# Dissipativity Analysis of Large-Scale Networked Systems
## 2 System Description and Some Preliminaries
\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{19}\]  
The characterization of the left term of Equation (19) shows that if the inequality has a solution, then there must be \(h>0\). So far, the necessity has been proved. Then, multiply the left and right sides of Equation (10) by the matrices \(M\) and \(M^{T}\) respectively, and direct algebraic operations can complete the sufficiency proof. \[ \begin{array}{c}(*)^{T}\left[\begin{array}{cc}\left[\begin{array}{cc}0&P \\ P&0\end{array}\right]\\ &-Q\end{array}\right]\left[\begin{array}{ccc}0&I&0\\ A_{\text{TS}}&A_{\text{TT}}&B_{\text{T}}\\ C_{\text{S}}&C_{\text{T}}&D_{\text{T}}\\ 0&0&I\end{array}\right]\left[\begin{array}{cc}\Phi(I-A_{\text{SS}}\Phi)^{-1} \left[\begin{array}{cc}A_{\text{ST}}&B_{\text{S}}\end{array}\right]\\ &\left[\begin{array}{cc}I&0\\ 0&I\end{array}\right]\end{array}\right]\\ -h\times(*)^{T}\left[\begin{array}{cc}I&-\Phi\\ -\Phi^{T}&\Sigma^{2}\end{array}\right]\left[\begin{array}{cc}I&\left[\begin{array} []{cc}0&0\end{array}\right]\\ A_{\text{SS}}&\left[\begin{array}{cc}A_{\text{ST}}&B_{\text{S}}\end{array} \right]\end{array}\right]\left[\begin{array}{cc}\Phi(I-A_{\text{SS}}\Phi)^{-1} \left[\begin{array}{cc}A_{\text{ST}}&B_{\text{S}}\end{array}\right]\\ &\left[\begin{array}{cc}I&0\\ 0&I\end{array}\right]\end{array}\right]<0.\end{array}  \tag{20}\]  
The proof is completed. It can be seen that the condition in Lemma 4 hides the connection relationship among subsystems inside the parameters, while the left side of Equation (10) in Theorem 1 linearly depends on the symmetric matrix \(P\), and the structure of the system is specifically reflected in it, which can effectively make use of the sparse structure of the subsystem connection matrix. Furthermore, the matrices \(A_{*\#}\), \(B_{*}\), \(C_{*}\) and \(D_{*}\) with \(*,\#=\text{T,S}\) are all block diagonal, and large-scale networked systems are sparse. Combined with the research on sparse semi-definite programming problems [31; 32; 33], when the system is relatively large, the computational complexity of solving the above sparse LMI is frequently lower than the condition in Lemma 4. This aspect can also be explained in subsequent numerical simulations. It is worth noting that the condition of Theorem 1 does not bring conservatism compared with Lemma 4, a dissipative criterion based on the lumped description. When there are a huge number of subsystems, the strict dissipativity analysis using the condition in Theorem 1 may still encounter computational difficulties. To overcome this difficulty, we further explore the structural characteristics of the subsystem connection matrix \(\Phi\), and put forward the conditions for strictly dissipative analysis based on the parameters of each subsystem. A simple derivation leads to the following relationship,  
\[\left[ \begin{array}{cc}I&-\Phi\\ -\Phi^{T}&\Phi^{T}\Phi\end{array} \right]\leq 2\bigg{(}\left[ \begin{array}{ cc}I\\ 0\end{array} \right]\left[ \begin{array}{cc}I&0\end{array} \right]+\left[ \begin{array} []{cc}0\\ \Phi^{T}\end{array} \right]\left[ \begin{array}{cc}0&\Phi\end{array} \right] \bigg{)}. \tag{21}\]  
Combined with Lemma 4 and the properties of the subsystem connection matrix, on the basis of Equation (10), the necessary condition for the strict dissipativity analysis that only depends on the parameters of a single subsystem can be obtained. Theorem 2: _Assume that the networked system \(\Gamma\) is controllable.