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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

user463102
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2 Answers2

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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.

renapagli
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This is nonlinear system so you can't put it in a matrix form. However, state space representation can model nonlinear systems by transforming higher order differential equations into first one. In the example you've posted, the state space form of your system is as done as follows: let $y_1=x, y_2=\dot{x}$, then

$$ \begin{align} \dot{y}_1 &= y_2 \\ \dot{y}_2 &= -\frac{c_1}{m_1} y_2 - \frac{k_1}{m_1} y_1 - \frac{k_{1,p}}{m_1} y^3_1 + \frac{F_0}{m_1} \cos\omega t \end{align} $$ Now the system can be rewritten in a general form as follows: $$ \begin{align} \dot{y} = f(y,t) \end{align} $$ Notice how state space representation maps the system from second-order to first-order. The system is non-autonomous due to the explicit dependency on time.

CroCo
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