Analysing nonlinear systems with higher order sinusoidal input describing functions

For high precision motion systems, modelling and control design specifically oriented at friction effects is instrumental. The Sinusoidal Input Describing Function theory represents a solid mathematical framework for analysing non-linear system behaviour. This theory however limits the description of the non-linear system behaviour to an approximated linear relation between sinusoidal excitation and sinusoidal response. An extension to Higher Order Describing Functions can be realised by calculating the corresponding Fourier coefficients. The resulting Higher Order Sinusoidal Input Describing Functions (HOSIDFs) relate the magnitude and phase of the higher harmonics of the periodic system response to the magnitude and phase of a sinusoidal excitation. This paper describes two techniques to measure HOSIDFs. The first technique is FFT based. The second technique is based on IQ (=in phase/quadrature phase) demodulation. In a case study both techniques are used to measure the changes in dynamics due to friction as function of drive level in an electric motor.