# Super Twisted Sliding Mode Observer for Enhancing Ventilation Drive Performance
## 2 System Modelling
### Super-Twisted Slide Mode Observer (ST-SMO) Speed Estimator
#### 2.2.3 Stability Analysis  
The stability of the observer is provided by the stability analysis of the current observer by calculating the observer gains \(a_{1}\) and \(a_{2}\). According to Section 2.2.2, (19) and (20) can be calculated as  
\[\Psi_{r}^{\alpha} = -a_{1}u_{\alpha} \tag{21}\] \[\Psi_{r}^{\beta} = -a_{3}u_{\beta} \tag{22}\]  
A Lyapunov function is given as  
\[V = \frac{1}{2}S_{n}^{T}S_{n} \tag{23}\]  
Lyapunov stability theory maintains that the initial condition mandates the Lyapunov function to be positive definite to fulfill the conditions for a stable current observer, but the succeeding condition demands a negative first derivative.  
\[\dot{V} = \frac{1}{2}S_{n}^{T}S_{n}<0 \tag{24}\] \[\dot{V} = S_{\alpha}\hat{S}_{\alpha}+S_{\beta}\hat{S}_{\beta} \tag{25}\]  
\[\dot{V} = \left(S_{\alpha}p\left(-a_{1}v_{s}^{\alpha}-\frac{1}{t_{r}}\hat{ \Psi}_{s}^{\alpha}-\hat{\omega}\hat{\Psi}_{r}^{\beta}+\frac{L_{m}^{2}}{L_{r}t_ {r}}\hat{\eta}_{s}^{\alpha}\right)-\varepsilon S_{\alpha}^{2}\right)+\left(S_ {\beta}p\left(-a_{2}v_{s}^{\beta}-\frac{1}{t_{r}}\hat{\Psi}_{s}^{\beta}-\hat{ \omega}\hat{\Psi}_{r}^{\alpha}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{\eta}_{s}^{ \beta}\right)-\varepsilon S_{\beta}^{2}\right) \tag{26}\]  
Equation (26) indicates that the two terms \(\varepsilon S_{\alpha}^{2}\) and \(\varepsilon S_{\beta}^{2}\) are negative definite. Consequently, the last two equations should provide negative values, \(V<0\).  
\[\left(S_{\alpha}p\left(-a_{1}v_{s}^{\alpha}-\frac{1}{t_{r}}\hat{\Psi}_{s}^{ \alpha}-\hat{\omega}\hat{\Psi}_{r}^{\beta}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{ \eta}_{s}^{\alpha}\right)\right)<0 \tag{27}\]  
\[\left(S_{\beta}p\left(-a_{2}v_{s}^{\beta}-\frac{1}{t_{r}}\hat{\Psi}_{s}^{ \beta}-\hat{\omega}\hat{\Psi}_{r}^{\alpha}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{ \eta}_{s}^{\beta}\right)\right)<0 \tag{28}\]  
Equations (26) and (27) suggest that the constructed sliding mode surface in (14) is based on the switching functions defined in (27) and (28). The alteration of the slide surface signals results in a modification of \(\hat{\Psi}_{s}^{\alpha}\) and \(\hat{\Psi}_{r}^{\beta}\), as delineated in (19) and (20). Therefore, the values of \(a_{1}\) and \(a_{2}\) are established as identical in the specified states. Consequently, (27) and (28) may be restructured as  
\[\left(-a_{1}v_{s}^{\alpha}-\frac{1}{t_{r}}\hat{\Psi}_{s}^{\alpha}-\hat{\omega }\hat{\Psi}_{r}^{\beta}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{\eta}_{s}^{\alpha}\right) <0 \tag{29}\]  
\[\left(-a_{2}v_{s}^{\beta}-\frac{1}{t_{r}}\hat{\Psi}_{s}^{\beta}-\hat{\omega} \hat{\Psi}_{r}^{\alpha}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{\eta}_{s}^{\beta}\right) <0 \tag{30}\]  
The subsequent inequalities can be attained by selecting the gains a\({}_{1}\) and a\({}_{2}\) as exceedingly large positive values. Therefore, the adaptive values of \(a_{1}\) and \(a_{2}\) may be formulated as follows:  
\[a_{1}>-\frac{1}{t_{r}}\hat{\Psi}_{s}^{\alpha}-\hat{\omega}\hat{\Psi}_{r}^{ \beta}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{\eta}_{s}^{\alpha} \tag{31}\]\[a_{2}>-\frac{1}{t_{r}}\Psi_{s}^{\beta}-\hat{\omega}\Psi_{r}^{\alpha}+\frac{L_{m}^{2 }}{L_{r}t_{r}}\hat{t}_{s}^{\beta} \tag{32}\]  
Therewith, the values of \(a_{1}\) and \(a_{2}\) can be designed to fulfill the equality, given as  
\[a_{1}\ =\ \Delta-\frac{1}{t_{r}}\hat{\Psi}_{s}^{\alpha}-\hat{\omega}\hat{ \Psi}_{r}^{\beta}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{\tau}_{s}^{\alpha} \tag{33}\]  
\[a_{2}\ =\ \Delta-\frac{1}{t_{r}}\hat{\Psi}_{s}^{\beta}-\hat{\omega}\hat{ \Psi}_{r}^{\alpha}+\frac{L_{m}^{2}}{L_{r}t_{r}}\hat{\tau}_{s}^{\beta} \tag{34}\]  
Here, \(\Delta\) represents a positive number, chosen to satisfy the inequalities (33) and (34) during high-speed operation. To achieve complete adaptation of the gains and the free chattering SMO design, delta can be designated as (\(\Delta\ =\ \hat{\omega}\)). Consequently, the values of gains \(a_{1}\) and \(a_{2}\) are decreased during low-speed operation, thereby eliminating chattering and ensuring stability across varied operating speeds.