# Dissipativity Analysis of Large-Scale Networked Systems
## 1 Introduction
The previously mentioned UAV formation flight refers to the arrangement of multiple UAVs in a certain formation so that they maintain in formation or change their relative positions within a limited scope during the flight. To maintain a certain formation shape, information interaction is required among the UAVs. In a centralized strategy, each UAV has to know information about the whole formation, demanding substantial information interaction. It is computationally intensive and requires the high performance of the airborne computer. In fact, each UAV can interact with its position, speed, attitude, and motion target with only the UAVs connected to it in the formation. In this way, the amount of computation is greatly reduced, and the system is relatively simple to implement. It is this sparse property or specific structural form among subsystem connections that we exploit to give more computationally efficient dissipativity criteria for large-scale networked systems. Dissipativity explains some of the energy losses and control problems of control systems and is a more general performance indicator of system performance. In this paper, our objective is to reduce the computational burden of dissipativity analysis for large-scale network systems with a large number of subsystems. In general, the large-scale networked systems studied in this paper have the following characteristics. The first is that the scale of the system is large, including many subsystems. The second and most important point is that the interaction among subsystems of large-scale networked systems is usually sparse or has a specific structural form. We introduce intermediate variables in networked systems to describe the relationship among subsystems, which is more general and explicitly characterizes the structural characteristics of large-scale systems. In this regard, this paper proposes several new LMI conditions, which effectively use the block diagonal structure of the system parameter matrix and the sparsity of the subsystem connection matrix, avoiding the inverse computation of high-dimensional matrices. The simulation results indicate that the conditions proposed in this paper are more efficient than the existing results. The following is the structure of the paper. The model and the definition of dissipativity for the networked system and some preliminary results are given in Section 2. In Section 3, some conditions for dissipative analysis of networked systems are given, and the relationship between these conditions and existing conditions is discussed. Some numerical simulation results are presented in Section 4. The research results of this paper are summarized in Section 5, and the direction of further research is also proposed here. **Notation 1**. : _The symbol \(R\) is used to denote the set of real numbers, and the vector space produced by real numbers of appropriate dimensions is denoted as \(R^{\#}\). \(col\left\{Z_{i}|_{i=1}^{L}\right\}\) denotes the vector/matrix stacked by \(Z_{i}(i=1,2,\ldots,L)\), and \(diag\{Z_{i}|_{i=1}^{L}\}\) signifies a block diagonal matrix with \(Z_{i}\) as the \(i\)-th diagonal block. \(\left\{Z_{ij}|_{i=1,j=1}^{i=M,j=N}\right\}\) stands a matrix with \(M\times N\) blocks, and its \(i\)-th row \(j\)-th column block matrix being \(Z_{ij}\). \(D_{n}\) and \(0_{n\times m}\) represent the \(n\) dimensional zero column vector and the \(n\times m\) dimensional zero matrix respectively, the dimension subscript is omitted if there is no ambiguity, and the identity matrix \(I\) is the same. The superscript \(T\) represents the transpose of a matrix or vector, and \((*)^{T}WZ\) or \(ZW(*)^{T}\) is shorthand for \(Z^{T}WZ\) or \(ZWZ^{T}\)._