# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
###### Abstract  
Motivated by the fact that electrical transients are rather fast compared with mechanical response, the traditional cascade control structure constituted by the inner current and outer speed loops is usually employed in the permanent magnet synchronous motors (PMSMs) servo control community. According to the above-mentioned time-scale characteristic of the PMSMs drive systems, this technique addresses the problems of the non-cascade sliding mode control (SMC) strategy for the surface-mounted PMSMs. Firstly, by appropriately introducing the singular perturbation theory, the corresponding mathematical equations are modeled as a singular perturbation system. Meanwhile, a composite sliding mode surface is constructed based on the Lyapunov equation, such that the system stability can be also guaranteed. Then, according to the exponential reaching law, a standard non-cascade SMC law is designed. Furthermore, an optimal nonlinear function-based tracking differentiator (TD) is presented to smooth the reference velocity value, while providing differential signals. As a result, a novel TD-based SMC strategy is synthesized by incorporating a nonlinear function, thus improving the inherent chattering phenomenon. Finally, a surface-mounted PMSM servo system is performed to illustrate the advantages and effectiveness of the proposed approaches. The main contribution of this paper is to present an alternative non-cascade SMC framework based on the singular perturbation approach, which provides a novel control structure for a PMSM speed regulation system. On the contrary, four high-speed PMSMs with different rotor topologies were comparatively designed and analyzed in details [6], which were aimed at driving the air compressor of a fuel cell vehicle. In the past decades, numerous literatures have been reported to develop the powerful approaches for the PMSMs drive systems (see [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] and the references therein). It should be emphasized that the advanced nonlinear control technologies can greatly improve the system performances and robustness against internal parametric perturbations and external load disturbances, simultaneously. With the series structure, an improved model predictive control (MPC) method was proposed in [7], thus improving the control performance under the conventional finite control set MPC strategy. The robust \(H_{\infty}\) control methodology was employed in the design of disturbance observer-based feedforward and feedback controllers [8], and thus a two degree of freedom (DOF) composite speed control framework was synthesized. The discrete-time active disturbance rejection control (ADRC) method was presented to suppress the disturbances [9], where incorporated the repetitive controller and operated in parallel with the extended state observer (ESO). In order to decrease the approximate error of the speed loop, a second-order model description was proposed in [10], and then a composite controller comprised of the SMC feedback law and ESO-based feedforward compensation was conducted. Among the above-mentioned nonlinear control strategies, the SMC technique is famous as its particularly strong robustness, and has attracted much more attention in recent years [11]. The traditional feedback control approaches usually suffer from the contradiction between the rapidity and overshoot, and thus the composite nonlinear feedback (CNF) technique was introduced in the designed integral SMC speed controller [12]. An ESO-based continuous terminal SMC speed regulation problem was addressed in [13], the proposed composite SMC method obtained the fast convergence and satisfactory tracking performances. By taking the time-varying disturbances into account, the generalized proportional integral observer (GPIO) was constructed to estimate various disturbances, whose estimation values were incorporated into the continuous SMC law.