Stability of Discrete Time Systems with Unilateral Nonlinearities

This paper considers the effect of unilateral nonlinearities on the stability of discrete time control systems, a problem of some importance in haptic display. The unilateral nonlinearity is a simple piecewise linear function: ξ(x) = x for x≥0, ξ(x) = 0 for x<0. This function plays an important role in the modeling of collisions, and is a part of essentially all implementations of virtual surfaces. The unilateral nonlinearity is, in part, responsible for the instability often seen when haptic display operators contact virtual surfaces. In this paper, it is shown that an operator contacting a virtual surface via a haptic display can reasonably be modeled as a linear, shift-invariant system (H(z)) in feedback with a unilateral nonlinearity. Conditions for the absence of oscillations in such a system are then derived. The derivation follows a method originally presented by Mitra in which the existence of periodic oscillations is first assumed, then conditions leading to a contradiction are found. This approach i s particularly attractive in that it exploits specific properties of the unilateral nonlinearity. The results developed here are presented graphically in the Nyquist plane, allowing direct comparison to other well-known criteria, such as Tsypkin’s Condition. It is shown that the new criterion is much less conservative than Tsypkin’s Condition.