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.

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} \quad (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^{(n-m)} \quad (A2)

The σ(x) is called switching function. Then the vital part of VSC 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.

These conditions are stated in the form of following theorems.

Theorem 1(condition of existence of sliding mode and reachability)
Consider a Lyapunov function

V(\sigma(x))=\frac{1}{2}\sigma^T(x)\sigma(x)\quad \quad(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.

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)\sigma(x)\;<0\;\forall t\}

These two theorems form the foundation of variable structure control.

[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 \quad (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)}}{x}(f(x,t)x+g(x,t)u) \end{matrix} \quad \quad (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.

    Filippov A.F. "Differential equations with discontinuous right-hand sides", Kluwer 1988

    Utkin V.I. "Sliding modes in control and optimization", Springer 1992

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