# A Survey on Machine-Learning Based Security Design for Cyber-Physical Systems
## 2 Hierarchical Structure of Cyber-Physical Systems
### Physical System Layer  
The physical system layer represents multiple physical systems, which are objects that operate in the real world. In the physical system layer, multiple physical systems have sensors and actuators in which sensors report states of the physical objects to the computing system in the cyber-world; the actuators operate the physical objects according to commands from the computing system [32]. From sensing and actuating in the physical system, the physical system fulfills the physical processes.  
Figure 1: A hierarchical cyber-physical system structure.  
The operation of the physical system is represented as system dynamics in a continuous-time domain. To simplify the expression of physical dynamics, we consider a single-input single-output (SISO) linear time-invariant (LTI) system as follows:  
\[\dot{x}(t) =Ax(t)+Bu(t) \tag{1}\] \[y(t) =Cx(t),\]  
where \(x\in\mathbb{R}^{n}\) is the state of the physical system with \(n\) state variables, \(A\in\mathbb{R}^{n\times n}\) is the system matrix, \(B\in\mathbb{R}^{n}\) is the input matrix, \(C\in\mathbb{R}^{1\times n}\), \(u\in\mathbb{R}\) is the control input signal, and \(y\in\mathbb{R}\) is the sensor measurement. The system matrix term \(Ax(t)\) represents the state transition by the characteristics of the physical system, the input matrix term \(Bu(t)\) represents the state transition by the control input signal \(u(t)\), and the output matrix term \(Cx(t)\) represents the measurement of the state by the sensor attached on the physical system, where all physical state variables in the state vector \(x(t)\) cannot be measured by the sensor. The physical systems periodically send a sensor measurement \(y(t)\) to the computing system in order to report the state information, and then, the computing system returns the control input signal \(u(t)\) to the physical systems to operate them as intended. We assume that matrix pairs \((A,B)\) and \((A,C)\) in (1) are controllable and observable, respectively.  
The actuating process, which determines the dynamics of physical systems, is executed by the control input signal \(u(t)\) on the computing system. The control input signal \(u(t)\) is calculated as follows:  
\[\dot{\hat{x}}(t) =A\hat{x}(t)+Bu(t)+L(y(t)-C\hat{x}(t)) \tag{2}\] \[u(t) =-K\hat{x}(t),\]  
where \(\hat{x}(t)\) is the state estimation of the physical system, \(L\) is the observer gain, and \(K\) is the controller gain. In general, the sensors cannot indicate full states of the physical systems; the computing system implements state estimation based on the system model (1) and sensor measurement \(y(t)\) with an observer. From the state estimation in (2), the control input signal is calculated by the state feedback controller with controller gain \(K\). We assume that the controller gain \(K\) and observer gain \(L\) are well-designed to stabilize the state of the physical systems.  
Unmanned vehicles [7] and autonomous trains [9] are typical examples of physical systems within intelligent transport systems. Unmanned mobile objects are controlled by centralized control stations on the ground, where the dynamics of the unmanned mobile objects are determined according to the control command from the ground station [16]. Heating, ventilation, and air conditioning (HVAC) systems [36] and production machines [33] are examples of physical systems in industrial areas, and these industrial facilities belong to physical system-layer components. Likewise, in a smart grid, the power plants and the electric equipment that physically transmit electricity are elements of the physical system layer [14; 15; 37].  
Most physical systems, especially unmanned aerial vehicles (UAVs), have power constraints because they have an external battery [38]. Therefore, it is difficult for the physical system to conduct tasks requiring complex computations. To overcome the power limitation, the physical system layer interacts with the computing system through the network layer.