Editing Sliding mode control (section)

==Control scheme==
Consider a [[nonlinear system|nonlinear dynamical system]] described by

{| border="0" width="75%"
|-
| align="left" |
{{NumBlk|:|<math>
\dot{\mathbf{x}}(t)=f(\mathbf{x},t) + B(\mathbf{x},t)\,\mathbf{u}(t)
</math>
|  {{EquationRef|1}}}}
|}
where
:<math>\mathbf{x}(t) \triangleq \begin{bmatrix}x_1(t)\\x_2(t)\\\vdots\\x_{n-1}(t)\\x_n(t)\end{bmatrix} \in \mathbb{R}^n</math>
is an {{mvar|n}}-dimensional [[state space (controls)|state]] [[column vector|vector]] and
:<math>\mathbf{u}(t) \triangleq \begin{bmatrix}u_1(t)\\u_2(t)\\\vdots\\u_{m-1}(t)\\u_m(t)\end{bmatrix} \in \mathbb{R}^m</math>
is an {{mvar|m}}-dimensional input vector that will be used for state [[feedback]]. The [[Function (mathematics)|function]]s <math>f: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n</math> and <math>B: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^{n \times m}</math> are assumed to be [[continuous function|continuous]] and sufficiently [[smooth function|smooth]] so that the [[Picard–Lindelöf theorem]] can be used to guarantee that solution <math>\mathbf{x}(t)</math> to Equation&nbsp;({{EquationNote|1}}) [[existence|exists]] and is [[unique (mathematics)|unique]].

A common task is to design a state-feedback [[control systems|control law]] <math>\mathbf{u}(\mathbf{x}(t))</math> (i.e., a mapping from current state <math>\mathbf{x}(t)</math> at time {{mvar|t}} to the input <math>\mathbf{u}</math>) to [[Lyapunov stability|stabilize]] the [[dynamical system]] in Equation&nbsp;({{EquationNote|1}}) around the [[origin (mathematics)|origin]] <math>\mathbf{x} = [0, 0, \ldots, 0]^\intercal</math>. That is, under the control law, whenever the system is started away from the origin, it will return to it. For example, the component <math>x_1</math> of the state vector <math>\mathbf{x}</math> may represent the difference some output is away from a known signal (e.g., a desirable sinusoidal signal); if the control <math>\mathbf{u}</math> can ensure that <math>x_1</math> quickly returns to <math>x_1 = 0</math>, then the output will track the desired sinusoid. In sliding-mode control, the designer knows that the system behaves desirably (e.g., it has a stable [[stationary point|equilibrium]]) provided that it is constrained to a subspace of its [[Configuration space (physics)|configuration space]]. Sliding mode control forces the system trajectories into this subspace and then holds them there so that they slide along it. This reduced-order subspace is referred to as a ''sliding (hyper)surface'', and when closed-loop feedback forces trajectories to slide along it, it is referred to as a ''sliding mode'' of the closed-loop system. Trajectories along this subspace can be likened to trajectories along eigenvectors (i.e., modes) of [[LTI system]]s; however, the sliding mode is enforced by creasing the vector field with high-gain feedback. Like a marble rolling along a crack, trajectories are confined to the sliding mode.

The sliding-mode control scheme involves
# Selection of a [[hypersurface]] or a manifold (i.e., the sliding surface) such that the system trajectory exhibits desirable behavior when confined to this manifold.
# Finding feedback gains so that the system trajectory intersects and stays on the manifold.
Because sliding mode control laws are not [[continuous function|continuous]], it has the ability to drive trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is better than asymptotic). However, once the trajectories reach the sliding surface, the system takes on the character of the sliding mode (e.g., the origin <math>\mathbf{x}=\mathbf{0}</math> may only have asymptotic stability on this surface).

The sliding-mode designer picks a ''switching function'' <math>\sigma: \mathbb{R}^n \to \mathbb{R}^m</math> that represents a kind of "distance" that the states <math>\mathbf{x}</math> are away from a sliding surface.
* A state <math>\mathbf{x}</math> that is outside of this sliding surface has <math>\sigma(\mathbf{x}) \neq 0</math>.
* A state that is on this sliding surface has <math>\sigma(\mathbf{x}) = 0</math>.
The sliding-mode-control law switches from one state to another based on the ''sign'' of this distance. So the sliding-mode control acts like a stiff pressure always pushing in the direction of the sliding mode where <math>\sigma(\mathbf{x}) = 0</math>.
Desirable <math>\mathbf{x}(t)</math> trajectories will approach the sliding surface, and because the control law is not [[continuous function|continuous]] (i.e., it switches from one state to another as trajectories move across this surface), the surface is reached in finite time. Once a trajectory reaches the surface, it will slide along it and may, for example, move toward the <math>\mathbf{x} = \mathbf{0}</math> origin. So the switching function is like a [[topographic map]] with a contour of constant height along which trajectories are forced to move.

The sliding (hyper)surface/manifold is typically of dimension <math>n-m </math> where {{mvar|n}} is the number of states in <math>\mathbf{x}</math> and {{mvar|m}} is the number of input signals (i.e., control signals) in <math>\mathbf{u}</math>. For each control index <math>1 \leq k \leq m</math>, there is an <math>(n-1)</math>-dimensional sliding surface given by
{| border="0" width="75%"
|-
| align="left" |
{{NumBlk|:|<math>
\left\{
\mathbf{x} \in \mathbb{R}^n : 
\sigma_k(\mathbf{x}) = 0
\right\}
</math>
|  {{EquationRef|2}}}}
|}
The vital part of SMC design is to choose a control law so that the sliding mode (i.e., this surface given by <math>\sigma(\mathbf{x})=\mathbf{0}</math>) exists and is reachable along system trajectories. The principle of sliding mode control is to forcibly constrain the system, by suitable control strategy, to stay on the sliding surface on which the system will exhibit desirable features. When the system is constrained by the sliding control to stay on the sliding surface, the system dynamics are governed by reduced-order system obtained from Equation&nbsp;({{EquationNote|2}}).

To force the system states <math>\mathbf{x}</math> to satisfy <math>\sigma(\mathbf{x}) = \mathbf{0}</math>, one must:
# Ensure that the system is capable of reaching <math>\sigma(\mathbf{x}) = \mathbf{0}</math> from any initial condition
# Having reached <math>\sigma(\mathbf{x})=\mathbf{0}</math>, the control action is capable of maintaining the system at <math>\sigma(\mathbf{x})=\mathbf{0}</math>

===Existence of closed-loop solutions===
Note that because the control law is not [[continuous function|continuous]], it is certainly not locally [[Lipschitz continuous]], and so existence and uniqueness of solutions to the [[Closed-loop transfer function|closed-loop system]] is ''not'' guaranteed by the [[Picard–Lindelöf theorem]]. Thus the solutions are to be understood in the [[Aleksei Fedorovich Filippov|Filippov]] sense.<ref name="Zinober1990"/><ref name="Filippov88">{{Cite book
 | last       = Filippov
 | first      = A.F.
 | title      = Differential Equations with Discontinuous Right-hand Sides
 | publisher  = Kluwer
 | year       = 1988
 | isbn       = 978-90-277-2699-5
}}</ref> Roughly speaking, the resulting closed-loop system moving along <math>\sigma(\mathbf{x}) = \mathbf{0}</math> is approximated by the smooth [[dynamic system|dynamics]] <math>\dot{\sigma}(\mathbf{x}) = \mathbf{0};</math> however, this smooth behavior may not be truly realizable. Similarly, high-speed [[pulse-width modulation]] or [[delta-sigma modulation]] produces outputs that only assume two states, but the effective output swings through a continuous range of motion. These complications can be avoided by using a different [[nonlinear control]] design method that produces a continuous controller. In some cases, sliding-mode control designs can be approximated by other continuous control designs.<ref name="Khalil02"/>