# Dissipativity Analysis of Large-Scale Networked Systems
## 2 System Description and Some Preliminaries  
The networked system \(\Gamma\) is consisted of \(N\) linear time-invariant subsystems, and the \(i\)-th subsystem \(\Gamma_{i}\) is defined by the state-space model below,  
\[\left[ \begin{array}{c}\dot{x}(t,i)\\ z(t,i)\\ y(t,i)\end{array} \right]=\left[ \begin{array}{cc}A_{\rm TT}(i)&A_{\rm TS}(i)& B_{\rm T}(i)\\ A_{\rm ST}(i)&A_{\rm SS}(i)&B_{\rm S}(i)\\ C_{\rm T}(i)&C_{\rm S}(i)&D_{\rm T}(i)\end{array} \right]\left[ \begin{array}{ c}x(t,i)\\ v(t,i)\\ u(t)\end{array} \right], \tag{1}\]  
in which \(t\) and \(i\) denote respectively for the temporal variable and the index number of a subsystem, \(i=1,2,\ldots,N\). Moreover, \(x(t,i)\) is the state vector of the \(i\)-th subsystem \(\Gamma_{i}\) at time \(t\). \(y(t,i)\) and \(u(t,i)\) represent the external output vector and external input vector of the \(\Gamma_{i}\), respectively. \(z(t,i)\) and \(v(t,i)\) are the output vector to other subsystems and input vector from others, which is also called internal output vector and input vector. The connection relationship among subsystems can be expressed as  
\[v(t)=\Phi z(t), \tag{2}\]  
here, \(v(t)=col\left\{v(t,i)|_{i=1}^{N}\right\}\) and \(z(t)=col\left\{z(t,i)|_{i=1}^{N}\right\}\). \(\Phi\) is called the subsystem connectivity matrix. We assume that each row of the matrix \(\Phi\) has only one non-zero element equal to one and there are no columns in which all of the items are equal to zero. That means the internal output channels of a subsystem can affect the internal inputs channels of other subsystems, and some subsystem internal input channels depend on the internal output of multiple subsystems. This assumption, as explained in [23], does not jeopardize the generality of the adopted system model. Approximate power-law degree distribution widely exists in engineering systems, such as protein interaction networks, gene regulatory networks, power systems, the Internet, etc. [23]. In these systems, the dimension of the subsystem connection matrix \(\Phi\) is usually much smaller than the state dimension of the system, and the interactions among subsystems are sparse. In this paper, we assume that the dimensions of vectors \(x(t,i)\), \(v(t,i)\), \(z(t,i)\), \(u(t,i)\) and \(y(t,i)\) are \(m_{xi}\), \(m_{vi}\), \(m_{zi}\), \(m_{ui}\) and \(m_{yi}\), respectively. Based on the above assumptions and Equation (2), the dimension of the matrix \(\Phi\) is \(\sum_{i=1}^{N}m_{vi}\times\sum_{i=1}^{N}m_{zi}\). Then we can get \(\Phi^{T}\Phi=\Sigma^{2}\),where \(\Sigma^{2}=\text{diag}\Big{\{}\Sigma_{j}^{2}\Big{\}}_{j=1}^{N}\Big{\}}\), \(\Sigma_{j}^{2}=\text{diag}\Big{\{}m(i)|_{i=M_{xi,i}\text{-}1}^{M_{xi}}\Big{\}}\), \(M_{z,i}=\sum_{k=1}^{j}m_{zk}\), \(m(i)\) indicates the number of subsystems directly affected by the \(i\)-th element of the vector \(z(t)\), \(i=1,\cdots,\sum_{k=1}^{N}m_{vk}\), \(j=1,\cdots,N\). To simplify the mathematical derivation, we define the following matrix, \(A_{*\#}=diag\Big{\{}A_{*\#}(i)|_{i=1}^{N}\Big{\}}\), \(B_{*}=diag\Big{\{}B_{*}(i)|_{i=1}^{N}\Big{\}}\),\(C_{*}=diag\Big{\{}C_{*}(i)|_{i=1}^{N}\Big{\}}\)and\(D_{*}=diag\Big{\{}D_{*}(i)|_{i=1}^{N}\Big{\}}\) in which \(*,\#=\mathbb{T},\mathbb{S}\). By exploiting the connection relationship among subsystems, the dynamic system \(\Gamma\) may be expressed equivalently in the following state-space form,  
\[\left[ \begin{array}{c}\dot{x}(t)\\ y(t)\end{array} \right]=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]\left[ \begin{array}{c}x(t)\\ u(t)\end{array} \right], \tag{3}\]  
where  
\[A{=}A_{\rm TT}+A_{\rm TS}\Phi(I-A_{\rm SS}\[PERSON])^{-1}A_{\rm ST},\]\[B =B_{\rm T}+A_{\rm TS}\Phi(I-A_{\rm SS}\[PERSON])^{-1}B_{\rm S},\] \[C =C_{\rm T}+C_{\rm S}\Phi(I-A_{\rm SS}\[PERSON])^{-1}A_{\rm ST},\] \[D =D_{\rm T}+C_{\rm S}\Phi(I-A_{\rm SS}\[PERSON])^{-1}B_{\rm S}.\]  
**Note:** Well -posedness is very important in system design, and ill-posed systems are usually difficult to control or impossible to estimate [25; 26; 27].