Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear System Transformations

  1. Nov 21, 2012 #1
    Hi there,

    I have a linear algebra question relating actually to control systems (applied differential equations)

    for the linear system

    [itex]

    {\dot{\vec{{x}}} = {\bf{A}}{\vec{{x}}} + {\bf{B}}}{\vec{{u}}}\\
    \\

    A \in \mathbb{R}^{ nxn }\\
    B \in \mathbb{R}^{ nx1 }\\
    [/itex]

    In class, we formed a transformation matrix P using the controllability matrix [itex] M_c [/itex] as a basis (assuming it is full rank).
    [itex]

    M_c = [ {\bf{B \;AB \;A^2B\;....\;A^{n-1}B}}]
    [/itex]

    and there is a second matrix with a less established name. Given that the characteristic equation of the system is [itex] |I\lambda -A| = \lambda^n + \alpha_1 \lambda^{n-1} +... + \alpha_{n-1}\lambda + \alpha_n= 0 [/itex], we then construct a second matrix, call it M_2, which is given below.

    [itex]
    {\bf{M}}_2 =
    \begin{bmatrix}
    \alpha_{n-1} & \alpha_{n-2} & \cdots & \alpha_1 & 1 \\
    \alpha_{n-2} & \cdots & \alpha_1 & 1 & 0 \\
    \vdots & \alpha_1 & 1 & 0 & 0\\
    \alpha_1 & 1 & 0 & \cdots & 0\\
    1 & 0 & 0& \cdots & 0 \\
    \end{bmatrix}

    [/itex]

    then the transformation matrix is then given by

    [itex]

    P^{-1} = M_c M_2

    [/itex]


    and then applying the transformation always gives.. and this is what I don't understand....

    [itex]
    {\overline{\bf{A}}} = {\bf{PAP}}^{-1} =

    \begin{bmatrix}
    0 & 1 & 0 & \cdots & 0 \\
    0 & 0 & 1 & \cdots & \vdots \\
    \vdots & \vdots & 0 & 1 & 0\\
    0 & 0 & \cdots &0& 1\\
    -\alpha_{1} & -\alpha_{2} & \cdots & -\alpha_{n-1}& -\alpha_{n}\\
    \end{bmatrix}

    [/itex]

    Now I'm just looking for intuition is to why this is true. I know that this only works if the controllability matrix is full rank, which can the be used as a basis for the new transformation, but I don't get how exactly the M_2 matrix is using it to transform into the canonical form.... Can someone explain this to me? thanks...

    Disclaimer: I posted this in another PF subforum, but I think I might do better in this section.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Linear System Transformations
Loading...