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Finding deriviative of lyapunov's equation (chain rule/linear algebra)

  1. Feb 8, 2010 #1
    Hi there!

    I am just trying to work though a simple deriviation presented in a txt book (feedback control of dynamic systems p 616). I am REALLY new to linear algebra and they have skipped too many steps!

    they have used the chain rule for differentiation of V(x) (a lyapunov function) where V = xTPx ; where P is a symetric positive matrix (i have not been able to find reference to a 'positive matrix'.. is a positive-definite matrix the same thing?)

    Thus we have;

    d[tex]/dt[/tex]xTPx = x[tex]\dot{}[/tex]TPx + xTPx[tex]\dot{}[/tex]

    from there we somehow get to

    xT(FTP + PF)x (where i believe F is the 'state' matrix usually represened as A ie X dot = Fx + bu )

    if someone could point me in the right direction it would really be appreciated!!!
     
    Last edited: Feb 8, 2010
  2. jcsd
  3. Feb 8, 2010 #2

    CompuChip

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    Science Advisor
    Homework Helper

    It doesn't have much to do with calculus, it's just plugging in the definition.
    The only step that you might be missing is that (AB...XYZ)T = ZT YT XT ... BT AT for some matrices / vectors A, ..., Z.
     
  4. Feb 9, 2010 #3
    hi CompuChip

    thankyou so much for for taking the time to reply!
    despite all the time I have spent pooring over it I just couldnt see it!
    (and yes I was missing that step!)

    thanks again! :)
     
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