## 3 Design Process of the PACFTB Controller  
So as to ensure the following effect of LIM speed, a PACFTB method is designed. First, corresponding to the backstepping method, the tracking errors could be written as:  
\[e_{1} =v-v^{d} \tag{16}\] \[e_{2} =i_{qs}-i_{qs}^{c}\] (17) \[e_{3} =i_{ds}-i_{ds}^{d} \tag{18}\]  
where \(v^{d}\) and \(i_{ds}^{d}\) are given velocity reference and d-axis primary current, \(i_{qs}^{c}\) the given reference from constrained command filter based on speed loop control. The command filter can reduce the differential expansion problem in backstepping control and reduce the computational complexity of the controller. Then, the mathematical form of command filter can be written as [8]:  
\[\left[ \begin{array}{c}z_{1}\\ z_{2}\end{array} \right]=\left[ \begin{array}{c}z_{2}\\ 2\varsigma\omega_{n}\left[S_{R}\left(\frac{\omega_{n}^{2}}{2\varsigma\omega_{ n}}\left(S_{M}(u)-z_{1}\right)\right)-z_{2}\right]\end{array} \right] \tag{19}\]  
where \(u=x^{d}\); \(\varsigma\) denotes the dampling of the system while \(\omega_{n}\) denotes the bandwidth; \(S_{M}(\cdot)\) and \(S_{R}(\cdot)\) represent the magnitude and rate boundedness, respectively. Through the application of command filters, differential signal \(\hat{x}^{c}\) could be obtained by integration, and this would eliminate the magnified interference problems caused by practical applications. Whereas, due to the use of command filters, there will be filtering error \(\mu=x^{c}-x^{d}\) which can interfere with the following performance and that must be taken into consideration. An effective method is by properly increasing \(\omega_{n}\), the output \(x^{c}\) can quickly coverage to \(x^{d}\), and the filtering errorcan be reduced. Furthermore, this paper uses a filter compensator signal to eliminate the filter error as follows:  
\[\dot{\epsilon}_{1}=-k_{1}\epsilon_{1}+\left(i_{qs}^{c}-i_{qs}^{d}\right) \tag{20}\]  
where \(k_{1}>0\) is a positive constant, \(\epsilon_{1}\) is the compensation signal of the filter. Then, a redefined tracking error signal \(\dot{\epsilon}_{1}\) is given as:  
\[\dot{\epsilon}_{1}=e_{1}-\epsilon_{1} \tag{21}\]  
Define the nonsingular terminal sliding surface as follows:  
\[S_{d}=e_{3}+k_{d}\left(\int_{0}^{t}e_{3}dt\right)^{p_{1}/q_{1}}=e_{3}+k_{d}I_{ d}{}^{p_{1}/q_{1}} \tag{22}\]  
\[S_{q}=e_{2}+k_{q}\left(\int_{0}^{t}e_{2}dt\right)^{p_{2}/q_{2}}=e_{2}+k_{q}I_{ q}{}^{p_{2}/q_{2}} \tag{23}\]  
where \(I_{d}=\int_{0}^{t}e_{3}dt\), \(I_{q}=\int_{0}^{t}e_{2}dt\), \(k_{d}\), \(k_{q}>0\) are the designed constant of the nonsingular terminal sliding surface, \(p_{1}\), \(p_{2}\), \(q_{1}\), \(q_{2}\) are the positive odd numbers and \(1<p_{1}/q_{1}<2\), \(1<p_{2}/q_{2}<2\). Suppose that at \(t_{r}\), \(S_{d}\) converges to zero, then, according to (22), \(\int_{0}^{t}e_{3}dt\) and \(e_{3}\) converges to zero within a limited time which is [32]:  
\[t_{s}=\frac{q_{1}}{k_{d}\left(p_{1}-q_{1}\right)}I_{d}(t_{r})^{\frac{q_{1}}{p_ {1}-q_{1}}} \tag{24}\]  
where \(I_{d}\left(t_{r}\right)=-\left[k_{d}^{-1}e_{2}\left(t_{r}\right)\right]^{q_{1} /p_{1}}\), thus, it is concluded that terminal sliding surface can make the controlled system converge from any state to origin point in finite time \(t_{s}+t_{r}\). Similarly, the convergence time can be calculated when the system reaches the equilibrium point along with terminal sliding surface \(S_{q}\). **Step 1:** So as to get the virtual control formula for \(i_{qs}^{d}\), the following Lyapunov function is designed:  
\[V_{1}=\frac{1}{2}\epsilon_{1}^{2} \tag{25}\]  
Then, it is further calculated as:  
\[ \begin{split}\dot{V}_{1}&=\xi_{1}\left(\frac{K_{T}} {M}i_{qs}+\Phi v+F-\bar{v}^{d}+k_{1}\epsilon_{1}-i_{qs}^{c}+i_{qs}^{d}\right) \\ &=\xi_{1}\left(i_{qs}-\bar{v}^{d}+k_{1}\epsilon_{1}-i_{qs}^{c}+ i_{qs}^{d}+f_{1}\left(X\right)+F\right)\\ &=-k_{1}\epsilon_{1}^{2}+\bar{\epsilon}_{1}\left(e_{2}-\bar{v}^{ d}+i_{qs}^{d}+f_{1}\left(X\right)+F+k_{1}\epsilon_{1}\right)\end{split}