Sliding mode control

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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.

Contents

[edit] Control scheme

This control scheme involves following two steps:

  1. selection of a hypersurface or a manifold such that the system trajectory exhibits desirable behaviour when confined to this manifold.
  2. 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

\dot{x}(t)=f(x,t)+B(x,t)u(t),\quad x\in R^n, B\in R^{n\times m}
(A1)\,

For existence and uniqueness of solution of above equation, assume that the functions f(.,.) and B(.,.) are continuous and sufficiently smooth.

The sliding surface is of dimension (n-m) given by

\sigma(x)=[\sigma_1(x),\ldots,\sigma_m(x)]^T=0,\quad \sigma(x) \in R^{m}
(A2)\,

The σ(x) is called switching function. Then the vital part of VSC (Variable Structure Control) design is to choose a control law so that the sliding mode exists and is reachable along σ=0.

The principle of sliding mode control is to forcibly constrain the system, by suitable control strategy, to stay on the sliding surface on which the system will exhibit desirable features. When the system is constrained by the sliding control to stay on the sliding surface, the system dynamics are governed by reduced order system obtained from (A2) as will be explained later.

To force the system states to satisfy σ=0, one must ensure that the system is capable of reaching the state σ=0 from any initial condition and, having reached σ=0, that the control action is capable of maintaining the system at σ=0. Note that the closed-loop system has no solutions in the traditional sense. Thus the solutions are to be understood in the Filippov sense.

[edit] Theoretical foundation

The following theorems form the foundation of variable structure control.

[edit] Theorem 1: existence of sliding mode and reachability

Consider a Lyapunov function

V(\sigma(x))=\frac{1}{2}\sigma^T(x)\sigma(x)
(A3)\,

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^T\dot{\sigma}\;<0

in a neighborhood of σ=0. This is also a condition for reachability.

[edit] 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\}


[edit] Theorem 3: sliding motion

Let : \frac{\partial\sigma}{\partial{x}}B be nonsingular. Then, when in the sliding mode σ = 0, the system trajectories satisfy the original system equation with the control replaced by its equivalent value found from the equation \dot\sigma=0 .

The same motion is approximately maintained, provided the equality σ = 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 .

[edit] Control design

Consider a plant with single input. The sliding surface σ(x) = 0 is defined as follows:

\sigma(x)\;=\;s_1x_1+s_2x_2+\ldots+s_{n-1}x_{n-1}+x_n
(A4)\,

Taking the derivative of Lyapunov function in (A3), we have

\begin{matrix}\dot{V}&=&\sigma(x)^T\dot{\sigma}(x)\\ &=&\sigma(x)^T\frac{\partial{\sigma(x)}}{\partial{x}}\dot{x} \\ &=& \sigma(x)^T\frac{\partial{\sigma(x)}}{\partial{x}}(f(x,t)x+B(x,t)u) \end{matrix}
(A5)\,

Now 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\{\begin{matrix} u^+(x), & \mbox{for}\;\sigma\;>0 \\ u^-(x),& \mbox{for}\;\sigma\;<0\end{matrix}\right.

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).

[edit] References

  • Filippov, A.F. (1988). Differential Equations with Discontinuous Right-hand Sides. Kluwer. ISBN 9789027726995. 
  • Utkin, V.I. (1992). "Sliding Modes in Control and Optimization". Springer-Verlag. ISBN 9780387535166.