# 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  
To demonstrate the advantages of the synergetic nonlinear state observer design procedure in the subsections above, we considered an alternative method for reconstructing a useful signal on the receiver side. The recovery methods described in the previous sections and given in Table 1 did \begin{table}
\begin{tabular}{c p{284.5 pt}} \hline \hline
**Design Step** & **Description** \\ \hline Step 1 & Deduce the system into a form suitable for transmitting a signal over a communication channel: for the initial nonlinear system of the 3 rd order, using the global reconstruction of a dynamic system [13] replace the variables for transmitting a useful signal to the communication channel (Equation (3)). \\ Step 2 & Perform multiplicative parametric modulation: select two constant coefficients in different equations of the original model and replace them with the sum of this coefficient and the time function that reflects the useful signal (for each of the coefficients) (System (5)). \\ Step 3 & Adopt these sums as new unknown parameters and and extend the generator’s initial model by dynamic models of these parameters \(\mathrm{i}_{1}(t)=\mathrm{o}_{1}(t)=\mathrm{o}_{2}(t)=0\). As a result we get a 5 th order system with 3 observable (\(x\), \(y\) and \(z\)) and 2 unobserved (\(w_{1}\) and \(w_{2}\)) variables (Equation (9)). \\ Step 4 & Introduce variables \(\mathrm{\hat{w}}_{1}\) and \(\mathrm{\hat{w}}_{2}\), which will be estimates of the parameters \(w_{1}\) and \(w_{2}\). \\ Step 5 & To find them we define the vector of the macro variable and write down the reduction equations with new unknown functions that will be defined on the final steps of the procedure (Matrix (10)). \\ Step 6 & Differentiating reduction equations with respect to time (System (12)). \\ Step 7 & Write down a main functional equation of the ADAR method [38; 40] that will define the final system desired dynamics. The \(L\) coefficients matrix will provide a desired system stability condition (Equation (13)). \\ Step 8 & From the main functional equation with respect to the extended initial mathematical model, the vector of the macro variable, the reduction equations and its derivative we obtain the draft equations of the observer (Equation (14)). \\ Step 9 & The observer equations should not contain unobservable state variables, therefore, from the draft observer equations it is necessary to write out all terms containing unobservable variables and equate them to zero. We thus get a system of the 4 th order (Equation (15)). \\ Step 10 & For the resulting system, we choose values of the coefficients \(L\) based on the stability condition of the observer and the main functional equation (Equations (16)–(18)). \\ Step 11 & We obtain the observer Equation (19) taking into account the found functions for the reduction Equation (11). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example of steps for our synergetic observer design procedure for Rössler generator. Figure 17: General block diagram of a synergetic observer communication system. Here: \(a\), \(b\), \(c\) are the known parameters of the chaotic generator model; \(a^{\star}(t)\) and \(c^{\star}(t)\) are the modulated parameters; \(Z(t)\) is the Rössler transformed system (2) variable; \(Z_{noise}(t)\) is the \(Z(t)\) variable with noises admixed in the transmission media; \(a^{\star}(t)\) and \(c^{\star}(t)\) are the estimated modulated parameters; \(\mu_{1}(t)\) and \(\mu_{2}(t)\) are the initial useful signals; \(\mu_{1}\)’\((t)\) and \(\mu_{2}\)’\((t)\) are the reconstructed useful signals; and \(\alpha\) is the observer tuning parameter. not provide applicability to nonlinear mathematical models of chaotic generators and simultaneous interdependent recovery of two parameters. The procedure for reconstructing a signal using an estimation of the least squares method (LS method) is described here. Let us identify the parameters \(a\) and \(c\) with the recursive least squares method.