## 3 Design of Voltage Detector
### Fractional Order Chaos Synchronization Dynamic Error  
In terms of signal processing, the chaos synchronization system is often used to check the power quality, voltage oscillation, and control application [20,21]. The dynamic behavior can make an abnormal signal track normal signal. In order to implement the synchronization of master signal (MS) and slave signal (SS), the SS uses the MS signal as a reference signal. In reference [19], [PERSON] _et al._ focused on the tracking time and the number of tracking cycle on MPPT. However, their study did not discuss the steady-state responses. In this study, we will use the same concept to discuss the steady-state behaviors of MPPT in this study. This paper also uses the Sprott Chaos System [22] to discuss it. The equation for the Sprott chaos system is:  
\[\dddot{x}+a\dddot{x}+b\dddot{x}=-1.2x+2\mathrm{sign}(x) \tag{7}\]  
where the symbolic function (\(\Delta\)) is defined as:  
\[\mathrm{sign}(\Delta)= \begin{cases}+1,\text{ if }\Delta\geq 0\\ -1,\text{ if }\Delta<0\end{cases}  \tag{8}\]  
The dynamic equation of MS and SS in Sprott Chaos System can be defined as:  
\[ \begin{cases}\dot{X}_{m_{1}}=X_{m_{2}}\\ \dot{X}_{m_{2}}=X_{m_{3}}\\ \dot{X}_{m_{3}}=-aX_{m_{3}}-bX_{m_{2}}-1.2X_{m_{3}}+2\mathrm{sign}(X_{m_{1}}) \end{cases}  \tag{9}\]  
\[I_{ph}=[I_{sc}+k_{sc}(T-T_{r})]\frac{S}{1000} \tag{10}\]  
In the state of dynamic error, the error variance is defined as \(e_{1}=Xm_{1}-Xs_{1}\), \(e_{2}=Xm_{2}-Xs_{2}\), \(e_{3}=Xm_{3}-Xs_{3}\), and \(e= \begin{bmatrix}e_{1},e_{2},e_{3}\end{bmatrix} \). Equations (9) and (10) are changed to a matrix pattern:  
\[ \begin{bmatrix}\dot{e}_{1}\\ \dot{e}_{2}\\ \dot{e}_{3}\end{bmatrix} = \begin{bmatrix}0&1&0\\ 0&0&1\\ -1.2&-b&-a\end{bmatrix} e+2 \begin{bmatrix}0\\ 0\\ \mathrm{sign}(Xm_{1})-\mathrm{sign}(Xs_{1})\end{bmatrix}  \tag{11}\]The system state variables \(Xm_{1}\) and \(Xs_{1}\) capture the sinusoid in positive period or negative period synchronization, so \(\operatorname{sign}(Xm_{1})-\operatorname{sign}(Xs_{1})=0\), according to [PERSON] _et al._[19], the system in Equation (11) can be reduced to the following second-order system:  
\[ \begin{bmatrix}\dot{e}_{2}\\ \dot{e}_{3}\end{bmatrix} = \begin{bmatrix}0&1\\ -b&-a\end{bmatrix}\begin{bmatrix}e_{2}\\ e_{3}\end{bmatrix} = \begin{bmatrix}A\end{bmatrix} \dot{e} \tag{12}\]  
As \(a>0\) and \(b>0\), Equation (12) is tenable. The fractional order general expression in [12] is used in this chaos system; the fractional order general expression can be expressed as:  
\[\frac{\mathrm{d}^{a}e(t)}{\mathrm{d}t^{a}}\approx\lim_{\Delta t \to 0}\frac{e(t)-ae(t-t_{0})}{\left(t-(t-t_{0})\right)^{a}}\approx\frac{e(t)- ae(t-t_{0})}{\left(\Delta t\right)^{a}} \tag{13}\]  
when \(\alpha=1\), the \(\alpha\) value is 0-1 in fractional order. The fractional order geometric interpretation of Equation (13) has been described [21]. Equation (12) can be expressed in fractional order:  
\[ \begin{bmatrix}D_{t}^{a}e_{2}\\ D_{t}^{a}e_{3}\end{bmatrix} \approx \begin{bmatrix}0&t^{\prime}\\ -b^{\prime}&-a^{\prime}\end{bmatrix}\begin{bmatrix}e_{2}(t)\\ e_{3}(t)\end{bmatrix} + \begin{bmatrix}0&-a^{\prime}\end{bmatrix}\begin{bmatrix}e_ {2}(t-t_{0})\\ -ab^{\prime}&-a^{\prime}\end{bmatrix}\begin{bmatrix}e_{2}(t-t_{0})\\ e_{3}(t-t_{0})\end{bmatrix} . \tag{14}\]  
If \(t_{0}\) is the initial time, the system parameter is changed to:  
\[t^{\prime}=\frac{1}{\left(\Delta t\right)^{a}},a^{\prime}=\frac{a}{\left( \Delta t\right)^{a}},b^{\prime}=\frac{b}{\left(\Delta t\right)^{a}} \tag{15}\]  
where \(0<\alpha<1\) is the fractional order. This fractional order dynamic error equation is used in the PV panel end to capture the different dynamic behaviors of transient voltage as the signal ends. Therefore, we define variables \(xm_{2}=V_{m}[i]\), \(xm_{3}=V_{m}[i+1]\), and \(i\in[1,n-1]\) ; these are the data for expected voltage \(V_{m}\) extracted from the signal end. \(xs_{2}=V_{s}[i]\), \(xs_{3}=V_{s}[i+1]\) provide the data for real-time test voltage \(V_{s}\) extracted from the signal end.