Editing Sliding mode control (section)

===== Consequences for sliding mode control =====
In the context of sliding mode control, this condition means that
:<math> \underbrace{ \overbrace{\sigma^\intercal}^{\tfrac{\partial V}{\partial \sigma}} \overbrace{\dot{\sigma}}^{\tfrac{\operatorname{d} \sigma}{\operatorname{d} t}} }_{\tfrac{\operatorname{d}V}{\operatorname{d}t}} \leq -\mu ( \mathord{\overbrace{\| \sigma \|_2}^{\sqrt{V}}} )^{\alpha}</math>
where <math>\|\mathord{\cdot}\|</math> is the [[Euclidean norm]]. For the case when switching function <math>\sigma</math> is scalar valued, the sufficient condition becomes
:<math> \sigma \dot{\sigma} \leq -\mu |\sigma|^{\alpha} </math>.
Taking <math>\alpha =1</math>, the scalar sufficient condition becomes
:<math> \operatorname{sgn}(\sigma) \dot{\sigma} \leq -\mu </math>
which is equivalent to the condition that
:<math> \operatorname{sgn}(\sigma) \neq \operatorname{sgn}(\dot{\sigma})
\qquad \text{and} \qquad
|\dot{\sigma}| \geq \mu > 0</math>.
That is, the system should always be moving toward the switching surface <math>\sigma = 0</math>, and its speed <math>|\dot{\sigma}|</math> toward the switching surface should have a non-zero lower bound. So, even though <math>\sigma</math> may become vanishingly small as <math>\mathbf{x}</math> approaches the <math>\sigma(\mathbf{x})=\mathbf{0}</math> surface, <math>\dot{\sigma}</math> must always be bounded firmly away from zero. To ensure this condition, sliding mode controllers are discontinuous across the <math>\sigma = 0</math> manifold; they ''switch'' from one non-zero value to another as trajectories cross the manifold.