# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 1 Introduction of Interconnected System
Therefore, a problem worthy of attention is whether it can be proved that, under certain conditions, the influence of the lower subsystem can be distinguished on the higher subsystem, so as to avoid the complete measurement of the local subsystem. This can be regarded as a problem of the system's invertibility because one of the important objectives of invertibility analysis is to prove that the input or unknown input of the control system is distinguished. In the past 50 years, due to its important theoretical and practical significance, the invertibility of the system has been widely studied. The research on the invertibility of nonlinear systems began in reference [18]. In this paper, [PERSON]'s structural algorithm was extended to multiple-input multiple-output (MIMO) nonlinear systems. After that, the algorithm was modified in reference [19], which aimed at covering more kinds of systems. Similar literature related to the extension of the algorithm can be found in papers [20; 21; 22; 23; 24; 25]. The invertibility of the system in the literature is related to the distinguishability of the system. The distinguishability of two variables refers to their ability to produce recognizable output for a given system. Some concepts of distinguishability or invertibility can be found in the literature, as in [18; 22; 26; 27]. For example, the problem of the invertibility of switched linear systems was produced by [PERSON] and [PERSON] in [28], in which they discussed the ability to determine the active mode of the system from the input and output data. The idea was further extended to a nonlinear system in [27] and applied to fault diagnosis in [29]. In the above studies, the analysis of cascaded nonlinear systems received less attention. However, it is usually very important to describe the properties of composite systems, especially when the subsystems are nonlinear [30]. This paper considered a cascaded interconnected system consisting of two dynamic subsystems for physical or analysis purposes. The interconnection of two physical devices means that some variables associated with the first device are also variables or impact variables associated with the second device. Specifically, the problem is to give a sufficient and/or necessary condition, under which, given initial states, a local input can produce a distinguishable output of an interconnected system constituting two nonlinear subsystems. The essence of the problem is whether it can be proved that, under certain conditions, the input of the lower subsystem has a significant impact on the output of the higher subsystem. In order to solve this problem, the invertibility of cascaded interconnected systems is derived. The left invertibility of the interconnected system is capable of ensuring that the impacts of local variables on the global level are distinguishable. The property of distinguishability of two inputs or parameters refers to their capacity to generate different output signals for a given input signal. The discussion of invertibility is of great significance in practical engineering. For example, for the problem of fault detection and isolation (FDI), a significant way is to treat the fault as an unknown input, and the motivation of invertibility is actually to detect and isolate the input, that is, to identify the possible location and time of the fault in the system; for example, in [29; 31; 32; 33]. The contribution of this paper mainly lies in that it emphasizes the importance of the influences of local internal dynamics (actuator) on the global dynamics of the control system. Thus, it provides a basis for allowing the analysis of less complex subcomponents to study the characteristics of the interconnected systems. The paper is organized as follows: Section 2 is devoted to the definition of invertibility of an interconnected dynamic system. In Section 3, conditions are given to validate involving definitions. Then, the procedure of computation of the inverse of the interconnected system is presented in Section 4.