On the Sufficiency Condition for Existence of a Stable Sliding Mode

In this paper we study sliding mode controllers, in particular those intended for tracking. It is well known that a sliding motion occurs if s
_i
s
_i
< 0 in the neighborhood of the sliding surface s
_i
= 0. Our studies appear to show that the guarantee of "stability" on the sliding manifold is quite delicate. Only simulation evidence was available at the time of writing this paper. We also give new results on sliding mode controllers by requiring s
_i
= -k
_i
sgn s
_i
where s
_i
is the i-th switching surface and k
_i
an appropriately chosen positive parameter. This choice makes the synthesis of sliding mode controllers quite simple. Several examples will be given to illustrate our findings on a potential instability and the efficacy of the latter approach.