I am studying Lyapunov stability for a nonlinear systems as follow: \begin{equation}\label{nonlinear_stab_changeCord} \begin{cases} \begin{split} L\dot{\widetilde{x_{1}}} &= c_1 \widetilde{x_{1}} + c_2\widetilde{x_{2}} + c_3 \widetilde{x_{9}} + c_4 + 2 k_{i1} \widetilde{x_{2}} \widetilde{x_{9}} - 2 k_{p1} \widetilde{x_{1}} \widetilde{x_{2}} \\&-\frac{b_4R}{\left(V_{in} - 2 \left(\widetilde{x_{2}} + x_{2}^*\right)\right)} \left(\left(c_6 \widetilde{x_{4}} + c_7\right) \left(c_8 + c_9 \widetilde{x_{3}} - c_{10} \widetilde{x_{6}} + c_{10} \left(k_{p4} \widetilde{x_{4}} + k_{p2} k_{p3} k_{p4} \widetilde{x_{2}} + k_{p3} k_{p4} \widetilde{x_{8}} \right.\right.\right. \\& \left.\left.\left.-k_{p3} k_{p4} k_{i2} \widetilde{x_{10}} - k_{i3} k_{p4} \widetilde{x_{11}} - k_{i4} \widetilde{x_{12}}\right)\right) + c_{14} \left(\widetilde{x_{3}} + x_{3}^*\right) \left(c_{11} + c_{12} \widetilde{x_{4}} + c_{13} \widetilde{x_{5}}\right.\right.\\&\left.\left. + c_{13} \left(-k_{p6} \widetilde{x_{3}} - k_{p5} k_{p6} \widetilde{x_{7}} + k_{i5} k_{p6} \widetilde{x_{13}} + k_{i6} \widetilde{x_{14}}\right)\right)\right) \end{split}\\[10pt] \begin{split} C\dot{\widetilde{x_{2}}} &= b_1 \widetilde{x_{1}} + b_2 \widetilde{x_{9}} + b_3 + 2 k_{p1} \widetilde{x_{1}}^2 - 2 k_{i1} \widetilde{x_{1}} \widetilde{x_{9}} \\&+\frac{b_4}{\left(V_{in} - 2 \left(\widetilde{x_{2}} + x_{2}^*\right)\right)} \left(\left(c_6 \widetilde{x_{4}} + c_7\right) \left(c_8 + c_9 \widetilde{x_{3}} - c_{10} \widetilde{x_{6}} + c_{10} \left(k_{p4} \widetilde{x_{4}} + k_{p2} k_{p3} k_{p4} \widetilde{x_{2}} + k_{p3} k_{p4} \widetilde{x_{8}} \right.\right.\right. \\& \left.\left.\left.-k_{p3} k_{p4} k_{i2} \widetilde{x_{10}} - k_{i3} k_{p4} \widetilde{x_{11}} - k_{i4} \widetilde{x_{12}}\right)\right) + c_{14} \left(\widetilde{x_{3}} + x_{3}^*\right) \left(c_{11} + c_{12} \widetilde{x_{4}} + c_{13} \widetilde{x_{5}}\right.\right.\\&\left.\left. + c_{13} \left(-k_{p6} \widetilde{x_{3}} - k_{p5} k_{p6} \widetilde{x_{7}} + k_{i5} k_{p6} \widetilde{x_{13}} + k_{i6} \widetilde{x_{14}}\right)\right)\right) \end{split}\\[10pt] \begin{split} L_f\dot{\widetilde{x_{3}}} &= k_{i6}\widetilde{x_{14}} - k_{p6}\widetilde{x_3} - r_f\widetilde{x_3} - k_{p5}k_{p6}\widetilde{x_7} + k_{i5}k_{p6}\widetilde{x_{13}} \end{split}\\[10pt] \begin{split} L_f\dot{\widetilde{x_{4}}} &= k_{i4}\widetilde{x_{12}} - k_{p4}\widetilde{x_4} - r_f\widetilde{x_4} - k_{p3}k_{p4}\widetilde{x_8} + k_{i3}k_{p4}\widetilde{x_{11}} - k_{p2}k_{p3}k_{p4}\widetilde{x_2} + k_{i2}k_{p3}k_{p4}\widetilde{x_{10}} \end{split}\\[10pt] C_f\dot{\widetilde{x_5}} = \widetilde{x_3} - \widetilde{x_7} - C_f\omega \widetilde{x_6} \\[10pt] C_f\dot{\widetilde{x_6}} = \widetilde{x_4} - \widetilde{x_8} + C_f\omega \widetilde{x_5} \\[10pt] L_g\dot{\widetilde{x_7}} = \widetilde{x_5} - r_g\widetilde{x_7} - L_g\omega \widetilde{x_8} \\[10pt] L_g\dot{\widetilde{x_8}} = \widetilde{x_6} - r_g\widetilde{x_8} + L_g\omega \widetilde{x_7}\\[10pt] \dot{\widetilde{x_9}} = -\widetilde{x_1}\\[10pt] \dot{\widetilde{x_{10}}} = -\widetilde{x_2}\\[10pt] \dot{\widetilde{x_{11}}} = k_{i2}\widetilde{x_{10}} - k_{p2}\widetilde{x_2} - \widetilde{x_8}\\[10pt] \dot{\widetilde{x_{12}}} = k_{i3}\widetilde{x_{11}} - k_{p3}\widetilde{x_8} - \widetilde{x_4} - k_{p2}k_{p3}\widetilde{x_2} + k_{i2}k_{p3}\widetilde{x_{10}}\\[10pt] \dot{\widetilde{x_{13}}} = -\widetilde{x_7}\\[10pt] \dot{\widetilde{x_{14}}} = k_{i5}\widetilde{x_{13}} - k_{p5}\widetilde{x_7} - \widetilde{x_3} \end{cases} \end{equation}
To shift the equilibrium point to the origin, the change of coordinates $ \widetilde{x} = x - x^* $ is introduced. which $\widetilde{x_1}$, $\widetilde{x_2}$, $\widetilde{x_3}$, $\widetilde{x_4}$, $\widetilde{x_5}$, $\widetilde{x_6}$, $\widetilde{x_7}$, $\widetilde{x_8}$, $\widetilde{x_9}$, $\widetilde{x_{10}}$, $\widetilde{x_{11}}$, $\widetilde{x_{12}}$, $\widetilde{x_{13}}$ and $\widetilde{x_{14}}$ are state variables and another parameter in this equation are constant.
We rewrite our system in a more compact form $\dot{\widetilde{x}}=A\widetilde{x}+B+f(x) $, where
\begin{equation} \begin{cases} \begin{split} f_1(\widetilde{X}) &= \left[k_{i1} \widetilde{x_{2}} \widetilde{x_{9}} - 2 k_{p1} \widetilde{x_{1}} \widetilde{x_{2}} \right.\\&\left.-\frac{b_4R}{\left(V_{in} - 2 \left(\widetilde{x_{2}} + x_{2}^*\right)\right)} \left(\left(c_6 \widetilde{x_{4}} + c_7\right) \left(c_8 + c_9 \widetilde{x_{3}} - c_{10} \widetilde{x_{6}} + c_{10} \left(k_{p4} \widetilde{x_{4}} + k_{p2} k_{p3} k_{p4} \widetilde{x_{2}} + k_{p3} k_{p4} \widetilde{x_{8}} \right.\right.\right.\right. \\& \left.\left.\left.\left.-k_{p3} k_{p4} k_{i2} \widetilde{x_{10}} - k_{i3} k_{p4} \widetilde{x_{11}} - k_{i4} \widetilde{x_{12}}\right)\right) + c_{14} \left(\widetilde{x_{3}} + x_{3}^*\right) \left(c_{11} + c_{12} \widetilde{x_{4}} + c_{13} \widetilde{x_{5}}\right.\right.\right.\\&\left.\left.\left. + c_{13} \left(-k_{p6} \widetilde{x_{3}} - k_{p5} k_{p6} \widetilde{x_{7}} + k_{i5} k_{p6} \widetilde{x_{13}} + k_{i6} \widetilde{x_{14}}\right)\right)\right)\right] \end{split}\\ \begin{split} f_2(\widetilde{X}) &= \bigl[2 k_{p1} \widetilde{x_{1}}^2 - 2 k_{i1} \widetilde{x_{1}} \widetilde{x_{9}} \bigr.\\&\bigl.+\frac{b_4}{\left(V_{in} - 2 \left(\widetilde{x_{2}} + x_{2}^*\right)\right)} \left(\left(c_6 \widetilde{x_{4}} + c_7\right) \left(c_8 + c_9 \widetilde{x_{3}} - c_{10} \widetilde{x_{6}} + c_{10} \left(k_{p4} \widetilde{x_{4}} + k_{p2} k_{p3} k_{p4} \widetilde{x_{2}} + k_{p3} k_{p4} \widetilde{x_{8}} \bigr.\right.\right.\right. \\& \left.\left.\left.\bigl.-k_{p3} k_{p4} k_{i2} \widetilde{x_{10}} - k_{i3} k_{p4} \widetilde{x_{11}} - k_{i4} \widetilde{x_{12}}\right)\right) + c_{14} \left(\widetilde{x_{3}} + x_{3}^*\right) \left(c_{11} + c_{12} \widetilde{x_{4}} + c_{13} \widetilde{x_{5}}\bigr.\right.\right.\\&\left.\left.\bigl. + c_{13} \left(-k_{p6} \widetilde{x_{3}} - k_{p5} k_{p6} \widetilde{x_{7}} + k_{i5} k_{p6} \widetilde{x_{13}} + k_{i6} \widetilde{x_{14}}\right)\right)\right)\bigr] \end{split} \end{cases} \end{equation}
\begin{equation} f(\widetilde{X}) = \begin{bmatrix} f_1(\widetilde{X})\\ f_2(\widetilde{X})\\ 0\\ \vdots \\ 0 \end{bmatrix} \end{equation} \begin{equation} \boldsymbol{B} = \begin{bmatrix} c_4\\ b_3\\ 0\\ \vdots \\ 0 \end{bmatrix} \end{equation}
The systems are composed of a linear part and an additive nonlinearity. I was wondering how to define a lyapunov function within the framework of Linear Matrix Inequalities approach for this nonlinear system?
some efforts:
we define lyapunov function as follow: \begin{equation} V(X) = V_1(X) + V_2(X) + V_3(X) + V_4(X) \end{equation} where \begin{equation} \begin{cases} V_1(X) = \frac{1}{2}\left(L\widetilde{x_1}+RC\widetilde{x_2}\right)^2\\ V_2(X) = \frac{1}{4}\left(V_{in}-2(\widetilde{x_2}+x_2^*)\right)^2 + \frac{1}{2}L_fb_4c_{13}c_{14}\left(\widetilde{x_3}+x_3^*\right)^2 -\frac{1}{2}\frac{L_fb_4c_{10}}{c_6}\left(c_6\widetilde{x_4}+c_7\right)^2\\ V_3(X) = \frac{1}{2}\left(C_f\widetilde{x_5}^2+C_f\widetilde{x_6}^2+L_g\widetilde{x_7}^2+L_g\widetilde{x_8}^2\right)\\ V_4(X) = ? \end{cases} \end{equation} where $ c_6 < 0 $ then \begin{equation} \begin{cases} \begin{split} \dot{V_1} &= \left(L\widetilde{x_1}+RC\widetilde{x_2}\right)\bigl(c_1 \widetilde{x_{1}} + c_2\widetilde{x_{2}} + c_3 \widetilde{x_{9}} + c_4 + 2 k_{i1} \widetilde{x_{2}} \widetilde{x_{9}} - 2 k_{p1} \widetilde{x_{1}} \widetilde{x_{2}} + b_1 R \widetilde{x_{1}} + b_2 R\widetilde{x_{9}}\bigr.\\&\bigl. + b_3 R + 2 R k_{p1} \widetilde{x_{1}}^2 - 2 k_{i1}R \widetilde{x_{1}} \widetilde{x_{9}}\bigr) \end{split}\\ \begin{split} \dot{V_2} &= -\left(V_{in}-2(\widetilde{x_2}+x_2^*)\right)\left(b_1 \widetilde{x_{1}} + b_2 \widetilde{x_{9}} + b_3 + 2 k_{p1} \widetilde{x_{1}}^2 - 2 k_{i1} \widetilde{x_{1}} \widetilde{x_{9}} \right) \\&-b_4\left(c_6\widetilde{x_4}+c_7\right)\left(c_8 + c_9 \widetilde{x_{3}} - c_{10} \widetilde{x_{6}}\right) - b_4c_{14}\left(\widetilde{x_3}+x_3^*\right)\left(c_{11} + c_{12} \widetilde{x_{4}} + c_{13} \widetilde{x_{5}}\right)\\&-b_4c_{13}c_{14}r_f\left(\widetilde{x_3}^2+\widetilde{x_3}x_3^*\right) + b_4c_{10}r_f\left(c_6\widetilde{x_4}^2+c_7\widetilde{x_4}\right) \end{split}\\ \dot{V_3} = \widetilde{x_3}\widetilde{x_5}+\widetilde{x_4}\widetilde{x_6}-r_g\widetilde{x_7}^2-r_g\widetilde{x_8}^2\\ \dot{V_4} = ? \end{cases} \end{equation}
We still needed to define an appropriate $V_4$ in lyapunov function!