By Xingjian Jing, Ziqiang Lang
This publication is a scientific precis of a few new advances within the quarter of nonlinear research and layout within the frequency area, targeting the appliance orientated idea and techniques in accordance with the GFRF proposal, that's almost always performed via the writer long ago eight years. the most effects are formulated uniformly with a parametric attribute strategy, which supplies a handy and novel perception into nonlinear impact on process output reaction when it comes to attribute parameters and therefore facilitate nonlinear research and layout within the frequency area. The ebook starts off with a quick advent to the history of nonlinear research within the frequency area, by way of recursive algorithms for computation of GFRFs for various parametric types, and nonlinear output frequency houses. Thereafter the parametric attribute research technique is brought, which results in the recent knowing and formula of the GFRFs, and nonlinear attribute output spectrum (nCOS) and the nCOS established research and layout process. in response to the parametric attribute process, nonlinear effect within the frequency area could be investigated with a singular perception, i.e., alternating sequence, that's through a few software leads to vibration keep an eye on. value bounds of frequency reaction capabilities of nonlinear platforms can be studied with a parametric attribute strategy, which bring about novel parametric convergence standards for any given parametric nonlinear version whose input-output courting permits a convergent Volterra sequence enlargement. This e-book pursuits these readers who're operating within the components with regards to nonlinear research and layout, nonlinear sign processing, nonlinear approach id, nonlinear vibration keep an eye on, etc. It fairly serves as an outstanding reference should you are learning frequency area tools for nonlinear systems.
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Extra info for Frequency Domain Analysis and Design of Nonlinear Systems based on Volterra Series Expansion: A Parametric Characteristic Approach
For example, 2 The Generalized Frequency Response Functions and Output Spectrum of. . 37). 2 Consider a nonlinear mechanical system shown in Fig. 1. _ 3x_ 3, and The output property of the spring satisfies A¼kx, the damper F¼a1x+a 2 the active unit is described by F¼a2x_ . u(t) is the external input force. 1. 38a,b). u(t) m A Fig. 4 23 The Frequency Response Functions for Block-Oriented Nonlinear Systems Block-oriented nonlinear systems such as Hammerstein and Wiener models are composed by a cascade combination of a linear dynamic model and a static (memoryless) nonlinear function.
Represents a natural characterand the function operator ωk1 þÁÁÁþωkn ¼ω ωk1 þÁÁÁþωkn ¼ω istic of the system which cannot be designed as mentioned. The first term Ln(jω) represents the influence from the linear part of the system and the second term represents the nonlinear influence from input nonlinearities. These two terms can be designed purposely in practice. 2 provide a useful insight into the design of input nonlinearities to achieve a specific output spectrum in practice. 2. 2 are satisfied for two nonlinear terms but they are not opposite at a frequency when c0,n(l1, Á Á Á, ln) ¼ c1 0 0 and c0,n(l1 , Á Á Á, ln ) ¼ c2, then they must be opposite at this frequency when 0 0 c0,n(l1, Á Á Á, ln) ¼ c1 and c0,n(l1 , Á Á Á, ln ) ¼ À sgn1(c2)c3 for a proper value of c3.
Consider the following SIDO NARX system, xðtÞ ¼ M1 X m K X X cp, mÀp ðk1 ; Á Á Á; km Þ m¼1 p¼0 k1 , km ¼0 yðtÞ ¼ M2 X m K X X m¼1 p¼0 k1 , km ¼0 p Y xðt À ki Þ p Y i¼1 uðt À ki Þ ð2:28aÞ uð t À k i Þ ð2:28bÞ i¼pþ1 i¼1 e c p, mÀp ðk1 ; Á Á Á; km Þ m Y x ðt À k i Þ m Y i¼pþ1 where M1, M2 and K are all positive integers, and x(t), y(t), u(t) 2 ℝ. 28b) represents the system output which is a nonlinear function of state x(t) and input u(t) in a general polynomial form. 28a,b), an alternative simple method would be adopted here, since the model structure and nonlinear types are known clearly.