LIMIT CYCLES IN FEEDBACK CONTROL SYSTEMS WITH HYSTERESIS Alberto Cavallo ∗ Giuseppe De Maria ∗ Ciro Natale ∗ ∗ Seconda Universit` a degli Studi di Napoli, Dip. di Ingegneria dell’Informazione, via Roma 29, 81031 Aversa, Italy Abstract: The paper presents a theoretical and numerical study of the existence problem of self-sustained periodic solutions in feedback control systems including a hysteresis nonlinearity with non-local memory. The presence of this type of nonlinearity in the feedback control system can cause undesired effects, e.g. limit cycles, instability, and thus must be taken into account in the design phase of the control system. In the present work the problem is addressed by resorting to functional analysis tools and to the Preisach operator theory.Copyright c  2005 IFAC Keywords: Hysteresis, Limit cycles, Feedback control, Preisach operators 1. INTRODUCTION In traditional control systems, the problem of foreseeing limit cycles has been usually tackled by modelling the nonlinearities of the components, e.g. the saturation of the power amplifier, or the backlash of a mechanical transmission, with a memory-less function, or with simple nonlinear models with memory, e.g. ideal relay with hystere- sis. In modern control systems the major source of nonlinearity can be imputed to the actuator when this device uses a so-called “smart mate- rial”, e.g. magnetostrictive materials, electroac- tive polymers, piezoelectric ceramics and shape memory alloys. The principal use of such materials is the development of actuators and sensors for a number of different applications in, e.g. aero- nautics both for fixed-wing and rotary-wing, naval and aerospace engineering, MEMS and nanoscale manufacturing, micropositioning systems, medical sensors. In this paper, the attention is focused on the problem of foreseeing limit cycles in control systems employing actuators based on Terfenol- D magnetostrictive materials (see (?)), which ex- hibit a strong hysteretic behaviour due to their magnetic nature caused by the loss phenomena taking place inside the active material. The prob- lem of modelling such a behaviour can be car- ried out either on a physical ground, describing processes on a mesoscopic scale (?) or, from the phenomenological viewpoint, by defining mathe- matical operators with memory able to describe input/output relationships of systems with hys- teresis (?). The existence problem of periodic solutions in a closed-loop system containing a hysteretic com- ponent has been tackled by resorting to different mathematical approaches, i.e. the ideal relay with hysteresis (?), the Poincar´ e maps, the harmonic balance (?), and, more in general frequency meth- ods (?). The relay hysteresis model is simple and describes the main characteristic of the oscillation phenomenon due to hysteresis. Also, more gen- eral relay models are adopted to study periodic oscillations in (?), where topological methods are applied. However, those models cannot accurately reconstruct real hysteretic behaviours, due to dis- continuous outputs or local memory mechanism. In order to overcome these limitations, in the present paper, the Preisach operator (?) is adopted to model the hysteresis, and the existence of a limit cycle is theoretically discussed by resorting to a suitable modification of the describing func- tion method (?; ?). Specifically, owing to the Lips-