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Math Help - Diophant equation

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    1

    Diophant equation

    Hi !

    I'm trying to solve a problem concerning a RST regulator. In this context, I need to solve a Diophant Equation.

    this equation is :
    A(z)*R(z) + B(z)*S(z) = C(z)

    where :
    A(z) = (z-1)*(z-0.513)
    B(z) = 0.648*(z+0.801)
    R(z) = r0*z + r1
    S(z) = (z-1)*(s0*z + s1)
    C(z) = z*(zē-0.7497*z+0.2432)

    I need to find r0, r1, s0 and s1 with a minimal solution for S(z)

    I have begun with the rewriting of the equation in a matrix :
    A*X = C

    where (Matlab code) :
    Code:
    A =
        1.0000         0    0.6480         0
       -1.5130    1.0000   -0.1290    0.6480
        0.5130   -1.5130   -0.5190   -0.1290
             0    0.5130         0   -0.5190
    C = 
        1.0000
       -0.7497
        0.2432
             0
    X =
        r0
        r1
        s0
        s1
    For information, A is a Sylvester matrix

    I don't know how to do. can you help me ?
    (if it is easyer for you, I have Matlab for my disposal)

    Thanks !
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  2. #2
    A Plied Mathematician
    Joined
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    CT, USA
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    I have a few questions, just for clarification:

    1. What is an RST Regulator? I've heard of an LQR (Linear Quadratic Regulator) - it shows up in control theory. Is this the same thing?

    2. Do you really mean a Diophantine equation?

    3. A clarification about your equations: you say that A is a matrix, for example. And yet you have A(z) = (z-1)*(z-0.513). What kind of multiplication is that? And what kind of addition? Or is your matrix A different from A(z)?

    4. The determinant of A is very close to zero. Is it supposed to be exactly zero? I would assume so, since otherwise your system would have only one solution, and there'd be no room for manoeuver. What do you get for the row reduction?

    Cheers.
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