Editing Sliding mode control (section)

====Reachability: Attaining sliding manifold in finite time====
To ensure that the sliding mode <math>\sigma(\mathbf{x})=\mathbf{0}</math> is attained in finite time, <math>\operatorname{d}V/{\operatorname{d}t}</math> must be more strongly bounded away from zero. That is, if it vanishes too quickly, the attraction to the sliding mode will only be asymptotic. To ensure that the sliding mode is entered in finite time,<ref>{{Cite book|title=Sliding Mode Control in Engineering|last1=Perruquetti|first1=W.|last2=Barbot|first2=J.P.|publisher=Marcel Dekker Hardcover|year=2002|isbn=978-0-8247-0671-5}}</ref>
:<math>\frac{\operatorname{d}V}{\operatorname{d}t} \leq -\mu (\sqrt{V})^{\alpha}</math>
where <math>\mu > 0</math> and <math>0 < \alpha \leq 1</math> are constants.

===== Explanation by comparison lemma =====
This condition ensures that for the neighborhood of the sliding mode <math>V \in [0,1]</math>,
:<math>\frac{\operatorname{d}V}{\operatorname{d}t} \leq -\mu (\sqrt{V})^{\alpha} \leq -\mu \sqrt{V}.</math>
So, for <math>V \in (0,1]</math>,
:<math>\frac{ 1 }{ \sqrt{V} } \frac{\operatorname{d}V}{\operatorname{d}t} \leq -\mu,</math>
which, by the [[chain rule]] (i.e., <math>\operatorname{d}W/{\operatorname{d}t}</math> with <math>W \triangleq 2 \sqrt{V}</math>), means
:<math>\mathord{\underbrace{D^+ \Bigl( \mathord{\underbrace{2 \mathord{\overbrace{\sqrt{V}}^{ {} \propto \|\sigma\|_2}}}_{W}} \Bigr)}_{D^+ W \, \triangleq \, \mathord{\text{Upper right-hand } \dot{W}}}} = \frac{ 1 }{ \sqrt{V} } \frac{\operatorname{d}V}{\operatorname{d}t} \leq -\mu</math>
where <math>D^+</math> is the [[upper right-hand derivative]] of <math>2 \sqrt{V}</math> and the symbol <math>\propto</math> denotes [[proportionality (mathematics)|proportionality]]. So, by comparison to the curve <math>z(t) = z_0 - \mu t</math> which is represented by differential equation <math>\dot{z} = -\mu</math> with initial condition <math>z(0)=z_0</math>, it must be the case that <math>2 \sqrt{V(t)} \leq V_0 - \mu t</math> for all {{mvar|t}}. Moreover, because <math>\sqrt{V} \geq 0</math>, <math>\sqrt{V}</math> must reach <math>\sqrt{V}=0</math> in finite time, which means that {{mvar|V}} must reach <math>V=0</math> (i.e., the system enters the sliding mode) in finite time.<ref name="Khalil02">{{Cite book
 | last = Khalil
 | first = H.K.
 | authorlink = Hassan K. Khalil
 | year = 2002
 | edition = 3rd
 | url = http://www.egr.msu.edu/~khalil/NonlinearSystems/
 | isbn = 978-0-13-067389-3
 | title = Nonlinear Systems
 | publisher = [[Prentice Hall]]
 | location = Upper Saddle River, NJ}}</ref> Because <math>\sqrt{V}</math> is proportional to the [[Euclidean norm]] <math>\|\mathord{\cdot}\|_2</math> of the switching function <math>\sigma</math>, this result implies that the rate of approach to the sliding mode must be firmly bounded away from zero.

===== 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.