In this paper, we propose a new hypernetwork model based on Tracy-Singh Product on the correlation matrix of hypergraph. Node degree, node hyperdegree, and hyperedge degree and their corresponding polynomials are introduced to describe this hypernetwork model. It is shown that this kind of hypernetworks is fractal as its correlation matrix is a fractal matrix. The fractal parameter is then given. What's more, this kind of hypernetworks is also small-world for the diameter won't exceed twice the diameter of primitive hypergraph. By a novel product of node degree polynomial, node hyperdegree polynomial and hyperedge degree polynomial, node degree, node hyperdegree and hyperedge degree are obtained.

This paper deals with the stability characteristics of zeros for sampled-data models with a class of triangle sample and hold realized by a traditional zero-order hold. For any controlled models in the modern industrial system, using a digital control strategy has been shown to provide the means to achieve the assigned objectives. In this process, one must utilize the sample and hold device to obtain the sampled-data models. Previous studies have shown that the triangle sample and hold can improve the stability properties of zeros of a sampled-data control system compared with zero-order hold. However, it is difficult to use triangle sample and hold in practice. In this paper, an approximated method of using triangle sample and hold is proposed. More importantly, on the basis of that method, we explicitly derive the corresponding accurate sampled-data model of controlled models. In addition, we also provide the expression for sampling zeros and the theorem for the stability of a linear control system in the fast sampling process. The results of this paper show that the proposed method has the same advantages as the accurate one. Finally, theoretical findings are validated through numerical simulations with different considerations.

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