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!
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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. $\endgroup$– AJNCommented Feb 7, 2022 at 12:05
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1$\begingroup$ hint: sin(x) ≈ x , for small x $\endgroup$– Pete WCommented Feb 7, 2022 at 14:01
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1$\begingroup$ here is an answer I wrote which shows linearisation steps, in case you want to go that way. $\endgroup$– AJNCommented Feb 7, 2022 at 15:34
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$\begingroup$ if it's a small angle you can put -g/l in there. $\endgroup$– kamranCommented Feb 7, 2022 at 16:42
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1 Answer
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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.
