IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 5, MAY 2009
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Robustness Issues of the Best Linear Approximation of a Nonlinear System Johan Schoukens, Fellow, IEEE, John Lataire, Rik Pintelon, Fellow, IEEE, Gerd Vandersteen, and Tadeusz Dobrowiecki
Abstract—In many engineering applications, linear models are preferred, even if it is known that the system is disturbed by nonlinear distortions. A large class of nonlinear systems, which are excited with a “Gaussian” random excitation, can be represented as a linear system GBLA plus a nonlinear noise source YS . The nonlinear noise source represents that part of the output that is not captured by the linear approximation. In this paper, it is shown that the best linear approximation GBLA and the power spectrum SY S of the nonlinear noise source YS are invariants for a wide class of excitations with a user-specified power spectrum. This shows that the alternative “linear representation” of a nonlinear system is robust, making its use in the daily engineering practice very attractive. This result also opens perspectives to a new generation of dynamic system analyzers that also provide information on the nonlinear behavior of the tested system without increasing the measurement time. Index Terms—Approximation, best linear approximation, excitation, nonlinear distortion, nonlinear system.
I. I NTRODUCTION
I
N MANY engineering applications, linear models are preferred, even if it is known that the system to be modeled is nonlinear. Typical examples are amplifiers with nonlinear distortions, vibrating mechanical structures, or the dynamic behavior of electrical machines. Most real-life systems are only approximately linear. Using the tools that are presented in this paper and in references [10] and [14], it turns out that very often, nonlinear distortions are present at a lower (e.g., −60 dB) or higher (−20 dB) level. Therefore, it is important not only to understand the properties of this linear approximation but to know whether this linearized result is valid for a broad class of excitations as well. Best Linear Approximation/Nonlinear Distortions: The output y(t) of a nonlinear system can be replaced by the output of a linear system + a nonlinear “noise source” (see Fig. 1) [10],
Manuscript received June 19, 2008; revised December 12, 2008. First published February 6, 2009; current version published April 7, 2009. This work was supported in part by the Fund for Scientific Research (FWO-Vlaanderen), in part by the Flemish Government (Methusalem METH1), and in part by the Belgian Federal Government (IUAP VI/4). The Associate Editor coordinating the review process for his paper was Dr. Jerome Blair. J. Schoukens, J. Lataire, R. Pintelon, and G. Vandersteen are with the Department Fundamental Electricity and Instrumentation (ELEC), Vrije Universiteit Brussel, 1050 Brussel, Belgium (e-mail:
[email protected];
[email protected]). T. Dobrowiecki is with the Department of Measurement and Information Systems, Budapest University of Technology and Economics, Budapest 1117, Hungary. Digital Object Identifier 10.1109/TIM.2009.2012948
Fig. 1. Equivalent representation of a nonlinear system.
[14]. The signal YBLA (t) captures that part of y(t) that can be described by the best linear approximation GBLA as GBLA (Ω) = SY U (Ω)/SU (Ω)
(1)
where SY U and SU are the cross spectrum and auto-spectrum, respectively, and Ω is the discrete or continuous time-frequency variable. The “nonlinear noise source” yS (t) with power spectrum SYS (Ω) captures all the remaining errors, which are also called “stochastic nonlinearities,” as Y (Ω) = YBLA (Ω) + YS (Ω) = GBLA (Ω)U (Ω) + YS (Ω). (2) The variable Ω stands for the discrete (Ω = e−j2πf /fs ) or continuous time Ω = 2πf j frequency. The properties of the linear approximation GBLA and the error term YS have extensively been studied [3]–[5], [7]–[11], [14]. For the purposes of this paper, it is enough to notice that YBLA is not correlated with YS , although it is not independent of YS ! Consider the very simple example of a static nonlinear system y(t) = u(t)3 driven by a Gaussian noise with zero mean and unit variance u(t) ∼ N (0, 1). In that simple case, it can be shown that GBLA (Ω) = 3, and yS (t) = u(t)3 − 3u(t), which is dependent upon u(t) but not correlated with it: E{u(t1 )yS (t2 )} = 0 ∀t1 , t2 . For random inputs, YS looks like noise, and it is very hard to distinguish it from disturbing (measurement) noise for an inexperienced user. For Gaussian noise excitations, GBLA (Ω) and SYS (Ω) are completely determined by the power spectrum of the input SU . Robustness Aspects of the Best Linear Approximation and the Nonlinear Noise Source: In this paper, we combine two previous results [2], [16] in one general framework. It was shown before in [2] that the best linear approximation [see (1)] is invariant with respect to some changes of the frequency grid of multisine excitations with frequency grids of increasing density. Then, in [16], a similar robustness of the power spectrum of the nonlinear noise source [see (2)] was studied. In this paper, a class of periodic excitation signals will be defined,
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 5, MAY 2009
which results in the same equivalent linear representation of the nonlinear system. For a signal belonging to this class, the best linear approximation GBLA and the power spectrum SYS (Ω) of the nonlinear noise source will be “equivalent”1 to those obtained with Gaussian noise. Impact on the Measurement Society: The results that will be obtained in this paper show that the alternative “linear representation” in (2) of a nonlinear system is robust. This is a very important message for the measurement society for many reasons. 1) In many applications, a linear transfer function model is used, even if it is known that, in practice, most systems are prone to nonlinear distortions. This paper gives a theoretic justification for this practice by showing that the obtained results are applicable to a whole class of excitation signals. Until now, this theoretic justification has been lacking. 2) Replacing the Gaussian excitation by well-designed periodic excitations allows considerable reduction of the measurement time and measurement in one experiment GBLA , SYS (Ω), together with the level of the even and odd nonlinearities. The discussion of these techniques is out of the scope of this paper. More information can be found in [11]. Towards a New Generation of Network Analyzers: Using the combined results of the previous section, it is clear that the classical framework for frequency response function (FRF) measurements that is used by the actual network analyzers can be extended to include nonlinear distortions. This allows for a new generation of instruments that is optimized to operate in the presence of nonlinear distortions. On the basis of this paper, the standard Gaussian noise excitation can be replaced by well-designed random phase multisines, which provides these instruments with better characteristics and new capabilities. 1) The measured FRF will be representative for all “Gaussian-like” signals with the same power spectrum (result of this paper). 2) Due to the use of these multisines, the measurement time can significantly be reduced. For systems with dominating even nonlinear distortions, the reduction can be a factor 10 or more (see, for example, [10] or [14]). 3) The user will get additional information on the level and nature (even or odd) of the nonlinear distortions. This information is nowadays inaccessible for the practicing engineer. 4) These nonlinear distortion levels provide natural bounds for the “linear” designer. It makes no sense to tune the design below the nonlinear distortion levels. Outline of the Paper: This paper is completely focused on the robustness of the best linear approximation: what excitations result in the same GBLA and the same power spectrum SYS for the nonlinear noise source? This paper consists of three parts. First, the theoretical setup is formally introduced: the class of equivalent excitation signals is defined, and the 1 It will be explained later how the discrete power spectrum of a periodic signal will be compared with the continuous power spectrum of a nonperiodic signal.
class of included nonlinear distortions is described. Next, the equivalences for GBLA , SYS (Ω) are discussed, and finally, an experimental verification on a lab experiment is made. II. T HEORETICAL S ETUP In this section, we first define the excitation signals that will be considered. Next, they are grouped in equivalence classes. Then, the major contribution of this paper is formulated and experimentally illustrated. The detailed theoretic justification is given in the Appendix to increase the readability for the practicing engineers. A. Definition of the Excitation Signals Introduction: Gaussian noise is a very popular test signal in many application fields. It will also be used as the reference signal in this paper. Next, we define the periodic excitations (which also have a random behavior) that will be “equivalent” to the Gaussian noise (the same GBLA , SYS are retrieved). The periodic signals are described as random multisines. These are the sum of periodically related sines, with a user-defined amplitude spectrum, and random phases. The major motivation to introduce random multisines is that the user has full control over the amplitude spectrum, even with a single realization. For random noise excitations, it is necessary to average over multiple realizations before the asymptotic properties are closely approached. This increases the required measurement time. The multisines also allow measurement of the level and nature (even or odd) of the nonlinear distortions, as shown in [10] and [14]. Filtered Gaussian Noise: The first class of excitations, used as reference for the others, consists of a zero-mean Gaussian noise with power spectrum SU (Ω). In many applications, these signals are generated as filtered white noise, with a filter H s.t. SU (Ω) ∼ |H(Ω)|2 . Periodic Random Signals: In this section, (periodic) noise and random phase multisines are briefly introduced [11], [14]. The period of the signals is N/fs , where N is the number of samples in a period, and fs is the sample frequency. Definition 1—(Periodic) Noise or Random Multisine: A signal u(t) is a (periodic) noise excitation (also called random multisine) if
N/2−1
u(t) = N −1/2
ˆk ej (2πk Nt +ϕk ) U
(3)
k=−N/2+1 k=0
ˆk ≥ 0, U ˆk = U ˆ−k , U ˆ0 = 0, and U ˆ±N/2 = with ϕ−k = −ϕk , U ˆk and phases ϕk are mutually indepen0. The amplitudes U dent and are also independent over the frequencies. The expected value E{ejϕk } = 0. The expected value of the squared ˆk |2 over multiple realizations of the excitation amplitude |U 2 ˆk | } = A(kfs /N )2 , where A(f ) is the amplitude funcE{|U tion. A(f ) is piecewise continuous with a finite number of discontinuities. In many applications, it is advantageous to use a deterministic amplitude spectrum. In each realization of the signal, the amplitude spectrum exactly equals what is defined by the user, which is not the case for random multisines or Gaussian noise.
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SCHOUKENS et al.: ROBUSTNESS ISSUES OF THE BEST LINEAR APPROXIMATION OF A NONLINEAR SYSTEM
This leads to the concept of “Random phase multisines,” where only the phase is a random variable. Definition 2—(Random Phase) Multisine: A signal u(t) is a random phase multisine excitation if
N/2−1
u(t) = N
−1/2
ˆk ej (2πk Nt +ϕk ) U
(4)
k=−N/2+1 k=0
where ϕ−k = −ϕk . The phases ϕk are independently distribˆk is either zero (the uted with E{ejϕk } = 0. The amplitude U ˆ harmonic is not excited) or Uk = A(k/N ) (see Definition 1). Remarks: 1) The difference between periodic noise and a random multisine is that, for the latter, the amplitude spectrum is a deterministic (frequency dependent) function, whereas it is a random variable for the periodic √ noise. 2) In (3) and (4), the scaling factor 1/ N is used to normalize the root mean square (RMS) value √ of the excitation. It would be better to scale by 1/ N − 2, but since all claims will be for N √ → ∞, we prefer to simplify the scaling factor to 1/ N to simplify the expressions without any change of the asymptotic results. 3) From here on, we do not make any more strict distinction between periodic and nonperiodic noise excitations. The reason for this is that a nonperiodic noise sequence with length Np N can be broken into Np subsequences with length N , and each of these subrecords can be considered to result from an experiment where this subrecord was periodically repeated. The differences between the response of a dynamic system to this “artificial” periodic sequence and the original nonperiodic subsequence are the transient effects (leakage√errors in the frequency domain) that disappear as 1/ N [7, pp. 201–202], [10, p. 57]. Since all claims made in this paper will asymptotically be valid for a period length growing to infinity (N → ∞), we can neglect these effects. (Random) Frequency Grid: We will consider multisines on different frequency grids, and not all frequencies on an equidistant grid will be used in the excitation. Some possibilities are discussed as follows. 1) Full: All harmonics in the frequency band of interest (the frequency band where the user wants to get information about the system) are excited, for example, k = [1, 2, 3, . . . , 100] in (4) is excited. 2) Odd: Only the odd harmonics in the frequency band of interest are excited, for example, k = [1, 3, 5, . . . , 99] in (4) is excited. 3) Random grid: Within each group of Nsub successive harmonics, a number of randomly selected harmonics is not excited. The probability that a harmonic is excited is called p. For example, p = 2/3 if one out of the three harmonics is not excited (Nsub = 3). In most applications, groups with a fixed number of grid lines (e.g., 4 or 8) are considered to guarantee a uniform coverage of the frequency grid. It is, of course, possible to consider only one group of frequencies that covers the complete frequency band of interest, but we do not select this as the standard
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situation because it can result in a very poor frequency resolution (random coverage of the frequency band of interest), which invalidates the claims in Theorem 1. In [2], it is mentioned that in such √ a case, the convergence is not O(1/N ) but only O(1/ N ). Remark: Combinations of these characteristics are possible, for example, an odd random grid random phase multisine where a number of randomly selected odd lines is not excited in groups with a fixed number of odd grid lines. B. Spectral Analysis All the signals of the previous section will be described in the frequency domain. For the periodic signals, we consider one period of length N . For the noise sequences, we considered a rectangular windowed signal also with length N . The discrete Fourier transform (DFT) X(k) of such a signal x(t), t = 0, . . . , N − 1 is defined as N −1 1 √ x(t)e−j2πkt/N . X(k) = N t=0
(5)
The frequency index k corresponds to the signal frequency kfs /N , where fs is the sampling frequency. All of the results in this paper will refer to the DFT coefficients. If x(t) is a stationary noise sequence with power spectrum SX (Ω), then we have 1 fs E |X(k)|2 = SX (Ωk ) + O(N −2 ) N N
(6)
and the expectation is taken over multiple realizations of the random excitations; we did not take out a factor 1/N to clearly show the relation with the frequency resolution of the DFT, which is fs /N . C. Riemann Equivalence Class of Excitation Signals ESU In this section, we define which signals belong to the equivalence class ESU that collects all signals that are (asymptotically) Gaussian distributed and have asymptotically (N → ∞) the same power on each finite-frequency interval. This is precisely defined in the next definition. Definition 3—Riemann Equivalence Class ESU of Excitation Signals: Consider a power spectrum SU (Ω), which is piecewise continuous, with a finite number of discontinuities. A random signal belongs to the equivalence class if it is any of the following. 1) It is a Gaussian noise excitation with power spectrum SU (Ω). 2) It is a random multisine or random phase multisine s.t. kω2 ω2 1 1 2 E | U (k)| = SU (ν)dν + O(N −1 ) N 2π k=kω1
with kωi = int
ω1
ωi N 2πfs
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∀kωi
and 0 < ω1 , ω2 < πfs . (7)
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Discussion: 1) From (7), it is possible to define a relation between the probability p(k) that a frequency line k is excited, the expected value of its DFT amplitude U (k), and the power spectrum SU (Ωk ) as 1 1 fs E | U (k)|2 = p(k)A(k/N )2 = SU (Ωk ). N N N (8) From (8), it follows that A(k/N ) =
fs
SU (Ωk ) . p(k)
(9)
2) For multisines that do not use all grid lines, for example, odd excitations, an equivalent probability can be defined. An odd multisine excites only one line out of two. In that case, we can use the factor p(k) as a measure for the “fill-factor” of the multisine (no longer a probability interpretation) and have that in (9) p(k) = 0.5 to get the correct amplitudes.
Definition 5—Riemann Equivalent Power Spectra: Two power spectra SY1 , SY2 are Riemann equivalent if ω2
ω2 SY1 (ν) dν =
ω1
ω1
0 < ω1 , ω2 < πfs . (10) Discussion: From this definition, it follows that both power spectra SY1 , SY2 asymptotically have the same power (continuous or discrete) in each finite frequency band. For discrete spectra, sums as in (7) should be considered. Note also that signals that have not the same periodicity can be Riemann equivalent. In addition, even signals (that excite only the even harmonics) can be Riemann equivalent to odd signals (that excite only the odd harmonics). All the excitations that belong to the Riemann equivalence class ESU have Riemann equivalent power spectra. With this background information, the following theorem can be formulated. Theorem 1: Consider a nonlinear system belonging to SG . All excitations belonging to the equivalence class ESU result 2α−1 in the same best linear approximation GBLA = ∞ α=1 GBLA −1 within O(N ), where fs /2
D. Class of Nonlinear Systems The class of nonlinear systems that are considered in this paper is specified in the next definition. Definition 4—Set of Systems SG : SG is the set of systems y(t) = g(u(t)) that can be approximated in mean square sense by a convergent Volterra system for u0 ∈ ESU , for which the multidimensional Fourier transforms of the Volterra kernels are continuous functions of the frequencies. Within the scope of this paper, it is not possible to discuss the Volterra systems in more detail. The interested reader is referred to the book of Schetzen [17], [18] that gives an excellent introduction. It also discusses in full detail the class of systems that can be approximated in mean square sense by the Volterra series for Gaussian excitations. Note that g(u(t)) also includes dynamic nonlinear systems. In addition to smooth nonlinear systems, discontinuous systems also belong to this set. The major property of these systems is that a periodic input results in a periodic output with the same period.
G2α−1 BLA (f )
In this section, we make the major contribution of this paper. The equivalence claims for the best linear approximation GBLA and the power spectrum SYS of the stochastic nonlinearity will precisely be formulated. To make a general statement, we first need to define the class of equivalent power spectra. This definition is very similar to that of the equivalence class of excitation signals. It is needed because we will compare periodic signals that have a discrete spectrum to noise signals that have a continuous power spectrum. It is clear that both spectra cannot be equal, but using Riemann integrals, they can be related to each other.
fs /2
···
= cα 0
G2α−1 f,−fk
,fk1,...,fkα−1 SU (f1 )
1
0
· · · SU (fα−1 ) df1 · · · dfα−1
(11)
where G2α−1 f,−fk1 ,fk1 ,...,fkα−1 is the Volterra kernel of order 2α − 1. The power spectrum SYS of the stochastic nonlinearities are Riemann equivalent within O(N −1 ) to S0 (the power spectrum of the stochastic nonlinearities for a filtered Gaussian noise excitation) with SYS (f ) =
∞ ∞
SYr,s (f ) S
r=1 s=1
with r + s = n even, and
s SYr,s (f ) = E YSr (f )Y S (f ) S fs /2
fs /2
···
= III. I NVARIANCES FOR GBLA AND SYS
SY2 (ν) dν + O(N −1 )
0
Γr,s (f, f1 , . . . , fn/2−1 )SU (f1 ) 0
· · · SU (fn/2−1 ) df1 · · · dfn/2−1
(12)
where Γr,s (f, f1 , . . . , fn/2−1 ) is a smooth function of f, f1 , . . . , fn/2−1 depending upon the Volterra kernels, and YSr (f ) is the contribution of the rth-degree Volterra kernel to the stochastic nonlinearity. Proof: See Appendix A. Loosely spoken, Theorem 1 guarantees that the measured GBLA and SYS are the same for all excitations in the equivalency class. The theorem makes no statement about the quality of the linear approximation, and it does not guarantee that most of the output power will be captured in the linear
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TABLE I SIGNALS OF THE EQUIVALENCE CLASS ESU AT FREQUENCY Ω = 2πfs kΩ /N
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multisines excitations that belong to the equivalence class ESU will be Riemann equivalent to S0 (Ω). C. Random Grid Multisines Multisines that do not excite all the frequencies should carefully be designed. For randomly selected lines with a fill factor p(k), it is necessary to design the amplitude spectrum properly as specified in (9) to belong to the equivalence class ESU . The power spectrum SYS (Ω) will be Riemann equivalent to S0 (Ω). D. Odd Random (Phase) Multisine For odd random (phase) multisines that put all power on the odd frequencies, the results should properly be interpreted because they split the stochastic nonlinearities at the output in two sets: the even nonlinear distortions are only present at the even frequencies, whereas the odd nonlinearities are only active at the odd frequencies. The resulting power spectrum
approximation. This is completely dependent on the system, and it will be one of the results of the measurements. The result of Theorem 1 is clear for the best linear approximation. However, the interpretation of the equivalent power spectra SYS (Ω) needs some further discussion for the different excitation signals. Table I gives an overview of this discussion. A. Gaussian Noise A Gaussian noise with a power spectrum SU (Ω) is used as the reference signal. The corresponding power spectrum S0 (Ω) of the stochastic nonlinearities will be used as the reference for the other signals. Notice that the power spectrum of the nonlinear noise source can be split in even (S0E ) and odd (S0O ) contributions that can be attributed to the even and odd nonlinearities in the system, e.g., y = x2 or y = x3 S0 (Ω) = S0E (Ω) + S0O (Ω).
if Ω is even
SYS (Ω) = SYS O (Ω),
if Ω is odd
(15)
is still the Riemann equivalent to S0 (Ω) = S0E (Ω) + S0O (Ω). However, for a noise excitation, the odd and even stochastic nonlinearities S0O (Ω) and S0E (Ω) appear at all frequencies (even and odd frequencies). For an odd excitation signal, the odd stochastic nonlinearities are only present at the odd frequencies, and YSO (Ωeven ) = 0. The even stochastic nonlinearities are only present at the even frequencies, and YSE (Ωodd ) = 0. Therefore, the “fill factor” equals p = 0.5 in this case, and hence 1 fs E |YSO (Ωodd )|2 = 2S0O (Ωodd ) + O(N −2 ) (16) N N or E |YSO (Ωodd )|2 = 2S0O (Ωodd )fs + O(N −1 ) (17) and similarly E |YS (Ωeven )|2 = 2S0E (Ωeven )fs + O(N −1 ).
(18)
(13)
From (6), it follows that for the DFT spectrum of the nonlinear noise source the following relation holds: fs 1 E |YS (k)|2 = S0 (Ωk ) + O(N −2 ). N N
SYS (Ω) = SYS E (Ω),
(14)
B. Multisine We first consider a full random multisine. “Full” stands for the fact that all the frequencies in the frequency band of interest are excited, and “random” indicates that the phases are random chosen, s.t. E{ejϕ } = 0. The power spectrum SYS (Ω) for full
IV. E XPERIMENTAL I LLUSTRATION In this section, we report an experimental verification of the proposed equivalence theory. The equivalence claims are tested on an electronic nonlinear system. All measurements were made with HP E1430a data acquisition cards that have a high linearity and have anti-alias filters on board. The electronic system was designed to illustrate many aspects of the developed theory: the best linear approximation (which can be equal to zero in some frequency bands) is measured, and even and odd nonlinearities (which dominate each in a frequency band) are analyzed. The system consists of two parallel branches (see Fig. 2). The upper branch is a low-pass filter, followed by
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Fig. 2. Nonlinear test system.
Fig. 4. Random grid multisine excitation (p = 40%) and noise excitation. (Dark gray and bold black) Amplitude YBLA for the noise and multisine excitation, respectively. (Light gray and thin black) SYS for the multisine and noise excitation, respectively.
Fig. 3. Random grid multisine excitation with varying probability of active excitation harmonic (p = 10%, 20%, 40%, 80%, 100%). For all p-values, (dark gray) amplitude YBLA , (black) SYS at the nonexcited lines, and (light gray) SYS at the excited lines.
an even nonlinearity. The lower branch is a high-pass filter followed by an odd nonlinearity. In the crossover of both frequency bands, the even and odd nonlinearities are equally important. The filter G1 is a low-pass filter with a bandwidth of 1 kHz, whereas G2 is a high-pass filter with a cutoff frequency of 2 kHz. The gains of both branches are tuned to get about the same total power in the even and odd nonlinear contributions at the output. In each experiment, the results were averaged over 200 realizations of the random excitation to get smooth estimates of the power spectra SYS (see Appendix D). For the periodic excitations, two periods with 3256 data points/period are processed per realization. For noise excitations, blocks with a length of 2 times 3256 data points are used. The sampling frequency is 39 062.5 Hz. Three experiments will be discussed. First, we show for each of these experiments the equivalence of the power spectra SYS , as explained before. At the end of this section, we discuss the behavior of the best linear approximation GBLA . A. Experiment 1: Random Grid Random Phase Multisine With p = 10%, 20%, 40%, 80%, 100% This experiment shows that changing the probability to excite a harmonic does not affect the expected value of SYS , calculated over the different realizations of the input. It should be noticed that in each realization another random set of grid lines was selected. This is important when only a few lines are excited. The result in Fig. 3 shows that SYS is indeed invariant to these variations. Notice that at the excitation lines a higher level is seen. This is the O(1/N ) effect that was neglected in the theory (see Theorem 1). Once the density of excited excitation lines is high enough, the gray and black results coincide (not shown on this figure). Note also that the amplitude of YBLA changes with the probability p of the excitation signal design.
Fig. 5. Full and odd random grid multisine excitation (p = 100%). (Dark gray) Odd multisine results. (Light gray) Even multisine results. (Thin black line) Estimated level SˆYs Full starting from SYSO oddMS and SYSE oddMS .
This is because the total power of the excitation is kept constant during the different experiments, as explained in the theory. Finally, it can be observed that the disturbing noise level in these measurements (below −70 dB) is far below the nonlinear effects. For that reason, it is no longer shown on the other figures. B. Experiment 2: Random Grid Random Phase Multisine With p = 40% Versus Gaussian Noise Excitation This experiment illustrates the equivalence between the random multisine and a noise excitation. From Fig. 4, a very nice agreement is observed between SYS for both excitations. C. Experiment 3: Full and Odd Random Grid Multisine With p = 100% In this experiment, the relations between SYS for a full and an odd multisine are checked. To do so, we measured and plotted the even and odd spectra E{|YSE |2 } and E{|YSO |2 } measured with the odd multisine. From these levels, the spectrum for the full random grid multisine is calculated as
2 (19) E |YSfull |2 = E |YSE |2 + E |YSO |2 combining in the sum of the neighboring frequency lines. The division by a factor 2 is needed to compensate for the scaling factor 2, as explained in Section III-D. From Fig. 5, the very nice confirmation of the theory can be seen. The predicted level (using the odd multisine) nicely coincides with that of
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disturbance, whereas all the results can still be carried over to the full class of excitations.
A PPENDIX A P ROOF OF T HEOREM 1
Fig. 6. Measured GBLA . White: noise excitation. Gray: Full multisine excitation (p = 10%, 20%, 40%, 80%, 100%). Black: Odd multisine excitation (p = 10%, 20%, 40%, 80%, 100%).
the full multisine. It can also be seen from this figure that the even nonlinearities are dominantly present in the low-frequency band, whereas the odd nonlinearities are present in the whole frequency band (below and above the excited band). This very well illustrates that with odd random grid multisines, a full characterization of the system can be given from one experiment. Not only is the FRF of the “best linear approximation” obtained (see Appendix D), but the nature (even or odd) and the level of nonlinear distortions are also accessible.
The proof is based on the general expressions for GBLA and SYS that are given in Appendices B and C. Next, the result of Theorem 1 immediately follows from the theory of Riemann integrals, as stated in Appendix C. To show the Riemann equivalence of the power spectra YS , it is important to consider the integral of YS over a finite frequency interval with a width that is independent of N , because the power spectra are not necessarily equal to each other.
A PPENDIX B T HEORETICAL E XPRESSIONS FOR GBLA AND SYS In this paper, we consider nonlinear systems that can be approximated in mean square sense by Volterra systems [17], [18]. The output of such systems can be written as the sum of the contributions of degree α as Y (k) =
D. Behavior of the Best Linear Approximation GBLA Fig. 6 plots the amplitude of the frequency response of the best linear approximation GBLA for different excitations. In the higher-frequency band, where the odd nonlinearity dominates, we retrieve the dynamic system G2 for all the excitations, in agreement with the theory. Because GBLA = 0 for even nonlinearities, we have that in the lower frequency band GBLA drops to zero (see description of the system). For the full excitations (noise, full multisine), the FRF measurements are disturbed by the even nonlinearities. This is not so for the odd excitations where the even nonlinearities are only present at the even frequencies that are not used for the FRF measurements. That explains why the “noise” floor is much lower for the latter one.
Y α (k).
(20)
α=1
1) GBLA : The systematic contributions are given by YBLA (k) =
∞
2α−1 YBLA (k)
=
α=1
∞
G2α−1 BLA (Ωk )
U (k)
α=1
= GBLA (k)U (k).
(21)
It is shown in [10] that under some regularity conditions the best linear approximation is given by G2α−1 BLA (Ωk ) =
cα N α−1
V. C ONCLUSION In this paper, we have defined a Riemann equivalency for periodic and nonperiodic signals. These are signals that asymptotically inject the same power in each finite frequency band. Next, it is experimentally verified that these excitations result in the same best linear approximation GBLA . The power spectra of the stochastic nonlinearities SYS are Riemann equivalent for excitation signals that are Riemann equivalent. This result is important for two reasons. First, it shows that the equivalent linear representation of a nonlinear system is robust. Second, the user can select within the equivalence class of excitation signals the signal that simplifies the measurements most, resulting, for example, in an important reduction of the experiment time. This can be obtained by replacing the random noise excitations by well-designed random grid odd multisines. This allows a significant reduction in the measurement time (for example, a factor 10 to 100) when the even nonlinearities are the dominant
∞
N
G2α−1 k,−k1 ,k1 ,...,−kα−1 ,kα−1
k1 ,...,kα−1 =1
×
α−1
E |U (ki )|2 (22)
i=1
for random multisines and by fs /2
G2α−1 BLA (Ωk )
fs /2
···
= cα 0
×
α−1
G2α−1 f −f1 ,f1 ,...,−fα−1 ,fα−1 0
SU (Ωki ) df1 · · · dfα−1
(23)
i=1
for Gaussian noise, where cα = 2α−1 (2α − 1)!!, and G2α−1 k,−k1 ,k1 ,...,−kα−1 ,kα−1 is the Volterra kernel of degree 2α − 1.
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 5, MAY 2009
2) SYS : It can be shown [15] that the power spectrum of the stochastic contributions YS is given by SYS (k) =
∞ ∞
SYr,s (k), S
with r + s = n even
(24)
r=1 s=1
with
s (k) = E YSr (k)Y S (k) SYr,s S 1 Γr,s (k1 , . . . , kn/2 ) = n/2 N k ,...,k 1
n/2
2 1 2 +O × E |U (k1 )| · · · E U (kn/2 ) N (25) with k1 + · · · + kn/2 = k for a random multisine, and
s SYr,s (k) = E YSr (k)Y S (k) S fs /2
fs /2
···
= 0
Γr,s (f, f1 , . . . , fn/2−1 )SU (f1 ) · · · 0
SU (fn/2−1 ) df1 · · · dfn/2−1
(26)
with f = kfs /N , for Gaussian noise, and with Γr,s as a smooth function that depends on the Volterra kernels. In (25), only those terms are considered to contribute to the output frequency k. Remark: The Riemann sums (22) and (25) asymptotically converge (N → ∞) to the integrals (23) and (26), respectively (assuming that the grids behave well from the Riemann theory point of view for increasing N ).
A PPENDIX C R IEMANN I NTEGRALS In this Appendix, we state, for the readers’ convenience, a well-known result based on Riemann integrals [6], [13]. Theorem 2: Consider a piecewise continuous function Γ(Ω) with a finite number of discontinuities. For periodic signals belonging to ESU and a frequency interval [Ω1 , Ω2 ] that is independent of N , the following equality then holds: 1 2π
Ω2 Γ(Ω)SU (Ω) dΩ Ω1
=
k2
Γ(Ωk )p(k)A(k/N )2 + O(N −1 ).
(27)
k=k1
This theorem is at the basis of all claims in this paper because the relation between GBLA or SYS and the input power spectrum is of the form (27).
A PPENDIX D M EASUREMENT OF THE P OWER S PECTRUM SYS Two possibilities to measure the power spectrum of the nonlinear noise source will be used. The first method is based on successive measurements, using each time a new realization of the random input [1]. The second method uses excitations that do not excite all the frequency lines. The nonlinear noise contributions are measured at the unexcited frequencies. In this paper, the first method is used. If, for a given phase realization, multiple periods are measured, then it is possible to separate the nonlinear noise source from the measurement noise [1], [12], because the nonlinear distortion does not vary from one period to the other, whereas the noise does. This will allow to verify that the nonlinear distortions are the dominating noise source in the experiments. R EFERENCES [1] T. D’haene, R. Pintelon, J. Schoukens, and E. Van Gheem, “Variance analysis of frequency response function measurements using periodic excitations,” IEEE Trans. Instrum. Meas., vol. 54, no. 4, pp. 1452–1456, Aug. 2005. [2] T. Dobrowiecki and J. Schoukens, “Measuring the best linear approximation of a nonlinear system with uniformly frequency-distributed periodic signals,” in Proc. IEEE Instrum. Meas. Technol. Conf., Warsaw, Poland, May 1–3, 2007, pp. 1–6. [3] M. Enqvist, Some Results on Linear Models of Nonlinear Systems. Linköping, Sweden: Inst. Technol., Linköping Univ., 2003. Licentiate Thesis 1046. [4] M. Enqvist, “Linear models of nonlinear systems,” Ph.D. dissertation No. 985, Inst. Technol., Linköping Univ., Linköping, Sweden, 2005. [5] M. Enqvist and L. Ljung, “Linear approximations of nonlinear FIR systems for separable input processes,” Automatica, vol. 41, no. 3, pp. 459– 473, Mar. 2005. [6] W. Kaplan, Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, 1991. [7] L. Ljung, “Estimating linear time-invariant models of nonlinear timevarying systems,” Eur. J. Control, vol. 7, no. 2/3, pp. 203–219, 2001. [8] P. M. Makila, “On optimal LTI approximation of nonlinear systems,” IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1178–1182, Jul. 2004. [9] P. M. Makila and J. R. Partington, “Least-squares LTI approximation of nonlinear systems and quasistationarity analysis,” Automatica, vol. 40, no. 7, pp. 1157–1169, Jul. 2004. [10] R. Pintelon and J. Schoukens, System Identification. A Frequency Domain Approach. Piscataway, NJ: IEEE Press, 2001. [11] R. Pintelon and J. Schoukens, “Measurement and modelling of linear systems in the presence of non-linear distortions,” Mech. Syst. Signal Process., vol. 16, no. 5, pp. 785–801, 2002. [12] R. Pintelon, G. Vandersteen, L. De Locht, Y. Rolain, and J. Schoukens, “Experimental characterization of operational amplifiers: A system identification approach—Part I: Theory and simulations,” IEEE Trans. Instrum. Meas., vol. 53, no. 3, pp. 854–862, Jun. 2004. [13] A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd ed. Auckland, New Zealand: McGraw-Hill, 1978. [14] J. Schoukens, R. Pintelon, T. Dobrowiecki, and Y. Rolain, “Identification of linear systems with nonlinear distortions,” Automatica, vol. 41, no. 3, pp. 451–504, 2005. [15] J. Schoukens, R. Pintelon, and M. Enqvist, “Study of the LTI relations between the outputs of two coupled Wiener systems and its application to the generation of initial estimates for Wiener–Hammerstein systems,” Automatica, vol. 44, no. 7, pp. 1654–1665, Jul. 2008. [16] J. Schoukens, J. Lataire, R. Pintelon, and G. Vandersteen, “Issues of the equivalent linear representation of a nonlinear system,” in Proc. IEEE Instrum. Meas. Technol. Conf., Victoria, BC, Canada, May 15–18, 2008, pp. 332–335. [17] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, 1st ed. New York: Wiley, 1980. [18] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, 2nd ed. Melbourne, FL: Krieger, 2006.
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SCHOUKENS et al.: ROBUSTNESS ISSUES OF THE BEST LINEAR APPROXIMATION OF A NONLINEAR SYSTEM
Johan Schoukens (M’90–SM’92–F’97) received the Engineer and the Doctor degrees in applied sciences from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1980 and 1985, respectively. He is currently a Professor with the Department Fundamental Electricity and Instrumentation (ELEC), VUB. The main interests of his research are in the field of system identification for linear and nonlinear systems. Dr. Schoukens received the Best Paper Award in 2002 and the Distinguished Service Award in 2003 from the IEEE Instrumentation and Measurement Society.
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Gerd Vandersteen received the Engineer and the Doctor degrees in applied sciences from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1991 and 1997, respectively. In 2000, he was a Principal Scientist with the Wireless Group, Inter-University Micro-Electronics Centre (IMEC), where he focused on the modeling, measurement, and simulation of electronic circuits in state-of-the-art silicon technologies. Since 2008, he has been a Professor with the Department Fundamental Electricity and Instrumentation (ELEC), VUB, where his interests are within the context of measuring, modeling, and analysis of complex linear and nonlinear systems.
John Lataire was born in Brussels, Belgium, in 1983. He received the Electrical Engineer degree in electronics and information processing from Vrije Universiteit Brussel (VUB), Brussels, in July 2006. Since August 2006, he has been a Ph.D. Researcher with the Department Fundamental Electricity and Instrumentation (ELEC), VUB. Since October 2007, he has been on a Ph.D. fellowship from the Research Foundation—Flanders (FWO). His main interests include the measurement and identification of slowly time-varying weakly nonlinear dynamic systems.
Rik Pintelon (M’90–SM’96–F’98) received the Engineer and the Doctor degrees in applied science from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1982 and 1988, respectively. He is currently a Professor with the Department Fundamental Electricity and Instrumentation (ELEC), VUB. His main research interests include the fields of parameter estimation/system identification and signal processing.
Tadeusz Dobrowiecki received the M.Sc. degree in electrical engineering from the Technical University of Budapest, Budapest, Hungary, in 1975 and the Ph.D. degree from the Hungarian Academy of Sciences, Budapest, in 1981. He is currently an Associate Professor with the Budapest University of Technology and Economics, Budapest. His research interests include advanced signal processing algorithms and technical applications of knowledge-based techniques, particularly measurement and system identification.
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