MEASURING THE HIGHER ORDER SINUSOIDAL INPUT DESCRIBING FUNCTIONS OF A NON-LINEAR PLANT OPERATING IN FEEDBACK Pieter Nuij 1 , Maarten Steinbuch, Okko Bosgra Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands. Fax: +31 40 2461418. Abstract: In this paper two measuring techniques are presented for measuring the Higher Order Sinusoidal Input Describing Functions (HOSIDF) of a non-linear plant operating in feedback. The HOSIDF relate the magnitude and phase of the harmonics of the periodic response of a non-linear system to the magnitude and phase of the sinusoidal excitation. In a controlled system the harmonics generated by the non-linear system will be fed back to the input changing the sinusoidal excitation into a harmonic excitation. The first method applies linear techniques to compensate the bias caused by the harmonic components in the excitation signal. The second method uses a modified repetitive control scheme to suppress the harmonic components in the excitation signal. The effectiveness of both methods is tested in simulations of a mass subjected to Coulomb friction, Stribeck-effect and hysteresis in the pre-sliding regime. The friction forces are modeled with the modified Leuven friction model. The results are compared with the HOSIDF simulated under open loop condition. Both methods prove able to produce reliable results. Keywords: Frequency domain analysis, non-linear systems, describing function, repetitive control, periodic disturbances, closed loop identification, friction modeling, machine condition monitoring. 1. INTRODUCTION In many high precision positioning systems, posi- tion accuracy is a key performance objective. Dur- ing the last three decades, accuracy requirements have changed from the micrometer range to the sub micron and even nanometer range. Examples of high precision systems are wafer scanners for lithographic applications, laser beam recorders for CD/DVD mastering, lathes for the production of optical components like contact lenses, electron 1 Corresponding author. E-mail : p.w.j.m.nuij@tue.nl microscopes and coordinate-measuring machines. But also in consumer products, subcomponents like hard disc drives and optical storage devices can not function without extreme positioning ac- curacy. The operating conditions under which these products have to perform are changing from laboratory environment to an environment as hos- tile as a car on an unmetalled road. This poses ex- tra challenges to both the mechanical and the con- trol design. For both the mechanical and the con- trol aspects, this can only be achieved by the use of advanced design techniques, often necessitating the incorporation of non-linear phenomena in the