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Math Help - Inverted pendulum with cart: unstable nonlinear system

  1. #1
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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!
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  2. #2
    Super Member
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    You have the system:

    \dot{x_1}=x_2=F_1(x_1,x_2,x_3,x_4)

    \dot{x_2}=\frac{u(\mathbf{x})}{v(\mathbf{x})}=F_2(  x_1,x_2,x_3,x_4)

    \dot{x_3}=x_4=F_3(x_1,x_2,x_3,x_4)

    \dot{x_4}=\frac{w(\mathbf{x})}{lv(\mathbf{x})}=F_4  (x_1,x_2,x_3,x_4)

    (1) The critical points are called equilibrium points or fixed points. That's where all the derivatives are zero. You can plug in (0,0,0,0) into each F and see that this makes them all zero.

    (2) You need to just start splitting out partials for the Jacobian. You know what the Jacobian is right? It's \frac{\partial(F_1,F_2,F_3,F_4)}{\partial(x_1,x_2,  x_3,x_4)}

    Now F_1 is just x_2 so that \frac{\partial(F_1)}{\partial(x_1,x_2,x_3,x_4)}=(0  ,1,0,0) right? How about \frac{\partial(F_2)}{\partial(x_1,x_2,x_3,x_4)}? Can you calculate the partials with respect to each x of F_2?. Do the same with the other two and you'll get the full Jacobian matrix. Once you get it, you need to evaluate it at the fixed point (0,0,0,0), that is \textbf{J}(0,0,0,0). You can do that, and you get the linearized matrix \textbf{A}. Now calculate all the eigenvalues of that matrix and then determine if the critical point at (0,0,0,0) is stable or not depending on the signs of the eigenvalues.

    Oh yea, all the secrets of the Universe can be found in differential equations.
    Last edited by shawsend; October 28th 2008 at 06:00 AM.
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