# A Data Secured Communication System Design Procedure with a Chaotic Carrier and Synergetic Observer
## 3 Methodology and the System Design Procedure
### Two-Channel Synergetic Observer Analytical Design Procedure and Two-Channel Receiver Block Diagramm  
Here we consider the design procedure for a synergetic state observer for the parameters of the Rossler chaotic generator [44]. The Rossler system is described by a system of nonlinear differential Equation (1). As control parameters adopted by the parameters \(a\) and \(c\), let us design the synergetic observer by setting parameter \(b\) as \"known\" [13; 26; 27; 28; 29]. According to overall design procedure [40], we replace the unknown parameters with their dynamic models  
\[ \begin{array}{l}\dot{w}_{1}(t)=0,\\ \dot{w}_{2}(t)=0,\end{array} \]  
the solutions of which are \(w_{1}(t)=\textit{const}\) and \(w_{2}(t)=\textit{const}\), and reflect the quick change in time of parameters \(a\) and \(c\). Then, we extend the system of Equations (1) to  
\[ \begin{array}{l}\dot{x}(t)=-y-z;\\ \dot{y}(t)=x+ay;\\ \dot{z}(t)=b+xz-cz;\\ \dot{w}_{1}=0;\\ \dot{w}_{2}=0,\end{array}  \tag{9}\]  
where \(w_{1}\). is the dynamic state variable model of parameter \(a\), and \(w_{2}\) is the dynamic state variable model of parameter \(c\). Thus, in System (9) the observable variables are \(x\), \(y\) and \(z\), and the unobservable variables are \(w_{1}\) and \(w_{2}\). We introduce variables \(\hat{w}_{1}\) and \(\hat{w}_{2}\), which are estimates of the parameters \(w_{1}\) and \(w_{2}\) respectively. To find them we define the vector of the macro variable \(\Psi(t)\) by the expression  
\[\psi=\left[ \begin{array}{c}w_{1}-\hat{w}_{1}\\ w_{2}-\hat{w}_{2}\end{array} \right]. \tag{10}\]  
Then, we write down the reduction equations  
\[ \begin{array}{l}\hat{w}_{1}=Q_{1}(x,\ y,\ z)+v_{1},\\ \hat{w}_{2}=Q_{2}(x,\ y,\ z)+v_{2},\end{array}  \tag{11}\]  
where \(Q_{1}(x,\ y,\ z)\) and \(Q_{2}(x,\ y,\ z)\) are the unknown functions of the observed system state variables in Equations (9), and \(v_{1}\) and \(v_{2}\) denote the state of the dynamic observer. Differentiating Equation (11) with respect to time produces  
\[ \begin{array}{l}\frac{d\hat{w}_{1}}{dt}=\frac{\partial Q_{1}(x,y,z)}{ \partial x}\frac{dx}{dt}+\frac{\partial Q_{1}(x,y,z)}{\partial y}\frac{dy}{ dt}+\frac{\partial Q_{1}(x,y,z)}{\partial z}\frac{dz}{dt}+\frac{d\hat{w}_{1}}{dt}, \\ \frac{d\hat{w}_{2}}{dt}=\frac{\partial Q_{2}(x,y,z)}{\partial x}\frac{dx}{ dt}+\frac{\partial Q_{2}(x,y,z)}{\partial y}\frac{dy}{dt}+\frac{\partial Q_{2}(x,y,z)}{ \partial z}\frac{dz}{dt}+\frac{d\hat{w}_{2}}{dt}.\end{array}  \tag{12}\]  
The macro variable in Equation (10) must satisfy the following functional equation  
\[\dot{\Psi}(t)+L\Psi=0, \tag{13}\]  
where \(L=\left[ \begin{array}{cc}L_{11}&L_{12}\\ L_{21}&L_{22}\end{array} \right]\) is the coefficient matrix, which provides the stability of Equations (13) and defines the observer's dynamics. Then, from Equation (13) with regard to Equations (9) to (12), we obtain  
\[ \begin{array}{l}-\frac{\partial Q_{1}(x,y,z)}{\partial x}(-y-z)-\frac{ \partial Q_{1}(x,y,z)}{\partial y}(x+w_{1}y)-\frac{\partial Q_{1}(x,y,z)}{ \partial z}(b+xz-w_{2}z)-\\ -\frac{d\hat{w}_{1}}{dt}+L_{11}(w_{1}-\hat{w}_{1})+L_{12}(w_{2}-\hat{w}_{2})= 0,\\ -\frac{\partial Q_{2}(x,y,z)}{\partial x}(-y-z)-\frac{\partial Q_{2}(x,y,z)}{ \partial y}(x+w_{1}y)-\frac{\partial Q_{2}(x,y,z)}{\partial z}(b+xz-w_{2}z)- \\ -\frac{d\hat{w}_{2}}{dt}+L_{21}(w_{1}-\hat{w}_{1})+L_{22}(w_{2}-\hat{w}_{2})= 0.\end{array}  \tag{14}\]  
Since the equations of the observer cannot contain the unobservable state variables, we have the following expressions  
\[\left\{ \begin{array}{l}-\frac{\partial Q_{1}(x,y,z)}{\partial y}y+L_{11}=0, \\ \frac{\partial Q_{1}(x,y,z)}{\partial z}z+L_{12}=0,\\ -\frac{\partial Q_{2}(x,y,z)}{\partial y}y+L_{21}=0,\\ \frac{\partial Q_{2}(x,y,z)}{\partial z}z+L_{22}=0.\end{array} \right.