# Finite-Time Consensus Tracking Control for Speed Sensorless Multi-Motor Systems
## 3 Finite-Time Convergent Observer Design  
In practical engineering, all state variables of the system are difficult to obtain because of the limitation of technical conditions. The lack of some state information hampers the achievement of the control protocol based on the full state feedback. Thus, the unknown compound interference must be accurately estimated and fed back to the controller to reduce the influence of high-frequency chattering. To this end, input \(u_{i}\) and output \(\theta_{i}\) of the system (2) are used to estimate the unmeasurable speed variable \(\omega_{i}\) and unknown compound interference \(d_{i}\) of the system in this section. Let \(\hat{\theta}_{i},\hat{\omega}_{i},\hat{d}_{i}\) denote the estimates associated with states \(\theta_{i}\), \(\omega_{i}\) and uncertainty \(d_{i}\), respectively. \(e_{11}\) and \(e_{i2}\) denote the state estimation errors, which are defined as \(e_{i}=\left[ \begin{array}{cc}e_{i1}&e_{i2}\end{array} \right]^{T}=\left[ \begin{array}{cc}\theta_{i}-\hat{\theta}_{i}&\omega_{i}-\hat{\omega}_{i} \end{array} \right]^{T}\). When the unknown state and interference is out of certain scope, the system will lose synchronization and produce high-frequency chattering. To precisely estimate unmeasured states \(\omega_{i}\) and uncertainty \(d_{i}\), we design the following finite-time observer for system (2):  
\[\left\{ \begin{array}{l}\dot{\theta}_{i}=\dot{\omega}_{i}+l_{1}|e_{11}|^{ \frac{1}{2}}\text{sgn}(e_{i1})\\ \dot{\omega}_{i}=\bar{k}_{i}\dot{\omega}_{i}+\bar{b}_{i}u_{i}+l_{2}\text{sgn} (e_{i1}),\end{array} \right. \tag{5}\]  
where \(l_{11}\) and \(l_{2}\) are the observer gains to be designed later. **Theorem 1**. : _In the multi-motor system, the parameters to be designed of the finite-time observers satisfy the following:_  
\[\left\{ \begin{array}{l}l_{i1}p_{11}+l_{i2}p_{12}>0\\ -p_{12}(l_{i1}p_{11}+l_{i2}p_{12})-(l_{i2}p_{22}+\frac{l_{i1}}{2}p_{12}-\frac{1} {2}p_{11})^{2}>0,\end{array} \right. \tag{6}\]  
_where \(p_{ij}(i,j=1,2)\) is the element of symmetrical positive definite matrix \(P\). The observer error will converge to an arbitrary small neighborhood of zero in finite time \(T_{1}\), and the estimated value of the composite disturbance can be described by \(\lim\limits_{i\to T_{1}}\hat{d}_{i}=l_{i2}\mathrm{sgn}(e_{i1})-e_{i2}\)._  
Proof. : Subtracting Formula (5) from (2), the error dynamics can be obtained as follows:  
\[\left\{ \begin{array}{l}\dot{e}_{i1}=e_{i2}-l_{i1}|e_{i1}|^{\frac{1}{2}} \mathrm{sgn}(e_{i1})\\ \dot{e}_{i2}=\ddot{k}_{i}e_{i2}-l_{i2}\mathrm{sgn}(e_{i1})+d_{i}.\end{array} \right. \tag{7}\]  
To examine the convergence of the observer, the Lyapunov function is selected as follows:  
\[U_{i}=\xi_{i}^{T}P\xi_{i}, \tag{8}\]  
where the auxiliary variable is \(\tilde{\varsigma}_{i}^{T}=\left[ \begin{array}{cc}|e_{i1}|^{\frac{1}{2}} \mathrm{sgn}(e_{i1})&e_{i2}\end{array} \right]^{T}\), and the symmetrical positive definite matrix is \(P=\left[ \begin{array}{cc}p_{11}&p_{12}\\ p_{12}&p_{22}\end{array} \right]\). If the parameters of matrix P satisfy \(p_{11}>0\), \(p_{11}p_{22}-p_{12}^{2}>0\), the Lyapunov function \(U_{i}\) is positive definite, as follows:  
\[\lambda_{\min}(P)\xi_{i}^{T}\tilde{\varsigma}_{i}\leq U_{i}\leq\lambda_{\max} (P)\xi_{i}^{T}\tilde{\varsigma}_{i}. \tag{9}\]  
Finding the derivation of \(U_{i}\) along the system trajectory (8) yields:  
\[\begin{split}\dot{U}_{i}&=\left[ \begin{array}{cc}2|e_{i1}|^ {\frac{1}{2}}\mathrm{sgn}(e_{i1})&2e_{i2}\end{array} \right]P\times\left[ \begin{array}{cc}\frac{1}{|2e_{i1}|^{\frac{1}{2}}}(e_{i2}-l_{i1}|e_{i1}|^{ \frac{1}{2}}\mathrm{sgn}(e_{i1}))\\ \ddot{k}_{i}e_{i2}-l_{i2}\mathrm{sgn}(e_{i1})+d_{i}\end{array}