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.


  • 2
    $\begingroup$ What you may be referring is the state space representation of a linear system where the coefficients can be separated from the state variables. $\dot x = A x$. Your system is a non linear system. It cannot be represented in this format. It can be linearised. The linear approximation of the system can be brought into the form you need. $\endgroup$
    – AJN
    Feb 7, 2022 at 12:05
  • 1
    $\begingroup$ hint: sin(x) ≈ x , for small x $\endgroup$
    – Pete W
    Feb 7, 2022 at 14:01
  • 1
    $\begingroup$ here is an answer I wrote which shows linearisation steps, in case you want to go that way. $\endgroup$
    – AJN
    Feb 7, 2022 at 15:34
  • $\begingroup$ if it's a small angle you can put -g/l in there. $\endgroup$
    – kamran
    Feb 7, 2022 at 16:42

1 Answer 1


Consider the Taylor series expansion

$$\sin \left(x_1\right) = x_1-\frac{x_1^3}{6}+O\left(x_1^4\right)$$

As $x_1$ gets further away from 0, the correctness of the approximation and that of the linear model with $\sin \left(x_1\right) \approx x_1 $ decreases.

enter image description here


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