# Finite-Time Consensus Tracking Control for Speed Sensorless Multi-Motor Systems
## 4 Finite-Time Consensus Tracking Protocol Design  
In this section, the unmeasured speed variable and estimated value of unknown compound interference are introduced into the design of the consensus protocol under the premise of speed sensorless. The control protocol does not need to use all status information. First, the consensus tracking error is defined as follows:  
\[\sigma_{i}=\sum_{j\in N_{i}}a_{ij}(\theta_{i}-\theta_{j})+a_{i0}(\theta_{i}- \theta_{0}), \tag{15}\]  
where \(N_{i}\) is the set of neighbors for the motor. For multi-motor system (2), the finite-time consensus tracking protocol based on the output feedback is designed as follows:  
\[u_{i}=\frac{1}{b_{i}}(-\bar{k}_{i}\hat{\omega}_{i}-\hat{d}_{i}-q_{1}[\omega_{i }^{p}+q_{2}^{p}\sigma_{i}^{2}]^{2/p-1}),i=1,2,\ldots,n, \tag{16}\]  
where \(1<p=p_{1}/\,p_{2}<2\), \(p_{1}\) and \(p_{2}\) are positive odd numbers; \(q_{1}\) and \(q_{2}\) are the positive constants to be designed; \(\hat{\omega}_{i}\) and \(\hat{d}_{i}\) are the estimated values \(\omega_{i}\) and \(d_{i}\). Theorem 4.1: _For multi-motor system (2), under the action of the consensus protocol (16), let parameters \(q_{1}\) and \(q_{2}\) satisfy the following inequality:_  
\[\left\{ \begin{array}{l}q_{1}\geq(2-1/p)2^{1-1/p}q_{2}^{1+p}(q_{5}+q_{6})\\ q_{2}=\frac{p2^{1-\frac{1}{p}}}{1+p}+\frac{\alpha+nT}{1+p}+q_{5},\end{array} \right. \tag{17}\]  
_where \(q_{3}=2^{1-\frac{1}{p}}\alpha+\frac{aq_{2}p}{1+p}+\frac{n\gamma p\cdot(2^{1-1 /p}+q_{2})}{1+p}\), \(q_{4}=\frac{1}{q_{2}}\gamma\frac{2^{1-1/p}}{1+p}\), \(q_{6}=\frac{q_{3}}{q_{2}}+nq_{4}+\frac{2^{1-\frac{1}{p}}}{1+p}\), and \(\alpha=\max\{\sum\limits_{j\in N_{i}}a_{ij}\}\), \(\gamma=\max\{a_{ij},\forall i,j\}\). Then, the consensus error will converge to an arbitrary small neighborhood of zero in finite time._  
Proof: The construction of the controller is mainly based on the inverse recursive design method, and the proof is divided into two steps. **Step 1**: A positive definite Lyapunov function is constructed as follows:  
\[V_{0}=\frac{1}{2}\sum_{i=1}^{n}\sum_{j\in N_{i}}a_{ij}(\theta_{i}-\theta_{j})^{2 }+\frac{1}{2}\sum_{i=1}^{n}a_{i0}(\theta_{i}-\theta_{0})^{2}. \tag{18}\]  
The communication topology is an undirected graph. Thus, the following equation is obtained by finding the derivation of \(V_{0}\):  
\[ \begin{split}\dot{V}_{0}&=\sum_{i=1}^{n}\sum_{j \in N_{i}}a_{ij}(\theta_{i}-\theta_{j})^{T}(\omega_{i}-\omega_{j})+\sum_{i=1}^ {n}a_{i0}(\theta_{i}-\theta_{0})^{T}\times\\ &(\omega_{i}-\omega_{0})\\ &=\sum_{i=1}^{n}\sum_{j=0}^{n}a_{ij}(\theta_{i}-\theta_{j})^{T} \varpi_{i}\\ &=\sum_{i=1}^{n}\sum_{j=0}^{n}a_{ij}(\theta_{i}-\theta_{j})^{T}( \omega_{i}^{*}+(\varpi_{i}-\omega_{i}^{*})),\end{split}  \tag{19}\]  
where \(\varpi_{i}=\omega_{i}-\omega_{j}\). The virtual control law is designed as \(\omega_{i}^{*}=-q_{2}\sigma_{i}^{1/p}\). Substituting the virtual control law into Formula (19) yields:  
\[ \begin{split}\dot{V}_{0}&=\sum_{i=1}^{n}\sum_{j=0 }^{n}a_{ij}(\theta_{i}-\theta_{j})^{T}(\omega_{i}^{*}+(\varpi_{i}-\omega_{i}^ {*}))\\ &=\sum_{i=1}^{n}\sigma_{i}\omega_{i}^{*}+\sum_{i=1}^{n}\sigma_{i} (\varpi_{i}-\omega_{i}^{*})\\ &\leq-q_{2}\sum_{i=1}^{n}\sigma_{i}^{1+1/p}+\sum_{i=1}^{n}| \sigma_{i}||\varpi_{i}-\omega_{i}^{*}|.\end{split}  \tag{20}\]  
Let \(\rho_{i}=\varpi_{i}^{p}-\omega_{i}^{*p}\). Then, according to Lemmas 2 and 4, one can obtain:  
\[ \begin{split}\dot{V}_{0}&\leq-q_{2}\sum_{i=1}^{n} \sigma_{i}^{1+1/p}+\sum_{i=1}^{n}|\sigma_{i}||\varpi_{i}-\omega_{i}^{*}|\\ &\leq-q_{2}\sum_{i=1}^{n}\sigma_{i}^{1+1/p}+\sum_{i=1}^{n}| \sigma_{i}|2^{1-1/p}\Big{|}\varpi_{i}^{p}-\omega_{i}^{*p}\Big{|}^{1/p}\\ &\leq-q_{2}\sum_{i=1}^{n}\sigma_{i}^{1+1/p}+\sum_{i=1}^{n}2^{1-1 /p}(\frac{p|\sigma_{i}|^{1+1/p}}{1+p}+\frac{|\rho_{i}|^{1+1/p}}{1+p}).\end{split}