# Dissipativity Analysis of Large-Scale Networked Systems
## 2 System Description and Some Preliminaries
Therefore, this paper assumes that System \(\Gamma\) is well-posed, which means that \((I-A_{\rm SS}\Phi)^{-1}\) exists. This paper intends to establish computationally effective conditions for the dissipativity analysis of large-scale networked systems \(\Gamma\). The concept of dissipativity is very important in the system, whether from the perspective of theoretical research or the perspective of practical application. Roughly speaking, dissipative systems can be described as such properties. At any time, the energy that the system may provide cannot exceed the energy already supplied. We first describe the definition of dissipativity for System \(\Gamma\). The definition is related to the supply function. For the \(i\)-th subsystem \(\Gamma_{i}\), its supply function is defined as  
\[s_{i}(u(t,i),y(t,i))=\left[ \begin{array}{c}y(t,i)\\ u(t,i)\end{array} \right]^{T}Q(i)\left[ \begin{array}{c}y(t,i)\\ u(t,i)\end{array} \right], \tag{4}\]  
where \(Q(i)\) is a symmetric matrix of suitable dimensions. Definition 1: The large-scale networked system (1) and (2) with \(x(0,i)=0\) is said to be dissipative with supply function \(s_{i}(u(t,i),y(t,i))\) if and only if there is a matrix \(P(i)\geq 0\), such that,  
\[\int\limits_{t_{0}}^{t_{1}}\sum\limits_{i=1}^{N}s_{i}(u(t,i),y(t,i))dt\geq\sum \limits_{i=1}^{N}x^{T}(t_{1},i)P(i)x(t_{1},i)-\sum\limits_{i=1}^{N}x^{T}(t_{0},i)P(i)x(t_{0},i) \tag{5}\]  
holds for all \(t_{0}\leq t_{1}\). According to the definition, the supply function can be interpreted as the energy transferred to the system, which means that within a period of time \([t_{0},t_{1}]\), as long as \(\int\limits_{t_{0}}^{t_{1}}\sum\limits_{i=1}^{N}s_{i}(u(t,i),y(t,i))dt\) is positive, the system will work normally, otherwise, the system will not work. \(\sum\limits_{i=1}^{N}x^{T}(t_{1},i)P(i)x(t_{1},i)-\sum\limits_{i=1}^{N}x^{T}( t_{0},i)P(i)x(t_{0},i)\) represents the actual energy consumption of the system after the time interval \(t_{1}-t_{0}\). Therefore, Equation (5) shows that in any time period \([t_{0},t_{1}]\), the energy change inside the system will not exceed the energy supplied by the outside. It can be seen from the following derivation that the definition of dissipativity for the networked system (1) and (2) are consistent with the one in [28] based on (3). The supply function based on the large-scale networked system (1) and (2) is as follows,  
\[s(u(t),y(t)){=}\sum\limits_{i=1}^{N}s_{i}(u(t,i),y(t,i)){=}\left[ \begin{array} []{c}y(t,1)\\ u(t,1)\\ \left[\begin{array}{c}y(t,2)\\ u(t,2)\end{array}\right]\\ \vdots\\ \left[\begin{array}{c}y(t,N)\\ u(t,N)\end{array}\right]\end{array} \right]^{T}Q\left[ \begin{array}{c}\left[ \begin{array}{c}y(t,1)\\ u(t,1)\end{array}\right]\\ \left[\begin{array}{c}y(t,2)\\ u(t,2)\end{array}\right]\\ \vdots\\ \left[\begin{array}{c}y(t,N)\\ u(t,N)\end{array}\right]\end{array} \right]. \tag{6}\]  
The supply function of System (3) is  
parameter \(A_{*\#}\), \(B_{*}\), \(C_{*}\) and \(D_{*}\) with \(*,\#=\text{T}\), S are block diagonal, the matrix \((I-A_{\text{SS}}\Phi)^{-1}\) is generally dense. When there are a large number of subsystems in large-scale networked systems, the calculation of matrices \(A\), \(B\), \(C\), and \(D\) involves the inversion of high-dimensional matrices. Therefore, when the scale of the networked system increases, the computational complexity of Equation (9) will become very high. Lemma 4 is a dissipative analysis condition based on lumped networked model. Due to the establishment of the lumped model, the connection relationship among subsystems is hidden inside the parameters, and its structural information is not effectively utilized. As a result, for large-scale networked systems, the use of this condition for dissipative testing will inevitably bring computational difficulties and even cannot be calculated. Then, to reduce the computational difficulty caused by the increase of system scale, we establish a computationally efficient sufficient, and necessary condition for the strict dissipativity analysis of large-scale networked systems.