# Improved Sliding Mode Control for Permanent Magnet Synchronous Motor Speed Regulation System
## 2 SMC Design for the PMSM
### SMC Design for the PMSM  
It is known that the major drawback of sliding mode control is the so-called chattering phenomenon, which is caused by the high-frequency switching of a sliding mode controller, exciting unmodeled dynamics in the closed loop. This phenomenon indicates that the control signal exhibits high frequency oscillations called chattering after the system state reaches the sliding surface. This chattering can give rise to oscillatory behaviors of unpredictable frequency and can be capable of exciting the system's resonance modes. Consequently, in sliding mode control applications, there is a need for control strategies to track periodic reference signals or reject periodic disturbances. Repetitive control, which builds on the well-known internal model principle, provides a specialized control scheme for asymptotic tracking and rejection of a periodic exogenous disturbance without steady-state error. Repetitive control has been successfully applied in real industrial applications in the past decades, for example, harmonic suppression of circulating current [28], suppression for exogenous disturbances and uncertainties in the dynamics of a plant [29], and a fast tool servo system [30]. Simultaneously, a repetitive control system can be applied in a plug-in style and has a simple structure formulated by an N-delay positive-feedback loop. Therefore, a repetitive control is one of the most promising control schemes for enabling perfect rejection of periodic disturbances if periodic chattering disturbances act on the sliding mode control system. To achieve robust and high-accuracy tracking performance, a sliding mode controller combined  
Figure 1: A typical vector control of permanent magnet synchronous motors (PMSMs). with a plug-in repetitive compensation controller is proposed in this control scheme. The integrated controller as shown in Figure 2 is designed to stabilize and compensate for the chattering, and the plug-in repetitive controller offers precise tracking performance and harmonic elimination:  
where a sliding mode control is used as the observer of the repetitive controller, and the chattering generated by the sliding mode control can be stabilized and compensated by the repetitive controller \(n_{given}\) is the given speed; \(n_{Fead}\) is the feedback speed; \(e(z)\) is the error speed; and \(D(z)\) is the external disturbance. As shown in Figure 2, the repetitive controller can be expressed as  
\[\frac{U_{Ro}}{U_{Ri}}=\frac{1}{1-L(z)Z^{-N}}\] (6a) The difference expression is rewritten as \[ \begin{array}{rl}U_{Ro}(k)=&U_{Ri}(k)+Z^{-1}[U_{Ro}(z)L(z)Z^{-N}]=U_{Ri}(k)+ Z^{-1}[U_{RoL}(z)Z^{-N}]\\ &=U_{Ri}(k)+U_{RoL}(k-N)\epsilon(k-N)\end{array}  \tag{6b}\]  
where \(U_{R0}\) and \(U_{Ri}\) are the output and the input of the repetitive controller, respectively, as shown in Figure 2. \(Z^{-1}[]\) is Z-inverse transformation symbol. \(\epsilon()\) is unit step function. The Equation (6b) indicates that the output \(U_{Ro}(k)\) will increase. For \(L(z)\) is a low-pass filter, it can attenuate the last output. \(S(z)\) is the regulating function for the control object. A detailed discussion on \(S(z)\) is beyond the scope of the article, here \(S(z)=1\). The configuration of the sliding mode controller is shown in Figure 3.  
Figure 3: Configuration of the sliding mode controller. Figure 2: Block diagram of the improved ding mode control (SMC) strategy. The state of the controller can be described as  
\[x_{1}=\omega_{ref}-\omega_{m} \tag{7}\]  
where \(x_{1}\) is the tracking error of speed.