## 2 Proposed Scheme and RF Transmission Algorithm
### Chaos Synchronization Design
The time derivative of \(\dot{V}(e)\) along the trajectories of (5) yields the following:  
\[\dot{V}(e)=e^{T}\Big{(}PA_{e}+A_{e}^{T}P\Big{)}e+2e^{T}PB\varphi(t). \tag{8}\]  
If we choose a solution \(P\) that satisfies the following Lyapunov equation,  
\[PA_{e}+A_{e}^{T}P=-Q, \tag{9}\]  
for any positive definite matrix \(Q\), subsequently, \(\dot{V}(e)\) can be stated as follows:  
\[\dot{V}(e)=-e^{T}Qe+2e^{T}PB\varphi(t). \tag{10}\]  
Then, based on Assumption (A2), the function \(\dot{V}(e)\) is reformulated as follows:  
\[\dot{V}(e)\leq-(\lambda_{\min}\{Q\}\|e\|-2\varphi^{+}\lambda_{\max}\{P\})\|e\|. \tag{11}\]Here, \(\dot{V}(e)\) is a negative definite function for all \(e(t)\) outside the set  
\[\mathcal{B}=\bigg{\{}e:\|e\|>\frac{2\varphi^{+}\lambda_{\text{max}}\{P\}}{\lambda_ {\text{min}}\{Q\}}\bigg{\}}. \tag{12}\]  
Thus, the trajectories of the error state dynamic (5) are uniformly bounded. As a result, the solution of the slave system (3) will be bounded around the solutions of the master system (1). The proposed algorithm has an important benefit in that it does not require knowledge of function \(f(x_{1},x_{2},x_{3})\) of the master system (1). This feature makes it especially desirable for synchronizing chaotic systems. In addition, similar to the instance of Refs. [35; 36], the observer gains may be chosen such that they minimize the estimation error. Example: To illustrate the effectiveness of the proposed scheme, let us consider the following multi-scroll chaotic system introduced in Ref. [37] as the master system:  
\[\dot{x}_{1m} =10x_{2m}, \tag{13a}\] \[\dot{x}_{2m} =10x_{3m},\] (13b) \[\dot{x}_{3m} =10\big{(}-ax_{2m}-cx_{3m}-d_{1}f(x_{1m})\big{)}. \tag{13c}\]  
with  
\[f(x_{1m})= \begin{cases}+\sin(2\pi bx_{1m}),&-\frac{k_{n}}{\text{P}}>x_{1m}, \\ -\sin(2\pi bx_{1m}),&-\frac{k_{n}}{\text{P}}\leq x_{1m}\leq\frac{k_{p}}{\text{ P}},\\ +\sin(2\pi bx_{1m}),&\frac{k_{p}}{\text{P}}<x_{1m},\end{cases}  \tag{14}\]  
where \(\{a,\ b,\ c,\ d_{1}\}\in\mathbb{R}\) and \(\{k_{n},\ k_{p}\}\in\mathbb{N}^{+}\). In system (13), the scroll numbers and the size of the attractor can be arbitrarily modified by a tuning of the parameters in function \(f(x_{1m})\). The system (13) has the following characteristics:  
1. [label=()]
2. It can present a chaotic, cyclical or stable dynamic. 3. By tuning the parameters \(k_{n}\) and \(k_{p}\), generating a chaotic attractor with several scrolls is possible. The number of scrolls can be distributed on the right and left side of the phase plane \((x_{1m},x_{2m})\). For study purposes, we only concentrated on the chaotic behavior of system (13). Now, the nonlinear function (14) is considered as an additional state of (13). Thus, the master system can be described as the following extended state system:  
\[\dot{x}_{1m} =10x_{2m}, \tag{15a}\] \[\dot{x}_{2m} =10x_{3m},\] (15b) \[\dot{x}_{3m} =10(-ax_{2m}-cx_{3m}+x_{4m}),\] (15c) \[\dot{x}_{4m} =10\varphi(t),\] (15d) \[y =x_{1m}, \tag{15e}\]  
where \(\varphi(t)\) is an unknown and bounded function. Meanwhile, the slave system is designed as follows:  
\[\dot{x}_{1s} =10\big{(}x_{2s}+l_{1}(y-x_{1s})\big{)}, \tag{16a}\] \[\dot{x}_{2s} =10\big{(}x_{3s}+l_{2}(y-x_{1s})\big{)},\] (16b) \[\dot{x}_{3s} =10\big{(}-ax_{2s}-cx_{3s}+x_{4s}+l_{3}(y-x_{1s})\big{)},\] (16c) \[\dot{x}_{4s} =10\big{(}\varphi(t)+l_{4}(y-x_{1s})\big{)}. \tag{16d}\]  
In the simulation, the implicit Euler method (with a step size \(h=1\times 10^{-3}\)) is used to solve the system (13) and its corresponding slave system (16). The parameter values for the master system are \(a=0.3,\ b=0.25,\ c=0.3,\ d=0.35,\ k_{n}=3,\ k_{p}=2\), and the observer gains \(l_{i}\) were tuned using the pole placement approach with \(l_{1}=\frac{2}{3}\omega_{0}-c\)