## 1 Introduction  
Recent advances in renewable energy and electric vehicle technology have prompted the development of multilevel inverters [1; 2; 3; 4; 5]. Compared with that of a two-level inverter, the switch of a multilevel inverter is subjected to lower voltage stress and is associated with a lower output voltage change rate (\(dv/dt\)). Accordingly, multilevel inverters are suitable for applications requiring high power consumption. Such applications require a relatively high number of power transistors, thus increasing the difficulty of fault detection in inverters. The traditional fault diagnosis procedure for inverters entails the use of information (numeric or waveform) directly measured by instruments. The information is examined by on-site personnel, who subsequently conduct fault diagnosis for the inverters, repair the inverters, and replace components on the basis of their experience. However, this traditional fault diagnosis procedure is prone to erroneous fault point detections, causing unnecessary waste of labor and time for component repair and replacement [6; 7; 8]. To improve the reliability of equipment using motor drive systems, researchers have invested considerable effort in exploring fault detection mechanisms [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19] and fault-tolerant control strategies [20; 21; 22; 23] for multilevel inverters in order to enable early fault detection when inverter power semiconductor components fail; this can thus maintain the inverter operation and minimize damage. Currently, primary techniques for smart fault diagnosis in inverters include model-based methods, expert systems, and machine learning systems. Although model-based methods are useful [9], they require the establishment of various fault-state models in advance. Moreover, such models should consider the values of snubber capacitance and balance resistor in inverters, which are difficult to obtain. The models comprise various parasitic elements that would inevitably require some assumptions and thus impose limitations to the general applicability of the models [10; 11]. In addition, expert systems can modify themselves. Nevertheless, the entire systems should still be established by experts; therefore, such systems are associated with relatively high costs [12]. Finally, machine learning systems [13] notably mimic the neural network of the human brain. Through such neural connections, data models can be updated using backpropagation, and hidden layers can reduce the dependence of machine learning algorithms on feature engineering. However, a large body of data is required for neural network learning to make accurate judgments, and the corresponding operating time is long [14; 15; 16; 17]. Therefore, developing new smart fault diagnosis systems with high accuracy, easy implementation, and fast response is imperative. The first author has previously used the extension theory to diagnose the type and number of fault modules in a photovoltaic module array [24]. This method measures the voltages, currents and powers of the photovoltaic module array at each moment, and then accordingly estimates the type and number of faulted modules in the photovoltaic module array. Although its fault diagnosis accuracy is very high, the input of this evaluation method must be a fixed single point value. Therefore, this method is not suitable for fault diagnosis where the input variables are continuous functions or waveforms. Based on this, we must first use another intelligent method like chaos theory to convert the three-phase voltage waveforms obtained in T-type multilevel inverters into equivalent characteristic values (chaos eye coordinates), where the fault diagnosis can be made by the extension theory. Multilevel inverters exhibit numerous advantages, such as various control modes, relatively low transistor withstand voltage levels, low current harmonics, and high frequency conversion efficiency.