# Dissipativity Analysis of Large-Scale Networked Systems
## 2 System Description and Some Preliminaries
This condition effectively utilizes the sparse structure of the subsystem connection matrix \(\Phi\) in the networked system, that is, each subsystem is only connected to a limited number of other subsystems. Assume that the networked system \(\Gamma\) is controllable. Then, System \(\Gamma\) is strictly dissipative with the supply function \(s_{i}(u(t,i),y(t,i))\) if and only if there exists a symmetric positive definite matrix \(P\) and a positive scalar \(h\) such that,  
\[ \begin{array}{c}(*)^{T}\Bigg{[}\begin{array}{cc}0&P\\ P&0\end{array}\end{array} \end{array} \tag{10}\] \[-h\times(*)^{T}\bigg{[} \begin{array}{cc}I&-\Phi\\ -\Phi^{T}&\Sigma^{2}\end{array} \bigg{]}\bigg{[} \begin{array}{cc}I&\begin{array} []{cc}0&0\\ A_{\text{SS}}&\begin{array}{cc}A_{\text{ST}}&B_{\text{S}}\end{array}\end{array} \end{array} \bigg{]}<0.\]  
**Proof of Theorem 1. ** Equation (9) can be expressed equivalently as follows,  
\[(*)^{T}\Bigg{[} \begin{array}{cc}\begin{array}{cc}0&P\\ P&0\end{array}\end{array} \end{array} \tag{11}\]  
We express Equation (11) in the following equivalent form,  
\[(*)^{T}\Bigg{[} \begin{array}{cc}\begin{array}{cc}0&P\\ P&0\end{array}\end{array} \end{array} \tag{12}\] \[-Q\Bigg{]}\left[ \begin{array}{ccc}I&0&0&0\\ 0&0&I&0\\ 0&0&0&I\\ 0&I&0&0\end{array} \right]\Bigg{[} \begin{array}{cc}\begin{array}{cc}I&0\\ 0&I\end{array}\\ \begin{array}{cc}\begin{array}{cc}A&B\\ C&D\end{array}\end{array}\end{array} \right]<0.\]  
Matrices \(A\), \(B\), \(C\), and \(D\) can be written as follows,  
\[ \begin{array}{c}\begin{bmatrix}A&B\\ C&D\end{array} \end{bmatrix}=\left[ \begin{array}{cc}A_{\text{TT}}&B_{\text{T }}\\ C_{\text{T}}&D_{\text{T}}\end{array} \right]+\left[ \begin{array}{c}A_{\text{ TS}}\\ C_{\text{S}}\end{array} \right]\Phi(I-A_{\text{SS}}\Phi)^{-1}\big{[} \begin{array}{ cc}A_{\text{ST}}&B_{\text{S}}\end{array} \big{]}. \tag{13}\]  
Substituting the above formula into Equation (12), we can get that,  
\[ \begin{array}{c}(*)^{T}\Bigg{[}\begin{array}{cc}\begin{array}{cc}0&P\\ P&0\end{array}\end{array}\end{array}  \tag{14}\] \[\times\Bigg{[} \begin{array}{cc}\Phi(I-A_{\text{SS}}\Phi)^{-1}A_ {\text{ST}}&\Phi(I-A_{\text{SS}}\Phi)^{-1}B_{\text{S}}\\ I&0\\ 0&I\end{array} \end{array}\]Then we define matrices \(F\), \(M\), and \(K\) as follows,  
\[F=(*)^{T}\left[ \begin{array}{cc}-\left[\begin{array}{cc}0&P\\ P&0\end{array}\right]\\ &Q\end{array} \right]\left[ \begin{array}{ccc}0&I&0\\ A_{\text{TS}}&A_{\text{TT}}&B_{\text{T}}\\ C_{\text{S}}&C_{\text{T}}&D_{\text{T}}\\ 0&0&I\end{array} \right], \tag{15}\]  
\[M=\left[ \begin{array}{cc}\Phi(I-A_{\text{SS}}\Phi)^{-1}\left[\begin{array}[] {cc}A_{\text{ST}}&B_{\text{S}}\end{array}\right]\\ &\left[\begin{array}{cc}I&0\\ 0&I\end{array}\right]\end{array} \right], \tag{16}\]  
\[K=\left[ \begin{array}{cc}I&-\Phi\end{array} \right]\left[ \begin{array}{cc}I &\left[\begin{array}{cc}0&0\end{array}\right]\\ A_{\text{SS}}&\left[\begin{array}{cc}A_{\text{ST}}&B_{\text{S}}\end{array} \right]\end{array} \right]. \tag{17}\]  
Obviously, \(M^{T}(-F)M<0\).When \(v=M_{\zeta}^{T}\), \(\zeta\in R^{\#}\), for any \(v\
eq 0\), we can get that \(Kv=0\), which means \(v^{T}Fv>0\). According to Lemma 2, there must be a real number \(h\) such that,  
\[ \begin{array}{c}(*)^{T}\left[\begin{array}{cc}-\left[\begin{array}{cc}0 &P\\ P&0\end{array}\right]\\ &Q\end{array}\right]\left[\begin{array}{ccc}0&I&0\\ A_{\text{TS}}&A_{\text{TT}}&B_{\text{T}}\\ C_{\text{S}}&C_{\text{T}}&D_{\text{T}}\\ 0&0&I\end{array}\right]\\ +h\times(*)^{T}\left[\begin{array}{cc}I&-\Phi\\ -\Phi^{T}&\Sigma^{2}\end{array}\right]\left[\begin{array}{cc}I&\left[\begin {array}[]{cc}0&0\end{array}\right]\\ A_{\text{SS}}&\left[\begin{array}{cc}A_{\text{ST}}&B_{\text{S}}\end{array} \right]\end{array} \right]>0.\end{array} \tag{18}\]  
That is,  
\[