## 3 Proposed SDRE Nonlinear Observer-Controller
### SDRE Controller  
Assume the following nonlinear system:  
\[\dot{x}(t)=f(x(t))+B(x(t))u(t),\ \ x(0)=x_{0} \tag{2}\]  
That \(x(t)\in R^{n}\) is the state vector, \(u(t)\in R^{m}\) is the control input and \(x_{0}\) is an initial condition. \(f(x(t)):R^{n}\to R^{n}\) and \(B(x(t)):R^{n}\to R^{n\times m}\) are two known functions of system states. We assume that this system has an equilibrium at the origin point, and therefore, the main idea used in SDRE is to show this nonlinear system in a quasi-linear manner [20]. That way, the factorization is performed as \(f(x(t))=A(x(t))x(t)\) that, \(A(x(t)):R^{n}\to R^{n\times n}\) is a state-dependent matrix. A remarkable point in this factorization is the numerous possibilities for decomposing \(f(x(t))\) as a coefficient of \(x(t)\). For example, if the function \(f(x(t))\) is \(f(x(t))=A_{1}(x(t))x(t)\) and \(f(x(t))=A_{2}(x(t))x(t)\), then any other linear combination of \(A_{1}(x(t))\) and \(A_{2}(x(t))\) also forms an alternate linear structure  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** & **Parameters** & **Value** \\ \hline
0.033 H & \(L_{3}\) & 0.1 \(\Omega\) & \(R_{7}\) & 0.1 F & C\({}_{1}\) \\
0.033 H & \(L_{6}\) & 0.01 \(\Omega\) & \(R_{01}\) & 0.01 F & C\({}_{2}\) \\
0.0033 H & \(L_{8}\) & 0.01 \(\Omega\) & \(R_{02}\) & 0.1 F & C\({}_{4}\) \\
0.1 \(\Omega\)in \(A(x(t))=\alpha(x(t))A_{1}(x(t))+(1-\alpha(x(t))A_{2}(x(t))\), in which \(\alpha(x(t))\) is a definite function of \(x(t)\). This issue is one of the advantages of the SDRE method, which provides the designer with an extra degree of freedom. In designing an optimal stabilizer, the aim is to find a control rule by which it can bring the system states to the equilibrium point at the system origin, see Equation (1), and minimize the predetermined cost function below [20]:  
\[J=\frac{1}{2}\int_{0}^{\infty}\Bigl{(}x^{T}(t)Q(x(t))x(t)+u^{T}(t)R(x(t))u(t) \Bigr{)}dt \tag{3}\]  
which in \(R(x(t))\ :R^{n}\to R^{m\times m}\) and \(Q(x(t))\ :R^{n}\ \to\ R^{n\times m}\) are the state dependent weighting matrices and are, respectively, a positive and a positive-semidefinite integrand point. The solution of the optimal control problem described by Equations (1) and (2) requires the finding of the Hamilton Jacobi Bellman equation (HJB) [20]. On the other hand, this HJB equation is very complex in general. In the SDRE method, it is shown that one can use the control rule \(u\) to find an approximate solution to this optimal control problem, in which the matrix \(\beta\) is the definite unique positive solution of the SDRE based on the following statement:  
\[\begin{array}{c}A^{T}(x(t))P(x(t))+P(x(t))A(x(t))-P(x(t))\\ B(x(t))R^{-1}(x(t))B^{T}(x(t))P(x(t))+Q(x(t))=0\end{array} \tag{4}\]  
The most important point is that Equation (4) has a definite unique positive symmetric solution if and only if pairs \(\bigl{(}A(x(t)),B(x(t))\bigr{)}\) and \(\bigl{(}A(x(t)),Q^{\frac{1}{2}}(x(t))\bigr{)}\) are point-to-point stable and point-to-point observer, respectively. In [20] it is shown that the equilibrium in the closed-loop system caused by the SDRE controller is local asymptotically stable.