ARTICLE IN PRESS International Journal of Non-Linear Mechanics 45 (2010) 681–690
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Output frequency properties of nonlinear systems Xing Jian Jing a,, Zi Qiang Lang b, Stephen A. Billings b a b
Department of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Department of Automatic Control and Systems Engineering, University of Sheffield Mappin Street, Sheffield S1 3JD, UK
a r t i c l e in fo
abstract
Article history: Received 29 September 2008 Received in revised form 3 February 2010 Accepted 6 April 2010
Nonlinear systems usually have complicated output frequencies. For the class of Volterra systems, some interesting properties of the output frequencies are studied in this paper. These properties show theoretically the periodicity of the output super-harmonic and inter-modulation frequencies and clearly demonstrate the mechanism of the interaction between different output harmonics incurred by different input nonlinearities in system output spectrum. These new results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities. Examples and discussions are given to illustrate these new results. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear systems Output frequency response Harmonics Signal processing
1. Introduction The frequency domain analysis of nonlinear systems has been studied for many years [1–3,37]. For a class of nonlinear systems, a frequency domain analysis can be conducted by using the Volterra series [3,4]. It is shown in [5,6] that a considerably large class of nonlinear systems (referred to as Volterra systems) have a convergent Volterra series expansion. Therefore, Volterra systems are extensively applied in modeling, identification, control and signal processing to various systems and engineering practice, for example, electrical, biological and mechanical systems, materials engineering and chemical engineering [3–12,18,26–30,32–36]. Based on Volterra series expansion, the study of nonlinear systems in the frequency domain was initiated by introduction of the concept of the generalized frequency response functions (GFRFs) [13]. Many results are thereafter achieved for the frequency domain analysis of nonlinear systems [3,12,14–19, 21–23,27,28,38]. An important phenomenon for nonlinear systems in the frequency domain is that there are always very complicated output frequencies, appearing as super-harmonics, sub-harmonics, inter-modulation and so on. This usually makes it rather difficult to analyze and design the output frequency response for nonlinear systems, compared with linear systems. Output frequencies of Volterra systems have been studied by several authors [18–23,27] using the frequency domain method above. These results provide different viewpoints for computation and prediction of output
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frequencies of nonlinear systems. The previous results indicate that Volterra systems can effectively be used to account for superharmonics and inter-modulation in the output spectrum of nonlinear systems. In this study, some interesting properties of the output frequencies of Volterra systems were investigated. These results provide a useful insight into the super-harmonic and intermodulation phenomena in output spectrum of nonlinear systems, with consideration of the effects incurred by different nonlinear components in the system. The new properties theoretically demonstrate several fundamental frequency characteristics of output spectrum of Volterra systems and reveal clearly the mechanism of the interaction (crossing effects) between different harmonic behavior in system output spectrum incurred by different nonlinear components related to system input. These results should have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities. They can provide an important guidance to modeling, identification, control and signal processing by using the Volterra series theory in practice. Examples and discussions are provided to illustrate the results.
2. Output frequencies of Volterra systems Consider nonlinear systems which have a Volterra series expansion up to order N [3–5] Z 1 N Z 1 n Y X hn ðt1 , . . . , tn Þ uðtti Þ dti ð1Þ yðtÞ ¼ n¼1
1
1
i¼1
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X.J. Jing et al. / International Journal of Non-Linear Mechanics 45 (2010) 681–690
where hn ðt1 , . . . , tn Þ is a real valued function of t1 , . . . , tn called the nth-order Volterra kernel. The nth-order GFRF of system (1) is defined as [13] Hn ðjo1 , . . . ,jon Þ Z 1 Z 1 hn ðt1 , . . . , tn Þ expðjðo1 t1 þ þ on tn ÞÞ dt1 dtn ¼ 1
1
ð2Þ The GFRF of a practical system can be obtained by the probing method [3]. The output spectrum of system (1) subject to a general input can be described as [16] N X
YðjoÞ ¼
Yn ðjoÞ
n¼1
1 Yn ðjoÞ ¼ pffiffiffi nð2pÞn1
Z o1 þ þ on ¼ o
Hn ðjo1 , . . . ,jon Þ
n Y
R where o1 þ þ on ¼ o ðÞ dso represents the integration on the super plane o1 þ þ on ¼ o. Yn ðjoÞ is referred to as the nth-order output spectrum. When the system is subject to a multi-tone input K X
jFi j cosðoi t þ +Fi Þ
ð4Þ
i¼1
the system output spectrum is N X
YðjoÞ ¼
Yn ðjoÞ
n¼1
Yn ðjoÞ ¼
X
1 2n ok
1
Hn ðjok1 , . . . ,jokn ÞFðok1 Þ Fðokn Þ
ð5Þ
þ þ okn ¼ o
where K ð 40Þ A Z, Fi A C, Fðoki Þ can be written explicitly as Fðoki Þ ¼ jFjki j jej+Fjki j sgn1ðki Þ for ki A f 7 1, . . . , 7K g and 8 > < 1 a40
sgn1ðaÞ ¼
> :
0
1
a¼0
for a A R:
ao0
As mentioned, nonlinear systems usually have complicated output frequencies, which are quite different from linear systems having output frequencies completely identical to the input frequencies. From Eqs. (3) and (5), it can be seen that the output frequencies in the nth-order output spectrum, denoted by Wn and simply referred to as the nth-order output frequencies, are completely determined by
o ¼ o1 þ o2 þ þ on or o ¼ ok1 þ ok2 þ þ okn which produce super-harmonics and inter-modulation in system output frequencies. In this study, the input u(t) in (3) is considered to be any continuous and bounded input function in t Z0 with Fourier transform U(jo) whose input range is denoted by V, i.e., oAV, and V can be regarded as any closed set in the real. The multi-tone function (4) is only a special case of this. This study focuses on this class of input signals. Therefore, for the input U(jo) defined in V, the nth-order output frequencies are Wn ¼ fo ¼ o1 þ o2 þ þ on joi A V ,i ¼ 1,2, . . . ,ng
ð6aÞ
or for the multi-tone input (4), Wn ¼ fo ¼ ok1 þ ok2 þ þ okn joki A V ,i ¼ 1,2, . . . ,ng
W ¼ W1 [ W2 [ [ WN
ð6cÞ
In Eqs. (6a)(6b) and (6c), V represents the input frequency range corresponding to the output spectrum of order larger than 1, V is the original input frequency range corresponding to the first order output spectrum and W1 represents the output frequencies incurred by the linear part of the system. For example, V may be a real set [a,b][[c,d], thus V ¼ ½d,c [ ½b,a [½a,b [ ½c,d, where d Z c Zb Z a 4 0. When the system is subject to the multi-tone input (4), then the input frequency range is V ¼ f 7 o1 , 7 o2 , . . . , 7 oK g, which is obviously a special case of the former one.
Uðjoi Þ dso
i¼1
ð3Þ
uðtÞ ¼
where V ¼ V [ V. This is an analytical expression for the superharmonics and inter-modulations in the nth-order output frequencies of nonlinear Volterra systems. All the output frequencies up to order N, denoted by W, can be written as
ð6bÞ
3. The periodicity In this section, some fundamental properties of the output frequencies of system (1) with assumption that V is any closed set of frequency points in real are developed. Especially, the periodicity of the output frequencies is revealed for the class of input signals as defined before. Although some results about the computation of the system output frequencies for the case with V¼[a,b] has been studied in [18,19], for the multi-tone case in [21,23,32] and for the multiple narrow-band signals in [27], and some of the properties discussed in this section can be partly concluded from some previous results for the case V¼[a,b] and multi-tone case V ¼ fo1 , o2 , . . . , oK g, all the properties of this section are established in a uniform manner based on the analytical expressions (6a), (6b) and (6c) for the input domain V which is any closed set in real. The following Property 1 is straightforward from (6a) and (6b). Property 1. Consider the nth-order output frequency Wn: (a) expansion, i.e., Wn 2 DWn, (b) symmetry, i.e., 8O A Wn , then O A Wn and (c) n-multiple, i.e., maxðWn Þ ¼ n maxðVÞ and minðWn Þ ¼ n maxðVÞ. Property 1 shows that the output frequency range will expand larger and larger with the increase of the nonlinear order (Property 1(a)), the expansion is symmetric around zero point (Property 1(b)) and its rate is n-multiple of the input frequency range (Property 1(c)). These are some fundamental properties which may be known in literature for some cases and clearly stated here for Volterra systems subject to the mentioned class of input signals. Property 1(a) shows that, the (n 2)th order output frequencies Wn 2 are completely included in the nth-order output frequencies Wn. This property can be used to facilitate the computation of output frequencies for nonlinear systems. That is, only the highest order of Wn in odd number and the highest order in even number, to which the corresponding GFRFs are not zero, are needed to be considered in Eq. (6c) (e.g., W¼WN 1[WN). For example, suppose the system maximum order N¼10, then only W10 and W9 are needed to be computed if H10(.) and H9(.) are not zero, and the system output frequencies are W¼W9[W10 (in case that H9(.) is zero, W9 should be replaced by the output frequencies corresponding to the highest odd order of nonzero GFRFs). For the case that V¼[a,b], Property 1(a) can also be concluded from the results in [19]. Here the conclusion is shown to hold for any V. The following proposition theoretically demonstrates another fundamental and useful property for the output frequencies of Volterra systems, and provides an interesting insight into system output frequency characteristics.
ARTICLE IN PRESS X.J. Jing et al. / International Journal of Non-Linear Mechanics 45 (2010) 681–690
Proposition 1 (Periodicity property). The frequencies in Wn can be generated periodically as follows: Wn ¼
n[ þ1
Pi ðnÞ
Corollary 1. When the system is subject to the class of input U(j,o) which is specially defined in Z [
ð7aÞ
½a þ ði1Þe,b þ ði1Þe
i¼1
i¼1
9 oj A V > > = Pi ðnÞ ¼ o ¼ o1 þ o2 þ þ on oj o0 for 1 r j r i1 > > > > oj 40 for j Zi ; : 8 > >
j > = Pi ðnÞ ¼ o ¼ ok1 þ ok2 þ þ okn okj o 0 for 1 rj r i1 > > > > ; : okj 4 0 for j Z i 8 > >
