Editing Sliding mode control (section)

==Control design examples==
* Consider a [[plant (control theory)|plant]] described by Equation&nbsp;({{EquationNote|1}}) with single input {{mvar|u}} (i.e., <math>m = 1</math>). The switching function is ''picked'' to be the linear combination
{| border="0" width="75%"
|-
| align="left" |
{{NumBlk|::|<math>
\sigma(\mathbf{x}) \triangleq s_1 x_1 + s_2 x_2 + \cdots + s_{n-1} x_{n-1} + s_n x_n
</math>
|  {{EquationRef|4}}}}
|}
:where the weight <math>s_i > 0</math> for all <math>1 \leq i \leq n</math>. The sliding surface is the [[simplex]] where <math>\sigma(\mathbf{x})=0</math>. When trajectories are forced to slide along this surface,
::<math>\dot{\sigma}(\mathbf{x}) = 0</math>
:and so
::<math>s_1 \dot{x}_1 + s_2 \dot{x}_2 + \cdots + s_{n-1} \dot{x}_{n-1} + s_n \dot{x}_n = 0</math>
:which is a reduced-order system (i.e., the new system is of order <math>n-1</math> because the system is constrained to this <math>(n-1)</math>-dimensional sliding mode simplex). This surface may have favorable properties (e.g., when the plant dynamics are forced to slide along this surface, they move toward the origin <math>\mathbf{x}=\mathbf{0}</math>). Taking the derivative of the [[Lyapunov function]] in Equation&nbsp;({{EquationNote|3}}), we have
::<math>
\dot{V}(\sigma(\mathbf{x}))
= \overbrace{\sigma(\mathbf{x})^{\text{T}}}^{\tfrac{\partial V}{\partial \mathbf{x}}} \overbrace{\dot{\sigma}(\mathbf{x})}^{\tfrac{\operatorname{d} \sigma}{\operatorname{d} t}}</math>
:To ensure <math>\dot{V} < 0</math>, the feedback control law <math>u(\mathbf{x})</math> must be chosen so that
::<math>\begin{cases}
\dot{\sigma} < 0 &\text{if } \sigma > 0\\
\dot{\sigma} > 0 &\text{if } \sigma < 0
\end{cases}</math>
:Hence, the product <math>\sigma \dot{\sigma} < 0</math> because it is the product of a negative and a positive number. Note that
{| border="0" width="75%"
|-
| align="left" |
{{NumBlk|::|<math>\dot{\sigma}(\mathbf{x})
=  \overbrace{\frac{\partial{\sigma(\mathbf{x})}}{\partial{\mathbf{x}}} \dot{\mathbf{x}}}^{\dot{\sigma}(\mathbf{x})}
= \frac{\partial{\sigma(\mathbf{x})}}{\partial{\mathbf{x}}} 
  \overbrace{\left( f(\mathbf{x},t) + B(\mathbf{x},t) u \right)}^{\dot{\mathbf{x}}}
= \overbrace{[s_1, s_2, \ldots, s_n]}^{\frac{\partial{\sigma(\mathbf{x})}}{\partial{\mathbf{x}}}} 
  \underbrace{\overbrace{\left( f(\mathbf{x},t) + B(\mathbf{x},t) u \right)}^{\dot{\mathbf{x}}}}_{\text{( i.e., an } n \times 1 \text{ vector )}}</math>
|  {{EquationRef|5}}}}
|}
:The control law <math>u(\mathbf{x})</math> is chosen so that
::<math>u(\mathbf{x})
=
\begin{cases}
u^+(\mathbf{x}) &\text{if } \sigma(\mathbf{x}) > 0 \\ 
u^-(\mathbf{x}) &\text{if } \sigma(\mathbf{x}) < 0
\end{cases}</math>
:where
:* <math>u^+(\mathbf{x})</math> is some control (e.g., possibly extreme, like "on" or "forward") that ensures Equation&nbsp;({{EquationNote|5}}) (i.e., <math>\dot{\sigma}</math>) is ''negative'' at <math>\mathbf{x}</math>
:* <math>u^-(\mathbf{x})</math> is some control (e.g., possibly extreme, like "off" or "reverse") that ensures Equation&nbsp;({{EquationNote|5}}) (i.e., <math>\dot{\sigma}</math>) is ''positive'' at <math>\mathbf{x}</math>
:The resulting trajectory should move toward the sliding surface where <math>\sigma(\mathbf{x})=0</math>. Because real systems have delay, sliding mode trajectories often ''chatter'' back and forth along this sliding surface (i.e., the true trajectory may not smoothly follow <math>\sigma(\mathbf{x})=0</math>, but it will always return to the sliding mode after leaving it).

* Consider the [[dynamic system]]
::<math>\ddot{x}=a(t,x,\dot{x})+u</math>
:which can be expressed in a 2-dimensional [[state space (controls)|state space]] (with <math>x_1 = x</math> and <math>x_2 = \dot{x}</math>) as
::<math>
\begin{cases}
\dot{x}_1 = x_2\\
\dot{x}_2 = a(t,x_1,x_2) + u
\end{cases}</math>
:Also assume that <math>\sup\{ |a(\cdot)| \} \leq k</math> (i.e., <math>|a|</math> has a finite upper bound {{mvar|k}} that is known). For this system, choose the switching function
::<math>\sigma(x_1,x_2)= x_1 + x_2 = x + \dot{x}</math>
:By the previous example, we must choose the feedback control law <math>u(x,\dot{x})</math> so that <math>\sigma \dot{\sigma} < 0</math>. Here,
::<math>\dot{\sigma} = \dot{x}_1 + \dot{x}_2 = \dot{x} + \ddot{x} = \dot{x}\,+\,\overbrace{a(t,x,\dot{x})+ u}^{\ddot{x}}</math>
:* When <math>x + \dot{x} < 0</math> (i.e., when <math>\sigma < 0</math>), to make <math>\dot{\sigma} > 0</math>, the control law should be picked so that <math>u > |\dot{x} + a(t,x,\dot{x})|</math>
:* When <math>x + \dot{x} > 0</math> (i.e., when <math>\sigma > 0</math>), to make <math>\dot{\sigma} < 0</math>, the control law should be picked so that <math>u < -|\dot{x} + a(t,x,\dot{x})|</math>
:However, by the [[triangle inequality]],
::<math>|\dot{x}| + |a(t,x,\dot{x})| \geq |\dot{x} + a(t,x,\dot{x})|</math>
:and by the assumption about <math>|a|</math>,
::<math>|\dot{x}| + k + 1 > |\dot{x}| + |a(t,x,\dot{x})|</math>
:So the system can be feedback stabilized (to return to the sliding mode) by means of the control law
::<math>u(x,\dot{x})
=
\begin{cases}
   |\dot{x}| + k + 1 &\text{if } \underbrace{x + \dot{x}} < 0,\\
   -\left(|\dot{x}| + k + 1\right) &\text{if } \overbrace{x + \dot{x}}^{\sigma} > 0
\end{cases}</math>
:which can be expressed in [[closed-form expression|closed form]] as
::<math>u(x,\dot{x}) = -(|\dot{x}|+k+1) \underbrace{\operatorname{sgn}(\overbrace{\dot{x}+x}^{\sigma})}_{\text{(i.e., tests } \sigma > 0 \text{)}}</math>
:Assuming that the system trajectories are forced to move so that <math>\sigma(\mathbf{x})=0</math>, then
::<math>\dot{x} = -x \qquad \text{(i.e., } \sigma(x,\dot{x}) = x + \dot{x} = 0 \text{)}</math>
:So once the system reaches the sliding mode, the system's 2-dimensional dynamics behave like this 1-dimensional system, which has a globally exponentially stable [[stationary point|equilibrium]] at <math>(x,\dot{x})=(0,0)</math>.

===Automated design solutions===
Although various theories exist for sliding mode control system design, there is a lack of a highly effective design methodology due to practical difficulties encountered in analytical and numerical methods. A reusable computing paradigm such as a [[genetic algorithm]] can, however, be utilized to transform a 'unsolvable problem' of optimal design into a practically solvable 'non-deterministic polynomial problem'. This results in computer-automated designs for sliding model control.<ref name="GA_SMC96">
{{cite journal
|last1=Li
|first1=Yun|title=Genetic algorithm automated approach to the design of sliding mode control systems
|journal=International Journal of Control
|year=1996
|volume=64
|issue=3
|pages=721–739
|doi=10.1080/00207179608921865
|url=https://www.researchgate.net/publication/230602763
|display-authors=etal|citeseerx=10.1.1.43.1654}}
</ref>