## 3 Current Decoupling Control Based on ITHESO for Torque Ripple Reduction
### ITHESO Stability of the Decoupled Current Control  
To confirm the stability of the decoupled current control for the VFRM system, a Lyapunov second theorem function for the closed loop system including the error dynamics of both the controller and the ITHESO should be established. The goal is to show that the Lyapunov function is positive definite and that its time derivative is negative semi-definite, indicating stability.  
The decoupled current control includes dynamics for the dq0-axis currents with disturbances due to torque ripple, which can be defined as  
\[\begin{array}{l}e_{d}=i_{\text{d}}-i_{\text{d}}^{*}\\ e_{q}=i_{\text{q}}-i_{\text{q}}^{*}\\ e_{0}=i_{0}-i_{0}^{*}\end{array} \tag{23}\]  
Figure 4: Diagram of decoupled current control based on ITHESO.  
For each axis, the ITHESO estimates current disturbances in the system, including position-dependent periodic disturbances.  
A Lyapunov candidate function that includes terms for the current tracking errors and the ITHESO estimation errors is proposed. Let us define a composite Lyapunov function \(V\) as:  
\[V=\frac{1}{2}\epsilon_{d}^{2}+\frac{1}{2}\epsilon_{q}^{2}+\frac{1}{2}\epsilon_{0 }^{2} \tag{24}\]  
This function \(V\) is positive definite, as each component is positive when the errors are non-zero.  
To ensure stability, the time derivative \(V\) should be negative semi-definite (or, ideally, negative definite). Taking the time derivative of \(V\):  
\[\overset{\bullet}{V}=e_{d}\overset{\bullet}{e}_{d}+e_{q}\overset{\bullet}{e} _{q}+e_{0}\overset{\bullet}{e}_{0} \tag{25}\]  
from Equation (22), the current error dynamics are influenced by the control inputs and the disturbances. This assumes that the primary control laws are designed to stabilize the system in the absence of disturbances and the disturbance compensations aim to cancel the harmonic disturbances.  
Therefore, the current error dynamics become:  
\[\begin{array}{rl}\overset{\bullet}{e}_{d}&=-k_{d}e_{d}+(d_{\text{d}}+T_{e \_{ripple}})\\ \overset{\bullet}{e}_{q}&=-k_{q}e_{q}+(d_{q}+T_{e\_{ripple}})\\ \overset{\bullet}{e}_{0}&=-k_{0}e_{0}+(d_{0}+T_{e\_{ripple}})\\ \end{array} \tag{26}\]  
The estimation errors have dynamics based on the ITHESO gains. Thus, substituting Equation (26) into (25) yields  
\[\overset{\bullet}{V}= e_{d}(-k_{d}e_{d}+(d_{\text{d}}+T_{e\_{ripple}}))+e_{q}(-k_{q}e_{q}+(d_{q} +T_{e\_{ripple}}))\] \[+e_{0}(-k_{0}e_{0}+(d_{0}+T_{e\_{ripple}}))\]  
Rearranging terms, this becomes  
\[\begin{array}{rl}\overset{\bullet}{V}=&-k_{d}e_{d}^{2}-k_{q}e_{q}^{2}-k_{0} e_{0}^{2}+e_{d}(d_{\text{d}}+T_{e\_{ripple}})\\ &+e_{q}(d_{q}+T_{e\_{ripple}})+e_{0}(d_{0}+T_{e\_{ripple}})\\ \end{array} \tag{28}\]  
The ITHESO effectively estimates and compensates for disturbances and torque ripple components. Assuming that the disturbance estimation error is bounded and can be reduced to negligible levels through proper observer design, the disturbance terms can be approximated as:  
\[(d_{\text{d}}+T_{e\_{ripple}})\approx 0,\ (d_{q}+T_{e\_{ripple}})\approx 0,\ (d_{0}+T_{e\_{ripple}})\approx 0 \tag{29}\]  
Thus, \(\overset{\bullet}{V}\) simplifies to:  
\[\overset{\bullet}{V}=-k_{d}e_{d}^{2}-k_{q}e_{q}^{2}-k_{0}e_{0}^{2} \tag{30}\]  
Since \(k_{\text{d}}\), \(k_{\text{q}}\), and \(k_{0}>0\), all of the terms in Equation (30) are negative or zero. This implies:  
\[\overset{\bullet}{V}\leq 0 \tag{31}\]  
Since \(V\) is positive definite and \(\overset{\bullet