# Fault Diagnosis in Partially Observable Petri Nets with Quantum Bayesian Learning
## 1 Introduction  
Reliable health monitoring is crucial for the safe operation of a discrete event dynamic system (DEDS). Highly constrained systems require minimal sensing and computing capacity, thus the prediction and diagnosis of the undetectable part of the fault are inevitable when only a part of the valid information is observed. The system behavior will highly change or may fail with the occurrence of some faults [1, 2]. Due to the limitation of the number of sensors, to overcome the problem of incomplete observation of the information and meet the real-time requirements, the diagnosis system generally adopts the diagnosis method based on the qualitative model [3]. Petri nets were first applied to fault identification and diagnosis, power systems [4], communication systems [5], etc. [PERSON] [5] proposed using Markov stochastic dynamics with the measured events and labeled sensors to represent the fault process and construct partially observable stochastic Petri nets. The probabilities of time and non-time marked trajectories consistent with the given time series are calculated systematically, and the diagnosis is based on the probability of failure. The authors of [6] proposed constructing a basis reachability graph, calculating the vector and the basis marker, identifying the fault class alarm level by introducing two kinds of fuzzy operators, and using bidirectional reasoning strategies (i.e., forward reverse reasoning) to determine the factor value of the diagnosis results. The problem of fault prediction for stochastic discrete event systems was studied, and an incremental algorithm was proposed. From the set of current possible states and their probabilities, the probability of consistent trajectories was evaluated based on the measured data, and the state estimation was obtained as a result. The probability model proposedthe use of a method for evaluating future fault probability [7]. Bayesian network machine learning methods are widely used in fault prediction and diagnosis. [PERSON] et al. [8] used the Grover quantum search algorithm to quickly search for the transformer fault, symptom type, and other target data. AHP deleted indicators with little impact on fault diagnosis and analyzed and determined the error classification rate of the variable precision rough set to obtain the minimum fault decision table to construct the Bayesian network fault reasoning model. It completed the diagnosis of transformer faults. The current research mainly regards traditional probability estimation plus Bayesian statistical methods, and the accuracy of probability estimation needs to be improved [9; 10; 11; 12; 13]. In this paper, we first propose a fault diagnosis method based on quantum Bayesian estimation for partially observable discrete event dynamic systems [14; 15; 16; 17; 18; 19; 20]. When the uncertainty level of the network is very high, the uncertain elements in the network cause uncertain paths. Quantum Bayesian inference is introduced, and then the interference term is added to the Bayesian formula via the quantum superposition principle, which brings the probability estimation closer to the real probability. Compared with traditional fault diagnosis methods, the quantum Bayesian estimation method can greatly improve the accuracy of fault diagnosis. The quantum parameters are manually selected to calculate the firing probability of fault transitions and judge the fault state of the system by establishing the quantum Bayesian network model of unobservable fault transition by using the firing relationship of observable transitions and the firing probability of unobservable transitions [21; 22; 23; 24; 25; 26; 27]. This paper is structured as follows: Section 2 presents a description of the quantum Bayesian fault diagnosis problem in partially observable Petri nets [28; 29; 30; 31]. Section 3 proposes a quantum Bayesian fault diagnosis method for Partially Observable Petri Nets. Section 4 illustrates the method with a case study and compares the proposed algorithm with the previous algorithm.