- Sliding mode control
control theory, sliding mode control is a type of variable structure controlwhere the dynamics of a nonlinear system is altered via application of a high-frequency switching control. This is a state feedbackcontrol scheme where the feedback is not a continuous function of time.
This control scheme involves following two steps:
# selection of a
hypersurfaceor 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-feedbacksliding 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 : be nonsingular.Then, when in the sliding mode , the system trajectories satisfy the original system equation with the control replaced by its "equivalent" value found from the equation .
The same motion is approximately maintained, provided the equality 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 in order to asymptotically stabilize any system of the form.
Consider a plant with single input. The sliding surface 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 , and let . Then it is asymptotically stabilized by means of the control .
last = Filippov
first = A.F.
title = Differential Equations with Discontinuous Right-hand Sides
publisher = Kluwer
date = 1988
isbn = 9789027726995
last = [http://www.ece.osu.edu/~utkin/ Utkin]
first = V.I.
title = "Sliding Modes in Control and Optimization"
publisher = Springer-Verlag
date = 1992
isbn = 9780387535166
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