Quantitative Analysis of Digital Chattering in Super-Twisting Sliding-Mode Speed Control of PMSM and Its Semi-Implicit Suppression

When the super-twisting sliding-mode algorithm (STA) is applied to the speed loop of a permanent magnet synchronous motor (PMSM), the chattering-free property predicted by continuous-time theory cannot be preserved in a digital controller, and explicit Euler discretization introduces a self-sustained oscillation. To address this problem, this paper quantitatively analyzes the digital-chattering mechanism of the explicitly discretized STA on a second-order speed-loop model that accounts for the current-loop dynamics: the peak-to-peak ripple of the integral term is derived to be exactly 2hk_{2}, the sliding-surface ripple is upper-bounded by {\left(hk_{1}/2\right)}^{2}, the current-reference ripple is upper-bounded by 2hk_{2}/b, and combining the robustness condition k_{2}\gt 1.1L yields the ripple lower bound 2.2hL/b, revealing the inherent conflict between disturbance rejection and ripple suppression in the explicit implementation. Furthermore, a semi-implicit discretization method is introduced, in which the control input is determined by the sign of the sliding surface one step ahead through the solution of a monotone scalar equation, so that the system reaches the discrete sliding surface exactly. Simulation results show that with a sampling period of 1 ms, the peak-to-peak sliding-surface ripple of the explicit STA is 17.6 and the current-reference ripple is 30.7 mA, whereas the semi-implicit scheme reduces them to the order of {10}^{-8} and {10}^{-10}, respectively, while the load-disturbance recovery time remains comparable to that of the explicit scheme. A gain-sweep experiment verifies that the ripple of the explicit scheme grows linearly with k_{2}, while no such constraint exists for the semi-implicit scheme. The results provide a reference for the digital implementation of the STA in PMSM servo systems.