Currently, I follow a course in modeling nonlinear engineering systems, where I encountered the topic of passivity in nonlinear systems. The following dynamics were given in an exercise:
$m\ddot{x} + \dot{x} + \dot{x}^7 = F$
I was asked to show that the input $F$ and the output $\dot{x}$ are a passive input-output pair. I understand that a system is passive if it is dissipative with respect to the supply rate s:
$s(t) = u^T(t)y(t) = F\dot{x}$
I also understand that the system is dissipative if we can find a (positive) storage function $S$ which satisfies
$S(x(t_1)) \leq S(x(t_0)) + \int_{t_0}^{t_1} s(t)dt$
Currently, I struggle with finding such a suitable storage function. I feel that I do not have the feeling yet for finding appropriate storage functions.
In my lecture notes, systems in the state-space form
$\dot{x} = f(x,u), \quad y=h(x,u)$
where primarily considered. Hence, I considered rewriting my dynamics as
$\begin{bmatrix}\dot{x_1} \\ \dot{x_2}\end{bmatrix} = \begin{bmatrix}x_2 \\ -\frac{1}{m}x_2 - \frac{1}{m}x_2^7\end{bmatrix} + \begin{bmatrix}0 \\ \frac{F}{m}\end{bmatrix}$
but this was not of any help for me yet. Can anyone help me in deriving some first clues for this problem?
$ \dot x (m \ddot x + \dot x + \dot x^7) = F \dot x$ or
$\frac{1}{2}m\frac{d}{dt}\dot x^2 + \dot x^2+\dot x^8 = F \dot x$ and now
$\frac{1}{2}m \dot x^2 + \phi(x,\dot x) = \int F \dot x dt$
Here
$\Phi(\dot x, x) = \frac{1}{2}m \dot x^2 + \phi(x,\dot x) \ge 0$