# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 3 Singular Perturbation Approach-Based SMC
### TD-Based Eventual SMC for a PMSM Servo System
: _There are three differences between the presented SMC (36) and the novel TD-based SMC (46): (1) The presented TD-based transition dynamic \(\hat{\alpha}_{\rm m}^{*}\) is employed to design the control law, rather than the actual reference velocity \(\omega_{\rm m}^{*}\). (2) The nominal disturbances \(\mathbf{f}_{\rm o}\) are adopted to reduce the switching gain value \(\gamma\), which is inspired by the upper bound relationship \(\|\mathbf{f}_{\rm T}\|\leq\|\mathbf{f}\|\leq M\). (3) A nonlinear function \(\text{fal}(\cdot)\) is introduced to replace the signum function, thus improving the chattering phenomenon. Such configurations can promote the regulation performances for a PMSM drive system, which can be demonstrated by the subsequent results._  
**Remark 6**. : _This study proposes the singular perturbation approach-based non-cascade SMC strategy for a surface-mounted PMSM drive system, which is symbolized by Figure 1. It should be emphasized that the conventional cascade control structure is popular among the PMSMs speed regulation systems [10; 11; 12; 13; 14; 15], which is usually comprised of the inner current and outer speed loops, respectively. Although some recent literature has presented a non-cascade control framework for a PMSM servo system [16], a PI controller for the d axis current loop is still required [17]. It can be concluded that the non-cascade control for PMSM is of great important significance. By employing the singular perturbation theory, a complete non-cascade SMC approach that obviously different from the traditional vector control method developed in this paper, whose effectiveness and feasibility are demonstrated by the following researches._In the last, the whole block control structure for a PMSM servo system is shown as Figure 2. In Figure 2, a DC voltage source with \(U_{\text{dc}}\) is firstly provided as the power supply, and then the three-phase inverter drives the PMSM based on the pulse signals. Meanwhile, the three-phase currents \(i_{\text{A}}\), \(i_{\text{B}}\) and \(i_{\text{C}}\) are individually measured by the sensors, which are transformed into the \(d\) and \(q\) axes stator currents \(i_{\text{d}}\) and \(i_{\text{q}}\) by the well-known Clark and Park components (where the \(i_{\text{a}}\) and \(i_{\text{g}}\) are intermediate current values without using the angle information). On the other hand, according to the mechanical encoder, the motor speed and electrical angle \(\theta_{\text{e}}\) will be calculated, simultaneously. Therefore, the different non-cascade control SMC strategy shown as Figure 1 can be implemented. Eventually, the controller output \(u_{\text{d}}\) and \(u_{\text{q}}\) will be transformed into \(u_{\text{a}}\) and \(u_{\text{g}}\) based on the Park inverse module, thus generating the modulating waves for the well-known space vector pulse width modulation (SVPWM) element. Finally, the close-loop control scheme for a PMSM servo system is realized. Figure 1: The schematic block diagrams of the different non-cascade SMC strategy.