How to model the state space matrix of a pendulum?

Question

Here's an image of the problem:

I'm trying to put the following problem in matrix form (state space) but I don't know how would I put -(g/L)sin(x1) in the A matrix since the x1 value is inside the sin() wave.

Thanks!

Answer 1

The right hand side of the state ODE is ok to have a sine value.

$$ \boldsymbol{Y} = \pmatrix{x_1 \\ x_2 } $$

$$ \dot{\boldsymbol{Y}} = \pmatrix{x_2 \\ - \tfrac{g}{\ell} \sin(x_1) } $$

Only when you are trying to linearize the problem to bring it onto the $\dot{\boldsymbol{Y}} = \mathbf{A} \boldsymbol{Y} + \boldsymbol{b}$ form that you need to use the small angle approximation and replace $\sin(x_1) \approx x_1$.

$$ \dot{\boldsymbol{Y}} = \begin{bmatrix} 0 & 1 \\ -\tfrac{g}{\ell} & 0 \end{bmatrix} \boldsymbol{Y} $$

Answer 2

If we assume small angular displacement, i.e. $\theta$ is small, then we can use the approximation $\sin(\theta) \approx \theta$. This will then allow you to complete your matrix.

Note that, without this assumption, the equations of motion have no closed form solution.