system identification in the presence of a...

SYSTEM IDENTIFICATION IN THE PRESENCE OF A SATURATION NONLINEARITY

W. D. Widanage1, K. R. Godfrey1, and A. H. Tan2

1: School of Engineering, University of Warwick, Coventry, CV4 7AL, U.K.

Fax: +44 2476 418922; Emails: [email protected]; [email protected]. 2: Faculty of Engineering, Multimedia University, 63100 Cyberjaya, Malaysia

Fax: +603 8318 3029; Email: [email protected].

ABSTRACT The paper deals with the identification of linear systems when they are in a pathway in series with a saturation nonlinearity. The objective is to estimate the parameters of the linear system and to obtain some characterisation of the nonlinearity, using only the input and output signals of the pathway. Both Wiener structures and Hammerstein structures are considered, and it is found that perturbation signal design plays a crucial role in achieving the objective. Keywords: Computer-optimised signals; Detection of nonlinearities; Multisine signals; Parameter estimation; Perturbation signals; Pseudo-random signals; System identification. 1. INTRODUCTION The type of perturbation signal required for the identification of a nonlinear system is usually highly dependent on the structure of the system. It is essential that any signal used should be persistently exciting in respect of the system being identified, and this has implications not only in terms of frequency (ensuring that the bandwidth of the system is adequately spanned), but also in terms of amplitude. For example, for a block-oriented system with a quadratic nonlinearity, it is essential to use signals with three or more levels in a Hammerstein structure (static nonlinearity preceding linear dynamics), while binary signals will suffice for a Wiener structure (linear dynamics preceding static nonlinearity) [3]. In a number of cases when multilevel signals are required, it is possible to determine optimal signal levels for the identification of the system. For example, optimal levels were recently determined for the identification of a block-oriented system with a nonlinearity of polynomial form [4], in which the condition number of a submatrix of the Vandermonde matrix of the signal levels vector was used as the measure of optimality. Results in [4] were presented for polynomial orders up to 6. In the present paper, the identification of block-oriented nonlinear systems with both Wiener and Hammerstein

structures is again considered, but this time, the nonlinearity is one that commonly occurs in practice – a saturation nonlinearity [1]. In a control system, such a nonlinearity can occur in either the control actuator (the input to the system to be controlled) or in the output of the system itself. A saturation nonlinearity cannot be approximated by a low-order polynomial, and it is also not readily invertible. There are two main objectives of the identification experiments described in this paper. The first is to identify the linear dynamics as accurately as possible, by processing the input and output signals using estimation routines in either the MATLAB System Identification Toolbox [8] or the Frequency Domain System Identification (FDIDENT) Toolbox [7]. The second objective is to detect the presence of nonlinear distortion and to obtain a characterisation of the static plot of the nonlinearity. This is done by perturbing the system with a signal u(t) with power at certain specified harmonics only, and then examining the output power spectrum at the non-specified harmonics [6, 9]. The suppressed harmonics are known as detection lines and the specified harmonics as the measurement lines. 2. DETAILS OF THE SIMULATIONS 2.1. Static Nonlinearity and Linear System The saturation nonlinearity had an input-output characteristic as shown in Fig. 1. Output +0.5 -0.5 +0.5 Input -0.5 Figure 1: Saturation nonlinearity.

Control 2004, University of Bath, UK, September 2004 ID-141

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The linear system was first order, with transfer function

1-0.3679z-11z)( =G (1)

The steady state gain of the system is thus 1.582. The signals were sampled at a sampling interval of 1s, so that the pole position corresponds to a continuous system with a time constant of 1s. 2.2. Perturbation Signals The perturbation signals used were periodic, and drawn from one of three readily accessible sources: • A Pseudo-Random Binary (PRB) signal with levels ±1

from the prs MATLAB routines [13]. The signal used had even harmonics suppressed, and so was inverse repeat (having the second half the inverse of the first half). With binary signals, no other patterns of harmonic suppression are possible. In this work, an inverse repeat Quadratic Residue Binary (QRB) signal was used; such signals have period N = 8k – 2, where k is a positive integer and 4k – 1 is prime.

• Pseudo-Random Ternary (PRT) signals with levels ±1 and 0, from the GALOIS package [2]. These were based on Maximum Length Ternary (MLT) sequences, which have period N = (qk – 1), where q is prime or a power of a prime (≠ 2) and k is a positive integer. In this work, q was chosen as 31, because it is then possible to design signals with harmonic multiples of 2, or of 2 and 3, or of 2, 3, and 5 suppressed. The value of k was set at 2, so that N = 960.

• Multisine (Sum of Harmonics) signals with RMS of 1, from the FDIDENT Toolbox [7]. Such signals can assume any level between their minimum and maximum values. Zero Order Hold Sum of Harmonics (ZOHSOH) signals were chosen. These have a power spectrum that pre-compensates for the shape of the spectrum of the zero order hold. ZOHSOH signals have a uniform power spectrum at the specified harmonics, but inevitably have some power at higher (unspecified) harmonics.

3. WIENER MODEL The Wiener structure examined is shown in Fig. 2. u(t) w(t) y(t) Figure 2: Wiener system. For linear system identification and the detection of nonlinearity, only the signals u(t) and y(t) were used. The intermediate signal w(t) was assumed not to be available for the identification.

One method of nonlinear identification is to perform a recursive operation on the accumulated input and output sampled data values. From prior knowledge of the linear subsystem and general behaviour of the nonlinearity, an estimate ŵ(t) of the intermediate signal can be obtained. Using this, the subsystems are re-evaluated giving better estimates of their parameters. However, this recursive technique presents a problem for the Wiener structure, in that the inverse of the nonlinearity cannot be found readily, since it is a many to one function. For the Wiener system, the shape of the nonlinearity can be obtained by plotting ŵ(t) against y(t); results of this are not included in this paper due to space restrictions. 3.1. Perturbation Signals Used The following perturbation signals were used:- • An inverse repeat QRB signal with N = 958 (odd

harmonics only). • Three PRT signals, all with N = 960, one having no

power at even harmonics, the second having no power at harmonic multiples of 2 and 3, and the third having no power at harmonic multiples of 2, 3 and 5.

• Four ZOHSOH multisines, all with N = 960, but with different harmonic specifications:- Multisine A: 128 harmonics, with harmonic multiples of 2 and 3 suppressed (i.e. harmonics 1, 5, 7, 11, 13,…., 383 present). Multisine B: 96 alternate odd harmonics present (1, 5, 9, 13,.…, 381). Multisine C: 96 ‘Special Odd’ harmonics (1, 3, 9, 11, 17, 19,…., 379) [10, Section 3.5]. Multisine D: An NID (No Interharmonic Distortion) multisine with 10 specified harmonics (1, 5, 13, 29, 49, 81, 119, 141, 207, 263). NID multisines are designed by computer search to eliminate distortion at the detection lines [5]. The drawback to such signals is that their measurement lines are relatively sparse. The period N of an NID multisine with a comparable number of specified harmonics to the other specifications is very large.

For all of the multisines, one set of signals was designed with random starting phases and a second set was designed with Schroeder starting phases [12]. Schroeder phases are designed to maximise the power in the specified harmonics, while reducing the peak–to–peak range of the signal, for either consecutive or consecutive odd harmonic specifications. The multisines were generated using an algorithm that, at each successive iteration, swaps between the time and frequency domain. In this work, signals with either 1 iteration or with 10,000 iterations were used.

Linear Dynamics G(z)

Saturation Nonlinearity

3.2. Detection of the Nonlinearity It was found that, for all of the multisine signals, random initial phases with 1 iteration gave better detection of the


nonlinearity than either random or Schroeder initial phases with 10,000 iterations. The following conclusions regarding detection of the nonlinearity were drawn from the simulation results:- • For the PRB and PRT signals with even harmonics

only suppressed, there was no measurable power in the output spectrum at the even harmonics.

• In all cases, the magnitude of the output power spectrum was considerably smaller than that of the input power spectrum at the same frequency, except at a few of the higher frequencies where nonlinear distortion can prove larger than the linear component.

The input and output power spectra for three of the perturbation signals are presented in Figs. 3 to 5. 3.3. Estimation of the Linear Dynamics The dynamics of the linear sub-system were estimated from several linear system estimation routines available in MATLAB Toolboxes using the signals u(t) and y(t) only (i.e. not using the intermediate signal w(t)). In all cases using multisine signals, it was found that the pole position estimated was closest to the theoretical value using random phased multisines with 1 iteration. This confirms the results in [11] in respect of estimation of the linear dynamics of systems with nonlinearities. The pole positions and steady state gains estimated from the Estimator for Linear Systems (ELiS) routine in the FDIDENT Toolbox [7] are shown in Table 1 for four different perturbation signals. Figure 3: Input and output power spectra for a PRT signal with harmonic multiples of 2, 3 and 5 suppressed.

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Figure 4: Input and output power spectra for a 1 iteration random phased multisine with Specification A. Figure 5: Input and output power spectra for a 1 iteration random phased NID multisine with Specification D. Results obtained using the Prediction Error Method (PEM) and Instrumental Variables (IV) routines (with model orders set to 1 and delay to 0) in the System Identification Toolbox [8] are shown in Table 2.

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TABLE 1 – Estimates of pole positions and gains using the routine ELiS from the FDIDENT Toolbox

Input Signal Frequency

Specification Pole

Position Steady

State Gain PRB

(N = 958) Odd harmonics only 0.017 0.501

PRT (N = 960)

Odd harmonics, not multiples of 3 or 5

0.347 0.766

Multisine A (N = 960)

Spec. A 0.341 0.558

Multisine D (N = 960)

NID, Spec. D 0.396 0.470

TABLE 2 – Estimates of pole positions using routines in

the System Identification Toolbox

State Space

ARX Input Signal Frequency Specification

PEM IV PRB

(N = 958) Odd harmonics only 0.663 0.014

PRT (N = 960)

Odd harmonics, not multiples of 3 or 5

0.352 0.353

Multisine A (N = 960)

Spec. A 0.293 0.339

Multisine D (N = 960)

NID, Spec. D 0.358 0.373

It can be seen from the Tables that, when using either ELiS or IV, the pole positions are reasonably close to the true value of 0.3679, except in the case of the binary signal. From the ELiS results, the steady state gains are not at all close to the true value of 1.582; this is because from examination of u(t) and y(t) only, the gain cannot be apportioned accurately between the static nonlinearity and the linear dynamics.

-2 -1 0 1 2

-0.4 -0.2

0 0.2 0.4 0.6

Intermediate Signal

O u

t p

u t

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a l

Figure 6: Static plot of the nonlinearity, for the QRB signal. For the binary signal input, none of estimated pole values is remotely near the true value. To see why this is so, the intermediate signal w(t) was examined. The static plot of y(t) versus ŵ(t) is shown in Fig. 6, from which it may be seen that very few of the data points lie on the linear part of the characteristic. This accounts for the poor linear dynamic estimates of the Wiener model with a binary

signal. Similar plots for the other perturbation signals showed a much better coverage of the linear portion. 4. HAMMERSTEIN MODEL The Hammerstein structure examined is shown in Fig. 7. u(t) w(t) y(t)

Saturation Nonlinearity

Linear Dynamics G(z)

Figure 7: Hammerstein system. The system examined consisted of the same saturation nonlinearity and first order linear dynamics as that of the Wiener model, and as before, the aim was to detect or identify the nonlinearity and estimate the linear dynamic parameters. For this system, the PRB signals and PRT signals could not be used as perturbation signals, since the saturation nonlinearity would simply reduce their amplitude, giving perfect linear estimation (with reduced amplitude due to the clipping), but no other characterisation of the nonlinearity. Therefore, only the multisine signals were used, and these were the same as for the Wiener system. 4.1. Recursive Estimation In contrast to the Wiener system, iterative estimation of the intermediate signal w(t) is now possible because the nonlinearity is at the front end, and its inverse function is not required to calculate ŵ(t). With this data set and the actual measured input and output data values, a recursive operation can be performed to refine the linear dynamic estimates and also identify the nonlinearity of the system. The recursive operation is as follows: • Step 1: Estimate linear dynamics, Ĝ(z) from the

measured input u(t) and output y(t); • Step 2: Estimate ŵ(t) using Ĝ(z) and y(t); • Step 3: Obtain the lower limit, gradient, and upper

limit for the static plot of ŵ(t) vs. u(t); • Step 4: Re-estimate ŵ(t) using u(t) and the

characterisation of the nonlinearity obtained in Step 3; • Step 5: Re-estimate Ĝ(z) using ŵ(t) and y(t); • Step 6: Go to Step 2. Using the measured output data values, the intermediate variable is theoretically evaluated by treating ŵ(t) as the input to the estimated linear dynamics. Results using this procedure are shown in Table 3. The static characteristic for a random phased multisine with Specification B is shown in Fig. 8, from which it can be seen that the shape of the saturation characteristic is very well estimated by this procedure. Its amplitude is too large, being just over 1.2, compared with the correct value of 0.5 (Fig. 1).


TABLE 3 – Recursion, and estimation using an ARX model in [8], with random phased single iteration multisines, all with N = 960

First Estimate Second Estimate Frequency Specification Pole

Value Steady

State Gain Pole

Value Steady

State Gain Spec. A 0.360 0.626 0.368 0.655 Spec. B 0.362 0.616 0.367 0.612 Spec. C 0.353 0.635 0.368 0.660

NID, Spec. D 0.405 0.622 0.369 0.605 The corresponding steady state gain of the linear dynamics is too small, being 0.612, compared with the correct value of 1.582. The estimated amplitude–gain product is approximately the same as the theoretical one, showing that, from examination of u(t) and y(t) only, it is possible to obtain a reasonable estimate for the overall pathway gain, but not for the gains of the individual blocks making up the pathway.

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Figure 8: Static plot of nonlinearity, for a random phased single iteration multisine with Specification B.

5. DISCUSSION AND CONCLUSIONS The results presented have illustrated that, through judicious choice of perturbation signals (all of which are readily available), it is possible to detect the presence of a saturation nonlinearity, and to obtain some characterisation of it. The perturbation signal needs to be chosen with care. For both types of structure examined, the nonlinear distortion occurred only at the odd harmonics, from which it may be concluded that the nonlinearity is an odd function. For the Hammerstein structure, it was possible to obtain a steady state characteristic using an iterative procedure, but this was not possible for the Wiener structure. Table 1 illustrates that the linear dynamics of a Wiener model are well estimated, when perturbed by NID multisines. This is expected since these signals are designed to eliminate distortion at the detection lines. From examination of u(t) and y(t) alone, it is only possible to obtain a reasonable estimate for the overall pathway gain, rather than for the two individual blocks (linear dynamics and static nonlinearity).

ACKNOWLEDGEMENT W.D. Widanage was supported by Undergraduate Research Bursary No. URB/01090/G from The Nuffield Foundation. REFERENCES [1] P. Airikka, “The PID controller: algorithm and

implementation”, IEE Computing and Control Engineering, 14 (6), pp. 6 – 11, (2003/2004).

[2] H.A. Barker, “GALOIS – a program for generating pseudo-random perturbation signals”, Proc. 12th IFAC Symp. Syst. Identification (SYSID 2000), R. Smith, Ed., pp. 505 – 508, Elsevier Science, Oxford, (2001).

[3] H.A. Barker, K.R. Godfrey and A.J. Tucker, “Nonlinear system identification with multilevel perturbation signals”, Proc. 12th IFAC Symp. Syst. Identification (SYSID 2000), R. Smith, Ed., pp. 1175 – 1178, Elsevier Science, Oxford, (2001).

[4] H.A. Barker, A.H. Tan and K.R. Godfrey, “The performance of multilevel perturbation signals for nonlinear system identification”, Proc. 13th IFAC Symp. Syst. Identification (SYSID 2003), Rotterdam, 27 – 29 August, pp. 683 – 688, (2003).

[5] C. Evans, D. Rees, L. Jones, and M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels”, IEEE Trans. – Instrum. Meas., IM-45, pp. 362 – 371, (1996).

[6] D.C. Evans, D. Rees, and L. Jones, “Identifying linear models of systems suffering nonlinear distortions”, Proc. IEE International Conference “Control ’94”, Coventry, pp. 288 – 296, (1994).

[7] I. Kollár, “Frequency Domain System Identification Toolbox for use with MATLAB”, The MathWorks, Natick, MA., (1994).

[8] L. Ljung, “System Identification Toolbox - Users Guide”, The MathWorks, Natick, MA., (1997).

[9] A.S. McCormack, K.R. Godfrey and J.O. Flower, “The detection of and compensation for nonlinear effects for frequency domain identification”, Proc. IEE International Conference “Control ’94”, Coventry, pp. 297 – 302, (1994).

[10] R. Pintelon, and J. Schoukens, “System Identification – a Frequency Domain Approach”, IEEE Press, (2001).

[11] J. Schoukens, R. Pintelon, Y. Rolain and T. Dobrowiecki, “Frequency response function measurements in the presence of nonlinear distortions”, Automatica, 37, pp. 939 – 946, (2001).

[12] M.R. Schroeder, “Synthesis of low peak factor signals and binary sequences with low autocorrelation”, IEEE Trans. – Inform. Theory, IT-16, pp. 85 – 89, (1970).

[13] A.H. Tan and K.R. Godfrey, “The generation of binary and near–binary pseudorandom signals: an overview”, IEEE Trans. – Instrum. Meas., IM-51 (4), pp. 583 – 588, (2002).


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