This control scheme involves following two steps:
Consider a NL system described by
The sliding surface is of dimension (n-m) given by
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.
These conditions are stated in the form of following theorems.
;Theorem 1(condition of existence of sliding mode and reachability): Consider a Lyapunov function
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
;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
in a neighborhood of σ=0. This is also a condition for reachability.
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 have
Now the control input u(t) is so chosen that time derivative of V is negative definite. The control input is chosen as follows:
to be continued.....