Sliding mode control

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{dV(sigma)}{dt}=sigma^Tdot{sigma};<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;=;{x:sigma^T(x)dot{sigma}(x);<0;forall t}

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::u(x,t)=left{egin{matrix} u^+(x), & mbox{for};sigma;>0 \ u^-(x),& mbox{for};sigma;<0end{matrix} ight.

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).


*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 = [ Utkin]
first = V.I.
title = "Sliding Modes in Control and Optimization"
publisher = Springer-Verlag
date = 1992
pages =
isbn = 9780387535166

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