## 3 Modified SDRE Method
Equating the adjoint differential Equation (30) and the differential form of Equation (32), we have  
\[\dot{\mathbf{x}}=\mathbf{A}(\mathbf{x})\mathbf{x}+\mathbf{B}\mathbf{u}= \mathbf{A}_{1}\mathbf{x}+\mathbf{A}_{2}(\mathbf{x})\mathbf{x}+\mathbf{B} \mathbf{u}, \tag{20}\]1Consequently, employing Equations (22) and (29), it is possible to obtain  
\[ \begin{split}&\frac{\partial}{\partial t}[\mathbf{K}_{1}+\mathbf{K}_{2}( \mathbf{x})]\mathbf{x}+[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{A}_{ 1}\mathbf{x}+[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{A}_{2}( \mathbf{x})\mathbf{x}\\ &-[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{B}\mathbf{R} ^{-1}\mathbf{B}^{T}[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{x}\\ &=-\bigg{[}\mathbf{A}_{1}+\mathbf{A}_{2}(\mathbf{x})+\frac{ \partial(\mathbf{A}_{2}(\mathbf{x}))\mathbf{x}}{\partial\mathbf{x}}\bigg{]}^{ T}[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{x}-\mathbf{Q} \mathbf{x},\end{split}  \tag{38}\]  
and rearranging terms in Equation (39), we find  
\[ \begin{split}&\Bigg{(}\dot{\mathbf{K}}_{2}(\mathbf{x})+\bigg{[} \frac{\partial(\mathbf{A}_{2}(\mathbf{x}))\mathbf{x}}{\partial\mathbf{x}} \bigg{]}^{T}[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\Bigg{)}\mathbf{x}+([ \mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{A}_{1}+[\mathbf{K}_{1}+ \mathbf{K}_{2}(\mathbf{x})]\mathbf{A}_{2}(\mathbf{x})\\ &-[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\mathbf{B}\mathbf{R} ^{-1}\mathbf{B}^{T}[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x})]\,+[\mathbf{A}_ {1}+\mathbf{A}_{2}(\mathbf{x})]^{T}[\mathbf{K}_{1}+\mathbf{K}_{2}(\mathbf{x}) ]+\mathbf{Q}\Big{)}\mathbf{x}=\mathbf{0}.\end{split}  \tag{39}\]  
Based on linearization Equations (34)-(36), let the matrix \(\mathbf{K}_{1}\) solve Equation (39), the algebraic Riccati equation (ARE):  
\[\mathbf{K}_{1}\mathbf{A}_{1}+\mathbf{A}_{1}^{T}\mathbf{K}_{1}-\mathbf{K}_{1} \mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{K}_{1}+\mathbf{Q}=\mathbf{0}; \tag{40}\]  
then, the optimality condition is given by  
\[ \begin{split}&\dot{\mathbf{K}}_{2}(\mathbf{x})+\bigg{[}\frac{ \partial\mathbf{A}_{2}(\mathbf{x})}{\partial\mathbf{x}}\mathbf{x}\bigg{]}^{T} \mathbf{K}_{1}+\bigg{[}\frac{\partial\mathbf{A}_{2}(\mathbf{x})}{\partial \mathbf{x}}\mathbf{x}\bigg{]}^{T}\mathbf{K}_{2}(\mathbf{x})\\ &+\mathbf{K}_{2}(\mathbf{x})\mathbf{A}_{1}+\mathbf{A}_{1}^{T} \mathbf{K}_{2}(\mathbf{x})+\mathbf{K}_{1}\mathbf{A}_{2}(\mathbf{x})+\mathbf{A }_{2}^{T}(\mathbf{x})\mathbf{K}_{1}\\ &+\mathbf{K}_{2}(\mathbf{x})\mathbf{A}_{2}(\mathbf{x})+\mathbf{A }_{2}^{T}(\mathbf{x})\mathbf{K}_{2}(\mathbf{x})-\mathbf{K}_{2}(\mathbf{x}) \mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{K}_{1}\\ &-\mathbf{K}_{1}\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{K} _{2}(\mathbf{x})-\mathbf{K}_{2}(\mathbf{x})\mathbf{B}\mathbf{R}^{-1}\mathbf{B }^{T}\mathbf{K}_{2}(\mathbf{x})=\mathbf{0}\end{split}  \tag{41}\]  
for \(\mathbf{K}_{2}(\mathbf{x})\) obtained from (36). So, the suboptimal control  
\[\mathbf{u}=-\mathbf{R}^{-1}\mathbf{B}^{T}[\mathbf{K}_{1}+\mathbf{K}_{2}( \mathbf{x})]\mathbf{x}. \tag{42}\]  
For index Equation (19), subject to Equation (18), it is possible to find the solution for the state-independent Riccati Equation (39). In general, it is difficult to find the solution of Equation (39) analytically, especially for complex systems [22; 25; 33; 37]. Symbolic software packages (such Matlab or Mathematica) are commonly used to solve such problems. However, for complex systems, the solution may become complicated, and then, it is necessary to approximate the solution. To approximate, an interpolation method or Taylor series method seems to be most suitable [15].