# Sliding mode control

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Sliding mode control

In control theory, sliding mode control is a type of variable structure control where the dynamics of a nonlinear system is altered via application of a high-frequency switching control. This is a state feedback control scheme where the feedback is not a continuous function of time.

Control scheme

This control scheme involves following two steps:
# selection of a hypersurface or a manifold such that the system trajectory exhibits desirable behaviour when confined to this manifold.
# Finding feed-back gains so that the system trajectory intersects and stays on the manifold.We will consider only state-feedback sliding mode control.

Consider a nonlinear system described by
For the system given by (A1), and the sliding surface given by (A2), a sufficient condition for the existence of a sliding mode is that:$frac\left\{dV\left(sigma\right)\right\}\left\{dt\right\}=sigma^Tdot\left\{sigma\right\};<0$in a neighborhood of &sigma;=0. This is also a condition for reachability.

Theorem 2: region of attraction

For the system given by (A1) and sliding surface given by (A2), the subspace for which &sigma;=0 is reachable is given by:$sigma;=;\left\{x:sigma^T\left(x\right)dot\left\{sigma\right\}\left(x\right);<0;forall t\right\}$

Theorem 3: sliding motion

Let :$frac\left\{partialsigma\right\}\left\{partial\left\{xB$ be nonsingular.Then, when in the sliding mode $sigma = 0$, the system trajectories satisfy the original system equation with the control replaced by its "equivalent" value found from the equation $dotsigma=0$.

The same motion is approximately maintained, provided the equality $sigma = 0$ only approximately holds.

It follows from Theorem 3 that the sliding motion is completely insensitive to any disturbances entering the system through the control channel. This establishes the most attractive sliding mode feature - its insensitivity to certain disturbances and model uncertainties. In particular, it is enough to keep the constraint $dot\left\{x\right\} + x = 0$ in order to asymptotically stabilize any system of the form$ddot\left\{x\right\}=a\left(t,x,dot\left\{x\right\}\right)+u$.

Control design

Consider a plant with single input. The sliding surface $sigma\left(x\right)=0$ is defined as follows:Taking the derivative of Lyapunov function in (A3), we haveNow the control input u(t) is so chosen that time derivative of V is negative definite. The control input is chosen as follows::

Consider once more the dynamic system $ddot\left\{x\right\}=a\left(t,x,dot\left\{x\right\}\right)+u$, and let $sup|a| leq k$. Then it is asymptotically stabilized by means of the control $u = -\left(|dot\left\{x\right\}|+k+1\right)sign\left(dot\left\{x\right\}+x\right)$.

References

*cite book
last = Filippov
first = A.F.
title = Differential Equations with Discontinuous Right-hand Sides
publisher = Kluwer
date = 1988
pages =
isbn = 9789027726995

*cite book
last = [http://www.ece.osu.edu/~utkin/ Utkin]
first = V.I.
title = "Sliding Modes in Control and Optimization"
publisher = Springer-Verlag
date = 1992
pages =
isbn = 9780387535166

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