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Frequency-Dependent Magnitude Bounds of the Generalized Frequency Response Functions for NARX Model
European Journal of Control (2009)1:68–83 # 2009 EUCA DOI:10.3166/EJC.15.68–83
Frequency-Dependent Magnitude Bounds of the Generalized Frequency Response Functions for NARX Model
Xing Jian Jing1, , Zi Qiang Lang2, Stephen A. Billings2 1 2
Institute of Sound and Vibration Research, University of Southampton, Southampton, UK Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield, UK
New magnitude bounds of the frequency response functions for the Nonlinear AutoRegressive model with eXogenous input (NARX) are investigated by exploiting the symmetry of the nth-order generalized frequency response function (GFRF) in its n frequency variables. The new magnitude bound of the nth-order symmetric GFRF is frequency-dependent, and is a polynomial function of the magnitude of the first order GFRF. The coefficients of this polynomial function are functions of model parameters. Based on this result, the system output spectrum can also be bounded by an analytical polynomial function of the magnitude of the first order GFRF. The conservatism in the bound evaluations is reduced compared with previous results. Several examples and necessary discussions illustrate the potential application and effectiveness of the new results. Keywords: Generalized Frequency Response Function (GFRF), Nonlinear Systems, Volterra Series, NARX, Frequency-domain analysis
1. Introduction The frequency-domain analysis of nonlinear systems has been studied for many years. It is known that a large class of nonlinear systems have a convergent
Correspondence to: X.J. Jing, E-mail: xingjian.jing@googlemail. com
Volterra series expansion of finite order [4, 18–21]. For this class of nonlinear systems, frequency-domain analysis can be conducted by using the concept of Generalized Frequency Response Function (GFRF) [7], which is the multi-dimensional Fourier transform of the Volterra kernels in the Volterra series expansion. The GFRF represents system underlying characteristics and extends the transfer function concept for linear systems to the nonlinear case, and thus provides a basis for the frequency-domain analysis of nonlinear systems. Based on this concept, many studies have been carried out on the estimation and analysis of the kernel function, GFRFs, and nonlinear system output frequency response for the past decades [1, 8, 11–17, 22]. Evaluation of the magnitude of the GFRFs for a nonlinear system in practice can usually reveal some important information about the system. It is very useful in measuring the significant order of nonlinearities, finding the significant nonlinear terms for system frequency response functions, and indicating system stability and truncation error [2, 3, 5, 9]. Hence, a proper method to evaluate the magnitude of the GFRFs can provide a useful technique in analysis and design of nonlinear systems in the frequency-domain. Several efforts to derive the magnitudes of the GFRFs and output frequency response have been attempted [2, 23]. However, the relationship between the magnitude of the system
Received 3 June 2007; Accepted 1 December 2008 Recommended by D. Nesic, A.J. van der Schaft
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frequency response functions and system timedomain model parameters is not revealed clearly in these existing magnitude bound evaluation algorithms, and the bound evaluation is conservative. For this reason, new bound characteristics of the frequency response functions for NARX (Nonlinear AutoRegressive model with eXogenous input) model were studied in [9] recently. The magnitude bound of the nth-order GFRF is expressed into an n-degree polynomial function of the magnitude bound of the first order GFRF. This should provide for the first time an explicitly analytical relationship between the magnitude of the GFRFs and model parameters, and the connection between the magnitude bounds and system stability or truncation error were discussed. However, the bound for the nth-order GFRF in [9] is only a constant with respect to different frequencies, which is the maximum evaluation on the GFRF for all the interested frequency points and is computed in terms of the magnitude bound of the first order GFRF. Hence the conservatism in the bound evaluation is obvious. For this reason, a better result was proposed in [10], where the magnitude bound of the nth-order GFRF is a function of frequency variables instead of a constant (when model parameters are constant) but still in terms of the magnitude bound of the first order GFRF. Especially, the internal parametric structure of the bound results in [10] was discussed in detail there. In addition, all the bound results developed above are for asymmetric GFRFs. In this study, some new frequency-dependent bound results are provided for the nth-order GFRF by exploiting its symmetry in frequency variables. The new bound results have the advantages of the previous results, which can reveal the relationship between the magnitude of GFRFs and any interested model parameters clearly. Moreover, they are functions of the magnitude of the first order GFRF instead of its magnitude bound as the previous results in [2, 9–10, 23]. It is shown that the new bound results achieved in this study can greatly reduce the conservatism in bound evaluations. For nonlinear systems which have only pure output nonlinearities, the bound evaluations can even be much better. These results provide a useful approach to the analysis and understanding of the relationship between system frequency response functions and model parameters which represent system nonlinearities at different frequency points. Potential applications can be found in output frequency response analysis, truncation error evaluation, and stability analysis. Theoretical analysis, necessary
discussions and examples demonstrate usefulness of these bound results.
2. The GFRFs of NARX Model Consider nonlinear systems, whose input output relationship can be approximated by the Volterra series of degree N as [4, 18–21] N Z X
yðtÞ ¼
1
1
n¼1
Z ...
1 1
hn ð1 ; ; n Þ
n Y
uðt i Þdi
i¼1
ð1Þ where hn ( 1, . . . , n)is a real valued function of 1, . . . , n called the nth-order Volterra kernel. The nth-order GFRF is defined as Z 1 Z 1 Hn ðj!1 ; ; j!n Þ ¼ hn ð1 ; ; n Þ 1
1
expðjð!1 1 þ þ !n n ÞÞ d1 dn
ð2Þ
Consider these nonlinear systems described by the following NARX model [6],
yðtÞ ¼
M X
ym ðtÞ
m¼1
ym ðtÞ ¼
m X
K X
cp;q ðk1 ; ; kpþq Þ
p¼0 k1 ; kpþq ¼1 p Y i¼1
yðt ki Þ
pþq Y
uðt ki Þ
ð3Þ
i¼pþ1
where ym(t) is the mth-order output of the NARX K K P P model, p þ q ¼ m, ki ¼ 1, . . . , K, and ðÞ ¼ ðÞ
K P kpþq ¼1
k1 ;kpþq ¼1
k1 ¼1
ðÞ. The nonlinear degree of parameter cp,q( ) is
defined as p þ q. Obviously, parameter cp,q( ) corresponds to the (p þ q)th degree nonlinear terms p pþq p q. Q Q yðt ki Þ uðt ki Þ, e.g., y(t 1) u(t 2) i¼1
i¼pþ1
Parameters c0,1(.) and c1,0(.) are of degree 1 and referred to as linear parameters, and all others are referred to as nonlinear parameters. The GFRFs for nonlinear Volterra systems described by (3) can be obtained through mapping this time-domain model into the frequency-domain using the probing method [17]. A recursive algorithm to
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compute the nth-order GFRF of system (3) is given as follows [16]: ! K X c1;0 ðk1 Þ expðjð!1 þ þ !n Þk1 Þ 1 k1 ¼1
Hn ðj!1 ; ; j!n Þ ¼
K X
Then (4) can be written for more simplicity as Hn ðj!1 ; ; j!n Þ 1 ¼ Ln ðj!1 ; ; j!n Þ nq n X K X X cp;q ðk1 ; ; kpþq Þ q¼0 p¼0 k1 ;kpþq ¼0
c0;n ðk1 ; ; kn Þ
k1 ;kn ¼1
j
e
expðjð!1 k1 þ þ !n kn ÞÞ þ
nq X n1 X K X
cp;q ðk1 ; ; kpþq Þ
expðjð!nqþ1 kpþ1 þ þ !n kpþq ÞÞ Hnq;p ðj!1 ; ; j!nq Þ n K X X
cp;0 ðk1 ; ; kp ÞHn;p ð j!1 ; ; j!n Þ
p¼2 k1 ;kp ¼1
ð4Þ Hn;p ðÞ ¼
npþ1 X
Hi ðj!1 ; ; j!i ÞHni;p1
i¼1
ðj!iþ1 ; ; j!n Þ expðjð!1 þ þ !i Þkp Þ ð5Þ Hn;1 ðj!1 ; ; j!n Þ ¼ Hn ðj!1 ; ; j!n Þ expðjð!1 þ þ !n Þk1 Þ ð6Þ Note that the subscripts of the term expðjð!nqþ1 kpþ1 þ þ !n kpþq ÞÞ in (4) have been corrected (compare with the results in [16]). The nth-order GFRF in (4) is asymmetric with respect to the frequency variables !1, . . . , !n, that is, the value of Hn(j!1, . . . , j!n) may differ with different permutation of !1, . . . , !n. The symmetric GFRFs, which are usually used in practice, can be obtained by symmetrising the asymmetric GFRFs as follows, Hsym n ðj!1 ; ; j!n Þ X 1 Hn ðj!1 ; ; j!n Þ ¼ n! all permutations of
ð7Þ
!1 ;;!n
The bound for the nth-order symmetric GFRF above is focused in this study. Moreover, for convenience in derivation, let H0;0 ðÞ ¼ 1; Hn;0 ðÞ ¼ 0 for n > 0; Hn;p ðÞ ¼ 0 for n < p; and q X 1 q ¼ 0; p > 1 expð ðpÞÞ ¼ 0 q ¼ 0; p 1 i¼1
i¼1
ð!nqþi kpþi Þ
Hnq;p ðj!1 ; ; j!nq Þ
where Ln ðj!1 ; ; j!n Þ ¼ 1
q¼1 p¼1 k1 ;kn ¼1
þ
q P
ð8Þ
K P k1 ¼1
ð9Þ
c1;0 ðk1 Þ expðjð!1 þ
þ !n Þk1 Þ: The recursive algorithm for the computation of GFRFs is (4–6) or (5–6, 8–9). Although the GFRFs above provide an important basis for the analysis of nonlinear systems, it can be seen from (2, 4–6, 8–9) that the GFRF is a multivariate function defined in high-dimensional frequency space. Therefore, the analytical computation of the nth-order GFRF in practice is very complicated in the high-dimensional space, especially when the order n is larger than 5, and the understanding of the analytical relationship between any interested model parameters and the GFRFs is not straightforward. In order to solve these problems, evaluation of the relationship between the magnitude of the GFRFs and system time-domain model parameters is an alternative approach. Some results have been achieved for these purposes recently [9–10]. As discussed before, the bound of the nth-order GFRF in [9] is computed in terms of the magnitude bound of the first order GFRF which is only a constant with respect to different frequencies corresponding to the maximum evaluation on the GFRF for all the interested frequency points. Although the bound result in [10] is a function of frequency variables instead of a constant, it is still a polynomial function of the magnitude bound of the first order GFRF and complicated in the coefficients of the polynomial. To overcome these conservatisms, a sharper evaluation method for the magnitude bound of the nth-order GFRF is expected. Therefore, new frequency-dependent magnitude bounds for the nthorder symmetric GFRF are established by exploiting the symmetry of the GFRFs in the frequency variables for the NARX model in this study. The new bound results are polynomial functions of the magnitude of the first order GFRF instead of its bound, which are frequency-dependent and less conservative compared with the existing results. The new results can reveal more clearly at different frequency points the relationship between the higher order GFRFs which represent system nonlinear effects, and the first order GFRF which represents the system linear component,
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and the relationship between any interested model parameters and system frequency response functions.
3. New Magnitude bounds for the GFRFs Some notations and definitions are introduced first, then the new frequency-dependent bound characteristics of the symmetric GFRFs of system (3) are derived and discussed. Denote L¼
inf
!1 þþ!n 2I!
where I! ¼
N S
fjLn ðj!1 ; ; j!n Þjg
ð10Þ
! ¼ !1 þ þ !k ; !i 2 R! g repref!j
k¼1
sents simply all the possible output frequency range when the system is stimulated by an input U(j!), which is defined in a frequency range R!, and N is the maximum order of the Volterra series. Let n ¼ 1 in (9), then gives K P k1 ¼1
H1 ðj!1 Þ ¼
1
¼
2) (a b)(k) ¼ a(k) þ b(k) for 0 k max (m, n), where if k > n or m, then a(k) ¼ 0 or b(k) ¼ 0. For similar definitions based on polynomials can be referred to [10]. Let ðÞ and ðÞ denote the multiple multiplicaðÞ
ðÞ
tion and addition by the operator ‘‘’’ and ‘‘’’ for the terms in (.) satisfying the conditions in (), respectively (which are similar to the operations P Q ðÞ and ðÞ). Moreover, denote Cn ¼ ½Cð0; nÞ, ðÞ
C(1, n-1), . . . ,C(n,0)].
c0;1 ðk1 Þ expðj!1 k1 Þ
k1 ¼1
k1 ¼1
iþj¼k 0in; 0jm
ðÞ
K P
K P
The following two operators ‘‘’’ and ‘‘’’ are needed, which define a multiplication and an addition between two vectors of different dimensions, respectively. Let a(i) denote the ith element of a vector a. Consider two vectors a and b of n and m dimensions, respectively. The two operators are defined as follows: P aðiÞbðjÞ for 0 k m þ n; 1) ða bÞðkÞ ¼
c1;0 ðk1 Þ expðj!1 k1 Þ 3.1. The Main Bound Characteristics
c0;1 ðk1 Þ expðj!1 k1 Þ L1 ðj!1 Þ
ð11Þ
Theorem 1: The magnitude of the nth-order symmetric GFRF of system (3) is bounded by a polynomial function with degree n of the magnitude of the first order GFRF, whose coefficients are functions of the model nonlinear parameters. That is, there exists a series of scalar positive real numbers bn;0 ; bn;1 . . . ; bn;n , such that sym H ðj!1 ; ; j!n Þ bn hT ðj!1 ; ; j!n Þ n n
This is the first order GFRF of the nonlinear system (3), which is the frequency response function of the linear elements of the system, representing the dynamics of the linear component in the system (i.e., the transfer function of the system when all the nonlinear parameters in the system model are set to be zero). Therefore, the relationship between the nthorder GFRF and the first order GFRF can reveal the dependence of high order GFRFs on the linear component of the system. Define 8 K P > > cp;q ðk1 ; ; kpþq Þ; > 1 q n 1; 1 p n q > > > k1 ;kpþq ¼1 > > K > > < P c0;n ðk1 ; ; kn Þ; q ¼ n; p ¼ 0 Cðp; qÞ ¼ k1 ;kn ¼1 > > K > P > > cp;0 ðk1 ; ; kp Þ; > q ¼ 0; 2 p n > > > > : k1 ;kp ¼1 0; else By definition, C(p, q) is a non-negative function of the parameter cp,q(.) defined on all 0 {p, q} n (which means 0 p n and 0 q n).
ð13aÞ
ð12Þ
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X.J. Jing et al.
where hn ðj!1 ; ; j!n Þ ¼ ½ 1 hn;1 ðj!1 ; ; j!n Þ hn;n ðj!1 ; ; j!n Þ, and for k ¼ 0; 1; . . . ; n: hn;k ðj!1 ; ; j!n Þ k! ¼ nðn 1Þ ðn kÞ X all the different combinations fr1 ;r2 ;;rk gtaken from f1;2;;ng with no repetition
1 ¼ sup jH1 ðj!Þj, then it can be Remark 2: Let H !2R! derived from (13b) that k! nðn 1Þ ðn kÞ Yk X i¼1 H1 ðj!ri Þ
hn;k ðj!1 ; ; j!n Þ ¼ Yk i¼1 H1 ðj!ri Þ
all the different combinations fr1 ;r2 ;;rk g taken from f1;2;;ngwith no repetition
k! nðn 1Þ ðn kÞ X k ¼ H k ð14Þ H 1 1
ð13bÞ bn ¼ ½bn;0 ; bn;1 . . . ; bn;n can be recursively determined as follows 0 bn ¼
0
0
B n1 1B BC n þ B @m¼2 pþq¼m L@ 0fp;qgm
B BCðp; qÞ @
P
ri ¼nq 1fr1 rp gnmþ1
b1 ¼ ½0; 1
ð13dÞ
p
n
i¼1
m¼2
all the different combinations fr1 ;r2 ;;rk g taken from f1;2;;ngwith no repetition
where bri ¼ 0 if p<1 and ðÞ ¼ 0 if n < 2.
Proof. See Appendix A. & Note that a combination {r1,r2, . . . ,rp} taken from {1,2, . . . ,n} without repetition in (13b) refer to p different numbers taken from {1,2, . . . ,n}. For example, {1,2, . . . ,p-1,p} and {1,2, . . . ,p-1,n} are such two different combinations, and {1,2, . . . ,p-1,p} and {1,2, . . . , p,p-1} are the same combination. Remark 1: Theorem 1 provides an explicit polynomial expression for the magnitude bounds of the GFRFs for system (3) in terms of the magnitude of the first order GFRF. The new result not only demonstrates an analytical relationship between the magnitude bound of the nth-order GFRF and system time-domain model parameters, but also reveals the relationship between the magnitudes of the higher order GFRFs, which represents the nonlinear effects of the model, and the magnitude of the first order GFRF, which represents the linear component of the model. Therefore, the effect of nonlinear components and the role of the linear component in the system described by (3) can be analyzed. This provides a useful insight into the frequency-domain analysis and design of nonlinear systems. &
1 11 p C CC CC br i C AAA for n > 1 i¼1
ð13cÞ
Using the above inequality and (13a) in Proposition 1, it can be shown that sym H ðj!1 ; ; j!n Þ bn hT ðj!1 ; ; j!n Þ n n ¼ bn;0 þ bn;1 hn;1 ðj!1 Þ þ þ bn;n hnn;n ðj!1 ; ; j!n Þ 1 þ þ bn;n H n bn;0 þ bn;1 H 1
ð15Þ
The second inequality in (15) is or is equivalent to the result obtained in [2,23,9]. From (15) it is obvious that the result in Theorem 1 is less conservative than those in [2,23,9], and shares the same advantage of an explicit relationship between the magnitude bound and model parameters as [9]. Moreover, compared with the previous results in [2,23,9], the magnitude bound of the nth-order GFRF in Theorem 1 is a function of the frequency variables, and can directly be computed by only using the model parameters and jH1(j!)j with no need to recursively compute the lower order GFRFs as the results in [2,23] or no need to evaluate the magnitude bound for H1(j!) as the results in [9]. Thus the conservatism and computational complexity for the bound evaluation are all reduced. Some existing algorithms in [2, 3, 5, 9] for evaluation of system truncation error and convergence where the GFRF bounds are involved can therefore be improved. It shall also be noted from (14) that, exploiting the symmetry in the frequency variables k to be relaxed to enables the magnitude boundQH P k1 i¼1 H1 ðj!ri Þ to reduce all the different combinations fr1 ;r2 ;;rk g taken from f1;2;;ngwith no repetition
73
Frequency-Dependent Magnitude Bounds
the conservatism of the previous results in [2,9,10,23], combinawith the only cost to compute the nðn1ÞðnkÞ k! tions but removing the cost to evaluate the bound for H1 ðj!Þ. &
Corollary 1: The magnitude of the nth-order GFRF can be bounded by sym H ðj!1 ; ; j!n Þ n (
Remark 3: In the derivation of the result in Theorem 1 (see Appendix A), if the maximum evaluation on the terms involving lower order GFRF is made p Q P (i.e., Hri ðj!Xþ1 ;;j!Xþri ÞÞ is made by 1fr P 1 rp gnmþ1 i¼1
min bn hTn ðj!1 ; ; j!n Þ; X
jHn ðj!1 ; ; j!n Þj bn ðj!1 ; ; j!n Þ hTn
ð16aÞ
n , and bn (!1, . . . , !n) is 1 H where hn ¼ ½ 1 H 1 a function of frequency variables and can be recursively determined by bn ðj!1 ; ; j!n Þ ¼
0
B B n1 B 1 BC n þ B B @Cðp; qÞ @ @ Ln ðj!1 ; ; j!n Þ m¼2 pþq¼m 0p;qm
ð17Þ
where bn is computed by (13cd), bn (!1, . . . , !n) is computed by (16b), hn ðj!1 ; ; j!n Þ is defined by (13b), 1 ¼ sup jH1 ðj!Þj . 1 H n , and H hn ¼ ½ 1 H 1
!2R!
This is another new bound evaluation for the nthorder GFRF, which has both the advantages of the two bound results in Theorem 1 and in [10] (i.e., (16a-b) ). Simulation studies in Section 4 will show that the bound result in Theorem 1 has obviously much better
0
0
bn ðj!1 ; ; j!n Þ
) hTn
all the permutations off!1 ;;!n g
ri ¼nq
using the magnitude bound of H1(.) and leave the other frequency-dependent terms (i.e., Ln ðj!11;;j!n Þ ) intact, (see inequality (A5) in the proof of Theorem 1), then the result in [10] can be achieved by following a similar proof process, i.e.,
1 n!
11 1 p CC C CC bri ð!XðiÞþ1 ; ; !XðiÞþri Þ C AA A i¼1
P
ri ¼nq 1fr1 rp gnmþ1
ð16bÞ It can be shown that sup bn;k ðj!1 ; ; j!k Þ
!1 ;;!n
hn;k ¼ bn;k ; sup hn;k ðj!1 ; ; j!k Þ ¼ !1 ;;!k
ð16cÞ
Although (16a-b) is also less conservative than the results in [2,23,9], it is not necessarily of obvious advantage compared with the result in Theorem 1 at any frequency point because the bound in [10], i.e., (16a-b), is still a function of the magnitude bound of the first order GFRF. Further study will show that, the bound result in Theorem 1 has less conservatism than (16a-b) at some frequency points but not for any frequency points. They are two different results from different viewpoints. Noticeably, the bound characteristics in Theorem 1 are in terms of the magnitude of the first order GFRF instead of its magnitude bound; the recursive computation of (16b) is clearly more complicated to determine the frequency variables in each recursion than (13c), while (13b) only involves some fixed combinations. In order to have both advantages of these two results in reducing conservatism of the bound evaluation, the following corollary is straightforward.
evaluation than the results in (16a-b) of [10] at some frequency points, and the bound result in Corollary 1 has remarkably improvement in bound evaluation. Moreover, the advantage of these two different results in relaxing the two parts bn (j!1, . . . , j!n) and hn (j!1, . . . , j!n) as mentioned in Remarks 2–3 can also reach together with much sharper performance for nonlinear systems with only nonlinearities in output. This will be discussed later in the following section. 3.2. A special case As discussed before, when nonlinear systems have only nonlinearities in output, the nth-order GFRF determined Q by (4–6,9) is obviously symmetrical for the part ni¼1 H1 ðj!ri Þ in the bound expression (13b). In this case, the bound evaluation by the results discussed in Section 3.1 can be improved to have much less conservatism so that both of the advantages of the results in Theorem 1 and (16a-b) of [10] can reach. The following result can be obtained.
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X.J. Jing et al.
Theorem 2: The magnitude of the GFRFs for system (3) with only pure input nonlinearities is bounded by sym H ðj!1 ; ; j!n Þ n
jH1 ðj!1 Þj jH1 ðj!n Þj n! X Bn ð!1 ; ; !n Þ
1 where n ¼ pffiffinð2Þ n1
N X
n
sup
!1 þþ!n ¼!
ðBn ð!1 ; ; !n Þ
!
ð18bÞ
1
m C Y C Bri ð!Xþ1 ;;!Xþri ÞC; A i¼1
ri ¼n 1fr1 rm gnmþ1
iP 1
jUðj!i Þjd! . For the
hn;n ðj!1 ; ; j!n ÞÞ
where Bn ð!1 ; ; !n Þ ¼ Ln ðj!11;;j!n Þ 0
B1 ¼ 1 and X ¼
!1 þþ!n ¼! i¼1
jYðj!Þj
n¼1
X
n Q
case of Theorem 2,
all the permutations off!1 ;;!n g
n B X B BCðm;0Þ P @ m¼2
R
When the system is subject to a multi-tone input K P described by uðtÞ ¼ jFi j cosð!i t þ ffFi Þ, then the i¼1
system output spectrum is bounded by
rx .
N X
jYðj$Þj
x¼1
n b n
n¼1
Proof. See Appendix B. &
sup
!k1 þþ!kn ¼$
hTn ðj!k1 ; ; j!kn Þ ð19aÞ
It can be verified that Bn (!1, . . . , !n) in Theorem 2 is bn,n (!1, . . . , !n) in (16a-b) corresponding to the jH1 ðj!n Þj bound result in [10], and jH1 ðj!1 Þj in Theorem 2 n! is hn;n (!1, . . . , !n) in (13b) of Theorem 1. Therefore, both the advantages of the two results are integrated here. The advantage of this result will be discussed in an example study in Section 4.
where n ¼ 2n Fð!Þ ¼
P !k1 þþ!kn ¼$
jFi jejffFi 0
jFð!k1 Þ Fð!kn Þj,
if ! 2 f!k ; k ¼ 1; ; Kg , else
$ ¼ !k1 þ þ !kn !k1 ; ; !kn 2 R! ¼ f!k ; k ¼ 1; ; Kg, and ! k ¼ !k . For the case of Theorem 2, jYðj$Þj N X
3.3. Bounds for Output Spectrum Based on Theorem 1 and Theorem 2, the magnitude bound of the output spectrum for the NARX 2 I! , where model (3) can also be derived. For any ! I! denotes the non-negative frequency range of the output spectrum of model (3), let sup
!1 þþ!n ¼!
hn ðj!1 ; ; j!n Þ
"1
sup
!1 þþ!n ¼!
¼
sup
# hn;n ðj!1 ; ; j!n Þ
Corollary 2: When the input of the NARX model (3) is a general input with spectrum U(j!) defined in R!, then the system output spectrum is bounded by jYðj!Þj
N X n¼1
n bn
sup
!1 þþ!n ¼!
n
sup
!1 þþ!n ¼!
ðBn ð!1 ; ; !n Þ
!
ð19bÞ
hn;n ðj!1 ; ; j!n ÞÞ Proof. See Appendix C. & 1 Note that n ¼ pffiffinð2Þ n1
R
n Q
!1 þþ!n ¼! i¼1
jUðj!i Þjd!
can be computed by using an algorithm proposed in [2].
hn;1 ðj!1 ; ; j!n Þ
!1 þþ!n ¼!
n¼1
Tn ðj!1 ; ; j!n Þ h ð18aÞ
Remark 4: Corollary 2 shows that the magnitude of the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order GFRF, whose coefficients are the functions of the model parameters and input magnitude. This reveals an analytical relationship between the magnitude of system output spectrum and model parameters, and also the relationship between system output spectrum and the linear component of the model. This polynomial expression can consequently facilitate the analysis of system output spectrum in terms of model parameters and the linear part of the model. In addition, the bound
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Frequency-Dependent Magnitude Bounds
results in Corollary 2 are also better than those in [2, 3, 23, 9] since the new bound of the nth-order GFRF is less conservative as discussed in Remarks 2–3. &
Hence, the magnitude bound of the nth-order GFRF for (20) is sym H ðj!1 ; ;j!n Þ bn hT ¼ bn;n hn;n n n ¼
4. Examples Two examples are given to further discuss and demonstrate the theoretical results in Theorems 1–2 and Corollaries 1–2. Example 1: To illustrate the results in Theorem 1 and Corollary 1, consider a simple case of model (3) as M X
K X
m¼p¼1
k1 ;kp ¼1
yðtÞ ¼
þ ðm 1Þ
cp;0 ðk1 ; . . . ; kp Þ
K X
p Y
yðt ki Þ
i¼1
!
c0;1 ðk1 Þuðt k1 Þ
ð20Þ
1 x¼0 . Compared with model (3), 0 else there are only nonlinearities in output in (20). In this case, all the other nonlinear parameters are zero except for the nonlinear parameters of the form cp ; 0ð:Þ. The bound polynomial coefficient bn (bn,0, b n,1, . . . , b n,n) up to the third order can be computed as where ðxÞ ¼
b2 ¼ ½0; 0; Cð2; 0Þ=L 0 1B b3 ¼ B C3 þ Cð2; 0Þ L@
P
ri ¼3 1fr1 ;r2 g2
2
br i
i¼1
1 C C A
1 ¼ ð½0; 0; 0; Cð3; 0Þ þ Cð2; 0Þ 2ðb1 b2 ÞÞ L ¼
1 ½0; 0; 0; Cð3; 0Þ L
! 2 þ Cð2; 0Þ ½0; 0; 0; Cð2; 0Þ L
¼ ½0; 0; 0;
1 2 Cð3; 0Þ þ 2 Cð2; 0Þ2 L L
It can be further verified that for n > 1 bn;k ¼ 0 for 0 k < n bn;n ¼
ri ¼n 1fr1 rp gnmþ1
Cð2; 0Þ jH1 ðj!1 ÞH1 ðj!2 Þj L
i¼1
ð21aÞ
Note that (21a) is the result obtained from Theorem 1. A similar result can be derived by using the results in [10] as sym H ðj!1 ; j!2 Þ 2
Cð2; 0Þ 2 H 1 L2 ðj!1 ; j!2 Þ
ð21bÞ
Use the results in [9], yielding sym H ðj!1 ; j!2 Þ Cð2; 0Þ H 2 1 2 L
ð21cÞ
According to Corollary 1, it can be obtained that sym H ðj!1 ; j!2 Þ 2 ( Cð2; 0Þ jH1 ðj!1 ÞH1 ðj!2 Þj; min L ) Cð2; 0Þ 2 H 1 L2 ðj!1 ; j!2 Þ To compare these results, consider the model parameters in (20) are: c1,0(1) ¼ -5, c1,0(2) ¼ -1, c0,1(1) ¼ 2 and C(2,0) ¼ c2,0(1,0) ¼ 10; all the other linear parameters are zero. That is, (20) can now be written as yðt 2Þ þ 5yðt 1Þ þ yðtÞ
n1 1 Cðn; 0Þ þ Cðm; 0Þ L m¼2 ! m bri ;ri P ri ¼n 1fr1 rp gnmþ1
i¼1
Note that in this case, the magnitude bound for the symmetric and asymmetric GFRFs are Q the same since changing the order of !r1 !rn in ni¼1 H1 ðj!ri Þ has no effect on the result. Consider the second order GFRF as an example, sym H ðj!1 ; j!2 Þ b2 hT ¼ b2;2 h2;2 2 2 ¼
k1 ¼1
n1 1 Cðn; 0Þ þ Cðm;0Þ L m¼2 ! Y m n b H ðj! Þ ri ;ri ri P i¼1 1
þ 10y2 ðt 1Þ ¼ 2uðt 1Þ Then from (9) and (11), L2 ðj!1 ; j!2 Þ ¼ ej2ð!1 þ!2 Þ þ 5ejð!1 þ!2 Þ þ 1 L ¼ 3
76
X.J. Jing et al.
H1 ðj!1 Þ ¼
ej2!1
2ej!1 1 ¼ 0:6667 H þ 5ej!1 þ 1
sym H ðj!1 ; j!2 Þ 2 Cð2; 0ÞH1 ðj!1 ÞH1 ðj!1 Þ j! jð! þ! Þ 2 1 2 ¼ e e L2 ðj!1 ; j!2 Þ 10H1 ðj!1 ÞH1 ðj!1 Þ ¼ ð21eÞ L2 ðj!1 ; j!2 Þ Clearly, (21e) is the true value of the magnitude of the second order GFRF. Comparisons of these different evaluations on the magnitude bounds are shown in Figs. 1–4. It can be obviously seen that the evaluation by (21a) is much better than those by (21b) and (21c), and (21d) is much better than all of (21a-c). In Fig. 4, it can be seen that only the peak in the middle of the diagram in Fig. 3 is cut off by the valley in the middle of the diagram in Fig. 2, while other places are kept as they are in Fig. 3. These show that the evaluation by (21a) is much better than (21b) at most frequency points in the studied case of this example. &
Fig. 2. Magnitude of the second order GFRF and its bound by (21b)
Thus,
Remark 5: For the simple case in Example 1, it can be derived from Theorem 2 that B2 ð!1 ; !2 Þ ¼ 2 X
1 L2 ðj!1 ; j!2 Þ
Cðm; 0Þ
m¼2
X P
m Y
! Bri ð!Xþ1 ; ; !Xþri Þ
i¼1 ri ¼2 1r1 rp 2mþ1
¼
Cð2; 0Þ L2 ðj!1 ; j!2 Þ
Fig. 1. Magnitude of the second order GFRF and its bound by (21c)
Fig. 3. Magnitude of the second order GFRF and its bound by (21a)
Fig. 4. Magnitude of the second order GFRF and its bound by (21d)
77
Frequency-Dependent Magnitude Bounds
sym H ðj!1 ; j!2 Þ jH1 ðj!1 Þj jH1 ðj!2 Þj 2 2! X Cð2; 0Þ L ðj!1 ; j!2 Þ all the permutations 2
jY2 ðjÞjC13 ð ¼
off!1 ;;!n g
Cð2; 0Þ ¼ jH1 ðj!1 Þj jH1 ð j!n Þj L2 ðj!1 ; j!2 Þ
ð22Þ
This is right the result in (21e), which is the real value of the magnitude of H2(.) for the studied case. Thus there is no conservatism in the bound (22) for H2(.). Moreover, the magnitude expression of the second order GFRF can be directly obtained by using Theorem 2 as shown above and the relationship between C(2,0), jH1(.)j and jH2(.)j is clear, while manual manipulation must be used if the recursive algorithm in (4–9) is adopted to compute the magnitude and the computation burden will be worse when high order (larger than 5) GFRFs are evaluated. This will facilitate the analysis of nonlinear systems based on GFRFs. & Example 2: To demonstrate the results in Corollary 2, still consider system (20) in Example 1, and additionally suppose it is subject to a single sinusoidal input signal described by uðtÞ ¼ Fd sinðtÞ ðFd > 0Þ . Then from (19a), it can be shown that
jYðjÞj
ðN1Þ=2 X
Cn2nþ1 ð
n¼0
sup
Fd 2nþ1 Þ b2nþ1;2nþ1 2
!1 þþ!2nþ1 ¼
Fd 3 Þ b3;3 sup h T3;3 ðj!k1 ;;j!k3 Þ 2 !1 þþ!3 ¼
3F3d 2Cð2;0Þ2 Cð3;0Þ ð Þ þ 8 L L2
supðH1 ðÞH1 ðÞH1 ðÞÞY2
The conditions for C(2,0), C(3,0) and H1() can therefore be determined. Note that the output spectrum is usually determined by its first several orders in practice. Therefore, when the requirements on these orders of output spectrum are known, the conditions on model parameters can then be determined simply by using this method. Compared with the direct analytical computations by using (4–6) and (C1-C2) in Appendix C, this method can determine the magnitude bound of the output spectrum up to any high order, while the analytical computation by using (4–6) and (C1-C2) in Appendix C will be hard to follow for the order larger than 5 in practice. For the case in this example, it is very easy to compute jY2(j)j by using the bound results of this study as shown but it will be much more complicated by using (4–6) and (C1-C2) in Appendix C. More examples can be seen in [9] where the bound was computed up to the 41st order. By using the bound results, the truncation error of the system can also be evaluated. For example, if the bound of the nth-order output spectrum Yn(jx03A9;) come down below a predefined threshold, the system can be regarded to have a sufficient Volterra series expansion at that order. More detailed discussions can be referred to [3]. &
hT2nþ1;2nþ1 ðj!k1 ; ; j!kn Þ ð23Þ
where bc is to take the integer part from (.), nðn1Þðnmþ1Þ . Cm n ¼ mðm1Þ2 By using inequality (23), analysis of the output frequency response of system (20) when subject to the input uðtÞ ¼ Fd sinðtÞ ðFd > 0Þ can be conducted. For instance, a sufficient convergent condition for the system output spectrum can be derived since the right side of inequality (23) can be regarded as a power series [4, 5, 9, 18–21], which consequently leads to an evaluation on system stability. Also given a specific requirement on output spectrum, the sufficient conditions on model parameters under which the requirement is achieved can be derived. For example, if jY2(j)j is to be designed such that jY2(j)j Y2 , then from (23) and Theorem 1, it can be achieved by designing the second order output spectrum to satisfy
5. Conclusions New frequency-dependent magnitude bounds for the GFRFs and output spectrum of nonlinear systems described by the NARX model have been derived in this paper. The new frequency-dependent bound results for the GFRFs are less conservative, and developed for the symmetric GFRFs, compared with the existing bound evaluation results. These new results can clearly demonstrate the relationship between the magnitude of higher order GFRF, which represents the nonlinear effect, and the magnitude of the first order GFRF, which represents the linear component in nonlinear systems. A much better result is also developed for nonlinear systems which have only nonlinearities in output which include all the advantages of the currently existing bound results. These results should be helpful to the analysis and design of nonlinear systems in the frequency-domain.
78
Further study will focus on application problems of the results established in this paper.
6. Acknowledgement The authors gratefully acknowledge the constructive comments and suggestions from anonymous reviewers and the support of the EPSRC-Hutchison Whampoa Dorothy Hodgkin Postgraduate award for this work.
References 1. Bedrosian E, Rice SO. The output properties of Volterra systems driven by harmonic and Gaussian inputs. Proc Inst Electr Electron Eng 1971; 59: 1688– 1707 2. Billings SA, Lang ZQ. A bound of the magnitude characteristics of nonlinear output frequency response functions. Int J Control 1996; 65(2): 309–328 3. Billings SA, Lang ZQ. Truncation of nonlinear system expansions in the frequency domain. Int J Control 1997; 68(5): 1019–1042 4. Boyd S, Chua LO. Fading memory and the problem of approximating nonlinear operators with Volterra series, IEEE Trans Circuits Syst 1985; CAS-32: 1150–1161 5. Chatterjee A, Vyas NS. Convergence analysis of Volterra series response of non-linear systems subject to harmonic excitation. J Sound Vibr 2000; 236: 339–358 6. Chen S, Billings SA. Representation of non-linear systems: the NARMAX model. Int J Control 1989; 49: 1012–1032 7. George DA. Continuous nonlinear systems. Technical Report 355. MIT Research Laboratory of Electronics: Cambridge, MA, 1959 8. Jing XJ, Lang ZQ, Billings SA. The parametric characteristic of frequency response functions for nonlinear systems. Int J Control 2006; 79(12) 1552–1564 9. Jing XJ, Lang ZQ, Billings SA. New bound characteristics of NARX model in the frequency domain. Int J Control 2007; 80(1): 140–149 10. Jing XJ, Lang ZQ, Billings SA. Magnitude bounds of generalized frequency response functions for nonlinear Volterra systems described by NARX model. Automatica 2008; 44: 838–845 11. Jing XJ, Lang ZQ, Billings SA. Mapping from parametric characteristics to generalized frequency response functions of non-linear systems. Int J Control 2008; 81 (7): 1071–1088 12. Jing XJ, Lang ZQ, Billings SA. Determination of the analytical parametric relationship for output spectrum of Volterra systems based on its parametric characteristics. J Math Anal Appl 2009; 351: 694–706 13. Kim KI, Powers EJ. A digital method of modelling quadratic nonlinear systems with a general random input. IEEE Trans Acoust Speech Signal Process 1988; 36: 1758–1769
X.J. Jing et al.
14. Kotsios S. Finite input/output representative of a class of Volterra polynomial systems. Automatica 1997; 33: 257–262 15. Lang ZQ, Billings SA. Output frequency characteristics of nonlinear systems. Int J Control 1996; 64(6): 1049– 1067 16. Peyton-Jones JC, Billings SA. Recursive algorithm for computing the frequency response of a class of nonlinear difference equation models. Int J Control 1989; 50(5): 1925–1940 17. Rugh WJ. Nonlinear System Theory—The Volterra/ Wiener Approach. Johns Hopkins University Press: Baltimore and London, 1981 18. Sandberg IW. Expansions for nonlinear systems. Bell Syst Tech J 1982; 61(2): 159–199 19. Sandberg IW. Volterra expansions for time-varying nonlinear systems. Bell Syst Tech J 1982; 61(2): 201–225 20. Sandberg IW. On Volterra expansions for time-varying nonlinear systems. IEEE Trans Circuits Syst 1983; CAS30: 61–67 21. Sandberg IW. The mathematical foundations of associated expansions for mildly nonlinear systems. IEEE Trans Circuits Syst 1983; CAS-30: 441–455 22. Worden K, Manson G, Volterra A. Series approximation to the coherence of the duffing oscillator. J Sound Vibr 2005; 286: 529–547 23. Zhang H, Billings SA. Gain bounds of higher order nonlinear transfer functions. Int J Control 1996; 64(4): 767–773
Appendix A: Proof of Theorem 1 Using the notations in (8, 10–12), it follows from (9) that jHn ðj!1 ; ; j!n Þj nq n X X
1 Ln ðj!1 ; ; j!n Þ
Cðp; qÞHnq;p ðj!1 ; ; j!nq Þ
ðA1Þ
q¼0 p¼0
It can be easily seen that n nq P P Cðp; qÞ Hnq;p ðj!1 ; ; j!nq Þ includes all the q¼0 p¼0
combinations of (p,q) satisfying p þ q ¼ m, 0 fp; qg m, and m ¼ 2, . . . , n. Hence, nq n X X
Cðp; qÞHnq;p ðj!1 ; ; j!nq Þ
q¼0 p¼0
¼
n X
X
Cðp; qÞHnq;p ðj!1 ; ; j!nq Þ
m¼2 pþq¼m 0fp;qgm
ðA2Þ
79
Frequency-Dependent Magnitude Bounds
Note that (5) can also be written as Hn;p ðj!1 ; ; j!n Þ ¼
npþ1 X
p Y
rP 1 rp ¼1 i¼1 ri ¼n
Hri ðj!Xþ1 ; ; j!Xþri Þ
expððj!Xþ1 þ þ j!Xþri Þki Þ; where X ¼
i1 X
ðA3Þ
rx
In the following, for a proof by induction, the first two steps, i.e., n ¼ 1 and 2 are checked first to show the results; then under the assumption that (13a-d) hold for all the order less than n, it is shown that (13a-d) hold for n. For n ¼ 1, it is a special case which is not included in (13c) and defined in (13d) specifically. Using (13d), the bound sym of jH1(j!1)j Tcan be expressed as jH1(j!1)j ¼ H ðj!1 Þ ¼ b1 h . For n ¼ 2, it is the beginning 1 1 step of the recursion in (13c). It can be shown from (A1-A2) or (A5) that
x¼1
1 Cð0; 2Þ þ Cð1; 1Þ L2 ðj!1 ; j!2 Þ H1;1 ðj!1 Þ þ Cð2; 0ÞH2;2 ðj!1 Þ 1 Cð0; 2Þ þ Cð1; 1ÞjH1 ðj!1 Þj L þCð2; 0ÞjH1 ðj!1 ÞH2 ðj!2 Þj
jH2 ðj!1 ; j!2 Þj
thus Hn;p ðj!1 ; ; j!n Þ p npþ1 X Y H ðj! ; ; j!Xþri Þ ¼ r r ¼1 i¼1 ri Xþ1 1 p P ri ¼n
exp jð!Xþ1
þ þ !Xþri Þki
Hence
p Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP 1 rp ¼1 i¼1 npþ1 X
sym H ðj!1 ; j!2 Þ 2 1 jH1 ðj!1 Þj þ jH1 ðj!2 Þj Cð0; 2Þ þ Cð1; 1Þ L 2 þ Cð2; 0ÞjH1 ðj!1 ÞH2 ðj!2 Þj
ri ¼n
which yields Hnq;p ðj!1 ; ; j!n Þ nqpþ1 p X Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP i¼1 1 rp ¼1
¼ ðA4Þ
Therefore, from (A1-A2) and (A4) it can be derived that jHn ðj!1 ; ; j!n Þj n X X 1 Ln ðj!1 ; ; j!n Þ m¼2 pþq¼m 0fp;qgm
0
1
Cð1; 1Þ
jH1 ðj!1 ÞjþjH1 ðj!2 Þj 2
Cð2; 0Þ
jH1 ðj!1 ÞH2 ðj!2 Þj
iT
¼ b2 hT2 ðj!1 ; j!2 Þ
ri ¼nq
h
1 ½ Cð0; 2Þ L
Cðp; qÞ 1
p C B Y X C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r gnmþ1 i¼1 A P 1 p ri ¼nq
ðA5Þ
Therefore, the first two steps hold for the theorem. More steps can be checked for the theorem. Since the nth-order GFRF will involve the GFRFs of lower order from 1 to n-1, now for a proof by induction, suppose that the theorem holds for all the GFRFs with order less than n for system (3), and prove the theorem hold for the nth-order P GFRF. Because 1 n m þ 1 n 1 and 0 ri ¼ n q n, it can be seen from the assumption above that each Hri ðj!Xþ1 ; ; j!Xþri Þ in (A5) is bounded by a polynomial function of jH1 ðÞj with degree ri n 1 . p Q Thus Hri ðj!Xþ1 ; ; j!Xþri Þ in (A5) is bounded by a i¼1
80
X.J. Jing et al.
polynomial function of jH1 ðÞj with degree n-q n. This further yields that, the bound of jHn ðj!1 ; ; j!n Þj can be expressed as a polynomial function of jH1 ðÞj with degree n. Moreover, note that for any a combination of (r1,r2, . . . ,rp), all the frequency variables !1, !2, . . . , !n completely appear in p Q Hri ðj!Xþ1 ; ; j!Xþri Þ without repetition (referring to i¼1
[16] for more discussions). Further investigation of (A3) shows that the kth-degree ( n-q) term in the p Q polynomial Hri ðj!rXþ1 ; ; j!rXþri Þ (after expansion i¼1
it should be a polynomial), which can be expressed as bn;k jH1 ð!r1 ÞH1 ð!r2 Þ H1 ð!rk Þj where 1 {r1 . . . rk} n, must have k different frequency variables. Hence, for a symmetric expression, jH1 ð!r1 ÞH1 ð!r2 Þ H1 ð!rk Þj can be written as
hn;k ¼
1 ðCð0; nÞ þ Cð1; n 1ÞjH1 ðÞj L þ Cð2; n 2ÞjH1 ðÞH1 ðÞj þ þ Cðn; 0Þ
jH1 ðj!1 Þ H1 ðj!n ÞjÞ n1 X 1X L m¼2 pþq¼m
þ
Cðp; qÞ
0fp;qgm
0
1
p C B Y X C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r gnmþ1 i¼1 A P 1 p ri ¼nq
ðA6Þ
Because the magnitude bound of the GFRFs of order less than n can all be expressed as a polynomial function in the following form
k! nðn 1Þ ðn kÞ Yk X i¼1 H1 ðj!ri Þ
Hri ðj!Xþ1 ; ; j!Xþri Þ bri ;0 þ bri ;1 jH1 ðÞj
all the different combinations fr1 ;r2 ;;rk gtaken from f1;2;;ng with no repetition
þ þ bri ;ri jH1 ðÞ H1 ðÞj ¼ bri hTri for |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ri
which has no relationship with the constant coefficient bnk . The detailed polynomial coefficients in the bound expression for jHn ðj!1 ; ; j!n Þj can now be derived by following the similar proofs in [9–10] as follows. It can be shown from inequality (A5) that
1 ri n m þ 1
where bri ¼ ½bri ;0 bri ;1 bri ;ri , hri ¼ ½ 1 jH1 ðÞj jH1 ðÞ H1 ðÞj , it can be derived for the last term |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ri
jHn ðj!1 ; ; j!n Þj
X
1 L pþq¼n
0fp;qgn
0
in inequality (A6) by using the new operators ‘‘’’ and ‘‘’’ that
Cðp; qÞ 1
X
p C B X Y C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r g1 i¼1 A P 1 p
1fr1 rp gnmþ1 i¼1 P ri ¼nq
ri ¼nq
n1 X 1X þ L m¼2 pþq¼m
0
0fp;qgm
¼ Cðp; qÞ
X
1fr P 1 rp gnmþ1 ri ¼nq
1
p C B Y X C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r gnmþ1 i¼1 A P 1 p ri ¼nq
p Y Hri ðj!Xþ1 ; ; j!Xþri Þ
0 B ¼B @
p
bri hnq
i¼1
P
ri ¼nq 1fr1 rp gnmþ1
1 p C br i C A hnq i¼1
81
Frequency-Dependent Magnitude Bounds
which further yields 0
1
p C B X Y C B Cðp; qÞB Hri ðj!Xþ1 ; ; j!Xþri ÞC @1fr r gnmþ1 i¼1 A m¼2 pþq¼m P 1 p 0fp;qgm n1 X
X
ri ¼nq
0
0 ¼
n1 X
X
m¼2 pþq¼m 0fp;qgm
B B BCðp; qÞB @ @
0
¼
n1 X
0
B B BCðp; qÞB @
@ pþq¼m m¼2 0fp;qgm 0
0
B n1 ¼B @
ri ¼nq 1fr1 rp gnmþ1
P
ri ¼nq 1fr1 rp gnmþ1
1
11
p CC C br i C A A hn i¼1
P
1
p C C hnq C br i C A A i¼1
P
B BCðp; qÞ
@ m¼2 pþq¼m 0fp;qgm
ri ¼nq 1fr1 rp gnmþ1
11 p CC C bri C A A hn i¼1
Therefore, it can be shown by substituting the above inequality into (A6) that jHn ðj!1 ; ; j!n Þj
þ
1 ðCð0; nÞ þ Cð1; n 1ÞjH1 ðÞj þ þ Cðn; 0ÞjH1 ðj!1 Þ H1 ðj!n ÞjÞ L 0 1 p C B Y X C B Cðp; qÞB Hri ðj!Xþ1 ; ; j!Xþri ÞC @1fr r gnmþ1 i¼1 A P 1 p 0fp;qgm
n1 X 1X L m¼2 pþq¼m
ri ¼nq
¼
1 ðCð0; nÞ þ Cð1; n 1ÞjH1 ðÞj þ þ Cðn; 0ÞjH1 ðj!1 Þ H1 ðj!n ÞjÞ L 0 0 11 p B CC 1 B n1 T C Cðp; qÞ P þ B B br i C @ @ AA h n ¼ bn hn L m¼2 pþq¼m i¼1 ri ¼nq 0fp;qgm
This proves that the theorem holds for the nth step. Therefore, the theorem follows by induction. &
Appendix B: Proof of Theorem 2 Because cp,q(.) ¼ 0 for all p þ q > 1 and q > 0, the GFRF in (4) for model (3) can be written as (for n > 1) ! K X c1;0 ðk1 Þ expðjð!1 þ þ !n Þk1 Þ 1 k1 ¼1
Hn ðj!1 ; ; j!n Þ ¼
n K X X p¼2 k1 ;kp ¼1
Hn;p ðj!1 ; ; j!n Þ
cp;0 ðk1 ; ; kp Þ
1fr1 rp gnmþ1
which yields 1 Ln ð!1 þ þ !n Þ n X Cðp; 0ÞHn;p ðj!1 ; ; j!n Þ
jHn ðj!1 ; ; j!n Þj
ðB1Þ
p¼2
It can be derived from (A5) that Hn;p ðj!1 ; ; j!n Þ npþ1 p X Y Hri ðj!rXþ1 ; ; j!rXþri Þ rP 1 rp ¼1 i¼1 ri ¼n
ðB2Þ
82
X.J. Jing et al.
Similarly, a proof by induction is adopted here. The first two steps are checked first. For n ¼ 1, it is obvious that jH1 ðj!1 Þj ¼ B1 jH1 ðj!1 Þj B1 H1 ð!Þ For n ¼ 2, jH2 ðj!1 ; j!2 Þj 1 Cð2; 0ÞH2;2 ðj!1 ; j!2 Þ L2 ðj!1 ; j!2 Þ 1 Cð2; 0Þ L2 ðj!1 ; j!2 Þ 1 2 X Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP 1 r2 ¼1 i¼1
When the input of the NARX model (3) is a general input, the output spectrum is [15] N X 1 Yðj!Þ ¼ pffiffiffi nð2Þn1 n¼1 Z n Y Hsym Uðj!i Þd! n ðj!1 ; ; j!n Þ
1 Cð2; 0ÞjH1 ðj!1 ÞjjH1 ðj!2 Þj L2 ðj!1 ; j!2 Þ
ðC1Þ using the mean value theorem and inequality (13a), which gives (18a), i.e., X N 1 jYðj!Þj ¼ pffiffiffi n¼1 nð2Þn1 Z Hsym n ðj!1 ; ; j!n Þ !1 þþ!n ¼!
Uðj!i Þd! i¼1 X N Hsym n ðj!1 ; ; j!n Þ ¼ pffiffiffi n1 n¼1 nð2Þ n Y
¼ B2 ð!1 ; !2 ÞjH1 ðj!1 ÞjjH1 ðj!2 Þj Therefore, the first two steps hold for the theorem. More steps can be checked for the theorem. Since the nth-order GFRF will involve the GFRFs of lower order from 1 to n-1, now for a proof by induction, suppose that the theorem holds for all the GFRFs of order less than n, and prove the theorem hold for the nth-order GFRF. For the nth-order GFRF, it can be derived from (B2) that Hn;p ðj!1 ; ; j!n Þ npþ1 p X Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP 1 rp ¼1 i¼1 ri ¼n
npþ1 X
p Y
rP 1 rp ¼1 ri ¼n
i¼1
i¼1
!1 þþ!n ¼!
ri ¼2
¼
Appendix C: Proof of Corollary 2
Uðj!i Þd! i¼1 !1 þþ!n ¼! N sym X Hn ðj!1 ; ; j!n Þ pffiffiffi nð2Þn1 n¼1 Z n Y jUðj!i Þjd! Z
!1 þþ!n ¼!
¼
!
N X
i¼1
n Hsym n ðj!1 ; ; j!n Þ
n¼1
Bri ð!Xþ1 ; ; !Xþri Þ
N X n¼1
jH1 ðj!1 Þj jH1 ðj!n Þj
¼
N X n¼1
Therefore, it follows from (B1) that 0 jHn ðj!1 ; ; j!n Þj
n Y
n
sup
!1 þþ!n ¼!
n b n
sup
sym H ðj!1 ; ; j!n Þ
!1 þþ!n ¼!
!
n
hTn ðj!1 ; ; j!n Þ 1
npþ1 p n B C X Y X 1 B C Bri ð!Xþ1 ; ; !Xþri Þ jH1 ðj!1 Þj jH1 ðj!n ÞjC BCðp; 0Þ A Ln ðj!1 ; ; j!n Þ p¼2 @ rP i¼1 1 rp ¼1 ri ¼n
¼ Bn ð!1 ; ; !n ÞjH1 ðj!1 Þj jH1 ðj!n Þj
Thus the theorem is proved by induction. &
83
Frequency-Dependent Magnitude Bounds
Similarly, when the system input is a multi-tone signal, the system output spectrum can be written as [15]
Yðj$Þ ¼
N X 1 n 2 !k n¼1
X þþ!kn ¼$ 1
Fð!k1 Þ Fð!kn Þ
Hsym n ðj!k1 ; ; j!kn Þ ðC2Þ
where jFi jejffFi if ! 2 f!k ; k ¼ 1; ; Kg Fol. Fð!Þ ¼ 0 else lowing a similar process, the magnitude bound can be obtained as (19a). When there are only nonlinearities in output, using the bound results in Theorem 2 in (18a) and (19a), the results in (18b) and (19b) can be derived readily. &
Frequency-Dependent Magnitude Bounds of the Generalized Frequency Response Functions for NARX Model
Xing Jian Jing1, , Zi Qiang Lang2, Stephen A. Billings2 1 2
Institute of Sound and Vibration Research, University of Southampton, Southampton, UK Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield, UK
New magnitude bounds of the frequency response functions for the Nonlinear AutoRegressive model with eXogenous input (NARX) are investigated by exploiting the symmetry of the nth-order generalized frequency response function (GFRF) in its n frequency variables. The new magnitude bound of the nth-order symmetric GFRF is frequency-dependent, and is a polynomial function of the magnitude of the first order GFRF. The coefficients of this polynomial function are functions of model parameters. Based on this result, the system output spectrum can also be bounded by an analytical polynomial function of the magnitude of the first order GFRF. The conservatism in the bound evaluations is reduced compared with previous results. Several examples and necessary discussions illustrate the potential application and effectiveness of the new results. Keywords: Generalized Frequency Response Function (GFRF), Nonlinear Systems, Volterra Series, NARX, Frequency-domain analysis
1. Introduction The frequency-domain analysis of nonlinear systems has been studied for many years. It is known that a large class of nonlinear systems have a convergent
Correspondence to: X.J. Jing, E-mail: xingjian.jing@googlemail. com
Volterra series expansion of finite order [4, 18–21]. For this class of nonlinear systems, frequency-domain analysis can be conducted by using the concept of Generalized Frequency Response Function (GFRF) [7], which is the multi-dimensional Fourier transform of the Volterra kernels in the Volterra series expansion. The GFRF represents system underlying characteristics and extends the transfer function concept for linear systems to the nonlinear case, and thus provides a basis for the frequency-domain analysis of nonlinear systems. Based on this concept, many studies have been carried out on the estimation and analysis of the kernel function, GFRFs, and nonlinear system output frequency response for the past decades [1, 8, 11–17, 22]. Evaluation of the magnitude of the GFRFs for a nonlinear system in practice can usually reveal some important information about the system. It is very useful in measuring the significant order of nonlinearities, finding the significant nonlinear terms for system frequency response functions, and indicating system stability and truncation error [2, 3, 5, 9]. Hence, a proper method to evaluate the magnitude of the GFRFs can provide a useful technique in analysis and design of nonlinear systems in the frequency-domain. Several efforts to derive the magnitudes of the GFRFs and output frequency response have been attempted [2, 23]. However, the relationship between the magnitude of the system
Received 3 June 2007; Accepted 1 December 2008 Recommended by D. Nesic, A.J. van der Schaft
69
Frequency-Dependent Magnitude Bounds
frequency response functions and system timedomain model parameters is not revealed clearly in these existing magnitude bound evaluation algorithms, and the bound evaluation is conservative. For this reason, new bound characteristics of the frequency response functions for NARX (Nonlinear AutoRegressive model with eXogenous input) model were studied in [9] recently. The magnitude bound of the nth-order GFRF is expressed into an n-degree polynomial function of the magnitude bound of the first order GFRF. This should provide for the first time an explicitly analytical relationship between the magnitude of the GFRFs and model parameters, and the connection between the magnitude bounds and system stability or truncation error were discussed. However, the bound for the nth-order GFRF in [9] is only a constant with respect to different frequencies, which is the maximum evaluation on the GFRF for all the interested frequency points and is computed in terms of the magnitude bound of the first order GFRF. Hence the conservatism in the bound evaluation is obvious. For this reason, a better result was proposed in [10], where the magnitude bound of the nth-order GFRF is a function of frequency variables instead of a constant (when model parameters are constant) but still in terms of the magnitude bound of the first order GFRF. Especially, the internal parametric structure of the bound results in [10] was discussed in detail there. In addition, all the bound results developed above are for asymmetric GFRFs. In this study, some new frequency-dependent bound results are provided for the nth-order GFRF by exploiting its symmetry in frequency variables. The new bound results have the advantages of the previous results, which can reveal the relationship between the magnitude of GFRFs and any interested model parameters clearly. Moreover, they are functions of the magnitude of the first order GFRF instead of its magnitude bound as the previous results in [2, 9–10, 23]. It is shown that the new bound results achieved in this study can greatly reduce the conservatism in bound evaluations. For nonlinear systems which have only pure output nonlinearities, the bound evaluations can even be much better. These results provide a useful approach to the analysis and understanding of the relationship between system frequency response functions and model parameters which represent system nonlinearities at different frequency points. Potential applications can be found in output frequency response analysis, truncation error evaluation, and stability analysis. Theoretical analysis, necessary
discussions and examples demonstrate usefulness of these bound results.
2. The GFRFs of NARX Model Consider nonlinear systems, whose input output relationship can be approximated by the Volterra series of degree N as [4, 18–21] N Z X
yðtÞ ¼
1
1
n¼1
Z ...
1 1
hn ð1 ; ; n Þ
n Y
uðt i Þdi
i¼1
ð1Þ where hn ( 1, . . . , n)is a real valued function of 1, . . . , n called the nth-order Volterra kernel. The nth-order GFRF is defined as Z 1 Z 1 Hn ðj!1 ; ; j!n Þ ¼ hn ð1 ; ; n Þ 1
1
expðjð!1 1 þ þ !n n ÞÞ d1 dn
ð2Þ
Consider these nonlinear systems described by the following NARX model [6],
yðtÞ ¼
M X
ym ðtÞ
m¼1
ym ðtÞ ¼
m X
K X
cp;q ðk1 ; ; kpþq Þ
p¼0 k1 ; kpþq ¼1 p Y i¼1
yðt ki Þ
pþq Y
uðt ki Þ
ð3Þ
i¼pþ1
where ym(t) is the mth-order output of the NARX K K P P model, p þ q ¼ m, ki ¼ 1, . . . , K, and ðÞ ¼ ðÞ
K P kpþq ¼1
k1 ;kpþq ¼1
k1 ¼1
ðÞ. The nonlinear degree of parameter cp,q( ) is
defined as p þ q. Obviously, parameter cp,q( ) corresponds to the (p þ q)th degree nonlinear terms p pþq p q. Q Q yðt ki Þ uðt ki Þ, e.g., y(t 1) u(t 2) i¼1
i¼pþ1
Parameters c0,1(.) and c1,0(.) are of degree 1 and referred to as linear parameters, and all others are referred to as nonlinear parameters. The GFRFs for nonlinear Volterra systems described by (3) can be obtained through mapping this time-domain model into the frequency-domain using the probing method [17]. A recursive algorithm to
70
X.J. Jing et al.
compute the nth-order GFRF of system (3) is given as follows [16]: ! K X c1;0 ðk1 Þ expðjð!1 þ þ !n Þk1 Þ 1 k1 ¼1
Hn ðj!1 ; ; j!n Þ ¼
K X
Then (4) can be written for more simplicity as Hn ðj!1 ; ; j!n Þ 1 ¼ Ln ðj!1 ; ; j!n Þ nq n X K X X cp;q ðk1 ; ; kpþq Þ q¼0 p¼0 k1 ;kpþq ¼0
c0;n ðk1 ; ; kn Þ
k1 ;kn ¼1
j
e
expðjð!1 k1 þ þ !n kn ÞÞ þ
nq X n1 X K X
cp;q ðk1 ; ; kpþq Þ
expðjð!nqþ1 kpþ1 þ þ !n kpþq ÞÞ Hnq;p ðj!1 ; ; j!nq Þ n K X X
cp;0 ðk1 ; ; kp ÞHn;p ð j!1 ; ; j!n Þ
p¼2 k1 ;kp ¼1
ð4Þ Hn;p ðÞ ¼
npþ1 X
Hi ðj!1 ; ; j!i ÞHni;p1
i¼1
ðj!iþ1 ; ; j!n Þ expðjð!1 þ þ !i Þkp Þ ð5Þ Hn;1 ðj!1 ; ; j!n Þ ¼ Hn ðj!1 ; ; j!n Þ expðjð!1 þ þ !n Þk1 Þ ð6Þ Note that the subscripts of the term expðjð!nqþ1 kpþ1 þ þ !n kpþq ÞÞ in (4) have been corrected (compare with the results in [16]). The nth-order GFRF in (4) is asymmetric with respect to the frequency variables !1, . . . , !n, that is, the value of Hn(j!1, . . . , j!n) may differ with different permutation of !1, . . . , !n. The symmetric GFRFs, which are usually used in practice, can be obtained by symmetrising the asymmetric GFRFs as follows, Hsym n ðj!1 ; ; j!n Þ X 1 Hn ðj!1 ; ; j!n Þ ¼ n! all permutations of
ð7Þ
!1 ;;!n
The bound for the nth-order symmetric GFRF above is focused in this study. Moreover, for convenience in derivation, let H0;0 ðÞ ¼ 1; Hn;0 ðÞ ¼ 0 for n > 0; Hn;p ðÞ ¼ 0 for n < p; and q X 1 q ¼ 0; p > 1 expð ðpÞÞ ¼ 0 q ¼ 0; p 1 i¼1
i¼1
ð!nqþi kpþi Þ
Hnq;p ðj!1 ; ; j!nq Þ
where Ln ðj!1 ; ; j!n Þ ¼ 1
q¼1 p¼1 k1 ;kn ¼1
þ
q P
ð8Þ
K P k1 ¼1
ð9Þ
c1;0 ðk1 Þ expðjð!1 þ
þ !n Þk1 Þ: The recursive algorithm for the computation of GFRFs is (4–6) or (5–6, 8–9). Although the GFRFs above provide an important basis for the analysis of nonlinear systems, it can be seen from (2, 4–6, 8–9) that the GFRF is a multivariate function defined in high-dimensional frequency space. Therefore, the analytical computation of the nth-order GFRF in practice is very complicated in the high-dimensional space, especially when the order n is larger than 5, and the understanding of the analytical relationship between any interested model parameters and the GFRFs is not straightforward. In order to solve these problems, evaluation of the relationship between the magnitude of the GFRFs and system time-domain model parameters is an alternative approach. Some results have been achieved for these purposes recently [9–10]. As discussed before, the bound of the nth-order GFRF in [9] is computed in terms of the magnitude bound of the first order GFRF which is only a constant with respect to different frequencies corresponding to the maximum evaluation on the GFRF for all the interested frequency points. Although the bound result in [10] is a function of frequency variables instead of a constant, it is still a polynomial function of the magnitude bound of the first order GFRF and complicated in the coefficients of the polynomial. To overcome these conservatisms, a sharper evaluation method for the magnitude bound of the nth-order GFRF is expected. Therefore, new frequency-dependent magnitude bounds for the nthorder symmetric GFRF are established by exploiting the symmetry of the GFRFs in the frequency variables for the NARX model in this study. The new bound results are polynomial functions of the magnitude of the first order GFRF instead of its bound, which are frequency-dependent and less conservative compared with the existing results. The new results can reveal more clearly at different frequency points the relationship between the higher order GFRFs which represent system nonlinear effects, and the first order GFRF which represents the system linear component,
71
Frequency-Dependent Magnitude Bounds
and the relationship between any interested model parameters and system frequency response functions.
3. New Magnitude bounds for the GFRFs Some notations and definitions are introduced first, then the new frequency-dependent bound characteristics of the symmetric GFRFs of system (3) are derived and discussed. Denote L¼
inf
!1 þþ!n 2I!
where I! ¼
N S
fjLn ðj!1 ; ; j!n Þjg
ð10Þ
! ¼ !1 þ þ !k ; !i 2 R! g repref!j
k¼1
sents simply all the possible output frequency range when the system is stimulated by an input U(j!), which is defined in a frequency range R!, and N is the maximum order of the Volterra series. Let n ¼ 1 in (9), then gives K P k1 ¼1
H1 ðj!1 Þ ¼
1
¼
2) (a b)(k) ¼ a(k) þ b(k) for 0 k max (m, n), where if k > n or m, then a(k) ¼ 0 or b(k) ¼ 0. For similar definitions based on polynomials can be referred to [10]. Let ðÞ and ðÞ denote the multiple multiplicaðÞ
ðÞ
tion and addition by the operator ‘‘’’ and ‘‘’’ for the terms in (.) satisfying the conditions in (), respectively (which are similar to the operations P Q ðÞ and ðÞ). Moreover, denote Cn ¼ ½Cð0; nÞ, ðÞ
C(1, n-1), . . . ,C(n,0)].
c0;1 ðk1 Þ expðj!1 k1 Þ
k1 ¼1
k1 ¼1
iþj¼k 0in; 0jm
ðÞ
K P
K P
The following two operators ‘‘’’ and ‘‘’’ are needed, which define a multiplication and an addition between two vectors of different dimensions, respectively. Let a(i) denote the ith element of a vector a. Consider two vectors a and b of n and m dimensions, respectively. The two operators are defined as follows: P aðiÞbðjÞ for 0 k m þ n; 1) ða bÞðkÞ ¼
c1;0 ðk1 Þ expðj!1 k1 Þ 3.1. The Main Bound Characteristics
c0;1 ðk1 Þ expðj!1 k1 Þ L1 ðj!1 Þ
ð11Þ
Theorem 1: The magnitude of the nth-order symmetric GFRF of system (3) is bounded by a polynomial function with degree n of the magnitude of the first order GFRF, whose coefficients are functions of the model nonlinear parameters. That is, there exists a series of scalar positive real numbers bn;0 ; bn;1 . . . ; bn;n , such that sym H ðj!1 ; ; j!n Þ bn hT ðj!1 ; ; j!n Þ n n
This is the first order GFRF of the nonlinear system (3), which is the frequency response function of the linear elements of the system, representing the dynamics of the linear component in the system (i.e., the transfer function of the system when all the nonlinear parameters in the system model are set to be zero). Therefore, the relationship between the nthorder GFRF and the first order GFRF can reveal the dependence of high order GFRFs on the linear component of the system. Define 8 K P > > cp;q ðk1 ; ; kpþq Þ; > 1 q n 1; 1 p n q > > > k1 ;kpþq ¼1 > > K > > < P c0;n ðk1 ; ; kn Þ; q ¼ n; p ¼ 0 Cðp; qÞ ¼ k1 ;kn ¼1 > > K > P > > cp;0 ðk1 ; ; kp Þ; > q ¼ 0; 2 p n > > > > : k1 ;kp ¼1 0; else By definition, C(p, q) is a non-negative function of the parameter cp,q(.) defined on all 0 {p, q} n (which means 0 p n and 0 q n).
ð13aÞ
ð12Þ
72
X.J. Jing et al.
where hn ðj!1 ; ; j!n Þ ¼ ½ 1 hn;1 ðj!1 ; ; j!n Þ hn;n ðj!1 ; ; j!n Þ, and for k ¼ 0; 1; . . . ; n: hn;k ðj!1 ; ; j!n Þ k! ¼ nðn 1Þ ðn kÞ X all the different combinations fr1 ;r2 ;;rk gtaken from f1;2;;ng with no repetition
1 ¼ sup jH1 ðj!Þj, then it can be Remark 2: Let H !2R! derived from (13b) that k! nðn 1Þ ðn kÞ Yk X i¼1 H1 ðj!ri Þ
hn;k ðj!1 ; ; j!n Þ ¼ Yk i¼1 H1 ðj!ri Þ
all the different combinations fr1 ;r2 ;;rk g taken from f1;2;;ngwith no repetition
k! nðn 1Þ ðn kÞ X k ¼ H k ð14Þ H 1 1
ð13bÞ bn ¼ ½bn;0 ; bn;1 . . . ; bn;n can be recursively determined as follows 0 bn ¼
0
0
B n1 1B BC n þ B @m¼2 pþq¼m L@ 0fp;qgm
B BCðp; qÞ @
P
ri ¼nq 1fr1 rp gnmþ1
b1 ¼ ½0; 1
ð13dÞ
p
n
i¼1
m¼2
all the different combinations fr1 ;r2 ;;rk g taken from f1;2;;ngwith no repetition
where bri ¼ 0 if p<1 and ðÞ ¼ 0 if n < 2.
Proof. See Appendix A. & Note that a combination {r1,r2, . . . ,rp} taken from {1,2, . . . ,n} without repetition in (13b) refer to p different numbers taken from {1,2, . . . ,n}. For example, {1,2, . . . ,p-1,p} and {1,2, . . . ,p-1,n} are such two different combinations, and {1,2, . . . ,p-1,p} and {1,2, . . . , p,p-1} are the same combination. Remark 1: Theorem 1 provides an explicit polynomial expression for the magnitude bounds of the GFRFs for system (3) in terms of the magnitude of the first order GFRF. The new result not only demonstrates an analytical relationship between the magnitude bound of the nth-order GFRF and system time-domain model parameters, but also reveals the relationship between the magnitudes of the higher order GFRFs, which represents the nonlinear effects of the model, and the magnitude of the first order GFRF, which represents the linear component of the model. Therefore, the effect of nonlinear components and the role of the linear component in the system described by (3) can be analyzed. This provides a useful insight into the frequency-domain analysis and design of nonlinear systems. &
1 11 p C CC CC br i C AAA for n > 1 i¼1
ð13cÞ
Using the above inequality and (13a) in Proposition 1, it can be shown that sym H ðj!1 ; ; j!n Þ bn hT ðj!1 ; ; j!n Þ n n ¼ bn;0 þ bn;1 hn;1 ðj!1 Þ þ þ bn;n hnn;n ðj!1 ; ; j!n Þ 1 þ þ bn;n H n bn;0 þ bn;1 H 1
ð15Þ
The second inequality in (15) is or is equivalent to the result obtained in [2,23,9]. From (15) it is obvious that the result in Theorem 1 is less conservative than those in [2,23,9], and shares the same advantage of an explicit relationship between the magnitude bound and model parameters as [9]. Moreover, compared with the previous results in [2,23,9], the magnitude bound of the nth-order GFRF in Theorem 1 is a function of the frequency variables, and can directly be computed by only using the model parameters and jH1(j!)j with no need to recursively compute the lower order GFRFs as the results in [2,23] or no need to evaluate the magnitude bound for H1(j!) as the results in [9]. Thus the conservatism and computational complexity for the bound evaluation are all reduced. Some existing algorithms in [2, 3, 5, 9] for evaluation of system truncation error and convergence where the GFRF bounds are involved can therefore be improved. It shall also be noted from (14) that, exploiting the symmetry in the frequency variables k to be relaxed to enables the magnitude boundQH P k1 i¼1 H1 ðj!ri Þ to reduce all the different combinations fr1 ;r2 ;;rk g taken from f1;2;;ngwith no repetition
73
Frequency-Dependent Magnitude Bounds
the conservatism of the previous results in [2,9,10,23], combinawith the only cost to compute the nðn1ÞðnkÞ k! tions but removing the cost to evaluate the bound for H1 ðj!Þ. &
Corollary 1: The magnitude of the nth-order GFRF can be bounded by sym H ðj!1 ; ; j!n Þ n (
Remark 3: In the derivation of the result in Theorem 1 (see Appendix A), if the maximum evaluation on the terms involving lower order GFRF is made p Q P (i.e., Hri ðj!Xþ1 ;;j!Xþri ÞÞ is made by 1fr P 1 rp gnmþ1 i¼1
min bn hTn ðj!1 ; ; j!n Þ; X
jHn ðj!1 ; ; j!n Þj bn ðj!1 ; ; j!n Þ hTn
ð16aÞ
n , and bn (!1, . . . , !n) is 1 H where hn ¼ ½ 1 H 1 a function of frequency variables and can be recursively determined by bn ðj!1 ; ; j!n Þ ¼
0
B B n1 B 1 BC n þ B B @Cðp; qÞ @ @ Ln ðj!1 ; ; j!n Þ m¼2 pþq¼m 0p;qm
ð17Þ
where bn is computed by (13cd), bn (!1, . . . , !n) is computed by (16b), hn ðj!1 ; ; j!n Þ is defined by (13b), 1 ¼ sup jH1 ðj!Þj . 1 H n , and H hn ¼ ½ 1 H 1
!2R!
This is another new bound evaluation for the nthorder GFRF, which has both the advantages of the two bound results in Theorem 1 and in [10] (i.e., (16a-b) ). Simulation studies in Section 4 will show that the bound result in Theorem 1 has obviously much better
0
0
bn ðj!1 ; ; j!n Þ
) hTn
all the permutations off!1 ;;!n g
ri ¼nq
using the magnitude bound of H1(.) and leave the other frequency-dependent terms (i.e., Ln ðj!11;;j!n Þ ) intact, (see inequality (A5) in the proof of Theorem 1), then the result in [10] can be achieved by following a similar proof process, i.e.,
1 n!
11 1 p CC C CC bri ð!XðiÞþ1 ; ; !XðiÞþri Þ C AA A i¼1
P
ri ¼nq 1fr1 rp gnmþ1
ð16bÞ It can be shown that sup bn;k ðj!1 ; ; j!k Þ
!1 ;;!n
hn;k ¼ bn;k ; sup hn;k ðj!1 ; ; j!k Þ ¼ !1 ;;!k
ð16cÞ
Although (16a-b) is also less conservative than the results in [2,23,9], it is not necessarily of obvious advantage compared with the result in Theorem 1 at any frequency point because the bound in [10], i.e., (16a-b), is still a function of the magnitude bound of the first order GFRF. Further study will show that, the bound result in Theorem 1 has less conservatism than (16a-b) at some frequency points but not for any frequency points. They are two different results from different viewpoints. Noticeably, the bound characteristics in Theorem 1 are in terms of the magnitude of the first order GFRF instead of its magnitude bound; the recursive computation of (16b) is clearly more complicated to determine the frequency variables in each recursion than (13c), while (13b) only involves some fixed combinations. In order to have both advantages of these two results in reducing conservatism of the bound evaluation, the following corollary is straightforward.
evaluation than the results in (16a-b) of [10] at some frequency points, and the bound result in Corollary 1 has remarkably improvement in bound evaluation. Moreover, the advantage of these two different results in relaxing the two parts bn (j!1, . . . , j!n) and hn (j!1, . . . , j!n) as mentioned in Remarks 2–3 can also reach together with much sharper performance for nonlinear systems with only nonlinearities in output. This will be discussed later in the following section. 3.2. A special case As discussed before, when nonlinear systems have only nonlinearities in output, the nth-order GFRF determined Q by (4–6,9) is obviously symmetrical for the part ni¼1 H1 ðj!ri Þ in the bound expression (13b). In this case, the bound evaluation by the results discussed in Section 3.1 can be improved to have much less conservatism so that both of the advantages of the results in Theorem 1 and (16a-b) of [10] can reach. The following result can be obtained.
74
X.J. Jing et al.
Theorem 2: The magnitude of the GFRFs for system (3) with only pure input nonlinearities is bounded by sym H ðj!1 ; ; j!n Þ n
jH1 ðj!1 Þj jH1 ðj!n Þj n! X Bn ð!1 ; ; !n Þ
1 where n ¼ pffiffinð2Þ n1
N X
n
sup
!1 þþ!n ¼!
ðBn ð!1 ; ; !n Þ
!
ð18bÞ
1
m C Y C Bri ð!Xþ1 ;;!Xþri ÞC; A i¼1
ri ¼n 1fr1 rm gnmþ1
iP 1
jUðj!i Þjd! . For the
hn;n ðj!1 ; ; j!n ÞÞ
where Bn ð!1 ; ; !n Þ ¼ Ln ðj!11;;j!n Þ 0
B1 ¼ 1 and X ¼
!1 þþ!n ¼! i¼1
jYðj!Þj
n¼1
X
n Q
case of Theorem 2,
all the permutations off!1 ;;!n g
n B X B BCðm;0Þ P @ m¼2
R
When the system is subject to a multi-tone input K P described by uðtÞ ¼ jFi j cosð!i t þ ffFi Þ, then the i¼1
system output spectrum is bounded by
rx .
N X
jYðj$Þj
x¼1
n b n
n¼1
Proof. See Appendix B. &
sup
!k1 þþ!kn ¼$
hTn ðj!k1 ; ; j!kn Þ ð19aÞ
It can be verified that Bn (!1, . . . , !n) in Theorem 2 is bn,n (!1, . . . , !n) in (16a-b) corresponding to the jH1 ðj!n Þj bound result in [10], and jH1 ðj!1 Þj in Theorem 2 n! is hn;n (!1, . . . , !n) in (13b) of Theorem 1. Therefore, both the advantages of the two results are integrated here. The advantage of this result will be discussed in an example study in Section 4.
where n ¼ 2n Fð!Þ ¼
P !k1 þþ!kn ¼$
jFi jejffFi 0
jFð!k1 Þ Fð!kn Þj,
if ! 2 f!k ; k ¼ 1; ; Kg , else
$ ¼ !k1 þ þ !kn !k1 ; ; !kn 2 R! ¼ f!k ; k ¼ 1; ; Kg, and ! k ¼ !k . For the case of Theorem 2, jYðj$Þj N X
3.3. Bounds for Output Spectrum Based on Theorem 1 and Theorem 2, the magnitude bound of the output spectrum for the NARX 2 I! , where model (3) can also be derived. For any ! I! denotes the non-negative frequency range of the output spectrum of model (3), let sup
!1 þþ!n ¼!
hn ðj!1 ; ; j!n Þ
"1
sup
!1 þþ!n ¼!
¼
sup
# hn;n ðj!1 ; ; j!n Þ
Corollary 2: When the input of the NARX model (3) is a general input with spectrum U(j!) defined in R!, then the system output spectrum is bounded by jYðj!Þj
N X n¼1
n bn
sup
!1 þþ!n ¼!
n
sup
!1 þþ!n ¼!
ðBn ð!1 ; ; !n Þ
!
ð19bÞ
hn;n ðj!1 ; ; j!n ÞÞ Proof. See Appendix C. & 1 Note that n ¼ pffiffinð2Þ n1
R
n Q
!1 þþ!n ¼! i¼1
jUðj!i Þjd!
can be computed by using an algorithm proposed in [2].
hn;1 ðj!1 ; ; j!n Þ
!1 þþ!n ¼!
n¼1
Tn ðj!1 ; ; j!n Þ h ð18aÞ
Remark 4: Corollary 2 shows that the magnitude of the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order GFRF, whose coefficients are the functions of the model parameters and input magnitude. This reveals an analytical relationship between the magnitude of system output spectrum and model parameters, and also the relationship between system output spectrum and the linear component of the model. This polynomial expression can consequently facilitate the analysis of system output spectrum in terms of model parameters and the linear part of the model. In addition, the bound
75
Frequency-Dependent Magnitude Bounds
results in Corollary 2 are also better than those in [2, 3, 23, 9] since the new bound of the nth-order GFRF is less conservative as discussed in Remarks 2–3. &
Hence, the magnitude bound of the nth-order GFRF for (20) is sym H ðj!1 ; ;j!n Þ bn hT ¼ bn;n hn;n n n ¼
4. Examples Two examples are given to further discuss and demonstrate the theoretical results in Theorems 1–2 and Corollaries 1–2. Example 1: To illustrate the results in Theorem 1 and Corollary 1, consider a simple case of model (3) as M X
K X
m¼p¼1
k1 ;kp ¼1
yðtÞ ¼
þ ðm 1Þ
cp;0 ðk1 ; . . . ; kp Þ
K X
p Y
yðt ki Þ
i¼1
!
c0;1 ðk1 Þuðt k1 Þ
ð20Þ
1 x¼0 . Compared with model (3), 0 else there are only nonlinearities in output in (20). In this case, all the other nonlinear parameters are zero except for the nonlinear parameters of the form cp ; 0ð:Þ. The bound polynomial coefficient bn (bn,0, b n,1, . . . , b n,n) up to the third order can be computed as where ðxÞ ¼
b2 ¼ ½0; 0; Cð2; 0Þ=L 0 1B b3 ¼ B C3 þ Cð2; 0Þ L@
P
ri ¼3 1fr1 ;r2 g2
2
br i
i¼1
1 C C A
1 ¼ ð½0; 0; 0; Cð3; 0Þ þ Cð2; 0Þ 2ðb1 b2 ÞÞ L ¼
1 ½0; 0; 0; Cð3; 0Þ L
! 2 þ Cð2; 0Þ ½0; 0; 0; Cð2; 0Þ L
¼ ½0; 0; 0;
1 2 Cð3; 0Þ þ 2 Cð2; 0Þ2 L L
It can be further verified that for n > 1 bn;k ¼ 0 for 0 k < n bn;n ¼
ri ¼n 1fr1 rp gnmþ1
Cð2; 0Þ jH1 ðj!1 ÞH1 ðj!2 Þj L
i¼1
ð21aÞ
Note that (21a) is the result obtained from Theorem 1. A similar result can be derived by using the results in [10] as sym H ðj!1 ; j!2 Þ 2
Cð2; 0Þ 2 H 1 L2 ðj!1 ; j!2 Þ
ð21bÞ
Use the results in [9], yielding sym H ðj!1 ; j!2 Þ Cð2; 0Þ H 2 1 2 L
ð21cÞ
According to Corollary 1, it can be obtained that sym H ðj!1 ; j!2 Þ 2 ( Cð2; 0Þ jH1 ðj!1 ÞH1 ðj!2 Þj; min L ) Cð2; 0Þ 2 H 1 L2 ðj!1 ; j!2 Þ To compare these results, consider the model parameters in (20) are: c1,0(1) ¼ -5, c1,0(2) ¼ -1, c0,1(1) ¼ 2 and C(2,0) ¼ c2,0(1,0) ¼ 10; all the other linear parameters are zero. That is, (20) can now be written as yðt 2Þ þ 5yðt 1Þ þ yðtÞ
n1 1 Cðn; 0Þ þ Cðm; 0Þ L m¼2 ! m bri ;ri P ri ¼n 1fr1 rp gnmþ1
i¼1
Note that in this case, the magnitude bound for the symmetric and asymmetric GFRFs are Q the same since changing the order of !r1 !rn in ni¼1 H1 ðj!ri Þ has no effect on the result. Consider the second order GFRF as an example, sym H ðj!1 ; j!2 Þ b2 hT ¼ b2;2 h2;2 2 2 ¼
k1 ¼1
n1 1 Cðn; 0Þ þ Cðm;0Þ L m¼2 ! Y m n b H ðj! Þ ri ;ri ri P i¼1 1
þ 10y2 ðt 1Þ ¼ 2uðt 1Þ Then from (9) and (11), L2 ðj!1 ; j!2 Þ ¼ ej2ð!1 þ!2 Þ þ 5ejð!1 þ!2 Þ þ 1 L ¼ 3
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X.J. Jing et al.
H1 ðj!1 Þ ¼
ej2!1
2ej!1 1 ¼ 0:6667 H þ 5ej!1 þ 1
sym H ðj!1 ; j!2 Þ 2 Cð2; 0ÞH1 ðj!1 ÞH1 ðj!1 Þ j! jð! þ! Þ 2 1 2 ¼ e e L2 ðj!1 ; j!2 Þ 10H1 ðj!1 ÞH1 ðj!1 Þ ¼ ð21eÞ L2 ðj!1 ; j!2 Þ Clearly, (21e) is the true value of the magnitude of the second order GFRF. Comparisons of these different evaluations on the magnitude bounds are shown in Figs. 1–4. It can be obviously seen that the evaluation by (21a) is much better than those by (21b) and (21c), and (21d) is much better than all of (21a-c). In Fig. 4, it can be seen that only the peak in the middle of the diagram in Fig. 3 is cut off by the valley in the middle of the diagram in Fig. 2, while other places are kept as they are in Fig. 3. These show that the evaluation by (21a) is much better than (21b) at most frequency points in the studied case of this example. &
Fig. 2. Magnitude of the second order GFRF and its bound by (21b)
Thus,
Remark 5: For the simple case in Example 1, it can be derived from Theorem 2 that B2 ð!1 ; !2 Þ ¼ 2 X
1 L2 ðj!1 ; j!2 Þ
Cðm; 0Þ
m¼2
X P
m Y
! Bri ð!Xþ1 ; ; !Xþri Þ
i¼1 ri ¼2 1r1 rp 2mþ1
¼
Cð2; 0Þ L2 ðj!1 ; j!2 Þ
Fig. 1. Magnitude of the second order GFRF and its bound by (21c)
Fig. 3. Magnitude of the second order GFRF and its bound by (21a)
Fig. 4. Magnitude of the second order GFRF and its bound by (21d)
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Frequency-Dependent Magnitude Bounds
sym H ðj!1 ; j!2 Þ jH1 ðj!1 Þj jH1 ðj!2 Þj 2 2! X Cð2; 0Þ L ðj!1 ; j!2 Þ all the permutations 2
jY2 ðjÞjC13 ð ¼
off!1 ;;!n g
Cð2; 0Þ ¼ jH1 ðj!1 Þj jH1 ð j!n Þj L2 ðj!1 ; j!2 Þ
ð22Þ
This is right the result in (21e), which is the real value of the magnitude of H2(.) for the studied case. Thus there is no conservatism in the bound (22) for H2(.). Moreover, the magnitude expression of the second order GFRF can be directly obtained by using Theorem 2 as shown above and the relationship between C(2,0), jH1(.)j and jH2(.)j is clear, while manual manipulation must be used if the recursive algorithm in (4–9) is adopted to compute the magnitude and the computation burden will be worse when high order (larger than 5) GFRFs are evaluated. This will facilitate the analysis of nonlinear systems based on GFRFs. & Example 2: To demonstrate the results in Corollary 2, still consider system (20) in Example 1, and additionally suppose it is subject to a single sinusoidal input signal described by uðtÞ ¼ Fd sinðtÞ ðFd > 0Þ . Then from (19a), it can be shown that
jYðjÞj
ðN1Þ=2 X
Cn2nþ1 ð
n¼0
sup
Fd 2nþ1 Þ b2nþ1;2nþ1 2
!1 þþ!2nþ1 ¼
Fd 3 Þ b3;3 sup h T3;3 ðj!k1 ;;j!k3 Þ 2 !1 þþ!3 ¼
3F3d 2Cð2;0Þ2 Cð3;0Þ ð Þ þ 8 L L2
supðH1 ðÞH1 ðÞH1 ðÞÞY2
The conditions for C(2,0), C(3,0) and H1() can therefore be determined. Note that the output spectrum is usually determined by its first several orders in practice. Therefore, when the requirements on these orders of output spectrum are known, the conditions on model parameters can then be determined simply by using this method. Compared with the direct analytical computations by using (4–6) and (C1-C2) in Appendix C, this method can determine the magnitude bound of the output spectrum up to any high order, while the analytical computation by using (4–6) and (C1-C2) in Appendix C will be hard to follow for the order larger than 5 in practice. For the case in this example, it is very easy to compute jY2(j)j by using the bound results of this study as shown but it will be much more complicated by using (4–6) and (C1-C2) in Appendix C. More examples can be seen in [9] where the bound was computed up to the 41st order. By using the bound results, the truncation error of the system can also be evaluated. For example, if the bound of the nth-order output spectrum Yn(jx03A9;) come down below a predefined threshold, the system can be regarded to have a sufficient Volterra series expansion at that order. More detailed discussions can be referred to [3]. &
hT2nþ1;2nþ1 ðj!k1 ; ; j!kn Þ ð23Þ
where bc is to take the integer part from (.), nðn1Þðnmþ1Þ . Cm n ¼ mðm1Þ2 By using inequality (23), analysis of the output frequency response of system (20) when subject to the input uðtÞ ¼ Fd sinðtÞ ðFd > 0Þ can be conducted. For instance, a sufficient convergent condition for the system output spectrum can be derived since the right side of inequality (23) can be regarded as a power series [4, 5, 9, 18–21], which consequently leads to an evaluation on system stability. Also given a specific requirement on output spectrum, the sufficient conditions on model parameters under which the requirement is achieved can be derived. For example, if jY2(j)j is to be designed such that jY2(j)j Y2 , then from (23) and Theorem 1, it can be achieved by designing the second order output spectrum to satisfy
5. Conclusions New frequency-dependent magnitude bounds for the GFRFs and output spectrum of nonlinear systems described by the NARX model have been derived in this paper. The new frequency-dependent bound results for the GFRFs are less conservative, and developed for the symmetric GFRFs, compared with the existing bound evaluation results. These new results can clearly demonstrate the relationship between the magnitude of higher order GFRF, which represents the nonlinear effect, and the magnitude of the first order GFRF, which represents the linear component in nonlinear systems. A much better result is also developed for nonlinear systems which have only nonlinearities in output which include all the advantages of the currently existing bound results. These results should be helpful to the analysis and design of nonlinear systems in the frequency-domain.
78
Further study will focus on application problems of the results established in this paper.
6. Acknowledgement The authors gratefully acknowledge the constructive comments and suggestions from anonymous reviewers and the support of the EPSRC-Hutchison Whampoa Dorothy Hodgkin Postgraduate award for this work.
References 1. Bedrosian E, Rice SO. The output properties of Volterra systems driven by harmonic and Gaussian inputs. Proc Inst Electr Electron Eng 1971; 59: 1688– 1707 2. Billings SA, Lang ZQ. A bound of the magnitude characteristics of nonlinear output frequency response functions. Int J Control 1996; 65(2): 309–328 3. Billings SA, Lang ZQ. Truncation of nonlinear system expansions in the frequency domain. Int J Control 1997; 68(5): 1019–1042 4. Boyd S, Chua LO. Fading memory and the problem of approximating nonlinear operators with Volterra series, IEEE Trans Circuits Syst 1985; CAS-32: 1150–1161 5. Chatterjee A, Vyas NS. Convergence analysis of Volterra series response of non-linear systems subject to harmonic excitation. J Sound Vibr 2000; 236: 339–358 6. Chen S, Billings SA. Representation of non-linear systems: the NARMAX model. Int J Control 1989; 49: 1012–1032 7. George DA. Continuous nonlinear systems. Technical Report 355. MIT Research Laboratory of Electronics: Cambridge, MA, 1959 8. Jing XJ, Lang ZQ, Billings SA. The parametric characteristic of frequency response functions for nonlinear systems. Int J Control 2006; 79(12) 1552–1564 9. Jing XJ, Lang ZQ, Billings SA. New bound characteristics of NARX model in the frequency domain. Int J Control 2007; 80(1): 140–149 10. Jing XJ, Lang ZQ, Billings SA. Magnitude bounds of generalized frequency response functions for nonlinear Volterra systems described by NARX model. Automatica 2008; 44: 838–845 11. Jing XJ, Lang ZQ, Billings SA. Mapping from parametric characteristics to generalized frequency response functions of non-linear systems. Int J Control 2008; 81 (7): 1071–1088 12. Jing XJ, Lang ZQ, Billings SA. Determination of the analytical parametric relationship for output spectrum of Volterra systems based on its parametric characteristics. J Math Anal Appl 2009; 351: 694–706 13. Kim KI, Powers EJ. A digital method of modelling quadratic nonlinear systems with a general random input. IEEE Trans Acoust Speech Signal Process 1988; 36: 1758–1769
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14. Kotsios S. Finite input/output representative of a class of Volterra polynomial systems. Automatica 1997; 33: 257–262 15. Lang ZQ, Billings SA. Output frequency characteristics of nonlinear systems. Int J Control 1996; 64(6): 1049– 1067 16. Peyton-Jones JC, Billings SA. Recursive algorithm for computing the frequency response of a class of nonlinear difference equation models. Int J Control 1989; 50(5): 1925–1940 17. Rugh WJ. Nonlinear System Theory—The Volterra/ Wiener Approach. Johns Hopkins University Press: Baltimore and London, 1981 18. Sandberg IW. Expansions for nonlinear systems. Bell Syst Tech J 1982; 61(2): 159–199 19. Sandberg IW. Volterra expansions for time-varying nonlinear systems. Bell Syst Tech J 1982; 61(2): 201–225 20. Sandberg IW. On Volterra expansions for time-varying nonlinear systems. IEEE Trans Circuits Syst 1983; CAS30: 61–67 21. Sandberg IW. The mathematical foundations of associated expansions for mildly nonlinear systems. IEEE Trans Circuits Syst 1983; CAS-30: 441–455 22. Worden K, Manson G, Volterra A. Series approximation to the coherence of the duffing oscillator. J Sound Vibr 2005; 286: 529–547 23. Zhang H, Billings SA. Gain bounds of higher order nonlinear transfer functions. Int J Control 1996; 64(4): 767–773
Appendix A: Proof of Theorem 1 Using the notations in (8, 10–12), it follows from (9) that jHn ðj!1 ; ; j!n Þj nq n X X
1 Ln ðj!1 ; ; j!n Þ
Cðp; qÞHnq;p ðj!1 ; ; j!nq Þ
ðA1Þ
q¼0 p¼0
It can be easily seen that n nq P P Cðp; qÞ Hnq;p ðj!1 ; ; j!nq Þ includes all the q¼0 p¼0
combinations of (p,q) satisfying p þ q ¼ m, 0 fp; qg m, and m ¼ 2, . . . , n. Hence, nq n X X
Cðp; qÞHnq;p ðj!1 ; ; j!nq Þ
q¼0 p¼0
¼
n X
X
Cðp; qÞHnq;p ðj!1 ; ; j!nq Þ
m¼2 pþq¼m 0fp;qgm
ðA2Þ
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Frequency-Dependent Magnitude Bounds
Note that (5) can also be written as Hn;p ðj!1 ; ; j!n Þ ¼
npþ1 X
p Y
rP 1 rp ¼1 i¼1 ri ¼n
Hri ðj!Xþ1 ; ; j!Xþri Þ
expððj!Xþ1 þ þ j!Xþri Þki Þ; where X ¼
i1 X
ðA3Þ
rx
In the following, for a proof by induction, the first two steps, i.e., n ¼ 1 and 2 are checked first to show the results; then under the assumption that (13a-d) hold for all the order less than n, it is shown that (13a-d) hold for n. For n ¼ 1, it is a special case which is not included in (13c) and defined in (13d) specifically. Using (13d), the bound sym of jH1(j!1)j Tcan be expressed as jH1(j!1)j ¼ H ðj!1 Þ ¼ b1 h . For n ¼ 2, it is the beginning 1 1 step of the recursion in (13c). It can be shown from (A1-A2) or (A5) that
x¼1
1 Cð0; 2Þ þ Cð1; 1Þ L2 ðj!1 ; j!2 Þ H1;1 ðj!1 Þ þ Cð2; 0ÞH2;2 ðj!1 Þ 1 Cð0; 2Þ þ Cð1; 1ÞjH1 ðj!1 Þj L þCð2; 0ÞjH1 ðj!1 ÞH2 ðj!2 Þj
jH2 ðj!1 ; j!2 Þj
thus Hn;p ðj!1 ; ; j!n Þ p npþ1 X Y H ðj! ; ; j!Xþri Þ ¼ r r ¼1 i¼1 ri Xþ1 1 p P ri ¼n
exp jð!Xþ1
þ þ !Xþri Þki
Hence
p Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP 1 rp ¼1 i¼1 npþ1 X
sym H ðj!1 ; j!2 Þ 2 1 jH1 ðj!1 Þj þ jH1 ðj!2 Þj Cð0; 2Þ þ Cð1; 1Þ L 2 þ Cð2; 0ÞjH1 ðj!1 ÞH2 ðj!2 Þj
ri ¼n
which yields Hnq;p ðj!1 ; ; j!n Þ nqpþ1 p X Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP i¼1 1 rp ¼1
¼ ðA4Þ
Therefore, from (A1-A2) and (A4) it can be derived that jHn ðj!1 ; ; j!n Þj n X X 1 Ln ðj!1 ; ; j!n Þ m¼2 pþq¼m 0fp;qgm
0
1
Cð1; 1Þ
jH1 ðj!1 ÞjþjH1 ðj!2 Þj 2
Cð2; 0Þ
jH1 ðj!1 ÞH2 ðj!2 Þj
iT
¼ b2 hT2 ðj!1 ; j!2 Þ
ri ¼nq
h
1 ½ Cð0; 2Þ L
Cðp; qÞ 1
p C B Y X C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r gnmþ1 i¼1 A P 1 p ri ¼nq
ðA5Þ
Therefore, the first two steps hold for the theorem. More steps can be checked for the theorem. Since the nth-order GFRF will involve the GFRFs of lower order from 1 to n-1, now for a proof by induction, suppose that the theorem holds for all the GFRFs with order less than n for system (3), and prove the theorem hold for the nth-order P GFRF. Because 1 n m þ 1 n 1 and 0 ri ¼ n q n, it can be seen from the assumption above that each Hri ðj!Xþ1 ; ; j!Xþri Þ in (A5) is bounded by a polynomial function of jH1 ðÞj with degree ri n 1 . p Q Thus Hri ðj!Xþ1 ; ; j!Xþri Þ in (A5) is bounded by a i¼1
80
X.J. Jing et al.
polynomial function of jH1 ðÞj with degree n-q n. This further yields that, the bound of jHn ðj!1 ; ; j!n Þj can be expressed as a polynomial function of jH1 ðÞj with degree n. Moreover, note that for any a combination of (r1,r2, . . . ,rp), all the frequency variables !1, !2, . . . , !n completely appear in p Q Hri ðj!Xþ1 ; ; j!Xþri Þ without repetition (referring to i¼1
[16] for more discussions). Further investigation of (A3) shows that the kth-degree ( n-q) term in the p Q polynomial Hri ðj!rXþ1 ; ; j!rXþri Þ (after expansion i¼1
it should be a polynomial), which can be expressed as bn;k jH1 ð!r1 ÞH1 ð!r2 Þ H1 ð!rk Þj where 1 {r1 . . . rk} n, must have k different frequency variables. Hence, for a symmetric expression, jH1 ð!r1 ÞH1 ð!r2 Þ H1 ð!rk Þj can be written as
hn;k ¼
1 ðCð0; nÞ þ Cð1; n 1ÞjH1 ðÞj L þ Cð2; n 2ÞjH1 ðÞH1 ðÞj þ þ Cðn; 0Þ
jH1 ðj!1 Þ H1 ðj!n ÞjÞ n1 X 1X L m¼2 pþq¼m
þ
Cðp; qÞ
0fp;qgm
0
1
p C B Y X C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r gnmþ1 i¼1 A P 1 p ri ¼nq
ðA6Þ
Because the magnitude bound of the GFRFs of order less than n can all be expressed as a polynomial function in the following form
k! nðn 1Þ ðn kÞ Yk X i¼1 H1 ðj!ri Þ
Hri ðj!Xþ1 ; ; j!Xþri Þ bri ;0 þ bri ;1 jH1 ðÞj
all the different combinations fr1 ;r2 ;;rk gtaken from f1;2;;ng with no repetition
þ þ bri ;ri jH1 ðÞ H1 ðÞj ¼ bri hTri for |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ri
which has no relationship with the constant coefficient bnk . The detailed polynomial coefficients in the bound expression for jHn ðj!1 ; ; j!n Þj can now be derived by following the similar proofs in [9–10] as follows. It can be shown from inequality (A5) that
1 ri n m þ 1
where bri ¼ ½bri ;0 bri ;1 bri ;ri , hri ¼ ½ 1 jH1 ðÞj jH1 ðÞ H1 ðÞj , it can be derived for the last term |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ri
jHn ðj!1 ; ; j!n Þj
X
1 L pþq¼n
0fp;qgn
0
in inequality (A6) by using the new operators ‘‘’’ and ‘‘’’ that
Cðp; qÞ 1
X
p C B X Y C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r g1 i¼1 A P 1 p
1fr1 rp gnmþ1 i¼1 P ri ¼nq
ri ¼nq
n1 X 1X þ L m¼2 pþq¼m
0
0fp;qgm
¼ Cðp; qÞ
X
1fr P 1 rp gnmþ1 ri ¼nq
1
p C B Y X C B Hri ðj!Xþ1 ; ; j!Xþri ÞC B @1fr r gnmþ1 i¼1 A P 1 p ri ¼nq
p Y Hri ðj!Xþ1 ; ; j!Xþri Þ
0 B ¼B @
p
bri hnq
i¼1
P
ri ¼nq 1fr1 rp gnmþ1
1 p C br i C A hnq i¼1
81
Frequency-Dependent Magnitude Bounds
which further yields 0
1
p C B X Y C B Cðp; qÞB Hri ðj!Xþ1 ; ; j!Xþri ÞC @1fr r gnmþ1 i¼1 A m¼2 pþq¼m P 1 p 0fp;qgm n1 X
X
ri ¼nq
0
0 ¼
n1 X
X
m¼2 pþq¼m 0fp;qgm
B B BCðp; qÞB @ @
0
¼
n1 X
0
B B BCðp; qÞB @
@ pþq¼m m¼2 0fp;qgm 0
0
B n1 ¼B @
ri ¼nq 1fr1 rp gnmþ1
P
ri ¼nq 1fr1 rp gnmþ1
1
11
p CC C br i C A A hn i¼1
P
1
p C C hnq C br i C A A i¼1
P
B BCðp; qÞ
@ m¼2 pþq¼m 0fp;qgm
ri ¼nq 1fr1 rp gnmþ1
11 p CC C bri C A A hn i¼1
Therefore, it can be shown by substituting the above inequality into (A6) that jHn ðj!1 ; ; j!n Þj
þ
1 ðCð0; nÞ þ Cð1; n 1ÞjH1 ðÞj þ þ Cðn; 0ÞjH1 ðj!1 Þ H1 ðj!n ÞjÞ L 0 1 p C B Y X C B Cðp; qÞB Hri ðj!Xþ1 ; ; j!Xþri ÞC @1fr r gnmþ1 i¼1 A P 1 p 0fp;qgm
n1 X 1X L m¼2 pþq¼m
ri ¼nq
¼
1 ðCð0; nÞ þ Cð1; n 1ÞjH1 ðÞj þ þ Cðn; 0ÞjH1 ðj!1 Þ H1 ðj!n ÞjÞ L 0 0 11 p B CC 1 B n1 T C Cðp; qÞ P þ B B br i C @ @ AA h n ¼ bn hn L m¼2 pþq¼m i¼1 ri ¼nq 0fp;qgm
This proves that the theorem holds for the nth step. Therefore, the theorem follows by induction. &
Appendix B: Proof of Theorem 2 Because cp,q(.) ¼ 0 for all p þ q > 1 and q > 0, the GFRF in (4) for model (3) can be written as (for n > 1) ! K X c1;0 ðk1 Þ expðjð!1 þ þ !n Þk1 Þ 1 k1 ¼1
Hn ðj!1 ; ; j!n Þ ¼
n K X X p¼2 k1 ;kp ¼1
Hn;p ðj!1 ; ; j!n Þ
cp;0 ðk1 ; ; kp Þ
1fr1 rp gnmþ1
which yields 1 Ln ð!1 þ þ !n Þ n X Cðp; 0ÞHn;p ðj!1 ; ; j!n Þ
jHn ðj!1 ; ; j!n Þj
ðB1Þ
p¼2
It can be derived from (A5) that Hn;p ðj!1 ; ; j!n Þ npþ1 p X Y Hri ðj!rXþ1 ; ; j!rXþri Þ rP 1 rp ¼1 i¼1 ri ¼n
ðB2Þ
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X.J. Jing et al.
Similarly, a proof by induction is adopted here. The first two steps are checked first. For n ¼ 1, it is obvious that jH1 ðj!1 Þj ¼ B1 jH1 ðj!1 Þj B1 H1 ð!Þ For n ¼ 2, jH2 ðj!1 ; j!2 Þj 1 Cð2; 0ÞH2;2 ðj!1 ; j!2 Þ L2 ðj!1 ; j!2 Þ 1 Cð2; 0Þ L2 ðj!1 ; j!2 Þ 1 2 X Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP 1 r2 ¼1 i¼1
When the input of the NARX model (3) is a general input, the output spectrum is [15] N X 1 Yðj!Þ ¼ pffiffiffi nð2Þn1 n¼1 Z n Y Hsym Uðj!i Þd! n ðj!1 ; ; j!n Þ
1 Cð2; 0ÞjH1 ðj!1 ÞjjH1 ðj!2 Þj L2 ðj!1 ; j!2 Þ
ðC1Þ using the mean value theorem and inequality (13a), which gives (18a), i.e., X N 1 jYðj!Þj ¼ pffiffiffi n¼1 nð2Þn1 Z Hsym n ðj!1 ; ; j!n Þ !1 þþ!n ¼!
Uðj!i Þd! i¼1 X N Hsym n ðj!1 ; ; j!n Þ ¼ pffiffiffi n1 n¼1 nð2Þ n Y
¼ B2 ð!1 ; !2 ÞjH1 ðj!1 ÞjjH1 ðj!2 Þj Therefore, the first two steps hold for the theorem. More steps can be checked for the theorem. Since the nth-order GFRF will involve the GFRFs of lower order from 1 to n-1, now for a proof by induction, suppose that the theorem holds for all the GFRFs of order less than n, and prove the theorem hold for the nth-order GFRF. For the nth-order GFRF, it can be derived from (B2) that Hn;p ðj!1 ; ; j!n Þ npþ1 p X Y Hri ðj!Xþ1 ; ; j!Xþri Þ rP 1 rp ¼1 i¼1 ri ¼n
npþ1 X
p Y
rP 1 rp ¼1 ri ¼n
i¼1
i¼1
!1 þþ!n ¼!
ri ¼2
¼
Appendix C: Proof of Corollary 2
Uðj!i Þd! i¼1 !1 þþ!n ¼! N sym X Hn ðj!1 ; ; j!n Þ pffiffiffi nð2Þn1 n¼1 Z n Y jUðj!i Þjd! Z
!1 þþ!n ¼!
¼
!
N X
i¼1
n Hsym n ðj!1 ; ; j!n Þ
n¼1
Bri ð!Xþ1 ; ; !Xþri Þ
N X n¼1
jH1 ðj!1 Þj jH1 ðj!n Þj
¼
N X n¼1
Therefore, it follows from (B1) that 0 jHn ðj!1 ; ; j!n Þj
n Y
n
sup
!1 þþ!n ¼!
n b n
sup
sym H ðj!1 ; ; j!n Þ
!1 þþ!n ¼!
!
n
hTn ðj!1 ; ; j!n Þ 1
npþ1 p n B C X Y X 1 B C Bri ð!Xþ1 ; ; !Xþri Þ jH1 ðj!1 Þj jH1 ðj!n ÞjC BCðp; 0Þ A Ln ðj!1 ; ; j!n Þ p¼2 @ rP i¼1 1 rp ¼1 ri ¼n
¼ Bn ð!1 ; ; !n ÞjH1 ðj!1 Þj jH1 ðj!n Þj
Thus the theorem is proved by induction. &
83
Frequency-Dependent Magnitude Bounds
Similarly, when the system input is a multi-tone signal, the system output spectrum can be written as [15]
Yðj$Þ ¼
N X 1 n 2 !k n¼1
X þþ!kn ¼$ 1
Fð!k1 Þ Fð!kn Þ
Hsym n ðj!k1 ; ; j!kn Þ ðC2Þ
where jFi jejffFi if ! 2 f!k ; k ¼ 1; ; Kg Fol. Fð!Þ ¼ 0 else lowing a similar process, the magnitude bound can be obtained as (19a). When there are only nonlinearities in output, using the bound results in Theorem 2 in (18a) and (19a), the results in (18b) and (19b) can be derived readily. &