Order reduction of nonlinear quasi-periodic systems subjected to external excitations

[Publication]

Abstract

This paper presents order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov-Perron (L-P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P Transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L-P Transformation. This approach is similar to the Lyapunov-Floquet (L-F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.

Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems’ essential dynamics. This work presents reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations. The methods proposed here use the L-P Transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.

The ‘quasi-periodic invariant manifold’ based technique yields ‘reducibility conditions’. These conditions (referred to in the perturbation literature as resonance conditions) help us understand the system’s various types of resonant interactions. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one must preserve all these ‘resonant’ states in the reduced-order model. Thus, if the ‘reducibility conditions’ are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.