Calculation of the Quasi-Linear System Matrix for Eigenvalue and Stability Analysis of Nonlinear Dynamic Systems

The method of quasi-linearization of nonlinear dynamic systems or nonlinear control systems is well known in the analysis of periodic bounded oscillations so-called limit cycles. At this, nonlinearities are approximated by the first term of a Fourier series, which are amplitude dependent describing functions for sinusoidal input, yielding a quasi-linear description. The eigenvalues of this quasi-linear system can be investigated to find all limit cycles by means of stability checking. In the presence of dynamic nonlinearities (e.g. hysteresis), the calculation of the quasi-linear system matrix is not quite simple. Concerning applicability and numerical efficiency, several ways are displayed and discussed. They are exemplified with shimmy osciallations of a basic aircraft nose gear model, where one dynamic and three different static nonlinearities are considered