# Dissipativity Analysis of Large-Scale Networked Systems
## 2 System Description and Some Preliminaries
A necessary condition for the strict dissipativity of System \(\Gamma\) with the supply function \(s_{i}(u(t,i),y(t,i))\) is that each subsystem has a symmetric positive definite matrix \(P(i)\) and a positive scalar \(h\) such that,_  
\[(*)^{T}\left[ \begin{array}{cc}\begin{array}{cc}0&P(i)\\ P(i)&0\end{array}\end{array} \right] \tag{22}\] \[-h\times(*)^{T}\left[ \begin{array}{cc}I&\\ &\Sigma_{i}^{2}\end{array} \right]\left[\begin{array}{cc}I&\\ A_{\text{SS}}(i)& \begin{array}{cc}\begin{array}{cc}A_{\text{ST}}(i)&B_{ \text{S}}(i)\end{array}\end{array} \right]\right]<0.\]  
For large-scale networked systems, sometimes the parameters of multiple subsystems are the same. In this case, using Theorem 2 is more efficient. A sufficient condition for strict dissipativity analysis which only depends on the parameters of a single subsystem is given below. Theorem 3: _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 there exists a symmetric positive definite matrix \(P(i)\) and real number \(h_{2}\geq h_{1}\geq 0\) (or \(h_{1}\leq h_{2}\leq 0\)) for each subsystem such that,_  
\[(*)^{T}\left[ \begin{array}{cc}\begin{array}{cc}0&P(i)\\ P(i)&0\end{array}\end{array} \right] \tag{23}\] \[-(*)^{T}\left[ \begin{array}{cc}h_{1}I&\\ &-h_{2}\Sigma_{i}^{2}\end{array} \right]\left[ \begin{array}{cc}I&\\ A_{\text{SS}}(i)&\begin{array}{cc}\begin{array}{cc}0&0\end{array}\end{array} \end{array} \right]<0.\]  
Proof of Theorem 3: From Lemma 1, we can get  
\[\left[ \begin{array}{cc}I&-\Phi\\ -\Phi^{T}&\Phi^{T}\Phi\end{array} \right]\geq(1-\alpha)\left[ \begin{array}{cc} 1\\ 0\end{array} \right](*)^{T}+\left(1-\frac{1}{\alpha}\right)\left[ \begin{array}{ cc}0\\ \Phi^{T}\end{array} \right](*)^{T}. \tag{24}\]  
Using the above formula and the conclusion in Theorem 1, one can obtain a sufficient condition for System \(\Gamma\) to be strictly dissipative is the existence of a symmetric positive definite matrix \(P\) and two real numbers \(h>0\), \(\alpha>0\), such that,\[(*)^{T}\Bigg{[}\begin{array}{cc} \begin{bmatrix}0&P\\ P&0\end{bmatrix} \\ &-Q\end{bmatrix}\Bigg{]} \begin{bmatrix}0&I&0\\ A_{\text{TS}}&A_{\text{TT}}&B_{\text{T}}\\ C_{\text{S}}&C_{\text{T}}&D_{\text{T}}\\ 0&0&I\end{bmatrix} \\ -h\times(*)^{T}\bigg{(}(1-\alpha)\bigg{[} \begin{array}{c}I\\ 0\end{array} \bigg{]}(*)^{T}+\Big{(}1-\frac{1}{\alpha}\Big{)}\bigg{[} \begin{array} []{cc}0\\ \Phi^{T}\end{array} \Big{]}(*)^{T}\bigg{)}\bigg{[} \begin{array}{cc}I&\begin{bmatrix} 0&0\\ A_{\text{SS}}&B_{\text{S}}\end{array} \end{bmatrix}\Bigg{]}<0. \tag{25}\]  
Let \(h_{1}=(1-\alpha)h\), \(h_{2}=-\big{(}1-\alpha^{-1}\big{)}h\), we can get \(h_{2}=\alpha^{-1}h_{1}\). Therefore, when \(\alpha\leq 1\), \(h_{2}\geq h_{1}\geq 0\); when \(\alpha\geq 1\), \(h_{1}\leq h_{2}\leq 0\). The proof can be completed by combining Lemma 3. Compared with Theorem 1, the left side of Equation (23) in Theorem 3 is linearly related to the matrix \(P(i)\), and its dimension is entirely governed by the dimension of the subsystem \(\Gamma_{i}\). When the state dimension of each subsystem is fixed, the computational complexity of Equation (23) only linearly depends on the number of subsystems \(N\). Therefore, Theorem 3 has a substantially higher computing efficiency than Theorem 1 for large-scale networked systems. However, it should be noted that Theorems 2 and 3 are conservative.