Editing Sliding mode control (section)

==Theoretical foundation==
The following theorems form the foundation of variable structure control.

===Theorem 1: Existence of sliding mode===
Consider a [[Lyapunov function]] candidate
{| border="0" width="75%"
|-
| align="left" |
{{NumBlk|:|<math>
V(\sigma(\mathbf{x}))=\frac{1}{2}\sigma^\intercal(\mathbf{x})\sigma(\mathbf{x})=\frac{1}{2}\|\sigma(\mathbf{x})\|_2^2
</math>
|  {{EquationRef|3}}}}
|}
where <math>\|\mathord{\cdot}\|</math> is the [[Euclidean norm]] (i.e., <math>\|\sigma(\mathbf{x})\|_2</math> is the distance away from the sliding manifold where <math>\sigma(\mathbf{x})=\mathbf{0}</math>). For the system given by Equation&nbsp;({{EquationNote|1}}) and the sliding surface given by Equation&nbsp;({{EquationNote|2}}), a sufficient condition for the existence of a sliding mode is 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}} < 0 \qquad \text{(i.e., } \tfrac{\operatorname{d}V}{\operatorname{d}t} < 0 \text{)} </math>
in a [[Neighbourhood (mathematics)|neighborhood]] of the surface given by <math>\sigma(\mathbf{x})=0</math>.

Roughly speaking (i.e., for the [[scalar (mathematics)|scalar]] control case when <math>m=1</math>), to achieve <math>\sigma^\intercal \dot{\sigma} < 0</math>, the feedback control law <math> u(\mathbf{x}) </math> is picked so that <math>\sigma</math> and <math>\dot{\sigma}</math> have opposite signs. That is,
* <math>u(\mathbf{x})</math> makes <math>\dot{\sigma}(\mathbf{x})</math> negative when <math>\sigma(\mathbf{x})</math> is positive.
* <math>u(\mathbf{x})</math> makes <math>\dot{\sigma}(\mathbf{x})</math> positive when <math>\sigma(\mathbf{x})</math> is negative.
Note that
:<math>\dot{\sigma} 
= \frac{\partial \sigma}{\partial \mathbf{x}} \overbrace{\dot{\mathbf{x}}}^{\tfrac{\operatorname{d} \mathbf{x}}{\operatorname{d} t}}
= \frac{\partial \sigma}{\partial \mathbf{x}} \overbrace{\left( f(\mathbf{x},t) + B(\mathbf{x},t) \mathbf{u} \right)}^{\dot{\mathbf{x}}}</math>
and so the feedback control law <math>\mathbf{u}(\mathbf{x})</math> has a direct impact on <math>\dot{\sigma}</math>.

====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.

===Theorem 2: Region of attraction===
For the system given by Equation&nbsp;({{EquationNote|1}}) and sliding surface given by Equation&nbsp;({{EquationNote|2}}), the subspace for which the <math>\{ \mathbf{x} \in \mathbb{R}^n : \sigma(\mathbf{x})=\mathbf{0} \}</math> surface is reachable is given by
:<math>\{ \mathbf{x} \in \mathbb{R}^n : \sigma^\intercal(\mathbf{x})\dot{\sigma}(\mathbf{x}) < 0 \}</math>
That is, when initial conditions come entirely from this space, the Lyapunov function candidate <math>V(\sigma)</math> is a [[Lyapunov function]] and <math>\mathbf{x}</math> trajectories are sure to move toward the sliding mode surface where <math>\sigma( \mathbf{x} ) = \mathbf{0}</math>. Moreover, if the reachability conditions from Theorem 1 are satisfied, the sliding mode will enter the region where <math>\dot{V}</math> is more strongly bounded away from zero in finite time. Hence, the sliding mode <math>\sigma = 0</math> will be attained in finite time.

===Theorem 3: Sliding motion===
Let
:<math> \frac{\partial \sigma}{\partial{\mathbf{x}}} B(\mathbf{x},t) </math>
be [[Algebraic curve#Singularities|nonsingular]]. That is, the system has a kind of [[controllability]] that ensures that there is always a control that can move a trajectory to move closer to the sliding mode. Then, once the sliding mode where <math> \sigma(\mathbf{x}) = \mathbf{0} </math> is achieved, the system will stay on that sliding mode. Along sliding mode trajectories, <math>\sigma(\mathbf{x})</math> is constant, and so sliding mode trajectories are described by the differential equation
:<math>\dot{\sigma} = \mathbf{0}</math>.
If an <math>\mathbf{x}</math>-[[stationary point|equilibrium]] is [[Lyapunov stability|stable]] with respect to this differential equation, then the system will slide along the sliding mode surface toward the equilibrium.

The ''equivalent control law'' on the sliding mode can be found by solving
:<math> \dot\sigma(\mathbf{x})=0 </math>
for the equivalent control law <math>\mathbf{u}(\mathbf{x})</math>. That is,
:<math>
\frac{\partial \sigma}{\partial \mathbf{x}} \overbrace{\left( f(\mathbf{x},t) + B(\mathbf{x},t) \mathbf{u} \right)}^{\dot{\mathbf{x}}} = \mathbf{0}</math>
and so the equivalent control
:<math>\mathbf{u} = -\left( \frac{\partial \sigma}{\partial \mathbf{x}} B(\mathbf{x},t) \right)^{-1} \frac{\partial \sigma}{\partial \mathbf{x}} f(\mathbf{x},t)</math>
That is, even though the actual control <math>\mathbf{u}</math> is not [[continuous function|continuous]], the rapid switching across the sliding mode where <math>\sigma(\mathbf{x})=\mathbf{0}</math> forces the system to ''act'' as if it were driven by this continuous control.

Likewise, the system trajectories on the sliding mode behave as if
:<math>\dot{\mathbf{x}} = \overbrace{f(\mathbf{x},t) - B(\mathbf{x},t) \left( \frac{\partial \sigma}{\partial \mathbf{x}} B(\mathbf{x},t) \right)^{-1} \frac{\partial \sigma}{\partial \mathbf{x}} f(\mathbf{x},t)}^{f(\mathbf{x},t) + B(\mathbf{x},t) u} = f(\mathbf{x},t)\left( \mathbf{I} - B(\mathbf{x},t) \left( \frac{\partial \sigma}{\partial \mathbf{x}} B(\mathbf{x},t) \right)^{-1} \frac{\partial \sigma}{\partial \mathbf{x}} \right)</math>
The resulting system matches the sliding mode differential equation
:<math>\dot{\sigma}(\mathbf{x}) = \mathbf{0}</math>
, the sliding mode surface <math>\sigma(\mathbf{x})=\mathbf{0}</math>, and the trajectory conditions from the reaching phase now reduce to the above derived simpler condition. Hence, the system can be assumed to follow the simpler <math>\dot{\sigma} = 0</math> condition after some initial transient during the period while the system finds the sliding mode. The same motion is approximately maintained when the equality <math> \sigma(\mathbf{x}) = \mathbf{0} </math> only approximately holds.

It follows from these theorems that the sliding motion is invariant (i.e., insensitive) to sufficiently small disturbances entering the system through the control channel. That is, as long as the control is large enough to ensure that <math>\sigma^\intercal \dot{\sigma} < 0</math> and <math>\dot{\sigma}</math> is uniformly bounded away from zero, the sliding mode will be maintained as if there was no disturbance. The invariance property of sliding mode control to certain disturbances and model uncertainties is its most attractive feature; it is strongly [[Robust control|robust]].

As discussed in an example below, a sliding mode control law can keep the constraint
:<math> \dot{x} + x = 0 </math>
in order to asymptotically stabilize any system of the form
:<math> \ddot{x}=a(t,x,\dot{x}) + u</math>
when <math>a(\cdot)</math> has a finite upper bound. In this case, the sliding mode is where
:<math>\dot{x} = -x</math>
(i.e., where <math>\dot{x}+x=0</math>). That is, when the system is constrained this way, it behaves like a simple [[BIBO stability|stable]] [[linear system]], and so it has a globally exponentially stable equilibrium at the <math>(x,\dot{x})=(0,0)</math> origin.