## 3 Open-Circuit Fault Diagnosis Strategy
### Design of IASMO  
In practical inverter systems where unavoidable unknown disturbances inherently exist, the model under normal working condition is formulated as:  
\[\left\{ \begin{array}{l}\dot{i}=Ai+Bu+Ev\\ y=Ci\end{array} \right. \tag{8}\]  
where \(i\in\Re^{3}\) represents the state variable, \(y\in\Re^{3}\) is the output variable, \(u\) is the three-phase output voltage, \(v\) represents unknown interference signal, bounded with known upper and lower bounds. It satisfies the constraint \(\|v\|<\lambda\), \(\lambda>0\). \(C\) and \(E\) are three-dimensional identity matrices. The IASMO is designed according to Equation (8) as follows:  
\[\left\{ \begin{array}{l}\dot{\hat{i}}=A\hat{i}+Bu+L\big{(}C\hat{i}-y\big{)}+f(S)\\ \hat{y}=C\hat{i}\end{array} \right. \tag{9}\]  
where \(\hat{i}\) represents the estimated current, \(L\) denotes the gain matrix, \(S=C\hat{i}-y\) represents the sliding mode surface. \(f(S)\) denotes the adaptive reaching law, defined as follows:  
\[f(S)=-\frac{k}{Q(S)}\tanh(\alpha S) \tag{10}\]  
where \(k\) denotes the observer gain. It satisfies \(k\geq\gamma\lambda+\gamma\rho\), where \(\rho\) is a positive constant. The switching function adopts the hyperbolic tangent function, expressed as follows:  
\[\tanh(\alpha S)=\frac{e^{aS}-e^{-\alpha S}}{e^{aS}+e^{-\alpha S}} \tag{11}\]  
where \(\alpha>0\). Traditional sliding mode observers typically utilize the sign function or the saturation function as the switching function. Among these, the sign function is discontinuous at the switching point and notoriously induces high-frequency chattering. Although the saturation function smooths the switching process through the boundary layer, it remains approximately equivalent to the sign function outside this region. Furthermore, designing the boundary layer thickness requires balancing convergence speed and chattering intensity. Figure 4 shows the curves of the conventional switching function and the hyperbolic tangent function. Compared with the above switching functions, the hyperbolic tangent function  
Figure 3: Schematic diagram of the proposed method. naturally reduces chattering due to its smooth transition characteristics. This reduces the effect of environmental changes on current tracking accuracy. The adaptive term in Equation (10) is represented as:  
\[Q(S)=\gamma+(1-\gamma)\bigg{(}1-\frac{2}{\pi}\arctan\lvert S\rvert\bigg{)} \tag{12}\]  
where \(0<\gamma<1\). When the trajectory of the state variable \(\hat{i}\) deviates significantly from the sliding mode surface, \(k/Q(S)\) asymptotically converges to \(k/\gamma\), thereby guaranteeing \(k/Q(S)>k\). This gain elevation in \(k/Q(S)\) accelerates the state trajectory's convergence toward the sliding mode surface, effectively minimizing the reaching time. As the trajectory of the state variable \(\hat{i}\) approaches the sliding mode surface, where \(\lvert S\rvert\) tends to zero, \(k/Q(S)\) converges to \(k\). This adaptive reduction in \(k/Q(S)\) effectively suppresses chattering in the observer. If there exists a positive definite symmetric matrix \(P\in\Re^{3\times 3}\) and a matrix \(Z\) such that the following inequality holds:  
\[A^{\mathrm{T}}P+PA-C^{\mathrm{T}}Z^{\mathrm{T}}-ZC<0 \tag{13}\]  
Under these conditions, the observer error dynamics are asymptotically stable, and the error dynamics are described by the following equation:  
\[\dot{e}=(A-LC)e+Ev-f(S) \tag{14}\]  
where \(e=i-\hat{i}\), and \(L=P^{-1}Z\). Proof of Theorem 1.: The Lyapunov function is constructed as follows:  
\[V=e^{\mathrm{T}}Pe \tag{15}\]  
Differentiating the above equation yields:  
\[\dot{V} =e^{\mathrm{T}}Pe+e^{\mathrm{T}}Pe \tag{16}\] \[=[e^{\mathrm{T}}(A-LC)^{\mathrm{T}}+v^{\mathrm{T}}E^{\mathrm{T}}- f^{T}(S)]Pe+e^{\mathrm{T}}P[(A-LC)e+Ev-f(S)]\] \[=e^{\mathrm{T}}[(A-LC)^{\mathrm{T}}P+P(A-LC)]e+2v^{\mathrm{T}}E^{ \mathrm{T}}Pe-2f^{T}(S)Pe\]  
\[\tanh(\alpha S)=\frac{e^{aS}-e^{-\alpha S}}{e^{aS}+e^{-\alpha S}} \tag{11}\]0 Conventional switching function versus hyperbolic tangent function curve.