INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2010; 20:1836–1851 Published online 21 December 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1551 Control of near-grazing dynamics and discontinuity-induced bifurcations in piecewise-smooth dynamical systems Sambit Misra and Harry Dankowicz ∗, † Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. SUMMARY This paper develops a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. In particular, it is shown that a discrete-in-time linear feedback correction to a parameter governing a state-space discontinuity surface can suppress discontinuity-induced fold bifurcations of limit cycles that achieve near-tangential intersections with the discontinuity surface. The methodology ensures a persistent branch of limit cycles over an interval of parameter values near the critical condition of tangential contact that is an order of magnitude larger than that in the absence of control. The theoretical treatment is illustrated with a harmonically excited damped harmonic oscillator with a piecewise-linear spring stiffness as well as with a piecewise-nonlinear model of a capacitively excited mechanical oscillator. Copyright 2009 John Wiley & Sons, Ltd. Received 7 April 2009; Revised 17 August 2009; Accepted 4 November 2009 KEY WORDS: discontinuity-induced bifurcations; delay feedback control; piecewise-smooth dynamical systems; grazing bifurcations 1. INTRODUCTION Piecewise-smooth dynamical systems are widely encountered in many engineering examples, including systems incorporating multibody dynamics, friction, contact, impact, hysteresis, saturation, and chatter. These systems are characterized by continuous dynamics governed by a collection of vector fields, which are chosen based on discrete logic events at ∗ Correspondence to: Harry Dankowicz, Department of Mechan- ical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. † E-mail: danko@illinois.edu Contract/grant sponsor: National Science Foundation; contract/grant numbers: 0510044, 0619028 specific locations in time and/or space and resulting in controlled or autonomous jumps in the vector field. Examples of such systems include switching circuits in power electronics, constrained robotic systems (e.g. bipedal robots), vehicle suspension systems, models of tapping-mode atomic-force microscopes, hard-disk- drive systems, and various micro-sensors and actuators (e.g. [1]). These systems fall under the general cate- gory of systems known as hybrid dynamical systems, which also include dynamical systems with logic interruptions, discrete-event dynamics, and systems with switching controllers. There have been several recent advances in understanding the controllability, stabilizability, and global dynamics of hybrid dynam- ical systems related to either the effect of switching between a set of controllers or piecewise-smooth vector fields. Copyright 2009 John Wiley & Sons, Ltd.