## 1 Introduction  
A modern system often consists of a series of interconnected dynamical units (sensors, actuators and system components) and, therefore, very complicated dynamics are exhibited [1]. Technological advances mean that these units themselves are dynamic systems and exhibit complicated dynamics. Therefore, a modern control system can be viewed as composed of dynamic subsystems connected in a series. In all situations, the global plant can be analyzed at different levels--down to the component level--to estimate the reliability of the whole plant. In recent years, the topic of observer design for separate nonlinear systems has been widely discussed in the literature, like high-gain observers [2; 3; 4], sliding mode observers [5; 6], adaptive observers [7; 8; 9], and unknown input observers (UIOs) [10; 11; 12]. These methodologies are typically centralized monitoring systems where intelligence is either at the system level or at the field device level of the processing plant. For the former, it aims at monitoring plant dynamics from the viewpoint of a global system, like in [13; 14; 15]. In these methods, the dynamics of subcomponents (i.e., actuators) are often neglected. They are, generally speaking, treated as constants in the input or output coefficient matrix (function) of the process system model. For the latter, it focuses on the field device level,aiming at analyzing the internal dynamics of a specific subcomponent while the influences of local internal dynamics on the global dynamics are neglected, like in [3; 16]. However, centralized observers may not be suitable for modern control systems. On one hand, a modern control system is in fact an interconnected system, while the centralized observer just enables an individual component to monitor internal dynamics locally. However, the dynamics of the field devices can cause significant disturbances to the global process and influence the quality of the final product [17; 18]. On the other hand, due to uneconomical measurement costs or physical environment factors such as high temperature, it is impossible to measure the state or partial state, such as in [19; 20]. To overcome these difficulties, an effective way is to decompose the system into several interconnected subsystems so that the observer can be decentralized in each subsystem. In this way, it allows the analysis of less complex subcomponents to study the characteristics of interconnected systems. The last few decades have also witnessed significant improvements in dynamic networks which consist of a number of interconnected units, like in [1; 21; 22; 23; 24]. A typical approach is to design distributed observers for each subsystem using the internal information of each subsystem, and then all the observers are aggregated to form the total estimator [25; 26; 27]. For instance, in [26], an observer was designed for the whole system from the separate synthesis of observers for each subsystem, assuming that for each of these separate designs, the states from the other subsystem were available. In [27], an observer for each subsystem was proposed, using the state estimation of the previous subsystem. In addition, a quasi-input-to-state stability and input-to-state dynamical stability (ISS/ISDS) reduced-order observer for the whole system was designed, considering the interconnections of quasi-ISS/ISDS reduced-order observers for each subsystem. A major challenge for these methods is the availability of the measurement of the interconnections between subsystems. Therefore, it is interesting to consider the problem of whether we can prove that, under some conditions, the effect in lower subsystems can be distinguished from higher subsystems, thus avoiding full measurements of the local subsystem. For example, in this work, the interconnection is the output of the actuator, and it is not economical or realistic to measure its output.