## 3 Current Decoupling Control Based on ITHESO for Torque Ripple Reduction
### Design of Decoupled Current Control with ITHESO for Torque Ripple Reduction  
According to the principle of ITHESO, the d-axis, q-axis, and 0-axis observers are designed for the current decoupling of VFRM while accounting for torque ripple disturbance, respectively. Equation (12) can be expressed as:  
\[\left\{ \begin{array}{l}\dot{i_{\text{d}}}=f_{\text{d}}+b_{\text{d}}u_{\text{ d}}\\ i_{\text{q}}=f_{\text{q}}+b_{\text{q}}u_{\text{q}}\\ i_{\text{0}}=f_{\text{q}}+b_{\text{q}}u_{\text{0}}\end{array} \right. \tag{13}\]  
Figure 3: Block diagram of the VFRM and its current controller. where \(f_{\rm d},f_{\rm q},f_{\rm q}\) and \(f_{0}\) are the total perturbations in each current loop and \(b_{\rm d}\), \(b_{\rm q}\), and \(b_{0}\) are the feedback coefficients of each reference voltage. These two parameters can be expressed as shown in Equations (14) and (15), respectively. It should be noted that in the proposed design, parameter mismatches such as variations in inductance, are captured as part of the total disturbance (\(f_{\rm d},f_{\rm q},f_{0}\)). The ITHESO dynamically estimates and compensates for these disturbances, meaning the performance of the observer is less affected when \(L_{\rm s}\) in Equation (15) changes with operating conditions. \[\left\{ \begin{array}{l}f_{\rm d}=\frac{1}{L_{\rm s}}\left(-R_{\rm s}i_{\rm d} +\omega_{\rm e}L_{\rm s}i_{\rm q}-L_{\rm s}\frac{{\rm d}i_{0}}{{\rm d}t}-(d_{ \rm d}+T_{e,ripple})\right)\\ f_{\rm q}=\frac{1}{L_{\rm s}}\left(-R_{\rm s}i_{\rm q}-\omega_{\rm e}L_{\rm s} i_{\rm d}-\omega_{\rm e}L_{\rm s}i_{0}-(d_{\rm q}+T_{e,ripple})\right)\\ f_{0}=-\frac{1}{L_{\rm s}}\left(R_{\rm s}i_{0}-\frac{L_{\rm s}}{2}\frac{{\rm d }i_{\rm d}}{{\rm d}t}-(d_{0}+T_{e,ripple})\right)\\ \end{array} \right. \tag{14}\]  
\[\left\{b_{\rm d}=b_{\rm q}=b_{0}=\frac{1}{L_{\rm s}}\right. \tag{15}\]  
According to Equation (13), the d, q, and 0-axis current loops are reconstructed as shown in (16), (17) and (18), respectively. \[\left\{ \begin{array}{l}\dot{i_{\rm d}}=f_{\rm d}+b_{\rm d}u_{\rm d}\\ f_{\rm d}=w_{\rm d}\\ y_{\rm d}=i_{\rm d}\end{array} \right. \tag{16}\]  
\[\left\{ \begin{array}{l}\dot{i_{\rm q}}=f_{\rm q}+b_{\rm q}u_{\rm q}\\ f_{\rm q}=w_{\rm q}\\ y_{\rm q}=i_{\rm q}\end{array} \right. \tag{17}\]  
\[\left\{ \begin{array}{l}\dot{i_{\rm 0}}=f_{\rm 0}+b_{\rm 0}u_{\rm 0}\\ f_{\rm 0}=w_{\rm 0}\\ y_{\rm 0}=i_{\rm 0}\end{array} \right. \tag{18}\]  
where \(y_{\rm d},y_{\rm q},y_{\rm 0}\) are the outputs of the current loop and \(w_{\rm d},w_{\rm q},w_{\rm 0}\) are bounded functions. Therefore, the dq0-axis ITHESO observers for observing the coupling and disturbance terms in the VFRM are designed as  
\[\left\{ \begin{array}{l}e_{\rm d}=\dot{i_{\rm d}}-i_{\rm d}\\ \dot{i_{\rm d}}=\dot{f_{\rm d}}-\beta_{\rm d1}e_{\rm d}+b_{\rm d}u_{\rm d}\\ \dot{f_{\rm d}}=-\beta_{\rm d2}e_{\rm d}\end{array} \right. \tag{19}\]  
\[\left\{ \begin{array}{l}e_{\rm q}=\dot{i_{\rm q}}-i_{\rm q}\\ \dot{i_{\rm q}}=\dot{f_{\rm q}}-\beta_{\rm q1}e_{\rm q}+b_{\rm q}u_{\rm q}\\ \dot{\bar{f_{\rm q}}}=-\beta_{\rm q2}e_{\rm q}\end{array} \right. \tag{20}\]  
\[\left\{ \begin{array}{l}e_{0}=\dot{i_{\rm 0}}-i_{\rm 0}\\ \dot{i_{\rm 0}}=\dot{f_{\rm 0}}-\beta_{\rm 01}e_{\rm 0}+b_{\rm 0}u_{\rm 0}\\ \dot{\bar{f_{\rm 0}}}=-\beta_{\rm 02}e_{\rm 0}\end{array} \right. \tag{21}\]  
where \(e_{\rm d},e_{\rm q},e_{\rm 0}\) are the errors of the estimated and real currents in the ITHESO, respectively, and \(\beta_{\rm d1},\bar{\beta}_{\rm d2},\bar{\beta}_{\rm q1},\bar{\beta}_{\rm q2}, \bar{\beta}_{\rm 01},\bar{\beta}_{\rm 02}\) are the gains of the errors, respectively. Further, the expression of the control input can be organized according to (13) as  
\[ \begin{cases}u_{\text{d}}+\frac{f_{\text{d}}}{b_{\text{d}}}=sLs_{\text{s}}i_{ \text{d}}\\ u_{\text{q}}+\frac{f_{\text{d}}}{b_{\text{d}}}=sLs_{\text{s}}i_{\text{q}}\\ u_{0}+\frac{f_{\text{d}}}{b_{\text{d}}}=sLs_{\text{s}}i_{0}\end{cases}