Parametric convergence bound of volterra series expansion and applications using nCOS-based analysis and design

Abstract

The Volterra series based nonlinear analysis and design methodology is a powerful tool that has been applied to various engineering practices. This study addresses several key issues of the Volterra series based methodology that have not been well developed in the literature, including its convergence, applications, and extensions. Two novel concepts, i.e., the parametric bound of convergence (PBoC) and parametric convergence margin (PCM), are proposed for nonlinear systems described by nonlinear auto-regressive with exogenous input (NARX) models. The proposed PBoC can calculate the convergence bound not only for the input magnitude, but also for the parameters of interest. The PCM is developed as a quantitative assessment to examine the distance from a given nonlinear system to the bound of a convergent Volterra series expansion. By applying the theoretical results above, the nonlinear characteristic output spectrum (nCOS) function can be well analysed and designed within a certain region of nonlinear parameters of interest. A nonlinear damping is proposed to overcome the well-known dilemma with respect to linear damping. The performance of the nonlinear damping is derived with the nCOS method, which also provides a straightforward and effective way to tackle the multiple-object nonlinear optimization problem. Linear components or linear controllers are usually easier to implement in practice, and are thus of considerable interest for analysis and design to achieve a better performance when simultaneously considering a system that is inherently nonlinear. The existing nCOS method is only available for nonlinear parameters, and thus is extended to those linear parameters of interest. A symbolic algorithm for calculating the new nCOS function is developed for single-input single-output (SISO) systems. In case that the built symbolic algorithm is complicated for MIMO systems, a numerical identification method is developed. The results above are established for nonlinear systems with polynomial nonlinearity. For those nonlinear systems with exponential-type nonlinearity, there would be too many parameters in the analysis and design because exponential nonlinearity is usually approximated by Taylor series expansions. An efficient algorithm with many fewer parameters for calculating the generalized frequency response function (GFRF) in the nonlinear analysis and design is then developed. The contributions of this thesis lie in the following points. The results of PBoC and PCM are notable extensions of those convergence results in the literature, and can provide a more straightforward and useful guidance for the parameter design or feedback design of nonlinear systems via the nCOS method. The new nCOS function can provide a straightforward understanding of the effect of the linear parameters of interest on the nonlinear output spectrum and thereby greatly facilitate the analysis and design of linear components or controllers for nonlinear systems. The extension of the nCOS method to exponential-type nonlinear system will considerably ease the analysis and design of systems with exponential nonlinearity, such as amplifier circuits and neural networks, in the frequency domain.