## 2 Coupling Correlation of Complex System
### Hierarchical Model and Description of Complex Systems
#### 2.1.2 Formal Description of Hierarchical Model  
For the task-function-resource hierarchy architecture of the system, from the perspective of the element set, the system's task element set, function element set, and resource element set can be defined separately. The task element \(t_{i}\) is a task unit established by the system requirement analysis. It is supported by a series of basic function elements. The task element set \(T\) can be expressed as a set of several task elements: \(T=\{t_{1},t_{2},\,\ldots,\,t_{k}\}\). Similarly, the function element \(f_{i}\) is the basic  
Figure 1: Typical task-function-resource architecture of complex systems. The straight lines represent the coupling correlation of elements in the same layer, and dashed lines represent coupling correlation across layers of elements in the different layers. Lines (connections) are bi-directional.  
function unit that supports the task implementation in the system. It is supported by a series of basic resource elements. The function element set \(F\) can be expressed as a set of several functional elements: \(F=\{f_{1},f_{2},\ \ldots,\ f_{m}\}\). The resource element \(r_{i}\) is a physical or logical unit that supports the realization of function in the system. The system resource element set \(R\) can be represented as a set of several resource elements: \(R=\{r_{1},r_{2},\ \ldots,\ r_{n}\}\).  
Through the analysis of the system hierarchy architecture, in order to describe specific relational information, adjacency matrix can be used for the most direct formal record, that is, the mapping correlation matrix between task-function elements can be expressed as shown in Matrix (1).  
\[M^{ff}=\left(M^{ff}_{ij}\right)_{kom}=\begin{array}{cccc}t_{1}&f_{2}&\ldots& f_{m}\\ t_{2}&\vdots&\left[\begin{array}{cccc}M^{ff}_{11}&M^{ff}_{12}&\ldots&M^{ff}_{1 m}\\ M^{ff}_{21}&M^{ff}_{22}&&M^{ff}_{2m}\\ \vdots&&\ddots&\vdots\\ M^{ff}_{k1}&M^{ff}_{k2}&\cdots&M^{ff}_{km}\end{array}\right]\end{array} \tag{1}\]  
where \(M^{ff}_{ij}=1\) means there is a direct correlation between task element \(t_{i}\) and function element \(f_{j}\)\(M^{ff}_{ij}=0\) means no direct correlation.  
Similarly, the function-resource element mapping correlation matrix can be expressed shown as Matrix (2).  
\[M^{fr}=\left(M^{fr}_{ij}\right)_{m\times tr}=\begin{array}{cccc}f_{1}&r_{2}& \ldots&r_{n}\\ f_{2}&\vdots&\left[\begin{array}{cccc}M^{ff}_{11}&M^{ff}_{12}&\ldots&M^{ff}_{1 n}\\ M^{ff}_{21}&M^{ff}_{22}&&M^{ff}_{2n}\\ \vdots&&\ddots&\vdots\\ M^{ff}_{m1}&M^{ff}_{m2}&\cdots&M^{ff}_{mn}\end{array}\right]\end{array} \tag{2}\]  
where \(M^{fr}_{ij}=1\) means there is a direct correlation between function element \(f_{i}\) and resource element \(r_{j}\)\(M^{fr}_{ij}=0\) means no direct correlation.  
If requiring to further record the cross-layer correlation relationship between task and resource elements, we can define and obtain it by matrix operation \(M^{tr}=M^{tf}\times M^{fr}\) shown in Matrix (3). However, in general, the practical significance of this cross-layer correlation is not obvious. It is the focus on the system design to clarify the software-hardware configuration mapping relationships from functions to resources. Therefore, this paper will focus on correlations from functions to resources.  
\[M^{tr}=M^{ff}\times M^{fr}=\begin{array}{cccc}t_{1}&\ldots&r_{n}\\ \vdots&\vdots\\ t_{n}&M^{ff}_{k1}+\ldots M^{ff}_{k1}\times M^{ff}_{km}\times M^{ff}_{mn}&\ldots &M^{ff}_{k1}\times M^{ff}_{k2}+\ldots M^{ff}_{km}\times M^{fr}_{mn}\end{array} \tag{3}\]  
where \(M^{tr}_{ij}\geq 1\) means there is a direct correlation between task element \(t_{i}\) and resource element \(r_{j}\)\(M^{tr}_{ij}=0\) means no direct correlation.