I could really use some help with the following:
When I have input $u$ and output $y$ of the following nonlinear dynamical system, How can I determine for which values of $r$ this system will be dissipative? And how to represent this in port-Hamiltonian form for the values of $r$?
$\ddot{q}+r\dot{q}-\dot{q}\cos(q)+\sin(q)=u$
$y=\dot{q}$
where $r \in \mathbb{R}$ is a scalar-valued parameter
Some extra info: The non linear system $\dot{x}=f(x,u), y=g(x,u)$ with input $u(t) \in\mathbb{R^p}$ and output$y(t) \in\mathbb{R^p}$ is dissipative if there exists a storage funtion $H: \mathbb{R^n} \rightarrow\mathbb{R}$ such that the dissipation inequality $H (x(t_1)) \leq H(x(t_0)) + \int_{t_0}^{t_1} y(t) u(t)dt$
Can I do something like this:
$x_1=q$
$x_2= \dot{q}$
$\dot{x_1}=x_2$ and $ \dot{x_2}+rx_2-x_2\cos(x_1)+\sin(x_1)=u$
$\begin{pmatrix} \dot{x_1}\\ \dot{x_2} \end{pmatrix} = \begin{pmatrix} x_2\\ -rx_2+x_2\cos(x_1)-\sin(x_1) \end{pmatrix} + \begin{pmatrix} 0\\ 1 \end{pmatrix}u$
Now: $H(x_1,x_2)=P(x_1)+K(x_2)$ where $P$ is the potential energy and $K$ is the kinetic energy.
What are the next steps to get to the full storage function? And how do I know if it is dissipative?
Thanks!!