[ \dot\mathbfx = \mathbff_0(\mathbfx) + \mathbfg(\mathbfx)\mathbfu + \mathbfY(\mathbfx)\theta ]
Why is this powerful? Because it captures internal dynamics, multiple equilibria, limit cycles, and chaos—phenomena invisible to linear transfer functions. A common first step is local linearization around an equilibrium point ((\mathbfx_0, \mathbfu_0)) where (\mathbff(\mathbfx_0, \mathbfu_0)=0). Defining (\delta\mathbfx = \mathbfx - \mathbfx_0), (\delta\mathbfu = \mathbfu - \mathbfu_0), we compute the Jacobian matrices: Defining (\delta\mathbfx = \mathbfx - \mathbfx_0)
[ \inf_\mathbfu \left[ \frac\partial V\partial \mathbfx \left( \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu \right) \right] < 0 ] (\delta\mathbfu = \mathbfu - \mathbfu_0)
[ \dot\mathbfx = \mathbff(\mathbfx) + \mathbfg(\mathbfx)\mathbfu + \Delta(\mathbfx) + \mathbfd(t) ] t) \ \mathbfy(t) &= \mathbfh(\mathbfx(t)
[ \beginalign* \dot\mathbfx(t) &= \mathbff(\mathbfx(t), \mathbfu(t), t) \ \mathbfy(t) &= \mathbfh(\mathbfx(t), \mathbfu(t), t) \endalign* ]
The approach introduces an extra robustifying term (\mathbfu_\textrob(\mathbfx)) such that:
This means there exists a control law that can decrease (V) at every point. The famous provides a universal stabilizing controller when a CLF is known: