Editing Controllability (section)

== Nonlinear systems ==
Nonlinear systems in the control-affine form

: <math>\dot{\mathbf{x}} = \mathbf{f(x)} + \sum_{i=1}^m \mathbf{g}_i(\mathbf{x})u_i</math>

are locally accessible about <math>x_0</math> if the accessibility distribution <math>R</math> spans <math>n</math> space, when <math>n</math> equals the dimension of <math>x</math> and R is given by:<ref>Isidori, Alberto (1989). ''Nonlinear Control Systems'', p. 92–3. Springer-Verlag, London. {{ISBN|3-540-19916-0}}.</ref>

:<math>R = \begin{bmatrix} \mathbf{g}_1 & \cdots & \mathbf{g}_m & [\mathrm{ad}^k_{\mathbf{g}_i}\mathbf{\mathbf{g}_j}] & \cdots & [\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}_i}] \end{bmatrix}.</math>

Here, <math>[\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}}]</math> is the repeated [[Lie bracket of vector fields|Lie bracket]] operation defined by

: <math>[\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}}] = \begin{bmatrix} \mathbf{f} & \cdots & j & \cdots & \mathbf{[\mathbf{f}, \mathbf{g}]} \end{bmatrix}. </math>

The controllability matrix for linear systems in the previous section can in fact be derived from this equation.