A shared digital twin of a manufacturing system is valuable to provide maintenance services in a collaborative manner. An accurate analytical model of the Bayesian network is crucial to depict endogenous failure mechanisms in the digital twin. Nodes’ connections in Bayesian networks correspond to a range of linear, bilinear or multilinear mappings over finite-state variables. The conditional probability table (CPT) of a child node can be represented as a k-dimensional tensor if it has (k-1) parent nodes. A new matrix analytic approach is proposed for Bayesian network inference based on the theory of semi-tensor product. The matrix representation of probabilistic networks is firstly studied, and a specific parameter training algorithm is constructed based on the matrix model. Bayesian network inference algorithms, including both forward and backward reasoning, are then presented for reliability analysis and fault diagnosis. A real manufacturing system is applied to verify the proposed approach. This matrix analytic approach helps to study the Bayesian network’s mathematical properties, and it is proved to be convenient and efficient in probability network modeling and inference.

制造系统的共享数字孪生对于以协作方式提供维护服务很有价值。贝叶斯网络的准确分析模型对于描述数字孪生中的内生故障机制至关重要。贝叶斯网络中的节点连接对应于有限状态变量上的一系列线性、双线性或多线性映射。子节点的条件概率表 (CPT) 可以表示为一个k维张量，如果它有 ( k-1) 父节点。提出了一种基于半张量积理论的贝叶斯网络推理新的矩阵解析方法。首先研究了概率网络的矩阵表示，并基于矩阵模型构建了具体的参数训练算法。贝叶斯网络推理算法，包括前向和后向推理，然后用于可靠性分析和故障诊断。一个真实的制造系统被应用于验证所提出的方法。这种矩阵分析方法有助于研究贝叶斯网络的数学性质，在概率网络建模和推理中被证明是方便和高效的。