- Sliding mode control
In

control theory ,**sliding mode control**is a type ofvariable structure control where the dynamics of a nonlinear system is altered via application of a high-frequency switching control. This is a statefeedback control scheme where the feedback is not a continuous function of time.**Control scheme**This control scheme involves following two steps:

# selection of ahypersurface 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 onlystate-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 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:$sigma;=;\{x:sigma^T(x)dot\{sigma\}(x);<0;forall\; t\}$

**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::$u(x,t)=left\{egin\{matrix\}\; u^+(x),\; mbox\{for\};sigma;0\; \backslash \; u^-(x),\; mbox\{for\};sigma;0end\{matrix\}\; ight.$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 $u\; =\; -(|dot\{x\}|+k+1)sign(dot\{x\}+x)$.

**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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