## 2 Motivations and Problem Formulations  
The problem of state observation is investigated for an interconnected nonlinear system, modeled by two cascaded nonlinear dynamical subsystems: the process and the actuator subsystems. As shown in Figure 1, an interconnected nonlinear system structure is proposed by considering both the actuator and the process as individual dynamic subsystems connected in cascade. The aim is to accurately estimate the state vector of both subsystems, as well as the interconnection. As a result, both local and global dynamics supervision are realized, and analysis of the influence of local internal dynamics on global dynamics is achieved.  
A dynamic process subsystem is proposed in an input affine form:  
\[\Sigma_{\mathbb{P}}:\begin{cases}\dot{\mathrm{x}}=f(x)+g(x)u_{a},&x(t_{0})=x_{0 }\\ y=x\end{cases} \tag{1}\]  
where \(x\in\Re^{n}\) is the state of the process subsystem, \(y\in\Re^{p}\) is the output of the global system, which is also the output of the process subsystem, \(u_{a}\in\Re^{m}\) is the input of the process subsystem, which is also the output of the actuator subsystem, \(u_{a}\) is assumed to be inaccessible, and f and g are smooth vector fields on \(\Re^{n}\).  
The actuator subsystem is described as  
\[\Sigma_{a}:\begin{cases}\dot{x}_{a}=f_{a}(x_{a},u)&x_{a}(t_{0})=x_{a0}\\ u_{a}=h_{a}(x_{a},u)\end{cases} \tag{2}\]  
where \(x_{a}\in\Re^{n}\) is the state, \(u\in\ \Re^{1}\) is the input and constant parameter to be monitored, \(u_{a}\in\Re^{m}\) is the output of the actuator subsystem, which is also the input of the process subsystem, \(f_{a}\) is the smooth vector field on \(\Re^{n}\), and \(h_{a}\) is the smooth vector field on \(\Re^{m}\).  
Considering the interconnected system described by Equations (1) and (2), it is required to monitor the performance of the interconnected system from the perspective of a single subsystem and the whole system; that is, it is required to describe the cause and effect relationships between the subsystem variables and global system output \(y\), thus providing advanced predictive maintenance techniques in an operating plant. The left invertibility of the interconnected system is then required for ensuring that the impact of local variables on the global level is distinguishable. The property of distinguishability of the two inputs or parameters refers to their capacity to generate different output signals for a given input signal.  
Figure 1: Structure of an interconnected system.  
One way to achieve this purpose is to propose observers for each of the subsystems and the whole network. However, the main difficulty is that the connection point between the two subsystems cannot be measured. This is because the interconnection point is the output of the actuator subsystem, and online measurement is difficult to achieve due to physical or uneconomical reasons. In addition, the measured value could be unreliable, due to its rough operation environment.  
Therefore, the state observation in this work can only rely on the global system output (i.e., the process state at the terminal boundary). As shown in Figure 1, the particular aim of our design is to accurately estimate the state vectors \(\mathrm{x}\) and \(\mathrm{x}_{\mathrm{a}}\) of each subsystem online, as well as the unmeasured interconnection vector \(\mathrm{u}_{\mathrm{a}}\).