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Math Help - Determine the general form of control sequence

  1. #1
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    Determine the general form of control sequence

    Hi, I have a question which relevant to controllability matrix. The question asks to determine the general form of sequence of control: u_0, u_1 and u_2,

    if a system of difference equation of the form X_{t+1}=AX_t+BU_t, where A=\begin{pmatrix}3 & 2 & 2 \\ -1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix} and B=\begin{pmatrix}0 & 0 \\ 0 & 1 \\ 1 & 0\end{pmatrix}, is to be controlled from X_0=0 to X_3=\begin{pmatrix}2 & 1 & 2\end{pmatrix},

    then show that, in fact, this target could have been achieved at X_2.


    What I did for this question is:

    First, it's simple, just find the controollability matrix, set it's C, then

    C=\begin{pmatrix}A & AB & A^{2}B\end{pmatrix}=\begin{pmatrix} 0 & 0 &|& 2 & 2 & |& 6 & 6\\ 0 & 1 & |& -1 & 0 & | & -3 & -2 \\ 1 & 0 & | & 1 & 0 &| & 1 & 0\end{pmatrix}

    In order to solve the system of equation CU=X_3-A^{3}X_0, I have chosen the first three column from above matrix, then

    the system has the matrix form:

    \begin{pmatrix}0 & 0 & 2\\ 0 & 1 & -1\\1 & 0 &1\end{pmatrix}\begin{pmatrix}u_2 \\ u_1 \\ u_0\end{pmatrix}=\begin{pmatrix}2\\1\\2\end{pmatri  x}-0

    for which the argumented matrix is:

    \begin{pmatrix}0&0&2&|&2\\0&1&-1&|&1\\1&0&1&|&2\end{pmatrix} ---> \begin{pmatrix}1&0&1&|&2\\0&1&-1&|&1\\0&0&2&|&2\end{pmatrix} ---> \begin{pmatrix}1&0&1&|&2\\0&1&-1&|&1\\0&0&1&|&1\end{pmatrix}

    then I found that
    u_0=1,
    u_1-u_0=1=>u_1=2,
    u_2+u_0=2=>u_2=1.

    This is the control sequence in a particular case, I'm not sure that how to find it's general case.

    And then substituting into the original system of difference equation to show that the target could have been achieved at X_2:

    X_{t+1}=AX_t+BU_t

    then
    X_1=AX_0+BU_0=0+\begin{pmatrix}0&0\\0&1\\1&0\end{p  matrix}*1=\begin{pmatrix}0&0\\0&1\\1&0\end{pmatrix  }

    X_2=AX_1+BU_1=\begin{pmatrix}3 & 2 & 2 \\ -1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix}0&0\\0&1\\1&0\end{pmat  rix}+\begin{pmatrix}0&0\\0&1\\1&0\end{pmatrix}*2=\  begin{pmatrix}2&2\\-1&0\\1&0\end{pmatrix}

    which does not match the target X_3=\begin{pmatrix}2 & 1 & 2\end{pmatrix}, so it's absolutely a wrong result. I guess the problem is from finding the control sequence, it should be in general form.

    I have the correct answer which shows
    u_0=\begin{pmatrix}M \\ R\end{pmatrix},
    u_1=\begin{pmatrix}1-S-3T-3R\\ S\end{pmatrix},
    u_2=\begin{pmatrix}1+S+2T+3R\\ 2-S-R\end{pmatrix}.

    But I really have no idea about how to obtain this results. Hope someone can help me this for a bit. Thanks a lot.
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  2. #2
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    Re: Determine the general form of control sequence

    Quote Originally Posted by jackw View Post
    Hi, I .

    Did you ever figure out how to do this question?

    We'vee got this EXACT same question for our courseowrk and we have the same problems as you, so we were wondering if you could throw any hints our way?

    pretty please!

    If anybody now has some hints they could give us, please reply.

    To Moderators: I know i may get told off for posting on old threads, so i apologise in advance!
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