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Inverted pendulum with cart: unstable nonlinear system

Hey guys,

Im really stuck when it comes to trying to solve this problem. It'll take some time to write up, so i'll upload a pdf file. I was given the following tip:

Regarding Question 2 of the assignment, you may use the following theorem to determine the stability of the nonlinear system:

Stability Theorem: Let (x_0,y_0) be a critical point of a nonlinear system

dot(x) = f(x)

and let A be the Jacobi matrix of the system, evaluated at (x_0,y_0). Then the critical point is

(i) asymptotically stable if Re(lambda_i) < 0 for all eigenvalues lambda_i of A

(ii) unstable if Re(lambda_i) > 0 for one or more eigenvalues lambda_i of A

(from Nonlinear Systems, by H. Khalil, Prentice Hall, Third edition. )

If anyone can tell me what to do i would greatly appreciate it!