## 4 Design Combination of SMC and NESO
### NESO Design  
The FPPMSM model (5) can be expressed as follows, taking into account the perturbation parameters, load, and unknown terms as disturbances:  
\[ \begin{cases}i_{dp}=f_{dp}+b_{1}v_{dp}+d_{1}(t)\\ i_{qp}=f_{qp}+b_{2}v_{qp}+d_{2}(t)\\ i_{ds}=f_{ds}+b_{3}v_{ds}+d_{3}(t)\\ i_{qs}=f_{qs}+b_{4}v_{qs}+d_{4}(t)\\ \Omega=a_{1}T_{e}+d_{5}(t)\end{cases}  \tag{19}\]  
In the faulty conditions model (9), the currents of \(i_{dp}\) and \(i_{qp}\) remain the same as in the model under healthy conditions, except for the \(\beta_{s}\)-axis, which is given by  
\[i_{\beta s}=f_{\beta s}+b_{\beta s}v_{\beta s}+d_{6}(t) \tag{20}\]  
With  
Figure 2: Structure of the MTO control criteria. \[f_{dp} =c_{1}i_{dp}+c_{2}\Omega i_{qp},\quad f_{qp}=c_{3}i_{qp}-c_{4}\Omega i _{dp}-c_{5}\Omega\] \[f_{ds} =c_{6}i_{ds}+c_{7}\Omega i_{qs},\quad f_{qs}=c_{8}i_{qs}-c_{9} \Omega i_{ds}-c_{10}\Omega\] \[f_{ps} =c_{11}i_{gs}-c_{12}\Omega\cos(3\theta)\]  
where \(d_{i}(t)\) (\(i=1,\ldots,6\)) represent the mismatched disturbances corresponding to models (19) and (20). It is assumed that the mismatched disturbances and their derivatives are bounded as follows:  
\[\rho_{i}>\max(|d_{i}(t)|)\,\delta_{i}>\max(|d_{i}(t)|) \tag{21}\]  
Assuming that the equation describing the state of a nonlinear system is given as  
\[\dot{x}=f(x,t)+bu+d(t) \tag{22}\]  
where \(f(x,t)\) is a known nonlinear function of \(x\), \(u\) is the control input, \(b\) is a constant input gain, and \(d(t)\) is the disturbance influencing the state. The estimation of disturbance \(d(t)\) in system (22) is conducted using a NESO designed as follows:  
\[ \begin{cases}\dot{z}_{1}=z_{2}-h(z_{1}-x)+f(x,t)+bu\\ \dot{z}_{2}=-h^{2}\tanh(z_{1}-x)\end{cases}  \tag{23}\]  
Figure 3 shows the NESO diagram, where \(z_{1}\) and \(z_{2}\) are the estimates of \(x\) and \(d(t)\), respectively; \(h\) is a positive design parameter. The estimation errors, denoted by \(e_{1}=z_{1}-x\) and \(e_{2}=z_{2}-d(t)\), are transformed into \(\zeta_{1}=e_{1}\) and \(\zeta_{2}=e_{2}-he_{1}\) considering the stable error dynamics at equilibrium. The dynamics of this transformation, derived from system (22) and the extended state observer (23), are as follows:  
\[ \begin{cases}\dot{\zeta}_{1}=\zeta_{2}\\ \dot{\zeta}_{2}=\dot{e}_{2}-h\zeta_{2}=-h^{2}\tanh(\zeta_{1})-\eta-h\zeta_{2} \end{cases}  \tag{24}\]  
where \(\eta=d(t)\). Consider the Lyapunov function shown below:  
\[V_{\zeta}=h^{2}\int_{0}^{\zeta_{1}}\tanh(\zeta_{1})d\zeta_{1}+\frac{1}{2}\zeta _{2}^{2} \tag{25}\]  
Equation (24) provides the Lyapunov function derivative, as follows:  
Figure 3: Diagram of the NESO. \[ \begin{split}\dot{V}_{\zeta}&=\frac{\partial V_{\zeta}}{ \partial\zeta_{1}}\zeta_{1}+\frac{\partial V_{\zeta}}{\partial\zeta_{2}}\zeta_ {2}\\ &=h^{2}\tanh(\zeta_{1})\zeta_{2}+\zeta_{2}(-h^{2}\tanh(\zeta_{1})- \eta-h\zeta_{2})\\ &=-\zeta_{2}(\eta+h\zeta_{2})\end{split}  \tag{26}\]  
If \(\eta=0\), then \(\dot{V}_{\zeta}=-h\zeta_{2}^{2}\leq 0\). Despite the derivative of the zero solution, the Lyapunov function remains stable due to its construction from the system (24). If \(\eta\
eq 0\) and assuming that \(|\eta|\leq\rho\), the derivative of the Lyapunov function (26) can be given as follows:  
\[\dot{V}_{\zeta}\leq-h\Big{(}\zeta_{2}+\frac{\eta}{2h}\Big{)}^{2}+\frac{\rho^{2 }}{4h} \tag{27}\]  
According to (27), it becomes evident that when \(|\zeta_{2}|\leq\frac{|\eta|}{h}\leq\frac{\rho}{h}\), then \(\dot{V}_{\zeta}<0\), indicating the stability of system (24), which will reduce at a steady state as follows:  
\[ \begin{cases}\dot{\zeta}_{1}=\zeta_{2}=0\\ \dot{\zeta}_{2}=-h^{2}\tanh(\zeta_{1})-\eta-h\zeta_{2}=0\end{cases}  \tag{28}\]  
As the HTF is monotonically increasing, we can deduce from Equation (28) that, by taking its absolute value, it will be upper bounded as \(|\zeta_{1}|\leq\tanh^{-1}(\frac{\rho}{h^{2}})\).