Editing Sliding mode control (section)

===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.