Analysis of Proportional—Derivative and Nonlinear Control of Mechanical Systems with Dynamic Backlash

Abstract

In this paper, we focus on the analysis and control of a simple rigid-body mechanical system with clearance. Contrary to most of the existing works in the literature concerning control, we explicitly treat all the nonlinear non-smooth characteristics of this system considered as a rigid-body mechanical system with unilateral constraints and impacts (dynamic backlash). The model is therefore a hybrid dynamical system, mixing discrete events as well as continuous states. The regulation and tracking capabilities of the proportional—derivative (PD) scheme are investigated. In particular, a complete proof of the existence of a limit cycle for non-collocated PD control is provided, including viability constraints. It is concluded that tracking requires the development of specific control schemes. Consequently, we propose a hybrid control that may be used to track some desired trajectories in conjunction with a PD input. Throughout the paper, the particular features of unilaterally constrained mechanical systems are taken into account, such as the fundamental viability property of closed-loop solutions and controls. This work is a new approach to be considered for application in several areas including the control of kinematic chains with joint clearance and vibro-impact systems, as well as liquid slosh control. Numerical results are presented to illustrate the possible performance of the proposed control scheme and its robustness properties.