# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 2 Inversion of Nonlinear Interconnected System  
### Modeling of the Interconnected System  
An interconnected system consisting of two dynamic subsystems was considered, as shown in Figure 1. The system decomposition could either be due to its physical structure or the purpose of analysis; for example, the local subsystem could represent field devices, like an actuator or a sensor. The high-level subsystem could then be the process dynamics, like a heat exchanger. In this way, it allowed the analysis of less complex subcomponents to study the characteristics of the interconnected system. Therefore, when analyzing the whole system, it was of great significance to treat the field equipment as an independent dynamic subsystem. The case of an inverted pendulum on a cart can be a typical example: if we view two subsystems physically as a cart and an inverted pendulum, then they constitute an interconnected system.  
Assume that the low-level subsystem is an input affine dynamic system and depicted by (1):  
\[\sum_{L}:\left\{\begin{array}{c}\dot{x}_{L}=f_{L}(x_{L})+g_{L}(x_{L})u,\;x_{ L}(t_{0})=x_{L0}\\ u_{L}=h_{L}(x_{L})\end{array}\right. \tag{1}\]  
where \(\mathsf{x}_{L}\in\mathcal{X}_{L}\subseteq\Re^{\mathsf{L}}\) is the state, \(\;\mathsf{u}_{L}\in\mathsf{U}_{L}\subseteq\mathbb{R}^{\mathsf{m}}\) is the output of the low-level subsystem, which is also the input of the high-level subsystem, and \(u\in\mathcal{U}\subseteq\;\Re^{l}\) is the local input. \(f_{L}\)and \(g_{L}\) are smooth vector field on \(\Re^{\mathsf{m}}\) and \(h\) is a smooth vector field on \(\Re^{\mathsf{m}}\). \(f_{L}\), \(g_{L}\), \(h_{L}\) are algebraic functions, respectively.  
An input affine dynamic system is also assumed for the higher level subsystem, and is described by (2):  
\[\sum_{H}:\left\{\begin{array}{c}\dot{x}_{H}=f_{H}(x_{H})+g_{H}(x_{H})u_{L},\; x_{H}(t_{0})=x_{H0}\\ y=h_{H}(x_{H})\end{array}\right. \tag{2}\]  
where \(x_{H}\in\mathcal{X}_{H}\subseteq\Re^{\mathsf{n}}\) is the state of the high-level subsystem, \(y\in\mathcal{Y}\subseteq\Re^{p}\) is the output of the high-level subsystem, which is also the output of the global system, and  
Figure 1: The structure of an interconnected system.  
\(\Re^{m}\) is the input of the high-level subsystem, which is also the output of the low-level subsystem. \(f_{H}\) and \(g_{H}\) are smooth vector field on \(\Re^{n}\) and \(h_{\rm H}\) is a smooth vector field on \(\Re^{p}\). \(f_{H},\ g_{H},\ h_{H}\) are algebraic functions, respectively.  
In this way, combined with subsystems \(\sum_{L}\) and \(\sum_{H}\) in (1) and (2), an interconnected system \(\sum\) was generated, where the input and output vectors were considered as \(u\) and \(y\), respectively.  
For the interconnected system described by (1) and (2), the main objective of this paper is to give conditions to prove its invertibility. In this paper, invertibility refers to the ability to recognize and recover the input \(u\) at the local level subsystem from the output \(y\) at the global level. Therefore, the essence of invertibility in this paper is to investigate the capability of distinguishing local inputs from the influences on global final products. That is actually a study of the distinguishability of two inputs and their capabilities of producing different outputs.