Analysis and Design of Nonlinear Systems in the Frequency Domain

Abstract

Nonlinear system analyses have been widely applied in engineering practice, where the frequency domain approaches have been developed to satisfy the requirement of the analysis and design of nonlinear systems. However, there exist many problems with current techniques including the challenges with the nonlinear system representation using physically meaningful models, and difficulties with the evaluation of the frequency properties of nonlinear systems. In the present work, some new approaches, that have potential to be used to systematically address these problems, are developed based on the NDE (Nonlinear Differential Equation) model and the NARX (Nonlinear Auto Regressive with eXegenous input) model of nonlinear systems. In this thesis, the background of the frequency domain analysis and design of nonlinear systems is introduced in Chapter 1, and the existing approaches are reviewed in Chapter 2. In general, the frequency analysis of nonlinear systems is conducted based on the Volterra series representation of nonlinear systems, and as basic issues, the evaluation of the Volterra series representation and its convergence are discussed in Chapters 3 and 4, respectively. An extension of the existing frequency analysis and design techniques is discussed in Chapter 5 to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems. An experimental study is conducted in Chapter 6 to show how a nonlinear component can benefit the engineering system, such to emphasis the significance of developing the analysis and design approaches of nonlinear systems. The main contributions are summarized as below. (1) The GALEs is proposed that can accurately evaluate the system Volterra series representation. By using the GALEs, the solution to the NDE model or the NARX model of nonlinear systems can be obtained by simply dealing with a series of linear differential or difference equations, which can facilitate a wide range of nonlinear system analyses and associated practical applications. (2) A new criterion is derived to determine the convergence of the Volterra series representation of nonlinear systems described by a NARX model. The analysis is performed based on a new function known as Generalized Output Bound Characteristic Function (GOBCF), which is defined in terms of the input, output and parameters of the NARX model of nonlinear systems. Compared to the existing results, the new criterion provides a much more rigorous and effective approach to the analysis of the convergence conditions and properties of the Volterra series representation of nonlinear systems. (3) The Output Frequency Response Function (OFRF) in terms of physical parameters of concern is introduced for the NARX Model with parameters of interest for Design (NARX-M-for-D). Moreover, a new concept known as the Associated Output Frequency Response Function (AOFRF) is introduced to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems. (4) Nonlinear damping can achieve desired isolation performance of a system over both low and high frequency regions and the optimal nonlinear damping force can be realized by closed loop controlled semi-active dampers. Both simulation and laboratory experiments are studied, demonstrating the advantages of the proposed nonlinear damping technologies over both traditional linear damping and more advanced Linear-Quadratic Gaussian (LQG) feedback control which have been used in practice to address building isolation system design and implementation problems.