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Finding the state space model of a nonlinear system

  1. Feb 5, 2016 #1
    1. The problem statement, all variables and given/known data
    The state space model of a nonlinear system is
    [tex]x'_1(t) = 2x^2_2(t) - 50[/tex]
    [tex]x'_2(t) = -x_1(t) - 3x_2(t) + u(t)[/tex]

    Where [tex]x_1(t)[/tex] and [tex]x_2(t)[/tex] are the states, and u(t) is the input. The output of the system is [tex]x_2(t)[/tex].

    Find the state space model of this system linearized at the equilibrium point (-15, 5)

    2. Relevant equations

    State space modeling, Jacobian Matrix(?)

    3. The attempt at a solution

    I think my attempt lies on false assumption, but here's what I did.

    Since we're linearizing at a known equilibrium point for a nonlinear system, and since [tex]x'_1(t) = 2x^2_2(t) - 50[/tex] contains non linear term, we must first take a derivative of it as denoted in Jacobian matrix. derivative of x_1(t) is

    [tex]x''_1(t) = 4x_2(t)[/tex]

    So the state space model we get is

    [x'_1] = [ 0 4 ] [x_1] + [ 0 ] u(t)
    [x'_2]aa[-1 -3 ] [x_2] a[ 1 ]

    (Sorry I have no clue how to put this matrix in Latex).

    But then how do we get the output from this state space model?
     
  2. jcsd
  3. Feb 6, 2016 #2

    pasmith

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    Homework Helper

    The linearization about (-15, 5) is obtained by evaluating the Jacobian matrix at (-15,5). Here the Jacobian matrix is [tex]
    J(x_1,x_2) = \begin{pmatrix} 0 & 4x_2 \\ -1 & -3 \end{pmatrix}
    [/tex] and the latex code is
    Code (LaTeX):

    J(x_1,x_2) = \begin{pmatrix} 0 & 4x_2 \\ -1 & -3 \end{pmatrix}
     
     
    Last edited: Feb 6, 2016
  4. Feb 6, 2016 #3

    So the evaluation at (-15,5) becomes

    [tex]y'= \begin{pmatrix} 0 & 20 \\ -1 & -3 \end{pmatrix}[/tex]

    ?
     
    Last edited: Feb 6, 2016
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