2})\frac{p}{1+p}\times|\rho_{i}|^{1+1/p }+\\ &\frac{2^{1-1/p}}{1+p}|\rho_{m}|^{1+1/p}+\frac{q_{2}}{1+p}|\sigma _{m}|^{1+1/p}).\end{split}  \tag{26}\]  
Substituting (25) and (26) into (24) yields the following equation:  
\[ \begin{split}\hat{V}_{1i}\leq&\frac{1}{q_{2}}(\alpha(2 ^{1-1/p}|\rho_{i}|^{1+1/p}+\frac{q_{2}}{1+p}|\sigma_{i}|^{1+1/p}+\frac{q_{2}p}{ 1+p}|\rho_{i}|^{1+1/p})+\\ &\gamma\sum\limits_{m=1}^{n}(\frac{2^{1-1/p}}{1+p}|\rho_{m}|^{1+ 1/p}+\frac{q_{2}}{1+p}|\sigma_{m}|^{1+1/p})+\\ & np\gamma\frac{(2^{1-1/p}+q_{2})}{1+p}|\rho_{i}|^{1+1/p})+\\ &\frac{1}{(2-1/p)2^{1-1/p}q_{2}^{1+p}}\rho_{i}^{2-1/p}(\bar{k}_{i }\omega_{i}+\bar{b}_{i}u_{i}+d_{i})\\ &\leq\frac{q_{3}}{q_{2}}|\rho_{i}|^{1+1/p}+\frac{\alpha}{1+p}| \sigma_{i}|^{1+1/p}+q_{4}\sum\limits_{m=1}^{n}|\rho_{m}|^{1+1/p}+\\ &\frac{\gamma}{1+p}\sum\limits_{m=1}^{n}|\sigma_{m}|^{1+1/p}+ \frac{(\bar{k}_{i}\omega_{i}+\bar{b}_{i}u_{i}+d_{i})}{(2-1/p)2^{1-1/p}q_{2}^{1 +p}}\rho_{i}^{2-1/p}.\end{split}  \tag{27}\]The following equation is obtained by combining Equations (21), (22) and (27).