# Improved Sliding Mode Control for Permanent Magnet Synchronous Motor Speed Regulation System
## 2 SMC Design for the PMSM
### Reachability Analysis  
In order to testify that the system state of this sliding mode control can reach the sliding mode surface form arbitrary initial state in limited time, the Lyapunov function candidate is defined as [31].  
\[V=\frac{1}{2}s^{2}\] (13a) then the derivation of Lyapunov can be set up with \[\dot{V}=s\dot{s}=s[-\varepsilon\text{sgn}(s)-qs]=-\varepsilon\text{sgn}(s)-qs ^{2} \tag{13b}\]  
If c in Equation (9) is a real number, then there will be \(\dot{V}\leq 0\). Therefore, according to [PERSON] theory, the speed for Equation (7) and can be converged to zero in limited time by designing the sliding mode surface of Equation (9) and control law of Equation (12).  
Figure 5 shows the different Simulink simulation results to the input with the system load superposed by ramp signal and step noise signal. As shown in Figure 5a, the speed response curves of the PMSM show that the proposed SMC can track the input more quickly and accurately. Simultaneously, the PMSM can effectively adjust the speed of the rotor to return to stable state according to a sudden load, and the proposed SMC can resume with less time and less speed error.  
The PMSM parameters used include the output power \(P_{e}\) = 0.75 kW, rated voltage \(U_{e}\) = 220 V, rated torque \(T_{e}\) = 2.4 N\(\cdot\)m, max torque \(T_{max}\) = 7.2 N\(\cdot\)m, rated current \(I_{e}\) = 4.2 A, rated speed \(n_{e}\) = 2000 r/min, stator phase resistance \(R_{s}\) = 0.901 \(\Omega\), pole pairs \(p\) = 4, and moment of inertia \(J\) = 1.2 \(\times\) 10\({}^{-4}\) kg\(\cdot\)m\({}^{2}\). The experimental prototype was built with the same design parameters used in the above simulation. The results of the output performance of the test bench are shown in Figure 6.  
Figure 4: Experiment platform.  
Figure 5: Numerical simulation results of different control. (**a**) Load signal; (**b**) the rotor speed response of the PMSM.  
Figure 6: Configuration of the traditional sliding mode control. (**a**) The traditional SMC; (**b**) the proposed SMC.  
The actual rotor speed for a traditional SMC during the process of startup and running of the motor is shown in Figure 6a, while Figure 6b shows the corresponding output torque under the same condition. As seen from the pictures, the overshoot is approximately 30% in the process of startup. Simultaneously, there is a large overshoot in the corresponding output torque. Figure 7a,b shows the actual rotor speed and the corresponding torque for the proposed SMC during the startup and running period of the motor, respectively. The results show the speed and torque errors are approximately 7% and 3%, respectively.  
Figure 8 displays the different results to the input with the system load during the process of running the motor. The deviation of the rotor speed in the traditional SMC is larger than the deviation observed with the proposed SMC. The contrast of the two results shows that the proposed SMC improves the steady state performance and enhances robust stability and disturbance rejection performance.  
Performance analysis and experimental results demonstrate that the proposed SMC, which combines SMC and repetitive control within an existing feedback control system based on an independent design philosophy, is a superior alternative for reducing the magnitude of transient error and dealing with complicated input signals.