# Machine Learning-Based Monitoring of DC-DC Converters in Photovoltaic Applications
## 2 Materials and Methods
### Testability Analysis  
Testability analysis represents a fundamental step in each diagnostic and prognostic system. Thanks to this study, the identifiable components are defined, thus facilitating the selection of test points and the definition of the achievable objectives.  
Since the Simulation After Test (SAT) techniques are usually used to detect parametric faults in analog circuits, the main step is the definition of the failure equations. These equations present the component values as unknowns and are obtained by comparing the circuit responses at specific points with their nominal forms. The solvability level of the failure equations corresponds to the testability of the circuit, and its maximum value is the total number of passive components. If the testability is less than the maximum value, one or more ambiguity groups can be defined: they are sets of components that cannot be uniquely determined starting from the selected measurements.  
Several methods have been developed in recent years in order to facilitate the calculation of testability in different types of electrical circuits [30]. The study of linear time-invariant circuits is widespread in the literature, and the calculation methods in the frequency domain can be easily adapted to different topologies [31]. Regarding the non-linear time-variant circuits, the testability evaluation presents many complexities due to the different topologies that the circuit assumes during operation. In the case of DC-DC converters, two different topologies are used in Continuous Conduction Mode (CCM), one for each switching period, while in Discontinuous Conduction Mode (DCM), the topologies become three due to the cancellation of the inductor current.  
In such cases, a time-domain method can be used [32]. The first step is to choose a specific switching period in steady-state conditions and to sample the inputs, which are the circuit power supply quantities. In this way, vector \(\mathbf{u_{0}}\) is obtained. The second step is the definition of the output vector, as shown in (8), where \(\mathbf{p}\) is the vector of the unknown parameters and \(\mathbf{y_{T}}\) is a vector of measurements obtainable from the circuit.  
\[\mathbf{y_{T}}(\mathbf{p},\mathbf{u_{0}})=\left[y(t_{1},\mathbf{p},\mathbf{u_{0}})^{tr},y(t_{2}, \mathbf{p},\mathbf{u_{0}})^{tr},\ldots,\;y(t_{n},\mathbf{p},\mathbf{u_{0}})^{tr}\right]^{tr} \tag{8}\]  
Then, the failure equations can be obtained by (9),  
\[\mathbf{y_{T}}(\mathbf{p},\mathbf{u_{0}})=\mathbf{y_{T}}^{*} \tag{9}\]  
where \(\mathbf{y_{T}}^{*}\) is the vector of measurements extracted from the circuit. Finally, testability is calculated as the rank of the Jacobian matrix obtained stating from these equations.  
The equivalence between the time-domain procedure proposed here and the testability analysis performed in the Laplace domain is presented in [33]. In this paper, the testability assessment of the Zeta converter is carried out through SapWinPE (SapWin for Power Electronics), which is a program for simulating analog switching circuits. A specific algorithm called TAPSLIN (Testability Analysis for Periodically switched Linear Networks) is implemented on SapWinPE to perform the testability analysis in symbolic form [32].