Comments on "A new controller design for a flexible one link manipulator"

In the original paper of W.T. Qian and C.C.H. Ma (ibid., vol. 37, p. 132-7, 1992), a variable structure sliding mode controller (VSSMC) design for the tip position control of a flexible one-link manipulator has been presented, where a switching line constructed from the tip position and its derivative was employed for the design. The claim was that if the slope of this line is chosen positive and the system variables are made to stay on this line, they will converge to zero exponentially, thus yielding a stable system in sliding mode (SM). The purpose of this comment is to show that the choice of a positive constant as the slope for this switching line will not guarantee the stability of the system in SM because, in view of the functional relationship of the tip position with the generalized coordinates of the system through the mode shape functions, what is presented as a switching line is in fact a switching hypersurface. Hence, the stability of the system in SM is guaranteed only if the motion on this hypersurface is asymptotically stable. A positive value for the slope of the switching line employed by Qian and Ma will not guarantee this stability because variations in the mode shape functions due to varying payload conditions or other disturbances at the tip will lead to a varying switching surface. These variations can be such that the resulting sliding motion becomes unstable. Also, the varying switching surface implies that the controller will fail to maintain sliding mode motion.