Lyapunov Function for Nonlinear System

Question

Which Lyapunov function should I choose to show the stability (or instability) of equilibrium points? With $k>0$, $K>0$, $\delta >0$. The system is Hurwitz (asymptotically stable) when $k>K$. Standard quadratic Lyapunov function \begin{equation} V(x_1,x_2,x_3,x_4) = \frac{k(x_1-x_2)^2}{2}+\frac{x_3^2}{2}+\frac{x_4^2}{2}-\frac{K(x_1-x_2)^2}{2} \end{equation} does not work. Linearized system \begin{equation} \begin{cases} \dot{x}_1=x_3,\\ \dot{x}_2=x_4,\\ \dot{x}_3=-k(x_1-x_2)-K x_2,\\ \dot{x}_4=k(x_1-x_2)-\delta x_4, \end{cases} \end{equation} gives \begin{equation} A = \begin{pmatrix} 0& 0& 1& 0\\ 0& 0& 0& 1\\ -k& k-K& 0& 0\\ k& -k& 0& -\delta \end{pmatrix}. \end{equation} By the Lyapunov's converse theorems I know that there exists a unique symmetric positive definite matrix $P$ as a solution to the Lyapunov equation with $\boldsymbol Q=\boldsymbol I$: $$\boldsymbol{PA}+\boldsymbol{A^TP}=-\boldsymbol{Q}$$ I have tried to solve it symbolically via Matlab, but did cope with it. I have also tried to differentiate Lyapunov function without knowing explicit form for $\boldsymbol {P}$ but the result of $\boldsymbol {\dot{V}} = \boldsymbol x^T \boldsymbol P \boldsymbol x$ gives the system of 16 equations which I could not solve as well.

Answer 1

With the help of MATHEMATICA.

$$ A = \left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -k & -k-K & 0 & 0 \\ k & -k & 0 & -\delta \\ \end{array} \right) $$

and the command

P = LyapunovSolve[A, Transpose[A], -IdentityMatrix[4]]

we obtain

$$ P = \left[ \begin{array}{cccc} -\frac{\delta ^2 \left(3 k K+k (3 k+2)+K^2+K\right)+K (k+K) (2 k+K+1)}{2 \delta k (k+K) (2 k+K)} & -\frac{\delta ^2 k+\delta ^2 K+K}{4 \delta k^2+2 \delta k K} & -\frac{1}{2} & -\frac{k+1}{2 (k+K)} \\ -\frac{\delta ^2 k+\delta ^2 K+K}{4 \delta k^2+2 \delta k K} & \frac{\delta ^2 (k+K)+K (2 k+K+1)}{2 \delta (k+K) (2 k+K)} & \frac{k+1}{2 (k+K)} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{k+1}{2 (k+K)} & -\frac{\frac{\delta ^2 \left(2 k^2+2 k K+k+K^2\right)}{k+K}+(k+1) K}{2 \delta k} & \frac{K}{2 \delta } \\ -\frac{k+1}{2 (k+K)} & -\frac{1}{2} & \frac{K}{2 \delta } & \frac{k K+K}{2 \delta k+2 \delta K} \\ \end{array} \right] $$