Design of Multilevel Signals for Identifying the Best ... - IEEE Xplore

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 2, FEBRUARY 2013

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Design of Multilevel Signals for Identifying the Best Linear Approximation of Nonlinear Systems Hin Kwan (Roland) Wong, Johan Schoukens, Fellow, IEEE, and Keith R. Godfrey

Abstract—The best linear approximation (BLA) of a nonlinear system minimizes the difference between the actual output of the system and the modeled output in a least square sense. It depends on the power and amplitude distribution of the excitation sequences used to identify it. The theory of the BLA for Gaussian input sequences (including random phased multisines) has been widely studied. It has recently been shown that the BLA when using a binary input sequence is biased with respect to that obtained using a Gaussian input sequence, and expressions for this bias have been obtained. In this paper, it is shown that it is possible to design discrete multilevel input sequences to mimic Gaussianity as closely as possible, thus reducing the bias, by adjusting sequence levels and the probability of the sequence being at these levels. Their performance is compared with true Gaussian sequences in simulation experiments. Index Terms—Linear approximation, multilevel systems, nonlinear systems, signal design, system identification.

I. I NTRODUCTION

I

N LINEAR system identification, both the frequency response function and the impulse response of a system contain information about the system, from which it is possible to model the system, make predictions about its behavior, and control the system to produce desired behavior. All systems are nonlinear to some extent, but linearizing a nonlinear system has merits in modeling and control. One of the most familiar ways of doing this is to use the best linear approximation (BLA) [1]–[5], which is a linear model which minimizes the difference between the actual output of the system and the modeled output in a least square sense. The BLA when using a Gaussian input sequence (including a random phased multisine) is well studied and has well-known properties [2], [6]. For Manuscript received June 20, 2012; revised August 14, 2012; accepted August 15, 2012. Date of publication October 24, 2012; date of current version December 29, 2012. This work was supported in part by the Fund for Scientific Research (FWO–Vlaanderen), by the Flemish Government (Methusalem), and by the Belgian Government through the Interuniversity Poles of Attraction (VI/4) Program. The work of H. K. Wong was supported by a doctoral training grant from the U.K. Engineering and Physical Sciences Research Council. The Associate Editor coordinating the review process for this paper was Dr. John Sheppard. H. K. Wong, (in a cotutelle with the Department of Fundamental Electricity and Instrumentation, Vrije Universiteit Brussel) and K. R. Godfrey are with the Systems, Measurement and Modelling Research Group, School of Engineering, University of Warwick, Coventry, CV4 7AL, U.K. (e-mail: [email protected]; [email protected]). J. Schoukens is with the Department of Fundamental Electricity and Instrumentation, Vrije Universiteit Brussel, 1050 Brussels (e-mail: Johan. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2012.2216471

example, in a cascaded block-structured system, the Gaussian BLA is directly proportional to the cascaded linear dynamics. For the identification of systems in practice, maximumlength sequences are often used, because of the ease of generation using, for example, the programs prs for binary sequences and G ALOIS for multilevel sequences [7]. Pseudorandom sequences are advantageous in situations where practical constraints limit the number of sequence levels that can be applied. Pseudorandom binary sequences have been widely used for many years, but there are also instances where multilevel sequences are necessary. In an example from the steel industry [8], ternary sequences were used to identify the frequency response between the applied force and strip position on a scale model of a hot-dip galvanizing process for steel strip. In this application, the strip could be moved by two electromagnets, one on either side. However, the associated power electronics only allowed one voltage level to be applied to each, thus limiting the input sequence to a maximum of three levels. An example from the field of communications is the identification of fiber-based wireless systems, where a simpler transmitter structure can be used if the input sequences are either binary or ternary [9]. A further example is the identification of an electronic nose described by Tan and Godfrey [10]. In this system, there were four compartments which could be filled with different chemicals or the same chemical but with different concentrations. A metal–oxide–semiconductor sensor was exposed to the content in one of the compartments at any particular time, and the input was implemented using four on–off valves. In [10], only two compartments were used, but the physical structure of the system allows a maximum of four input levels to be applied. When non-Gaussian inputs are used, the BLA obtained is biased with respect to the Gaussian case. The amount of bias depends on both the form of the nonlinearities and the higher order moments of the input sequence. Theoretical expressions for this bias have recently been developed for a binary input sequence [11]. Expressions for the bias have also been obtained in the case of a Wiener–Hammerstein system with a cubic or a quintic nonlinearity when a white noise input with an arbitrary probability distribution is used [12]. This paper considers the bias when a multilevel discrete input sequence is used. It is shown that it is possible to design discrete multilevel sequences to mimic Gaussianity as close as possible (hence reducing the bias) by adjusting sequence levels and the probabilities of the sequence being at these levels. Their performance has been compared with the Gaussian case in simulation experiments.

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II. BLA AND G AUSSIAN S EQUENCES The BLA of a nonlinear system minimizes, in a least square sense, the difference between the output of the nonlinear system and the output of the linear approximation of the system. It is often expressed as an impulse response, in which case, for a system with impulse response g, with a set of system inputs {u} and corresponding set of outputs {y}, the BLA is given by (1) gBLA = arg min E |y − g ∗ u|2 g

where ∗ denotes convolution and E denotes the expectation operator acting across all possible sequences from which the set {u} was generated. The subscript “g” denotes the variable to be tuned to minimize the expected value. This expression is valid for random noise and for periodic random sequences. For any given periodic input sequence, the nonlinear distortions appear in the estimated impulse response as similar to the effect of noise, the difference being that averaging over several periods of any given realization of u does not reduce the apparent noise, as it would in the case of a linear system with additive noise at the output. For the nonlinear case, to achieve a good estimate of the BLA, it is necessary to do multiple experiments, averaging over several different realizations of u. The BLA depends on the power and the amplitude distributions of the inputs {u}. As noted earlier, the BLA when using a Gaussian input sequence has been well studied [2], [6]. The objective in this paper will be to design multilevel sequences with moments as close as possible to those of a zero-mean Gaussian sequence u ∼ N (0, σ 2 ), for which the mth moment Mm ≡ E[um ] = (m − 1)!! for even m and zero for odd m. Here, (m − 1)!! is the double factorial of (m − 1) given by (m − 1)(m − 3)(m − 5) · · · 1. The closer the moments of an arbitrary sequence match the moments of N (0, σ 2 ), the closer the estimated BLA will be to that estimated using a Gaussian input sequence. A. Gaussian BLA for a Volterra System For a discrete Volterra system, the output y (q) (t) contains only the contributions from the q th Volterra kernel h(q) so that y (q) (t) =

∞

···

k1 =0

∞

h(q) (k1 , . . . , kq )

q

x(t − kα ). (2)

(3)

Inserting (2) in (3) and assuming that the input sequence (Gaussian or otherwise) is white, the kernel arguments form pairs such that =

∞ k1

···

∞ kp

pairing

h

(q)

∞ ∞

h(k1 , k1 , k2 , k2 , k2 , k2 , r)

k1 =0 k2 =0

× [x(t − k1 )]2 [x(t − k2 )]4 [x(t − r)]2 for k1 = k2 . Within this summation, there are 105 ways of permuting the dimension arguments for k1 = k2 = r, 35 ways for k1 = r and k2 = r, and, finally, 21 ways for k1 = r and k2 = r. For a zero-mean Gaussian sequence with standard deviation of one, the relevant combinations of moments of the sequence are given by M22 M4 = 3, M42 = 9, and M2 M6 = 15, so that the constants of proportionality for all three possible combinations of argument are equal to 105 × 3 = 35 × 9 = 21 × 15 = 315. Thus, for a zero-mean Gaussian input sequence, the combinatorial nature of the Gaussian moments allows (4) to be simplified to ∞ ∞ (q) gBLA,N (r) = Mq+1 · · · h(q)(k1 , k1 , k2 , k2 , . . . , kp , kp , r). kp

(5)

From [12] and [13], the BLA is given by (q) gBLA (r) = E y (q) (t)x(t − r) .

(q) gBLA (r)

S2,4,1 =

k1

α=1

kq =0

where q is odd, p = (q − 1)/2. A similar expression can be written for continuous-time systems by replacing the sums of kn with integrals. While only the q th kernel is shown(q)earlier, , then the law of superposition applies—since y = ∞ q=1 y ∞ (q) gBLA = q=1 gBLA . Note that there are (q − 1) terms and p pairs of kn in (4). The reason for the pairwise arguments in (4) is that, for any zero-mean input sequences (Gaussian or otherwise), in which all odd-order higher moments are zero, even-order higher moment terms must be constructed from the combination of dimension arguments (kn ) in pairs. For the particular case of a zero-mean Gaussian sequence, the double factorial form of the even-numbered higher moments results in the summation of the Volterra kernel dimensions with the same constant of proportionality, regardless of which argument pair (or pairs) is numerically equal to r, the element of interest. Further, Volterra kernels are symmetric, for example, h(a, b, c) = h(b, c, a) = h(c, b, a), and this results in a constant of proportionality dependent on multinomial coefficients. This is best understood with an example. Consider the contribution to the BLA of a seventh-degree Volterra kernel h. Using (4), part of the BLA includes a summation term of the form

( k1 , k1 , k2 , k2 , . . . , kp , kp , r)

q−1 terms,p pairs

×E x (t − r) 2

p α=1

x (t − kα ) 2

This is not the case for a non-Gaussian input sequence. For example, for a zero-mean random binary sequence with all higher order moments equal to unity, the contributions from the different Volterra kernel slices are summed in different proportions, resulting in a bias. The extent of the bias depends on several factors, including the linearity, the nonlinearity, and how much the input sequence differs from Gaussianity. B. Discrepancy Factor

(4)

It is useful to be able to quantify the difference between BLAs estimated using Gaussian input sequences and those estimated using non-Gaussian input sequences (in both cases,

WONG et al.: DESIGN OF MULTILEVEL SIGNALS FOR IDENTIFYING THE BLA OF NONLINEAR SYSTEMS

with power normalized to unity), and a convenient way of doing this is to use the discrepancy factor D [12] defined by N −1 |ˆ gBLA Gaussian (k) − gˆBLA non−Gaussian (k)|2 Δ D = k=0 (6) N −1 gBLA Gaussian (k)|2 k=0 |ˆ where N is the period of the input sequence (assumed to be longer than the settling time of the system) and gˆBLA Gaussian (k) and gˆBLA non−Gaussian (k) are the Gaussian and non-Gaussian BLA estimates, respectively.

The discrepancy factor can be minimized by making as many moments Mm of a multilevel sequence as possible equal those of a Gaussian sequence with the same power. This will be developed here for ternary, quaternary, and quinary sequences, all with levels symmetrical around zero and with zero mean. This results in all odd moments being zero, so that it is possible to concentrate on matching the even moments. The power of the sequences will be normalized to unity, which results in the moment M2 being matched in each case. It is then possible to match one further moment for a ternary sequence, two for a quaternary sequence, three for a quinary sequence, and so on. A. Ternary Sequences Let the levels be ±a (with equal probability of 0.5p) and zero, with probability (1 − p). Normalizing the power to unity and matching M4 with that of a Gaussian sequence give Δ

M2 = E[u2 ] = a2 p = 1 Δ M4 = E[u4 ] = a4 p = 3!! = 3. Solving these for a and p gives √ a= 3

p=

1 3

(7a) (7b)

(8)

so that the sequence levels are ±1.732, each with probability of 0.1667, and zero, with probability of 0.6667. B. Quaternary Sequences Let the levels be ±a1 , with equal probability of 0.5p, and ±a2 , with equal probability of 0.5(1 − p). Normalizing the power to unity and matching M4 and M6 with those of a Gaussian sequence give Δ

M2 = E[u2 ] = a21 p + a22 (1 − p) = 1 Δ M4 = E[u4 ] = a41 p + a42 (1 − p) = 3!! = 3 Δ M6 = E[u6 ] = a61 p + a62 (1 − p) = 5!! = 15. Solving these gives

√ a1 = 3 − 6 a2 = √ 1 p = (3 + 6). 6

3+

C. Quinary Sequences Let the levels be ±a1 , with equal probability of 0.5p1 ; ±a2 , with equal probability of 0.5p2 ; and zero, with probability of (1 − p1 − p2 ). Normalizing the power to unity and matching M4 , M6 , and M8 with those of a Gaussian sequence give Δ

M2 = E[u2 ] = a21 p1 + a22 p2 = 1 Δ

4

Δ

6

Δ

8

M4 = E[u ] = M6 = E[u ] = M8 = E[u ] =

III. M ULTILEVEL S EQUENCE D ESIGN FOR M INIMIZING D ISCREPANCY FACTOR

(9a) (9b) (9c)

√ 6 (10)

so that the sequence levels are ±0.742, each with probability of 0.4541, and ±2.334, each with probability of 0.0459.

521

a41 p1 a61 p1 a81 p1

(11a)

= 3!! = 3

(11b)

+

a42 p2 a62 p2

= 5!! = 15

(11c)

+

a82 p2

= 7!! = 105.

(11d)

+

Solving the simultaneous equations gives √ √ a1 = 5 − 10 a2 = 5 + 10 √ √ 1 1 (7 − 2 10). p1 = (7 + 2 10) p2 = 30 30

(12)

IV. S IMULATION E XPERIMENTS Two sets of experiments on systems with a Wiener structure were conducted to confirm the theory developed in Section III. In both experiments, measurements were taken at steady state. A. Experiment 1 In this experiment, ternary input sequences were compared with Gaussian input sequences; both had a period N = 1024. The outputs were noise free and were measured for M = 1024 independent realizations of each input. The 1024 input and output spectra were averaged, with a nonparametric estimate of the BLA then being obtained by direct division. The averaging is needed even though the system is noise free because the contribution of the nonlinear terms is different for each realization. The linear part of the Wiener system was a digital firstorder low-pass filter with time constant four sampling intervals, i.e., with a transfer function given by G(z) = z/(z − e−0.25 ). Two different (static) nonlinearities were used, as described as follows. Although the full analysis could be performed with nonparametric estimates, in order to obtain a more robust estimate of the discrepancy factor D, parametric estimates of the BLA were obtained using Estimator for Linear Systems (ELiS), an iterative weighted least square estimator in the Frequency Domain System Identification Toolbox [14]. The variances of the complex frequency response estimates were supplied to ELiS for weighting The cost function used internally −1 purposes. 2 W |e | , where |ek |2 denotes the squared was C = N k k k=0 errors between modeled output and actual output in terms of harmonic k and Wk denotes the weighting factors proportional to the reciprocal of the variance at each k. For the Gaussian input sequences, a parametric model of order 1/1 (i.e., one zero, one pole) gave satisfactory fitting, but for the ternary input sequences, a model of order 8/8 was needed because the BLA then contains higher order dynamics than those of the underlying linear system; this is due to nonlinear effects (see [13, Example 4.2 on p. 42]). For the purposes of finding the

522


obtained for this nonlinearity when p0 = 2/3 than for the more widely used values of p0 = 0 (binary input) and 1/3 (uniform ternary input). B. Experiment 2

Fig. 1. Discrepancy factors of random ternary sequences with various zerolevel probabilities.

discrepancy factor D, the high order of the parametric model with a ternary input sequence is not of concern. The ternary sequence experiments were done for several different values of p0 , the probability that the sequence is at zero [note that p0 = (1 − p)], including p0 = 0 (a binary sequence), p0 = 1/3 (a uniform ternary sequence), and p0 = 2/3 (from (8), the value for which D should be minimized). 1) Cubic Nonlinearity: The first nonlinearity was a pure cubic, i.e., f (x) = x3 , for which it is possible to obtain a theoretical expression for D. From (7b), the deviation of the fourth moment of the ternary sequence with an arbitrary value of p from that of a Gaussian sequence is given by δ4 = 3!! − a4 p = 3 − a4 (1 − p0 ).

(13) 2

For the ternary sequence to have a power of unity, a p = 1, so that a2 = (1/p) = (1/(1 − p0 )). Substituting in (13) 1 δ4 = 3 − . (14) 1 − p0 By writing [12, eqs. (17) and (22)] in terms of δ4 , the theoretical expression for D is given by 2 ∞ 6 g (k) 1 (15) Dtheory = 3 − k=0 3. ∞ 1 − p0 9 [ k=0 g 2 (k)] The results are shown in Fig. 1. For the cubic nonlinearity, the theoretical discrepancy factor Dtheory is shown as a solid line, and the simulation results are shown as circles; it can be seen that there is excellent agreement between the two. 2) Ideal Saturation Nonlinearity: The second nonlinearity was an ideal saturation characteristic with limits (i.e., clipping levels) set at ±0.75 and a slope of one between these limits. For this, there is currently no theory for finding D so that only simulation results are shown. From these, it can be seen that the minimum value of D is obtained with a lower value of p0 than for the cubic nonlinearity. The difference is not unexpected, because there is no reason to expect that the contributions across multiple higher order terms will be minimized at the same value. It is worth noting that, despite this, a lower value of D is

In this experiment, Gaussian, binary, ternary, quaternary, and quinary input sequences were used; all had a period N = 2048. The linearity was a three-tap finite-impulse-response filter with transfer function given by G(z) = (1 + 0.6z −1 + 0.1z −2 ), with the relatively short impulse response magnifying the levels of discrepancy [12]. For the ternary, quaternary, and quinary sequences, two versions were created, one with a uniform distribution of the levels and the other with levels and probabilities optimized for Gaussianity, using (8) (ternary), (10) (quaternary), or (12) (quinary). The outputs were noise free and were measured for 5000 independent realizations of each input. The 5000 input and output spectra were averaged (because the contribution of the nonlinear terms is different for each realization) with a nonparametric estimate of the BLA then being obtained by direct division. The nonlinearity was a static polynomial function f (x) = x7 + x5 + x3 + x; the degree was chosen to be higher than that in Experiment 1 to emphasize performance differences between sequences with different numbers of levels and between sequences with uniform probability distributions and those optimized for Gaussianity. The estimated (frequency domain) BLA magnitude is plotted against frequency line for most of these sequences in Fig. 2. The top band shows the magnitude estimated using the Gaussian sequences, while the next two show the estimates using the optimized quaternary and ternary sequences. The other three bands show the estimates obtained using quinary, ternary, and binary uniform probability sequences. The estimates using the quaternary uniform distribution sequences are not shown, because they come between those for the quinary and ternary uniform bands, which are already relatively close to each other. The results for the optimized quinary sequences are also not shown in Fig. 2, because the BLA estimated from it is indistinguishable from the Gaussian BLA. This is as expected, since from (11a)–(11d), it is possible to match moments Mm up to m = 8, which is higher than seven, the highest degree contained in the polynomial nonlinearity. It can be seen that increasing the number of levels of the input sequences decreases the difference between the estimated BLA and that estimated using a Gaussian sequence and that the difference is considerably smaller using an optimized sequence than it is using a uniform distribution sequence. V. D IFFERENT I DENTIFICATION R EQUIREMENTS With their probability distribution approaching as close as possible to that of a Gaussian sequence, the resulting multilevel sequences are inevitably less suitable for identification of a noisy linear system when there is a limit to the maximum input magnitude (as in many practical situations). For such systems, an input signal with a low crest factor (CF) [2], correspondingly, a high value of Performance Index for Perturbation Sequences

WONG et al.: DESIGN OF MULTILEVEL SIGNALS FOR IDENTIFYING THE BLA OF NONLINEAR SYSTEMS

Fig. 2.

523

Deviations of multilevel uniform and optimized discrete sequences relative to the Gaussian BLA.

(PIPS) [15], is desirable. PIPS is a compromise between high input sequence power, to maximize the sequence-to-noise ratio, and low input sequence amplitude, to minimize the effects of nonlinearities. Since the moment matching results in the probability distribution of the multilevel sequence becoming more Gaussian in shape, this inevitably reduces the probability of the sequence being at its extreme values, so that the CF of the multilevel sequence is increased; correspondingly, its PIPS is reduced. For example, consider a ternary sequence with levels ±a and zero; then, PIPS is reduced from 100 2/3% = 81.7% for a zero-mean sequence with equal probabilities of the three levels to 100 1/3% = 57.7% for a sequence with the fourth moment matched, as in (7b). An acceptable range of values for PIPS for identification of a noisy linear system is from 70% to 100% [16]; so, for such an application, moment matching would not be used. This illustrates that an input sequence designed for one purpose may not be the best for a different purpose. This is similar to the choice for multisine signals, where a low value of CF (high value of PIPS) is desirable for the identification of noisy linear systems, whereas a Gaussian multisine [2] is desirable for BLA estimation. VI. C ONCLUSION It has been shown that it is possible to reduce the bias between a BLA estimated from a zero-mean multilevel sequence with levels symmetrical around zero and that estimated from a Gaussian sequence, by matching moments of the multilevel sequence to those of a zero-mean Gaussian sequence. The odd moments of both types of sequences are automatically zero, and if the power of the multilevel sequence is made the same as that of the Gaussian sequence (so matching the second moments), it is possible to match one further moment for a ternary sequence, two for a quaternary sequence, and three for a quinary sequence.

R EFERENCES [1] M. Enqvist and L. Ljung, “Linear approximations of nonlinear FIR systems for separable input processes,” Dept. Elect. Eng., Linköping Univ., Linköping, Sweden, Tech. Rep. LiTH-ISY-R-2718, Dec. 2005. [2] R. Pintelon and J. Schoukens, System Identification: A Frequency Domain Approach, 2nd ed. Hoboken, NJ: Wiley-IEEE Press, Apr. 2012. [3] P. M. Mäkilä and J. R. Partington, “Least-squares LTI approximation of nonlinear systems and quasistationarity analysis,” Automatica, vol. 40, no. 7, pp. 1157–1169, Jul. 2004. [4] P. M. Mäkilä, “On optimal LTI approximation of nonlinear systems,” IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1178–1182, Jul. 2004. [5] P. M. Mäkilä, “LTI approximation of nonlinear systems via signal distribution theory,” Automatica, vol. 42, no. 6, pp. 917–928, Jun. 2006. [6] J. Schoukens, J. Lataire, R. Pintelon, G. Vandersteen, and T. Dobrowiecki, “Robustness issues of the best linear approximation of a nonlinear system,” IEEE Trans. Instrum. Meas., vol. 58, no. 5, pp. 1737–1745, May 2009. [7] K. R. Godfrey, A. H. Tan, H. A. Barker, and B. Chong, “A survey of readily accessible perturbation signals for system identification in the frequency domain,” Control Eng. Pract., vol. 13, no. 11, pp. 1391–1402, Nov. 2005. [8] H. A. Barker and K. R. Godfrey, “System identification with multi-level periodic perturbation signals,” Control Eng. Pract., vol. 7, no. 6, pp. 717– 726, Jun. 1999. [9] Y. H. Ng, A. H. Tan, and T. C. Chuah, “Channel identification of concatenated fiber-wireless uplink using ternary signals,” IEEE Trans. Veh. Technol., vol. 60, no. 7, pp. 3207–3217, Sep. 2011. [10] A. H. Tan and K. R. Godfrey, “Modeling of direction-dependent processes using Wiener models and neural networks with nonlinear output error structure,” IEEE Trans. Instrum. Meas., vol. 53, no. 3, pp. 744–753, Jun. 2004. [11] H. K. Wong, J. Schoukens, and K. R. Godfrey, “The use of binary sequences in determining the best linear approximation of nonlinear systems,” in Proc. 16th IFAC Symp. Syst. Identification, Brussels, Belgium, Jul. 10–13, 2012, pp. 1323–1328. [12] H. K. Wong, J. Schoukens, and K. R. Godfrey, “Analysis of best linear approximation of a Wiener–Hammerstein system for arbitrary amplitude distributions,” IEEE Trans. Instrum. Meas., vol. 61, no. 3, pp. 645–654, Mar. 2012. [13] M. Enqvist, “Linear models of nonlinear systems,” Ph.D. dissertation, Linköping Univ., Linköping, Sweden, 2005. [14] I. Kollár, Frequency Domain System Identification Toolbox for Use With MATLAB. Natick, MA: MathWorks Inc., 1994. [15] K. R. Godfrey, H. A. Barker, and A. J. Tucker, “Comparison of perturbation signals for linear system identification in the frequency domain,” Proc. Inst. Elect. Eng.—Control Theory Appl., vol. 146, no. 6, pp. 535– 548, Nov. 1999. [16] H. A. Barker, A. H. Tan, and K. R. Godfrey, “Object-oriented input signal design for system identification,” in Proc. 15th IFAC Symp. Syst. Ident., St. Malo, France, Jul. 6–8, 2009, pp. 180–185.

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Hin Kwan (Roland) Wong received the M.Eng. degree in electrical and electronic engineering with communication from the University of Warwick, Coventry, U.K., in 2009, where he is currently working toward the Ph.D. degree in a cotutelle agreement with the Vrije Universiteit Brussel, Brussels, Belgium, on the behavior, identification, and measurement of nonlinear systems. Mr. Wong was the recipient of the IBM Student Recognition Award in 2007, the Institution of Engineering and Technology Prize and the Dean’s Award for Academic Excellence in 2009, and the Best Research Student Poster Prize at the 2010 United Kingdom Automatic Control Council International Conference on Control.

Johan Schoukens (F’97) received the M.S. degree in electrical engineering and the Ph.D. degree in engineering from the Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1980 and 1985, respectively, and the Doctor Honoris Causa degree from the Budapest University of Technology and Economics, Budapest, Hungary, in 2011. From 1981 to 2000, he was a Researcher with the Belgian National Fund for Scientific Research (Fonds Wetenschappelijk Onderzoek–Vlaanderen) with the Department of Fundamental Electricity and Instrumentation, VUB, where he is currently a full-time Professor in electrical engineering. His main research interests include system identification, signal processing, and measurement techniques. Dr. Schoukens was the recipient of the 2002 Andrew R. Chi Best Paper Award of the IEEE T RANSACTIONS ON I NSTRUMENTATION AND M EA SUREMENT , the 2002 Society Distinguished Service Award from the IEEE Instrumentation and Measurement Society, and the 2007 Belgian Francqui Chair at the Université Libre de Bruxelles, Brussels. Since 2010, he has been a member of Royal Flemish Academy of Belgium for Sciences and the Arts.

Keith R. Godfrey received the D.Sc. degree from the University of Warwick, Coventry, U.K., in 1990, for publications with the collective title “Applications of Modelling, Identification and Parameter Estimation in Engineering and Biomedicine.” He is an Emeritus Professor with the Systems, Measurement and Modelling Research Group, School of Engineering, University of Warwick. He is the author, or coauthor, of more than 200 papers. Dr. Godfrey is a member of the International Federation of Automatic Control Technical Committees on Biomedical Engineering and Control and on Modelling, Identification and Signal Processing. He was a recipient of the Honeywell International Medal (2000/2001) of the Institute of Measurement and Control for distinguished contributions to control engineering.