# Controllability of non-linear systems via Lie Brackets

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1. Nov 19, 2014

### phys_student1

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. Nov 25, 2014