State-space representation

Question

I have 2 differential equations. I need to find state-space representation of the equations. Input should be $\theta(t)$ and outputs should be $x(t)$ and $y(t)$. $M,m,D_{x},D_{y},K_{x}$ and $K_{y}$ are constants.

$M\ddot{x}(t)$+$D_{x}\dot{x}(t)$+$K_{x}x(t)$= $m\sin(\theta(t))$

$M\ddot{y}(t)$+$D_{y}\dot{y}(t)$+$K_{y}y(t)$= $m\cos(\theta(t))$

Answer 1

Let the states be $\begin{bmatrix}{x_1 \\ x_2 \\ x_3\\ x_4}\end{bmatrix} = \begin{bmatrix}{x \\ \dot{x} \\ y\\ \dot{y}}\end{bmatrix}$

Then the state-space representation will be $\begin{bmatrix}{\dot{x_1} \\ \dot{x_2} \\ \dot{x_3}\\ \dot{x_4}}\end{bmatrix} = f(x_1,x_2,x_3,x_4,u)$

where,

$f(x_1,x_2,x_3,x_4,u) = \begin{bmatrix}{x_2 \\ \ddot{x} \\ x_4\\ \ddot{y}}\end{bmatrix} = \begin{bmatrix}{x_2 \\ \frac{1}{M}[m \sin(u) - D_x x_2 - K_x x_1] \\ x_4\\ \frac{1}{M}[m \cos(u) - D_y x_4 - K_y x_3]}\end{bmatrix}$

This is the basics of non-linear systems and the details can be found in any related book.