# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 1 Introduction of Interconnected System  
With the rapid developments of modern techniques, interconnections are becoming very common in modern control systems. The interconnected system here refers to the system composed of interacting subsystems either due to physical system structure or purposes of convenient analysis. The system or process itself may be the result of a series connection, parallel connection, and feedback interconnection of various subsystems [1]. For instance, a modern system is usually interfaced with multiple sensors, actuators, and process components. Therefore, a typical system can be viewed as composed of at least three interconnecting subsystems--actuator, sensor, and process subsystems. In order to ensure the normal operation of the whole system, the functions of these three parts must be normal. In addition, each subsystem can also be regarded as a series of dynamic subsystems, because each subsystem itself can be viewed as a dynamic system. In all cases, the global plant, as well as each subsystem, can be analyzed at different levels, up to the component level, to assess the reliability of the entire plant. It is becoming increasingly important to study the interconnected systems in analyzing dynamic systems, due to the fact that it allows to investigating less complicated components to study the properties of a complex system [2]. However, because of the requirement of a large amount of identification data or deep physical insight, the modeling and analysis of interconnected systems are challenging [3]. The concerning problems of interconnected systems are of great importance from a theoretical and practical viewpoint and have been studied extensively. The published research results involve the problems of stability, observability, controllability, and invertibility of interconnected systems; for example, in [4; 5; 6; 7; 8]. As shown in [8], the study provided a method to distinguish several dynamic subsystems, and it proved that there are usually delays during information transmission. This will lead to the instability and oscillation ofthese systems. Therefore, many investigations are devoted to the stability of these systems. As illustrated in [4], the study derives a condition to ensure input to the stable state of an interconnected system, that is, to guarantee input to the stable state of both subsystems. In addition to stability, the controllability of interconnected systems was also addressed. Networked control systems are widely used in various fields of engineering; for example, in power generation and distribution systems [9], automotive control systems [10], cooperative control of unmanned vehicles [11], etc. The forms of interconnected systems also vary in the existing research. In most studies on tackling power series, the form of analytic functions interconnections was employed, e.g., [12]. In [13], an interconnected system constituted with a nonlinear followed by a linear time-invariant dynamical system was analyzed. The observability of an interconnected system composed of a partial differential equation (PDE) and an ordinary differential equation (ODE) was discussed in [14]. Two Fliess operators' compositions were found in [15]. In [16], the interconnections of bilinear subsystems were discussed. When studying the characteristics of interconnected systems, the interconnected system composed of cascaded subsystems has received extensive attention. Generally speaking, due to the limitation of computational availability, system complexity, or communication bandwidth in the practical engineering world, it is very difficult to analyze cascaded interconnected systems with centralized structures [17]. Therefore, increasing attention has been paid in recent years to the research of distributed or decentralized methods. However, distributed analysis becomes more challenging due to the interaction between subsystems and the limited information available to each subsystem.