Elsevier

Automatica

Volume 50, Issue 3, March 2014, Pages 768-783
Automatica

Combined frequency-prediction error identification approach for Wiener systems with backlash and backlash-inverse operators

https://doi.org/10.1016/j.automatica.2013.12.030Get rights and content

Abstract

Wiener systems identification is studied in the presence of possibly infinite-order linear dynamics and memory nonlinear operators of backlash and backlash-inverse types. The latter is laterally bordered with polynomial lines of arbitrary-shape. It turns out that the borders are allowed to be noninvertible and crossing making possible to account, within a unified theoretical framework, for memory and memoryless nonlinearities. Moreover, the prior knowledge of the nonlinearity type, being backlash or backlash-inverse or memoryless, is not required. Using sine excitations, and getting benefit from model plurality, the initial complex identification problem is made equivalent to two tractable (though still nonlinear) prediction-error problems. These are coped with using linear and nonlinear least squares estimators which all are shown to be consistent.

Introduction

The Wiener model is a series connection of a linear dynamic bloc and a memoryless nonlinearity (Fig. 1). When both parts are parametric, the identification problem can be dealt with using stochastic methods (e.g. Vanbeylen & Pintelon, 2010; Vanbeylen, Pintelon, & Schoukens, 2009; Wigren, 1993, Wigren, 1994; Wills & Ljung, 2010) as well as deterministic methods (e.g. Bruls, Chou, Heverkamp, & Verhaegen, 1999; Vörös, 1997, Vörös, 2010). The stochastic methods enjoy local or global convergence properties under various assumptions, e.g. the system inputs should be persistently exciting (PE) or Gaussian and the system nonlinearity is invertible. The last limitation has recently been overcome by Wills, Schön, Ljung, and Ninness (2011). Multi-stage methods have been proposed in (e.g. Lovera, Gustafsson, & Verhaegen, 2000; Westwick & Verhaegen, 1996) and their consistency was ensured in the presence of Gaussian inputs provided the nonlinearity is odd. Deterministic parameter identification methods consist in reformulating the problem as an optimization task that is generally coped with using various relaxation techniques. Then, local convergence is ensured with PE inputs.

Nonparametric Wiener systems (where none of the linear subsystem and the nonlinear element assumes a priori known structure) have been approached using both stochastic and frequency methods. In stochastic methods (e.g. Greblicki & Pawlak, 2008; Mzyk, 2010), the nonlinearity is generally determined using variants of the kernel regression estimation technique while the (unknown) coefficients of a FIR/IIR approximation of the linear part are estimated using cross-correlation analysis. Several assumptions are needed e.g. Gaussian inputs, FIR linear dynamics, and Lipschitzian nonlinearity. In frequency methods, the linear subsystem frequency response and the nonlinearity map are determined in two or several stages (e.g. Bai, 2003, Crama and Schoukens, 2001, Crama and Schoukens, 2005 and Giri, Rochdi, & Chaoui, 2009). The case of series–parallel Wiener systems is dealt with in Schoukens and Rolain (2012) using, among others, the best linear approximation approach. Wiener systems where only the linear part is parametric have been considered in many places and dealt with under various assumptions. In Hu and Chen (2008), the parameters of the linear subsystem, supposed to be FIR, are estimated together with points of the nonlinearity using the stochastic approximation algorithm and Gaussian inputs. Estimating the linear subsystem parameters without estimating the system nonlinearity is possible if the inputs are separable stochastic processes and the system nonlinearity enjoys the invariance property (Enqvist, 2010). This approach is known as dimension reduction and more recent results, including extension to series–parallel Wiener systems, can be found in Lyzell and Enqvist (2012) and Lyzell, Andersen, and Enqvist (2012). Similar results are shown to be achievable in the case of non-Gaussian inputs provided the nonlinearity is partly known, e.g. it lies in the first and the third quadrant or is locally invertible on a known interval (Bai & Reyland, 2009). In Pelckmans (2011), the identification problem is cast as a convex quadratic programation procedure achieving almost consistent estimates (i.e. accuracy of the estimates is only guaranteed up to a small approximation term) provided the linear subsystem is FIR, the nonlinearity is monotone, the input is PE.

In the light of the above discussion, it is seen that most Wiener system identification methods were designed on the basis of several assumptions e.g. the nonlinearity is invertible, monotone or odd; the linear subsystem is FIR; the input signals are generally assumed to be Gaussian or PE. One more common limitation of most previous studies is that the system nonlinearity is supposed to be memoryless. A few exceptions are (Cerone et al., 2009, Dong et al., 2009, Giri et al., 2013) where backlash nonlinearities have been considered.

In this study, the identification problem is addressed for Wiener systems with parametric nonlinear operator of backlash and backlash-inverse types (Fig. 2a, Fig. 2b). The linear subsystem G(s) may be parametric or not, finite order or not. Backlash operators are generally met in gear transmission systems: a gear system without backlash cannot work! They are also used in damper and valve modelling and control (e.g. Shoukat Choudhury, Thornhill, & Shah, 2005). The backlash-inverse behaviour is met when Coulomb friction is involved (e.g. in leaf spring suspension). Presently, both types of nonlinear operators are bordered by polynomial lines. The latter are allowed to be noninvertible and crossing which makes possible to handle, using the same identification method, both memory and memoryless nonlinearities. The identification problem amounts to determining an accurate estimate of the (nonparametric) frequency response G(jω), for a set of frequencies (ω1ωm), and the coefficients of the (parametric) nonlinearity borders. While, it is not clear whether all unknown quantities can simultaneously be determined using a one-stage identification method (involving a single PE input signal), it is shown in this study that a two-stage determination in possible using sine input excitations. The key point is that, when an ωi-frequency excitation is applied, the linear part of the system boils down to a simple complex gain (i.e. G(jωi)). Then, getting benefit of the model plurality, one lets G(jω1)=1 limiting thus the system uncertainty to the nonlinear operator parameters which can then be estimated using nonlinear prediction error methods. In the second stage, the linear subsystem complex frequency gains G(jωi)(i2) are computed in parallel, using the data obtained in the remaining ωi-frequency experiments. Interestingly, all involved estimators are shown to be consistent. Furthermore, the identification method, thus constructed, applies indifferently to all considered nonlinearity types. Moreover, the prior knowledge of the nonlinearity type (being backlash, backlash-inverse or memoryless) is not a priori required. Some of the present results partly rely on the work of Giri, Rochdi, Ikhouane, Brouri, and Chaoui (2012). Also, this study features several differences and novelties compared to Giri et al. (2013). Specifically:

  • (i)

    The present identification method is a two-stage: the system nonlinear operator is identified first and based upon in the second stage to identify the linear subsystem. The method in Giri et al. (2013) is a three-step: the linear subsystem phase G(jω) is estimated first; then, the nonlinearity is determined based on the phase estimates; finally, the gain modulus |G(jω)| is estimated using the previously obtained estimates.

  • (ii)

    While the focus is limited to backlash nonlinearities in Giri et al. (2013), the present study is covering backlash, backlash-inverse and memoryless nonlinearities and the identification method indifferently applies to all categories and the user does not need to a priori knows which nonlinearity type he is facing.

  • (iii)

    The identification method in Giri et al. (2013) requires that the (backlash operator) borders contain at least one affine portion. Consequently, that method cannot apply to backlash operators with polynomial borders (of higher degree that one) while the present identification approach does.

  • (iv)

    The identification method of Giri et al. (2013) relies on specific analytic geometry notions e.g. spread/orientation compatibility, function affinization. It contrasts with the present identification method which instead relies on the optimization of prediction-error cost functions.

The paper is organized as follows: the identification problem is formulated in Section  2; data acquisition is discussed in Section  3; the nonlinear operator identification is coped with in Section  4; the linear frequency response determination is investigated in Section  5; simulations are presented in Section  6.

Section snippets

Identification problem formulation

Standard Wiener systems consist of a linear dynamic subsystem G(s) followed in series by a memoryless nonlinear operator F[] (Fig. 1). Presently, both memory and memoryless nonlinearities are considered and handled within a unified framework. More specifically, the Wiener system under study is analytically described by the following equations: x(t)=g(t)u(t)with  g(t)=L1(G(s))y(t)=w(t)+ξ(t)with  w(t)=F[x](t) where u(t) and y(t) denote the control input and the measured output; x(t) and w(t)

System frequency analysis and data acquisition

The frequency identification approach that is developed in this paper relies on the investigation of the system response to sine excitations u(t)=Ucos(ωt). These have already proved to be quite useful in the identification of Wiener systems (Giri et al., 2013, Giri et al., 2009, Rijlaarsdam, Oomen et al., 2012) and Wiener–Hammerstein systems (Rijlaarsdam, Nuij, Schoukens, & Steinbuch, 2012).

Fundamental system parameterizations

Throughout this subsection, the Wiener system (1), (2), , , (5) is excited by a sine input u(t)=Ucos(ωt) for some informative couple (U,ω), belonging to the set {(Uh,ωh);h=1m} found with Table 1. Using (6a), (6b) and (3), it readily follows from (8b), (8d) that: yU,ω(t)=i=0nai(U|G(jω)|)i(cos(ωtφ(ω)))i+ξ(t)=i=0nαi(ω)(cos(ωtφ(ω)))i+ξ(t),for all  t[tk+πω+τatk+2πω),kNandyU,ω(t)=i=0ndi(U|G(jω)|)i(cos(ωtφ(ω)))i+ξ(t)=i=0nδi(ω)(cos(ωtφ(ω)))i+ξ(t),for all  t[tk+τdtk+πω),kN where αi(U,ω)

Linear subsystem estimation

In this subsection, the aim is to get accurate estimates of the frequency gains G(jωh) where the frequencies ωh(h=1m) are those selected in Table 1. First, notice that by (18) one has |G(jω1)|=1/U1. Also, a consistent estimate φˆ(ω1,N) of φ(ω1) is obtained using Table 3, based on the analysis of Proposition 4. Therefore, it only remains to determine accurate estimates of (|G(jωh)|,φ(ωh)) for h=2m. This problem is presently dealt with by considering Optimization Problem 2 and the corresponding

Wiener system with backlash nonlinear operator

Presently, the system (1), (2), , , (5) is characterized by G(s)=0.5/(s2+0.7s+0.1) and a backlash operator F[] bordered by the functions: fa(x)=1.05+1.15x0.15x2+0.05x3andfd(x)=1.05+1.15x+0.15x2+0.05x3. At this stage, the structure of G(s) (being a second order) and the nature of F[] (being a backlash) are not known to the user. The noise ξ(t) is a sequence of normally distributed (pseudo) random numbers, with zero-mean and standard deviation σξ=0.35. According to the identification method

Conclusion

The problem of system identification is addressed for Wiener systems where the linear subsystem, described by (1)–(2), may be parametric or not, finite order or not. The nonlinear element is any memory operator of backlash or backlash-inverse type, that is bordered by polynomial lines. The latter are allowed to be noninvertible and crossing so that memory and memoryless polynomial nonlinearities can be handled within a unified theoretical framework. The identification problem is dealt with

Fouad Giri received a Ph.D. from the Institut National Polytechnique of Grenoble, France, in 1988 and is now Distinguished Professor at the University of Caen Basse-Normandie, Caen, France. He is currently serving as the General Chair of the IFAC Int. Workshops ALCOSP 2013 and PSYCO 2013. He is the Vice-Chair of the IFAC TC ‘Adaptive and Learning Systems’ and is holding membership positions in the IFAC TCs on ‘Modelling, Identification and Signal Processing’ and ‘Power Plants and Power

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    Fouad Giri received a Ph.D. from the Institut National Polytechnique of Grenoble, France, in 1988 and is now Distinguished Professor at the University of Caen Basse-Normandie, Caen, France. He is currently serving as the General Chair of the IFAC Int. Workshops ALCOSP 2013 and PSYCO 2013. He is the Vice-Chair of the IFAC TC ‘Adaptive and Learning Systems’ and is holding membership positions in the IFAC TCs on ‘Modelling, Identification and Signal Processing’ and ‘Power Plants and Power Systems’, and the IEEE CSS TC ‘System Identification and Adaptive Control’. He is Associate Editor of the IFAC Journal Control Engineering Practice, of the IEEE Transactions on Control Systems Technology and of the Editorial Board of IEEE CSS conferences. His research interests include Nonlinear System Identification, Adaptive Nonlinear Control, Constrained Control, and control applications to Power Converters, Electric Machines and Power Systems. He has supervised 20 Ph.D.s and has co-authored three books and over 250 journal and conference papers on these topics.

    Abdelhadi Radouane was born in 1969. He received the Agrégation degree in Electrical Engineering from the ENSET of Rabat-Morocco in 1996. He is currently completing his Ph.D. in Automatic Control under the supervision of Professor F. Giri. His research focuses on nonlinear system identification based on multi-model representations.

    Adil Brouri was born in 1973. He received the Agrégation degree in Electrical Engineering from the ENSET of Rabat-Morocco and completed a Ph.D. in Automatic Control under the supervision of Professor F. Giri. He is conducting a research activity on nonlinear system identification and has published several journal and conference papers on this topic.

    Fatima-Zahra Chaoui obtained a Ph.D. degree in Automatic Control from the Institut National Polytechnique of Grenoble-France, in 2000. She has spent long-term visits at the Laboratoire d’Automatique de Grenoble and the GREYC, University of Caen, both in France. Since 1995, she has been successively Assistant Professor and Professor at the Ecole Normale Supérieure d’Enseignement Technique (ENSET) of Rabat-Morocco. Her research interest includes nonlinear system identification and constrained control. She published over 100 journal and conference papers on these topics and has supervised 9 Ph.D.s.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Er-Wei Bai under the direction of Editor Torsten Söderström.

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