## 3 Design of Voltage Detector
### Chaos Theory and Chaos Synchronization Dynamic Error  
Chaos theory [16,17] is a nonlinear system theory. It is characterized by unpredictability and irregular motion. The chaos system makes dramatic change in micro variation. The chaos synchronization synchronizes the kinematic trajectories of two chaos systems, and the controller makes the error zero [18,19]. In the chaos synchronization system, the two chaos systems are called Master System (MS) and Slave System (SS). When the two systems have different initial values, the kinematic trajectories of the two systems have different chaos phenomena. However, when a controller is added to the SS to track the MS, and the MS and SS act in synchronization, the kinematic trajectories of two chaos systems have the same change at the same time. The MS and SS are defined as follows:  
\[MS=\begin{cases}\dot{X}m_{1}(t)=F_{1}(t,[Xm_{1},Xm_{2}\cdots Xm_{n}])\\ \dot{X}m_{2}(t)=F_{2}(t,[Xm_{1},Xm_{2}\cdots Xm_{n}])\\ \dot{X}m_{n}(t)=F_{n}(t,[Xm_{1},Xm_{2}\cdots Xm_{n}])\end{cases} \tag{4}\]  
\[SS=\begin{cases}\dot{X}s_{1}(t)=F_{1}(t,[Xs_{1},Xs_{2}\cdots Xs_{n}])\\ \dot{X}s_{2}(t)=F_{2}(t,[Xs_{1},Xs_{2}\cdots Xs_{n}])\\ \dot{X}s_{n}(t)=F_{n}(t,[Xs_{1},Xs_{2}\cdots Xs_{n}])\end{cases} \tag{5}\]  
\begin{table}
\begin{tabular}{c l} \hline
**Notation** & **Definition** \\ \hline \(Xm_{i}\) & The system states of master system \\ \(Xs_{i}\) & The system states of slave system \\ \(F_{i}\) & Nonlinear functions \\ \(E_{i}=Xm_{i}-Xs_{i}\) & Error states of chaos synchronization system \\ \(x\) & System state of Sprott Chaos System \\ \(\text{sign}(\Delta)\) & Symbolic function is defined as \(\text{sign}(\Delta)=\begin{cases}+1,\text{ if }\Delta\geq 0\\ -1,\text{ if }\Delta<0\end{cases}\) \\ \(a,b\) & System parameters of Sprott Chaos System \\ \(e_{i}=Xm_{i}-Xs_{i}\); \(i=1,2,3\) & Error states of Sprott Chaos System \\ \(\alpha\) & The value of fractional order \\ \(t_{0}\) & Initial time \\ \(D_{t}\) & Differential with respect to time \(t\) \\ \(t^{\ast}=\dfrac{1}{\left(\Delta t\right)^{a}},a^{\ast}=\dfrac{a}{\left(\Delta t \right)^{a}},b^{\ast}=\dfrac{b}{\left(\Delta t\right)^{a}}\) & System parameters of fractional order system \\ \(\Delta t\) & Sampling time \\ \(V_{m}\) & The data of expected voltage \\ \(V_{s}\) & The data of real-time test voltage \\ \(\varphi_{1}\), \(\varphi_{2}\) & Dynamic error equation as the variables of error judgment of chaos \\  & synchronization dynamic error detector \\ \(\varepsilon_{1}\), \(\varepsilon_{2}\) & The set values of error magnitude \\ \hline \end{tabular}
\end{table}
Table 1: Notation and definitions.  
The characteristic quantity extracted in this paper is the naturally tracked dynamic error amount of MS and SS in the master-slave synchronization system. Therefore, there is no controller for the master-slave synchronization system in this paper; as long as the master and slave chaos systems are subtracted from each other, the dynamic error status can be obtained.  
\[E=MS-SS=\begin{cases}\dot{E}_{1}(t)=F_{1}(t,[E_{1},E_{2}\cdots E_{n}])\\ \dot{E}_{2}(t)=F_{2}(t,[E_{1},E_{2}\cdots E_{n}])\\ \dot{E}_{n}(t)=F_{n}(t,[E_{1},E_{2}\cdots E_{n}])\end{cases} \tag{6}\]  
where \(E_{1}=Xm_{1}-Xs_{1}\), \(E_{2}=Xm_{2}-Xs_{2}\), \(E_{n}=Xm_{n}-Xs_{n}\), the final dynamic error status has chaos phenomenon, and the controller makes a decision based on a dynamic analysis of the error status.