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:in a neighborhood of σ=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 σ=0 is reachable is given by:

Theorem 3: sliding motion

Let :$frac\{partialsigma\}\{partial\{xB$ 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 $dot\{x\}\; +\; x\; =\; 0$ in order to asymptotically stabilize any system of the form$ddot\{x\}=a(t,x,dot\{x\})+u$.

Control design

Consider a plant with single input. The sliding surface $sigma(x)=0$ 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 $ddot\{x\}=a(t,x,dot\{x\})+u$, and let $sup|a|\; leq\; k$. Then it is asymptotically stabilized by means of the control $u\; =\; -(|dot\{x\}|+k+1)sign(dot\{x\}+x)$.

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