Output frequency response function-based analysis for nonlinear ...

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Based on the Volterra series expansion, the generalized frequency response function (GFRF) is proposed in George [16]. Thereafter, many researches on the frequency domain analysis of nonlinear Volterra systems are carried out [17–21]. By using this kind of frequency domain analysis method, nonlinear behaviour of a nonlinear Volterra system driven by a general input can be studied through its GFRFs and output spectrum. Therefore, nonlinear characteristics in the frequency domain of a Volterra system can be revealed from a novel and effective perspective. A limitation of this method may lie in that estimation and computation of the GFRFs and Volterra kernels for a nonlinear Volterra system usually involve some complicated computation and symbolic operations [22]. It can be seen that a straightforward analytical expression for the relationship between system time domain model parameters and system output frequency response can considerably facilitate the analysis and design of nonlinear Volterra systems in the frequency domain. For this purpose, a concept of the output frequency response function (OFRF) for nonlinear Volterra systems was proposed in Lang et al. [23], which defines an analytical relationship between system output spectrum and model parameters. The detailed structure of this relationship was studied in Jing et al. [24]. By using this explicit relationship, system model parameters of interest can be directly related to an engineering analysis and design objective which is dependent on system output frequency response, and the system output frequency response can be analysed in terms of some model parameters of interest. Therefore, based on the theoretical results established recently by the authors, a novel systematic approach to the frequency domain analysis of nonlinear Volterra systems is proposed in this paper. The new approach, referred to as OFRF-based analysis, allows analysis and design of output frequency response of nonlinear Volterra systems to be conducted in terms of system time domain model parameters with respect to any a specific input signal. A general procedure is proposed for the new frequency domain analysis method, and some fundamental results and practical techniques are developed to support and facilitate the application of this new method. The paper is structured as follows. Section 2 introduces the background of this study and provides some theoretical results established recently. Based on these theoretical results, a general procedure and some algorithms for the determination of the OFRF for a given NDE or NARX model are proposed and discussed in Section 3. Some useful results and techniques are developed in Section 4.1, especially for the analysis of nonlinear Volterra systems with respect to any specific model parameters, and what the OFRF-based analysis can deal with are summarized in Section 4.2. Section 5 provides an example to illustrate the new method. A summary is given in Section 6. 2. Fundamental concept of the OFRF-based analysis for nonlinear systems Nonlinear systems considered in this paper can be described by the following nonlinear differential equation (NDE) model: X M X m X K m¼1 p¼0 l1;lpþq¼0 where d 1 xðtÞ=dt 1 l¼0 cp;qðl1; ...; lpþqÞ Yp i¼1 d li yðtÞ dt li ¼ xðtÞ, p+q ¼ m, PK Y pþq i¼pþ1 ¼ l1;lpþq¼0 PK l1¼0 d li uðtÞ dt li ¼ 0, (1) P K lpþq¼0 ð Þ, M is the maximum degree of nonlinearity in terms of y(t) and u(t), and K is the maximum order of the derivative. In this model, the parameters such as c0,1(.) and c1,0(.) are linear parameters, which correspond to the linear terms in the model, i.e., d l yðtÞ=dt l and d l uðtÞ=dt l for k ¼ 0,1, y, K, and cpqð Þ for p+q41 are nonlinear parameters corresponding to nonlinear terms in the model of the form: Y p i¼1 d li yðtÞ dt li Y pþq i¼pþ1 d li uðtÞ dtli , ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 103 e.g., y(t) p u(t) q . p+q is called the nonlinear degree of the nonlinear parameter cpqð Þ. Moreover, cpqð Þ and cp0q0ð Þ are referred to as different type of nonlinearity if p6¼p0 or q6¼q0 . Similar to the NDE model (1), a discrete nonlinear model known as NARX model is often used for practical nonlinear system identification from
104 experimental data, which is given by yðtÞ ¼ XM X m X K m¼1 p¼0 k1;kpþq¼1 cp;qðk1; ...; kpþqÞ Yp i¼1 yðt kiÞ Ypþq i¼pþ1 uðt kiÞ. (2) In engineering practise, the NDE or NARX model can represent a wide class of nonlinear dynamical systems and include several well-known nonlinear input–output models as special cases [25]. The input–output relationship of model (1) or (2) can both be approximated in the neighbourhood of the equilibrium by a Volterra series of sufficient orders [15]. In order to conduct a frequency domain analysis for this class of nonlinear systems, the GFRF was proposed in [16] Z 1 Z 1 Hnðjo1; ...; jonÞ ¼ ... hnðt1; ...; tnÞ exp ð jðo1t1 þ þontnÞÞdt1 dtn, (3) 1 1 where hnðt1; ...; tnÞ is a real valued function of t1; ...; tn called the nth-order Volterra kernel of the system. By probing method [10], the recursive algorithms for the computation of the GFRFs for model (1) and (2) were developed in [26,27]. The system output frequency response was studied in Lang and Billings [28]. Although these results provide an important basis for frequency domain analysis of nonlinear systems, it can be seen that the direct analytical computation of system output spectrum involves very complicated integral and symbolic operations in multi-dimensional complex space when the involved nonlinearity order is high, and the analytical relationship between model parameters and output spectrum cannot be demonstrated clearly by the recursive algorithms [22,24]. This inhibits the application of these theoretical results to a certain extent. For these reasons, the concept of OFRF was proposed in Lang et al. [23], which was also studied in Jing et al. [24] through the parametric characteristic analysis. The result in [24] explicitly reveals the analytical polynomial relationship between system model parameters and system output spectrum, and allows the detailed structure of the OFRF to be determined up to any high orders without complicated symbolic computations in multi-dimensional complex space. Based on the results in Jing et al. [24], the nth-order GFRF in Eq. (3) can be written as Hnðjo1; ...; jonÞ ¼CEðHnðjo1; ...; jonÞÞfnðjo1; ...; jonÞ (4) and the system output spectrum can therefore be written into a more explicit polynomial form as follows, which is referred to as the OFRF in Lang et al. [23]: YðjoÞ ¼ XN n¼1 CEðHnð ÞÞj nðjoÞ T , (5a) where 1 jnðjoÞ ¼pffiffi n nð2pÞ Z 1 o1þ f nðjo1; ...; jonÞ þon¼o Yn UðjoiÞdso. i¼1 CE(.) is a novel coefficient extraction operator which has two fundamental operations ‘‘ ‘‘and ‘‘ ‘‘. The detailed definitions can refer to Jing et al. [24], and CE(Hn( )) is the parametric characteristics of the nth-order GFRF Hnð Þ, which can be recursively determined by CEðHnðjo1; ...; jonÞÞ ¼ C0;n n 1 n q q¼1 p¼1 Cp;q CEðHn q pþ1ð ÞÞ ARTICLE IN PRESS X.J. Jing et al. / Mechanical Systems and Signal Processing 22 (2008) 102–120 n p¼2 Cp;0 CE Hn pþ1ð Þ , ð5bÞ where Cp;q ¼½cp;qð0; ...; 0Þ; cp;qð0; ...; 1Þ; ...; cp;qðK; ...; KÞŠ for some p, q. The terminating condition is |fflfflfflfflfflffl{zfflfflfflfflfflffl} pþq CE(H 1(.)) ¼ 1. Obviously, the elements of CE(H n( )) are functions of the system time domain model parameters which define nonlinearities. For more clarity, Eq. (5a) can be rewritten as YðjoÞ ¼cFðjoÞ T , (6)
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Page 5 and 6: 106 Alternatively, CE(Hn( )) can al
Page 7 and 8: 108 proposed method above as demons
Page 9 and 10: 110 except that Cp,q ¼ c is non-ze
Page 11 and 12: 112 For clarity in discussion, let
Page 13 and 14: 114 Magnitude Magnitude of the sens
Page 15 and 16: 116 ARTICLE IN PRESS X.J. Jing et a
Page 17 and 18: 118 fundamental results, techniques
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