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Observability Analysis of Parallel Combining Automata based on Algebraic State Space

Via the semi-tensor product of matrices, this paper analyses the observability problem on the combined automata constructed in a parallel manner and presents a new approach to differentiate four kinds of observability that contain initial state observability and current state observability for with and without information, respectively. Firstly, by defining transition structure matrix and output structure matrix, we have the matrix expression of parallel combining automata, based on which, we succeed in structuring two polynomial matrices to follow the tracks of input and output string, the significant breakthrough to carry out our work. Meanwhile, we propose four sufficient and necessary theorems to explain the relationship between polynomial matrices and their corresponding observability. In fact, the polynomial matrices are two modifications of the matrix expression the former referred, because they have a large advantage in reducing dimensionality and analysis. Finally, the obtained results are applied to an illustrative example.