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State Vector element interdependency

  1. Feb 20, 2012 #1
    Hi Everyone,

    I am working on a state space model for an electromechanical system. Part of the issue is the internal state vector ends up having interdependencies.
    Take the following state vector as an example:

    \begin{equation}
    X =
    \begin{bmatrix}
    x_1\\ x_2\\ x_3\\ x_4\\ x_5 \\ x_6
    \end{bmatrix}=
    \begin{bmatrix}
    x \\ \dot{x} \\ (x_\text{max}+x_\text{pre} - x) \\ i \\ \Bigl(\frac{i}{x}\Bigl)^2 \\ \Bigl(\frac{i}{x}\Bigl)^2 \dot{x}
    \end{bmatrix}
    \end{equation}

    This would be the vector in the dX/dt = AX + BU


    \begin{equation}
    \dot{X} = AX +BU =
    \begin{bmatrix} %MATRIX A
    0 & 1 & 0 & 0 & 0 & 0 \\
    0 & \frac{-B}{(m + m_\text{thumb})} & \frac{-k_s}{(m + m_\text{thumb})} & 0 & \frac{N^2(\pi r^2)\mu_o}{2(m + m_\text{thumb})} & 0 \\
    0 & 0 & 0 & 0 & 0 & 0 \\
    0 & 0 & 0 & \frac{-iR_c}{L_c} & 0 & \frac{-K_c}{L_c} \\
    0 & 0 & 0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 & 0 & 0
    \end{bmatrix}
    \begin{bmatrix} % VECTOR X
    x_1\\ x_2\\ x_3\\ x_4\\ x_5 \\ x_6
    \end{bmatrix}+
    \begin{bmatrix} % MATRIX B
    0 & 0 \\
    \frac{1}{(m + m_\text{thumb})} & 0 \\
    0 & 0 \\
    0 & \frac{1}{L_c} \\
    0 & 0 \\
    0 & 0 \\
    \end{bmatrix}
    \begin{bmatrix} % VECTOR U
    F_\text{thumb} \\ V_\text{in}
    \end{bmatrix}
    \end{equation}

    Ultimately my problem is how can a find an equilibrium point with a change of coordinates if the last part of the above vector is dependent on the first. The change of coordinates needs to happen to allow for Lyapunov stability analysis around the equilibrium point of the system. Is there a way to work around this issue? Is it an issue at all?

    Also my A matrix has a 3 rows of zeros in it, may I remove these without effecting the lyapunov analysis?

    Any input would be greatly appreciated.
    Thank You.
     
  2. jcsd
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