Controllability of non-linear systems via Lie Brackets

[phys_student1]

In http://www.me.berkeley.edu/ME237/6_cont_obs.pdf , 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?
$$C=[g_1,g_2,g'_1,g'_2,[g_1,g_2],[g'_1,g'_2],[f_1,g_1],[f_1,g_2],[f_2,g_1],[f_2,g_2]]$$

 

[jvicens]

Thank you for your question. The definition of the controllability matrix given in the paper you referenced is correct. However, the way you have spanned C for your specific system is not entirely correct.

The controllability matrix is used to determine whether a system is controllable, i.e. can be driven from any initial state to any desired state in finite time using a control input. In order to span C, you need to consider all possible combinations of the control inputs $\mu_i$. In your case, you have two control inputs, $\mu_1$ and $\mu_2$, so you need to include all possible combinations of these two inputs in C. This means that your controllability matrix should be:

$$C=[g_1,g_2,g'_1,g'_2,[g_1,g_2],[g'_1,g'_2],[f_1,g_1],[f_1,g_2],[f_2,g_1],[f_2,g_2],[g_1,\mu_1],[g_1,\mu_2],[g_2,\mu_1],[g_2,\mu_2],[g'_1,\mu_1],[g'_1,\mu_2],[g'_2,\mu_1],[g'_2,\mu_2]]$$

This includes all possible combinations of the control inputs and will give you the correct controllability matrix for your system.

I hope this helps clarify any confusion. If you have any further questions, please don't hesitate to ask.