## 3 Velocity Estimator  
LEMAs use the motor effect to generate force and motion on the moving coil in the magnetic field. Velocity can be estimated by exploiting the generator effect, i.e., a coil moving in a magnetic field generates a back electromotive force that is proportional to velocity. A velocity estimator is built based on Equation (1), and the position is obtained by integrating the high-precision estimated velocity. This method avoids using the complete system model that contains mechanical uncertainties.  
When input current is measured, from Equation (1), the velocity can be calculated as follows:  
\[v(t)=\frac{U(t)-I(t)R-L\frac{dt(t)}{dt}}{k_{e}} \tag{5}\]  
The velocity estimation error is defined as \(e_{v}(t)=v(t)-\vartheta(t)\), suppose the change rate of the estimated velocity is proportional to the error between the actual and the estimated velocity, thus a velocity estimator is designed as follows:  
\[\frac{d\vartheta(t)}{dt}=H[v(t)-\vartheta(t)]=He_{v}(t) \tag{6}\]  
where \(H\) is the estimator parameter, \(v(t)\) is the actual velocity, and \(\vartheta(t)\) is the estimated velocity.  
If \(H\gg 0\), then the dynamics of the estimated velocity is faster than that of the physical system,  
\[\|\frac{d\vartheta(t)}{dt}\|\gg\|\frac{dv(t)}{dt}\| \tag{7}\]  
Taking the differentiation of \(e_{v}(t)\), then  
\[\frac{de_{v}(t)}{dt}=\frac{dv(t)}{dt}-\frac{d\vartheta(t)}{dt} \tag{8}\]  
Combining (6) with (8) gives  
\[\frac{de_{v}(t)}{dt}+He_{v}(t)=\frac{dv(t)}{dt} \tag{9}\]  
If the following Lyapunov function is considered:  
\[V=\frac{1}{2}e_{v}^{2}(t) \tag{10}\]  
Also considering that  
\[V=e_{v}(t)\dot{e}_{v}(t) \tag{11}\]  
Combining (8) with (11) yields  
\[V=e_{v}(t)[\frac{dv(t)}{dt}-\frac{d\vartheta(t)}{dt}] \tag{12}\]  
After inserting (6) into (12), then  
\[V=\frac{1}{H}\left[\frac{d\vartheta(t)}{dt}\frac{dv(t)}{dt}-\left(\frac{d \vartheta(t)}{dt}\right)^{2}\right] \tag{13}\]If \(H\gg 0\), combining (7) with (13), then  
\[V<0. \tag{14}\]  
Hence the velocity estimator is asymptotically stable,  
\[\lim_{t\rightarrow+\infty}e_{\eta}(t)=0 \tag{15}\]  
Since the measured current contains noise, the differential process of the current will introduce undesirable spikes. Therefore, the function term containing the derivative of the measured current should be removed from the velocity estimator.  
Define the supplementary variable is as follows:  
\[\eta(t)=\hat{v}(t)+N(I(t))=\hat{v}(t)+k_{a}I(t) \tag{16}\]  
where \(N(I(t))\) is a function of current, \(k_{a}\) is the parameter.  
Taking the derivative on both sides of (16), then  
\[\frac{d\eta(t)}{dt}=\frac{d\hat{v}(t)}{dt}+k_{a}\frac{dI(t)}{dt} \tag{17}\]  
Combining (5) and (6) with (17), then  
\[\begin{array}{ll}\frac{d\eta(t)}{dt}&=H[v(t)-\hat{v}(t)]+k_{a}\frac{dI(t)}{ dt}\\ &=H\frac{U(t)-I(t)R-L\frac{dI(t)}{dt}}{k_{e}}-H\hat{v}(t)+k_{a}\frac{dI(t)}{ dt}\\ &=\frac{H}{k_{e}}(U(t)-I(t)R)-H\hat{v}(t)+(k_{a}-\frac{HL}{k_{e}})\frac{dI(t)}{ dt}\end{array} \tag{18}\]  
Therefore, if only  
\[k_{a}=\frac{HL}{k_{e}} \tag{19}\]  
Then, the velocity estimator can be changed into the expression as follows:  
\[\frac{d\eta(t)}{dt}=\frac{H}{k_{e}}[U(t)-I(t)R]-H\hat{v}(t) \tag{20}\]  
\[\hat{v}(t)=\eta(t)-k_{a}I(t) \tag{21}\]