# A Data Secured Communication System Design Procedure with a Chaotic Carrier and Synergetic Observer
## 3 Methodology and the System Design Procedure
### Alternative Reconstruction Method Demonstration
For the system:  
\[\mathbf{y}=\mathbf{B}\mathbf{x} \tag{20}\]  
the formula for determining unknown parameters in discrete form is the following matrix expression:  
\[\mathbf{B}_{\mathbf{N+1}}=\mathbf{B}_{\mathbf{N}}+\frac{\mathbf{P}_{\mathbf{N}}^{-1}\mathbf{x}}{\mathbf{1}+\mathbf{ x}^{T}\mathbf{P}_{\mathbf{N}}^{-1}\mathbf{x}}\Big{(}\mathbf{y}_{\mathbf{N}+1}-\mathbf{x}^{T}\mathbf{B}_{\mathbf{N}} \Big{)} \tag{21}\]  
where \(\mathbf{B}\) is the vector of parameters to be defined; \(\mathbf{x}\) is the vector of input variables; \(\mathbf{y}\) is the vector of output variables. Next, we move from the discrete form of Equation (21) to the continuous one:  
\[\dot{\mathbf{B}}(t)=\frac{\mathbf{R}\mathbf{x}}{(\mathbf{1}+\mathbf{x}^{T}\mathbf{R}\mathbf{x})}\Big{(}\bm {y}-\mathbf{x}^{T}\mathbf{B}\Big{)}\frac{1}{\Delta}, \tag{22}\]  
where \(\mathbf{R}=\mathbf{P}^{-1}\), \(\mathbf{P}\) is the arbitrary matrix, \(\Delta\) is the step of the system solution. To build the \(a\) and \(c\) parameter identifiers of the mathematical model (1), we consider each of the equations separately. To design a parameter for observer \(a\), we extract from model (1) an equation containing the following parameter:  
\[\frac{dy}{dt}=x+ay.\]  
In this case, the input and output vectors will look like  
\[\mathbf{x}=[\mathbf{y}];\;\mathbf{y}=\Big{[}\dot{\mathbf{y}}(t)-\mathbf{x}\Big{]}, \tag{23}\]  
where \(\dot{\mathbf{y}}(t)\) is the evaluation of the derived variable \(y\), which can be obtained by a control loop with two integrators:  
\[ \begin{array}{l}\dot{\mathbf{\mu}}_{\mathbf{1}}(t)=\mathbf{\mu}_{2}+\mathbf{k}_{\mathbf{1}}(\mathbf{y }-\mathbf{\mu}_{\mathbf{1}}),\\ \dot{\mathbf{\mu}}_{\mathbf{2}}(t)=\mathbf{k}_{\mathbf{2}}(\mathbf{y}-\mathbf{\mu}_{\mathbf{1}}),\end{array}  \tag{24}\]  
where \(\mu_{2}=\dot{\mathbf{y}}(t)\) is the evaluation of the derived variable \(y\); \(k_{1}\), and \(k_{2}\) are the constant coefficients. The vector of unknown coefficients is formed from estimates of the corresponding parameters, in our case:  
\[\mathbf{B}=[a]. \tag{25}\]  
As the initial values of the matrix \(\mathbf{R}\), we can take any numerical nonsingular matrix, but for better convergence of the parameter estimates we choose the following one:  
\[\mathbf{R}=\mathbf{X}^{T}\mathbf{X}=[R_{1}], \tag{26}\]  
where \(\mathbf{X}=\left[ \begin{array}{l}y(t_{k})\\ y(t_{k+1})\\ y(t_{k+2})\end{array} \right]\) is the matrix formed by \(t_{k}\), \(t_{k+1}\) and \(t_{k+2}\) countdowns. Thus, from Equation (22) while taking into account Equations (23), (25), (26), we have:  
\[\dot{\mathbf{a}}=\frac{\mathbf{R_{1}}\mathbf{y}}{\mathbf{1}+\mathbf{R_{1}}\mathbf{y}^{2}}\Big{(}\dot{ \mathbf{y}}-\mathbf{x}-\hat{\mathbf{a}}\mathbf{y}\Big{)}\frac{1}{\Delta}\;. \tag{27}\]In the same way for the parameter \(c\) we get the following estimation:  
\[\dot{\hat{\mathbf{\varepsilon}}}=\frac{-R_{2}\mathbf{z}}{\mathbf{1}+R_{2}\mathbf{z}^{2}}\Big{(} \dot{\mathbf{z}}-\mathbf{b}-\mathbf{xz}+\hat{\mathbf{\varepsilon}}\mathbf{z}\Big{)}\frac{\mathbf{1}}{ \Delta}. \tag{28}\]  
The simulation plots of system of Equation (1) alongside with Equations (24), (27), (28) and the reconstructed information signal are shown in Figures 18 and 19. The simulation results show a low efficiency of this estimation method for reconstructing a signal with a chaotic carrier. In particular, in Figures 18 and 19, an unacceptably high recovery error was observed for parameter \(a\). In general, based on the results of a qualitative comparison of Figure 13, Figure 15 with Figure 18, we can outline the advantages of a synerge  
Figure 19: Transmitted signal LS method reconstruction error: (a) signal \(a\) reconstruction error; and (b) signal \(c\) reconstruction error.