# Development of a Novel Open Control System Implementation Method under Industrial IoT
## 3 Implementation of Open Control Systems Under Industrial IoT
### A Hierarchical Semantic Model of Control Domain
Moreover, this paper does not aim to define a control task granularity standard; instead, it mainly concentrates on the relationship and intimacy between control tasks. In other words, the control task's granularity size is not fixed, but can be adjusted according to the requirements. For instance, as Figure 4(b) shows, the interpretation task is divided into a compilation task and a tool compensation task. The former mainly deals with lexical, syntactic, and semantic analysis, while the latter processes the computation commands in the part program. The latter task depends on the former. Similarly, the interpolation task is decomposed into a velocity planning task and an interpretation task, both having a self-dependence labeled \(T_{1}\). Based on the above discussion, therefore, this paper suggests a hierarchical semantic model in light of an extended FSM and a dependency network model with the time property to describe the control domain. Figure 6 illustrates this hierarchical semantic model from an object-oriented paradigm perspective. To be specific, there are seven classes: ControlSystem, State, Transition, Event, Action, Task, and Dependence. Here, Action is the abstraction of continuous behavior. As for the relationship between them, a ControlSystem object associates with a State object as its initial state. A State object associates with zero or more Transition objects. Each Transition object associates with two State objects, which indicate its source state and target state, respectively. It associates with an Event object since a transition is triggered by an event. It is noted that the Transition class has a priority property. Moreover, a State object associates with zero or more Action objects, which represents the corresponding continuous behavior that will be executed when the State object is activated by a specific Transition object. Each Action object associates with a Task object as its start task. A Task object associates with zero or more Dependence objects, while each Dependence object associates with two Task objects as its source task and target task. In addition, the Dependence class has a time property and a priority property. From a mathematical perspective, based on this hierarchical semantic model, a control system can be described by the following septuple:  
\[C=(Q,q_{0},\Sigma,\delta,F,P_{\Sigma},\alpha) \tag{1}\]  
where  
\(Q\): the set of states;  
\(q_{0}\): the initial state;  
\(\Sigma\): the set of events;  
\(\delta\): state transition function that satisfies \(\delta:Q\times\Sigma\to Q\);  
\(F\): the set of terminal states, which can be empty;  
\(P_{\Sigma}\): transition priority function that satisfies \(P_{\Sigma}:Q\times\Sigma\to\mathbb{N}\);  
Figure 5: Continuous control domain described by dependency network model. \(\alpha\): state action function that satisfies \(\alpha:Q\to Actions\). Action is described by the following quintuple:  
\[Action=(T,t_{0},\gamma,P_{\gamma},\tau) \tag{2}\]  
where  
\(T\): the set of tasks;  
\(t_{0}\): the start task;  
\(\gamma\): task dependence function that satisfies \(\gamma:T\to T\);  
\(P_{\gamma}\): dependence priority function that satisfies \(P_{\gamma}:T\to\mathbb{N}\);  
\(\tau\): dependence time function that satisfies \(\tau:Q\to\mathbb{R}\).