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Controllability of non-linear systems via Lie Brackets

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  1. Nov 19, 2014 #1
    In http://www.me.berkeley.edu/ME237/6_cont_obs.pdf [Broken], page 65, the controllability matrix is defined as:
    $$C=[g_1, g_m,\dots,[g_i,g_j],[ad_{g_i}^k,g_j],\dots,[f,g_i],\dots,[ad_f^k,g_i],\dots]$$
    where the systems is in general given by
    $$\dot{x}=f(x)+\sum_i^m{g_i(x)\mu_i}$$
    Lets say you have a system given by
    $$[\dot{x}_1,\dot{x}_2]^T=[f_1, f_2]^T[x_1,x_2]^T+[g_1,g_2]^T\mu_1+[g'_1,g'_2]^T\mu_2$$
    How will I span C? It seems that different sources have different definitions. Is the what I write below correct?
    $$C=[g_1,g_2,g'_1,g'_2,[g_1,g_2],[g'_1,g'_2],[f_1,g_1],[f_1,g_2],[f_2,g_1],[f_2,g_2]]$$
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Nov 25, 2014 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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