I am looking for the state space representation for the following differential equation:
$m_{1}\ddot{x}+c_{1}\dot{x}+(k_1+k_{1,p}x^{2})x=F_{0} \ cos \ \omega t$
Rewriting this gives: $\ddot{x} = -\frac{c_1}{m_1}\dot{x}-\frac{k_1}{m_1}x-\frac{k_{1,p}}{m_{1}}x^{3}+\frac{F_{0}}{m_1} \cos \ \omega t$
This is what I've come up with:
$\begin{bmatrix} \dot{x}\\ \ddot{x} \end{bmatrix}$ = $\begin{bmatrix} 0&1\\ \frac{-k_{1}-k_{1,p}x^{2}}{m_{1}} & \frac{c_{1}}{m_{1}} \end{bmatrix}$ $\begin{bmatrix} x\\ \dot{x} \end{bmatrix} $ + $\begin{bmatrix} 0\\ \frac{cos \ \omega t}{m_{1}} \end{bmatrix}$ $F_{0}$
But I'm having doubts about the $x^{2}$ in the A matrix. Is it allowed for a state variable to be present in the $A$ matrix?
Thanks in advance,
Mike
The state space representation $\dot{x} = A x + B u$, where $A$ and $B$ do not depend on state variables is used to represent linear systems. In your case, the system is not linear because of the $x^3$ term. What you wrote is not wrong, but you cannot interpret your system as a linear system in state space representation.
You could linearize around an equilibrium point or around a trajectory to get a linear system.