# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
###### Abstract
Based on the novel reaching law, an extended sliding mode disturbance observer-based SMC speed controller was implemented in [15], thus compensating for the adverse influence of the lumped uncertainties. It is worth mentioning that the above-mentioned approaches mainly concentrate on the design and analysis of the speed controller [10; 11; 12; 13; 14; 15], thus improving the PMSM drive system performance. In addition, the traditional cascade control structure is employed in these researches, where a PI-type controller will usually act as the inner current loop. A speed current single loop control scheme was proposed to a PMSM drive system [16], which also involved the nonlinear disturbance observer. The single loop non-cascade control simplifying the system structure was presented in [17], where the dual disturbance observers-based feedforward and the integral SMC feedback were individually designed and eventually synthesized. However, the presented non-cascade control structure still requires a PI controller for the direct axis current regulation [16; 17]. Based on the feedback linearization technology and nonlinear disturbance observer, the composite single-loop terminal SMC structure was investigated to realize the speed and the current tracking regulations, simultaneously, thus replacing the conventional cascade control framework [18]. Inspiring by the above discussions, the investigation about non-cascade SMC design for PMSMs has the important significance. For the large-scale industrial processes, there usually exist small time constants, parasitic inductances and capacitance, thus separating the eigenvalues of system state space variables into the different regions [19]. As a result, the multi-time-scale plants can be modeled as the singular perturbation systems (SPSs), while the powerful singular perturbation theory is employed to controller design and stability analysis [20]. The advanced heavy water reactor characterized by 38 slow, 35 fast, and 17 fastest state variables was presented in [21], which was eventually modeled as a three-time-scale SPS. An extended high gain observer-based output feedback control strategy for the nonlinear SPSs was addressed [22], and the single link manipulator with several uncertain terms were discussed, respectively. In addition, the continuous SMC for compliant robot arms was formulated as a SPS [23],which was comprised by the slow rigid robot and a fast series elastic actuator dynamics. For an electric motor, the singular perturbation theory-based rigorous analysis for the augmented system was presented in [24], and the dual PI observers-based cascade control scheme was conducted in details. By constructing the disturbance observer and modeling the PMSM as a dual-time-scale SPS [25], the non-cascade SMC strategy was exhibited. It can be concluded from the reported literatures that the singular perturbation theory is a powerful tool to design and analyze the controlled system, and by modeling the PMSM as an SPS is still an open research field. Motivated by the above-mentioned researches, this paper considers the problems of the non-cascade SMC for surface-mounted PMSMs speed regulation systems. By selecting the singular perturbation parameter as the electrical time constant, a SPS is eventually modeled. According to eigenvalue placement technique, the exact decoupled subsystem dynamics are stable, thus resulting a composite sliding mode surface. Then, a standard non-cascade SMC law is designed based on the exponential reaching law. Furthermore, a novel tracking differentiator (TD)-based SMC is synthesized by incorporating a nonlinear function. Finally, the advantages and effectiveness of the proposed approaches are illustrated by research results. The contributions of this study can be summarized as follows. (1) The singular perturbation decomposition approach-based composite sliding mode surface is constructed, which involves the Lyapunov equation and rigorous theoretical analyses.