## 3 Current Decoupling Control Based on ITHESO for Torque Ripple Reduction
### Current Coupling Factors in VFRM  
Equation (10) shows the voltage equation of the VFRM in a transient state.  
\[\left\{\begin{array}{l}u_{\rm d}=R_{s}i_{\rm d}-\omega_{e}L_{s}i_{\rm d}+L_{ \rm s}\frac{\mathrm{d}i_{\rm d}}{\mathrm{d}t}+L_{\rm s}\frac{\mathrm{d}i_{\rm d }}{\mathrm{d}t}\\ u_{\rm q}=R_{s}i_{\rm q}+\omega_{e}L_{s}i_{\rm d}+\omega_{e}L_{s}i_{\rm e}i_{ \rm e}+L_{s}\frac{\mathrm{d}i_{\rm d}}{\mathrm{d}t}\\ u_{0}=R_{s}i_{0}+L_{\rm s}\frac{\mathrm{d}i_{\rm q}}{\mathrm{d}t}+\frac{L_{\rm s }}{2}\frac{\mathrm{d}i_{\rm d}}{\mathrm{d}t}\end{array}\right. \tag{10}\]  
Figure 3 illustrates the block diagram of the VFRM and its current controller drawn according to Equation (10), where delays due to sampling and computational errors are ignored and \(G_{\rm c}\) is the transfer function of the current controller; \(u_{\rm d}\)*, \(u_{\rm q}\)*, and \(u_{\rm 0}\)* are the reference voltages output from the d-axis, q-axis, and 0-axis current controllers, respectively. In the VFRM, there is a cross-coupling between the d-axis and the q-axis, a differential term coupling between the d-axis and the 0-axis, and a perturbation of the 0-axis to the q-axis. Therefore, the VFRM is a system containing complex inter-axis couplings.  
The differential term coupling is difficult to access directly, and there are other unknown perturbations in the system, such as inverter nonlinearities and motor parameter variations, which affect the effectiveness of feed-forward decoupling control. Therefore, in this paper, an ITHESO is used to observe the total current loop perturbations caused by factors such as coupling, perturbation terms, and a torque ripple component. The ITHESO requires little information about the system model and can perform state reconstruction for unknown nonlinear systems.  
The input signals of the ITHESO are the d-axis, q-axis, and 0-axis reference voltages, while the voltage references include the inverter nonlinear voltages in addition to the actual voltages acting on the motor windings. During the operation of the motor, both the motor parameters and the inverter nonlinearities change, and these factors are considered as uncertain disturbances, denoted by \(d\). Then the voltage equation can be expressed as:  
\[\left\{\begin{array}{l}u_{\text{d}}=R_{\text{s}}j_{\text{d}}+L_{\text{s}} \frac{\text{d}j_{\text{d}}}{\text{d}t}-\omega_{\text{e}}L_{\text{s}}j_{\text{ d}}+L_{\text{s}}\frac{\text{d}j_{\text{d}}}{\text{d}t}+d_{\text{d}}\\ u_{\text{q}}=R_{\text{s}}i_{\text{q}}+L_{\text{s}}\frac{\text{d}j_{\text{d}}}{ \text{d}t}+\omega_{\text{e}}L_{\text{s}}j_{\text{d}}+\omega_{\text{e}}L_{ \text{s}}j_{\text{o}}+d_{\text{q}}\\ u_{\text{0}}=R_{\text{s}}i_{\text{0}}+L_{\text{s}}\frac{\text{d}i_{\text{0}}}{ \text{d}t}+L_{\text{s}}\frac{\text{d}i_{\text{0}}}{\text{d}t}+\frac{L_{\text{ s}}}{\text{d}t}+d_{\text{0}}\end{array}\right. \tag{11}\]  
The transformation gives  
\[\left\{\begin{array}{l}\frac{\text{d}j_{\text{d}}}{\text{d}t}=\frac{u_{\text{ d}}}{L_{\text{s}}}-\frac{R_{\text{s}}}{L_{\text{s}}}j_{\text{d}}+\omega_{\text{e}} i_{\text{d}}-\frac{L_{\text{s}}}{L_{\text{s}}}\frac{\text{d}i_{\text{0}}}{ \text{d}t}-\frac{d_{\text{d}}}{L_{\text{s}}}\\ \frac{\text{d}i_{\text{0}}}{\text{d}t}=\frac{u_{\text{q}}}{L_{\text{s}}}-\frac {R_{\text{s}}}{L_{\text{s}}}i_{\text{q}}-\omega_{\text{e}}i_{\text{d}}-\omega _{\text{e}}\frac{L_{\text{s}}}{L_{\text{s}}}i_{\text{0}}-\frac{d_{\text{d}}}{L _{\text{s}}}\\ \frac{\text{d}i_{\text{0}}}{\text{d}t}=\frac{u_{\text{q}}}{L_{\text{s}}}-\frac {R_{\text{s}}}{L_{\text{s}}}i_{\text{0}}-\frac{L_{\text{s}}}{L_{\text{s}}} \frac{\text{d}i_{\text{d}}}{\text{d}t}-\frac{d_{\text{0}}}{L_{\text{s}}} \end{array}\right. \tag{12}\]  
where \(d_{\text{d}}\), \(d_{\text{q}}\), and \(d_{\text{0}}\) denote the uncertainty perturbations caused by parameter variations and inverter nonlinearities in the dq0-axis current loops.