On the existence of linear sliding surfaces for a class of uncertain dynamic systems with mismatched uncertainties

Recently, Choi has considered the problem of designing a variable structure control (VSC) law for uncertain systems where the uncertainty in the state model does not satisfy the matching condition. He has given a sufficient condition for the existence of linear sliding surfaces in terms of LMIs. In this note, we show that the LMI existence condition given in the above paper is a necessary and sufficient one. And, we give LMI existence conditions of a sliding surface guaranteeing general LMI regional pole placement constraints while in the above paper pole placement only in a vertical half-plane and a disk has been considered. Finally, we give simulation results for controlling an inverted pendulum mounted on a cart.