# Dissipativity Analysis of Large-Scale Networked Systems
## 1 Introduction  
In recent years, the research of large-scale networked systems has attracted great attentions [1; 2; 3; 4]. The system can be considered as composed of a large number of subsystems with different spatial locations connected in a certain way [5]. Generally, subsystems exchange information with their neighbors straightforwardly and predictably, but the system often exhibits complicated dynamic behavior when seen as a whole. Such systems have extensive application background, including airplane formation flight [6], power network distributed system [7], automated highways [8], multi-agent formation systems [9], and so on. For such a complex system, the classic method of bringing all the subsystems together and analyzing it as a single large-scale system has very strict requirements on the calculation speed and accuracy of the computer, which will inevitably bring computational difficulties. Therefore, using the system structure to find more efficient computational conditions is of great engineering significance for the dissipativity analysis of large-scale networked systems. There are already many results on the performance analysis of networked systems, but the research on dissipativity is not mature enough and needs to be further developed and improved. In 1972, the famous scholar [PERSON] first put forward the concept of dissipativity [10; 11]. Dissipativity describes the equilibrium relationship between the system's internal energy, which is a vital concept in theoretical research and practical application. Its essential meaning is that there is a non-negative energy function (called storage function) so that the energy supply rate of the system is always greater than the loss of energy inside the dynamic system. Based on [PERSON] work, many scholars have done in-depth studies on dissipativity and obtained rich results, which have played a significant role in the field of circuit, system, and control theory. [PERSON]. [12; 13] respectively studied the dissipative control problems of linear continuous-time and discrete systems based on linear matrix inequality (LMI) methods. A simplified mathematical model of the interconnected two-machine power system was established in [14], and its non-linear dynamic behaviorsuch as dissipativity properties was analyzed. In [15], the analysis and improvement of the dissipativity performance of interconnected passive systems are studied. For networked control systems, [PERSON]. [16] obtained some new sufficient conditions by utilizing Lyapunov stability theory and LMI technology to ensure that the closed-loop system is finite-time limited and dissipative. In [17], a distributed controller was created to ensure the dissipativity of a networked system made up of dynamically coupled subsystems. Its control synthesis is done locally at the subsystem level and doesn't involve the relationship among subsystems, hence it has certain drawbacks. The linear dynamic system with the interconnection structure specified by the directed graph is studied in [18]. Based on the dissipativity inequality, an LMI for calculating system performance is established and the concept of local dissipativity is defined. Using the knowledge of graph theory to analyze large-scale networked systems has certain constraints on the dynamic characteristics and connection modes of subsystems, which has certain limitations in practical application [19; 20]. Considering that the system structure of large-scale networked systems usually has sparse characteristics or specific structural forms [21; 22; 23], in the large-scale connected systems discussed in [24], the concept of internal input and output is introduced to represent the connections and functions among subsystems, and the connection relationship among sub-units of the entire system is described by subsystem connection matrix. This description method takes into account the situation where the dynamic characteristics of the subsystems are different and the connection relationship of the subsystems is arbitrary.