Automatica 50 (2014) 768–783

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Automatica

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Combined frequency-prediction error identification approach forWiener systems with backlash and backlash-inverse operators✩

Fouad Giri a,1, Abdelhadi Radouane b, Adil Brouri c, Fatima-Zahra Chaoui ba Université de Caen Basse-Normandie, GREYC Lab UMR CNRS, 14032 Caen, Franceb Ecole Normale Supérieure de l’Enseignement Technique, UM5S, Rabat, Moroccoc École Nationale Supérieure d’Arts et Métiers, Meknes, Morocco

a r t i c l e i n f o

Article history:Received 1 September 2012Received in revised form16 June 2013Accepted 30 November 2013Available online 10 January 2014

Keywords:System identificationWiener systemsBacklach nonlinearitiesFrequency approachPrediction error method

a b s t r a c t

Wiener systems identification is studied in the presence of possibly infinite-order linear dynamics andmemory nonlinear operators of backlash and backlash-inverse types. The latter is laterally bordered withpolynomial lines of arbitrary-shape. It turns out that the borders are allowed to be noninvertible andcrossingmaking possible to account, within a unified theoretical framework, formemory andmemorylessnonlinearities.Moreover, the prior knowledge of the nonlinearity type, being backlash or backlash-inverseormemoryless, is not required. Using sine excitations, and getting benefit frommodel plurality, the initialcomplex 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 areshown to be consistent.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The Wiener model is a series connection of a linear dynamicbloc and a memoryless nonlinearity (Fig. 1). When both parts areparametric, the identification problem can be dealt with usingstochastic methods (e.g. Vanbeylen & Pintelon, 2010; Vanbeylen,Pintelon, & Schoukens, 2009; Wigren, 1993, 1994; Wills & Ljung,2010) as well as deterministic methods (e.g. Bruls, Chou, Hev-erkamp, & Verhaegen, 1999; Vörös, 1997, 2010). The stochasticmethods enjoy local or global convergence properties under var-ious assumptions, e.g. the system inputs should be persistently ex-citing (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 pro-posed in (e.g. Lovera, Gustafsson, & Verhaegen, 2000; Westwick& Verhaegen, 1996) and their consistency was ensured in thepresence of Gaussian inputs provided the nonlinearity is odd. De-terministic parameter identification methods consist in reformu-lating the problem as an optimization task that is generally coped

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

E-mail addresses: fouad.giri@unicaen.fr (F. Giri),radouane_abdelhadi2@yahoo.fr (A. Radouane), girfou@yahoo.fr (A. Brouri),chaouifatima@yahoo.fr (F.-Z. Chaoui).1 Tel.: +33 231567287; fax: +33 231567340.

0005-1098/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.automatica.2013.12.030

with using various relaxation techniques. Then, local convergenceis ensured with PE inputs.

Nonparametric Wiener systems (where none of the linearsubsystem and the nonlinear element assumes a priori knownstructure) have been approached using both stochastic andfrequencymethods. In stochasticmethods (e.g. Greblicki & Pawlak,2008; Mzyk, 2010), the nonlinearity is generally determined usingvariants of the kernel regression estimation technique whilethe (unknown) coefficients of a FIR/IIR approximation of thelinear part are estimated using cross-correlation analysis. Severalassumptions are needed e.g. Gaussian inputs, FIR linear dynamics,and Lipschitzian nonlinearity. In frequency methods, the linearsubsystem frequency response and the nonlinearity map aredetermined in two or several stages (e.g. Bai, 2003, Crama &Schoukens, 2001, 2005 and Giri, Rochdi, & Chaoui, 2009). The caseof series–parallel Wiener systems is dealt with in Schoukens andRolain (2012) using, among others, the best linear approximationapproach.Wiener systemswhere only the linear part is parametrichave been considered inmany places and dealt with under variousassumptions. In Hu and Chen (2008), the parameters of the linearsubsystem, supposed to be FIR, are estimated together with pointsof the nonlinearity using the stochastic approximation algorithmand Gaussian inputs. Estimating the linear subsystem parameterswithout estimating the systemnonlinearity is possible if the inputsare separable stochastic processes and the system nonlinearityenjoys the invariance property (Enqvist, 2010). This approach isknown as dimension reduction and more recent results, including

F. Giri et al. / Automatica 50 (2014) 768–783 769

Fig. 1. Wiener model with memory or memoryless operator F [·].

extension to series–parallelWiener systems, can be found in Lyzelland 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. itlies in the first and the third quadrant or is locally invertible ona known interval (Bai & Reyland, 2009). In Pelckmans (2011), theidentification problem is cast as a convex quadratic programationprocedure achieving almost consistent estimates (i.e. accuracy ofthe estimates is only guaranteed up to a small approximation term)provided the linear subsystem is FIR, the nonlinearity ismonotone,the input is PE.

In the light of the above discussion, it is seen that most Wienersystem identification methods were designed on the basis of sev-eral assumptions e.g. the nonlinearity is invertible, monotone orodd; the linear subsystem is FIR; the input signals are generally as-sumed to be Gaussian or PE. One more common limitation of mostprevious studies is that the system nonlinearity is supposed to bememoryless. A few exceptions are (Cerone, Piga, & Regrunto, 2009;Dong, Tan, & Tan, 2009; Giri, Rochdi, Brouri, Radouane, & Chaoui,2013) where backlash nonlinearities have been considered.

In this study, the identification problem is addressed forWiener systems with parametric nonlinear operator of backlashand backlash-inverse types (Figs. 2a and 2b). The linear subsys-tem G(s) may be parametric or not, finite order or not. Back-lash operators are generally met in gear transmission systems: agear system without backlash cannot work! They are also used indamper and valve modelling and control (e.g. Shoukat Choudhury,Thornhill, & Shah, 2005). The backlash-inverse behaviour is metwhen Coulomb friction is involved (e.g. in leaf spring suspension).Presently, both types of nonlinear operators are bordered by poly-nomial lines. The latter are allowed to be noninvertible and cross-ing which makes possible to handle, using the same identificationmethod, both memory and memoryless nonlinearities. The identi-fication problem amounts to determining an accurate estimate ofthe (nonparametric) frequency responseG(jω), for a set of frequen-cies (ω1 . . . ωm), and the coefficients of the (parametric) nonlinear-ity borders. While, it is not clear whether all unknown quantitiescan simultaneously be determined using a one-stage identificationmethod (involving a single PE input signal), it is shown in this studythat a two-stage determination in possible using sine input exci-tations. The key point is that, when an ωi-frequency excitation isapplied, the linear part of the system boils down to a simple com-plex gain (i.e. G(jωi)). Then, getting benefit of the model plurality,one lets G(jω1) = 1 limiting thus the system uncertainty to thenonlinear operator parameters which can then be estimated us-ing nonlinear prediction error methods. In the second stage, thelinear subsystem complex frequency gains G(jωi) (i ≥ 2) arecomputed in parallel, using the data obtained in the remaining ωi-frequency experiments. Interestingly, all involved estimators areshown to be consistent. Furthermore, the identification method,thus constructed, applies indifferently to all considered nonlinear-ity types. Moreover, the prior knowledge of the nonlinearity type(being backlash, backlash-inverse or memoryless) is not a priorirequired. Some of the present results partly rely on the work ofGiri, Rochdi, Ikhouane, Brouri, and Chaoui (2012). Also, this studyfeatures several differences and novelties compared to Giri et al.(2013). Specifically:

Fig. 2a. Backlash operator with arbitrary-shape borders satisfying (4).

Fig. 2b. Backlash-inverse operator with arbitrary shape borders.

(i) The present identification method is a two-stage: the systemnonlinear operator is identified first and based upon in thesecond stage to identify the linear subsystem. The methodin Giri et al. (2013) is a three-step: the linear subsystemphase G(jω) is estimated first; then, the nonlinearity isdetermined based on the phase estimates; finally, the gainmodulus |G(jω)| is estimated using the previously obtainedestimates.

(ii) While the focus is limited to backlash nonlinearities in Giriet al. (2013), the present study is covering backlash, backlash-inverse and memoryless nonlinearities and the identificationmethod indifferently applies to all categories and the userdoes not need to a priori knows which nonlinearity type heis facing.

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

(iv) The identification method of Giri et al. (2013) relies onspecific analytic geometry notions e.g. spread/orientationcompatibility, function affinization. It contrasts with thepresent identification method which instead relies on theoptimization of prediction-error cost functions.

The paper is organized as follows: the identification problem isformulated 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 inSection 5; simulations are presented in Section 6.

2. Identification problem formulation

StandardWiener systems consist of a linear dynamic subsystemG(s) followed in series by a memoryless nonlinear operator F [·]

(Fig. 1). Presently, both memory and memoryless nonlinearitiesare considered and handled within a unified framework. Morespecifically, the Wiener system under study is analyticallydescribed by the following equations:

x(t) = g(t)∗u(t) with g(t) = L−1(G(s)) (1)y(t) = w(t)+ ξ(t) withw(t) = F [x](t) (2)

770 F. Giri et al. / Automatica 50 (2014) 768–783

Fig. 3. Equivalent representation of the backlash-inverse based Wiener system.

where u(t) and y(t) denote the control input and the measuredoutput; x(t) and w(t) are inner signals not accessible tomeasurement. The extra input ξ(t) accounts for measurementnoise and other modelling effects. It is supposed to be zero-meanergodic and uncorrelated with the control input u(t). The symbol∗ in (1) refers to the convolution operator and L−1 to the Laplacetransform-inverse. Accordingly, g(t) denotes the impulse responseof the linear subsystem and G(s) its transfer function. It is justsupposed that g ∈ L1 so that the whole system is BIBO stable,making possible open-loop system identification. Interestingly,G(s) is allowed to be infinite order in which case it assumes aninfinite number of isolated poles and zeros. The nonlinear operatorF [·] is a backlash or a backlash-inverse operator, characterizedby its ascendant and descendent bordering functions, fa(·) andfd(·). By definition, the ascendant (resp. descendant) border isthe lateral line along which the working point (x(t), w(t)) moveswhen x(t) > 0 (resp. x(t) < 0). Presently, (fa(·), fd(·)) arepolynomials of the form:

fa(x) = a0 + a1x + · · · + anxn, (3a)

fd(x) = d0 + d1x + · · · + dnxn (3b)

where the degree n is known while the coefficients are not.Some of the coefficients may be null but none of the two vectors[a1 . . . an]Tand [d1 . . . dn]T is identically null. It turns outthat the integer n is not necessarily the exact degree of theprevious polynomials, it is just an upper bound of them. A detaileddescription of how backlash operators work can be found in Giriet al. (2013). In particular, it is there pointed out that, not any pair(fa(·), fd(·)) defines a backlash operator; the only eligible pairs arethose satisfying the following operational property (Fig. 2a):

∀x, ∃z < x : fd(z) = fa(x) (4a)and ∀x, ∃z > x : fa(z) = fd(x). (4b)

Then, depending on the sign of x(t), the working point (x(t), w(t))can move on the ascendant border or on the descendant border oron horizontal lines connecting both borders.

Backlash-inverse operators are analytically described by thefollowing simple equation:

w(t) =

fa(x(t)) if x(t) > 0fd(x(t)) if x(t) < 0. (5)

Accordingly, the working point (x(t), w(t)) moves along theascendant (resp. descendant) border whenever x(t) ≥ 0 (resp.x(t) ≤ 0) and instantaneously jumps from one border to the otherwhen x changes sign. It turns out that, the only eligible functions(fa(·), fd(·)) are those which satisfy the operational property:|fd(x)| < ∞ ⇔ |fa(x)| < ∞, for all x (Fig. 2b).

The identification problem at hand consists in determining,as accurately as possible, estimates of the nonlinear operatorparameters (ai, di; i = 1 . . . n) as well as the linear subsystemfrequency gain G(jωh) (h = 1 . . .m), where the ωh’s and thenumber m are, to some extent, arbitrarily chosen by the user.Interestingly, the user needs not to a priori knows the exact typeof the nonlinear operator F [·], being backlash or backlash-inverse.It is only required that F [·] is one of these two types.

Remarks 1. (1) A couple (fa, fd) not satisfying (4) cannot define abacklash, see illustrations in Giri et al. (2013). That is, (4) is nota limitative assumption but a necessary operational condition.

(2) It is readily seen from (5) that, if fa = fd = f the backlash-inverse operator boils down to the memoryless nonlinearityi.e. w(t) = f (x(t)). Furthermore, the case fa = fd = f isnot ruled out because the borders are allowed to be crossing.Therefore, the present problem statement makes possible toindifferently handle memory and memoryless nonlinearities.

(3) Consider a backlash-inverse operator with invertible borders(fa(·), fd(·)) satisfying the additional property: fa(x) ≤

fd(x), ∀x. It can be checked that this operator is the rightinverse of the backlash operator with ascendant border f −1

a (·)

and descendent border f −1d (·) (e.g. Tao & Kokotovic, 1996). This

motives the reference to the class of operators (5) as ‘backlash-inverse’.

(4) A typical situation leading to backlash (resp. backlash-inverse)with arbitrary-shape borders is when a standard straight-linebordered backlash (resp. backlash-inverse) operator is placedin series with an arbitrary-shape memoryless nonlinearity(Giri et al., 2013).

(5) Wiener systems involving backlash-inverse nonlinearities alsoassume a series–parallel Wiener system interpretation withmemoryless nonlinearity. Indeed, (5) can be given the compactform:

w(t) =1 + sgn(x(t))

2fa(x(t))+

1 − sgn(x(t))2

fd(x(t))

def= h (x(t), x(t)) .

It turns out that, the Wiener system (1)–(2) assumes theequivalent structure of Fig. 3. Identification methods forseries–parallel Wiener systems have recently been developedin Lyzell et al. (2012) and Schoukens and Rolain (2012).However, the context of the present paper is quite differentsince the involved system nonlinearity may be a backlash,a backlash-inverse or a memoryless operator and the exacttype (being backlash, backlash-inverse, or memoryless) is nota priori known.

3. System frequency analysis and data acquisition

The frequency identification approach that is developed in thispaper relies on the investigation of the system response to sine ex-citations u(t) = U cos(ω t). These have already proved to be quiteuseful in the identification of Wiener systems (Giri et al., 2013,2009; Rijlaarsdam, Oomen, Nuij, Schoukens, & Steinbuch, 2012)andWiener–Hammerstein systems (Rijlaarsdam, Nuij, Schoukens,& Steinbuch, 2012).

3.1. Signal expressions

When the Wiener system (1)–(2) is excited with a sine in-put u(t) = U cos(ω t), all resulting signals depend on the am-plitude/frequency couple (U, ω). In steady state, these signals

F. Giri et al. / Automatica 50 (2014) 768–783 771

Fig. 4a. Hysteresis cycle described by the working point (xU,ω(t), wU,ω(t)) of abacklash operator being excited with a sine.

Fig. 4b. Steady-state output wU,ω(t) of the backlash operator of Fig. 4a. Theconstant stages thatwU,ω(t) exhibits are useful characteristics making it relativelyeasier to recognize the backlash nature of nonlinearity and determine the borderingfunctions (fa(x), fd(x)) and the phase ϕ(ω) using Proposition 1 (Part 1a).

write:

xU,ω(t) = U|G(jω)| cos (ωt − ϕ(ω)) (6a)

wU,ω(t) = F [xU,ω](t) (6b)

yU,ω(t) = wU,ω(t)+ ξ(t) (6c)

with ϕ(ω) = − G(jω). One fundamental property of all rate-independent hysteresis operators, including backlash andbacklash-inverse, is that their response to a periodic signal is in turnperiodic with the same period (e.g. Ikhouane & Rodellar, 2007).It turns out that, the signal wU,ω(t) is periodic with period 2π/ωand, accordingly, the working point (xU,ω(t), wU,ω(t)) describes aclosed hysteresis cycle (Figs. 4a and 5a). The rate-independenceproperty means that the shape of the resulting hysteresis cycle isindependent of the frequency ω, it only depends on the amplitudeU . Now, let us establish the analytical expression ofwU,ω(t). To thisend, consider the time sequence tk defined by:

ω tk − ϕ(ω) = 2kπ, k ∈ N. (7)

Of course, the tk’s depend on ω. But, this dependence is not ex-plicitly emphasized to keep simpler the forthcoming presentation.Also, it is readily seen that, tk − tk−1 = 2π/ω (k ∈ N) and theintervals

tk tk +

πω

(resp.

tk +

πω

tk+1) are the half periods on

which the function cos (ωt − ϕ(ω)) is monotonically decreasing(resp. increasing). Then, by definition of the operator F [·], the timestk and tk+π/ω correspond to the instantswhere theworking point(xU,ω(t), wU,ω(t)) leaves one border and:

(i) in the backlash case, it starts moving horizontally towards theopposite border (Figs. 4a and 4b);

(ii) in the backlash-inverse case, it instantaneously jumps to theopposite border ( Figs. 5a and 5b).

Fig. 5a. Hysteresis cycle described by the working point (xU,ω(t), wU,ω(t)) of abacklash-inverse operator being excited with a sine signal xU,ω(t).

Fig. 5b. Steady-state output wU,ω(t) of the backlash-inverse operator of Fig. 5a.The discontinuities that wU,ω(t) exhibits at instants tk and tk + π/ω are usefulcharacteristics making it relatively easier to recognize the nonlinearity type, beingbacklash-inverse, identify the bordering functions (fa(x), fd(x)) and the phase ϕ(ω).

Fig. 6a. Backlash or backlash-inverse operators when the operation conditions aresuch that condition (9) is not fulfilled.

More specifically, there exists a pair of real numbers 0 ≤ τa <π/ω and 0 ≤ τd < π/ω such that for all k ∈ N:

wU,ω(t) = fa(xU,ω(tk)), for t ∈ [tk tk + τd) , (8a)

wU,ω(t) = fd(xU,ω(t)), for t ∈

tk + τd tk +

π

ω

, (8b)

wU,ω(t) = fdxU,ω

tk +

π

ω

,

for t ∈

tk +

π

ωtk +

π

ω+ τa

, (8c)

wU,ω(t) = fa(xU,ω(t)), t ∈

tk +

π

ω+ τa tk +

2πω

. (8d)

Referring to Fig. 4a, Eq. (8a) describes the portion AB, (8b)describes BC, (8c) describes CD, and (8d) describes CD. In the caseof backlash-inverse operators and memoryless nonlinearities, onehas:

τa = τd = 0. (8e)The inverse is not true i.e. it may happen that τa = τd = 0 even

for backlash operators (Figs. 6a and 6b). Note that, in turn the pair(τa, τd) is dependent on (U, ω) but this is not emphasized alleviatenotations.

3.2. Informative inputs and corresponding output characteristics

When the amplitude/frequency couple (U, ω) is such that,

fa (U|G(jω)|) = fd (U|G(jω)|) or

fa (−U|G(jω)|) = fd (−U|G(jω)|) (9)

772 F. Giri et al. / Automatica 50 (2014) 768–783

Fig. 6b. Steady-state output wU,ω(t) of the backlash or backlash-inverse operatorof Fig. 6a. Since (9) does not hold,wU,ω(t) exhibits no useful output characteristicsat tk and tk + π/ω. Furthermore, in case the operator in Fig. 6a is a backlash typethen, one has τa = τd = 0.

the system (undisturbed) outputwU,ω(t) features some character-istics (discontinuities and constant stages)making system identifi-cation easier. These characteristics are defined in Definition 1 andthe benefit one can get from them is explained in Proposition 1.

Definition 1. Consider the Wiener system (1)–(5) being excitedwith u(t) = U cos(ω t).

(1) The amplitude/frequency couple (U, ω) is said to be informa-tive if the resulting (undisturbed) output wU,ω(t) is not a con-stant signal in steady-state.

(2) The resulting steady-state (undisturbed) outputwU,ω(t) is saidto exhibit a useful output characteristic at some time t if, eitherit is discontinuous at t , or it starts a constant stage at t . �

Proposition 1. Consider the Wiener system (1)–(5) being excitedwith u(t) = U cos(ω t).

(1) If the (undisturbed) output wU,ω(t) exhibits in steady-state auseful output characteristic (occurring or starting) at time t then,the nonlinear operator F [·] is a memory type and:(a) t ∈

tk, tk +

πω; k ∈ N

and ϕ(ω) = ω t (modulo π ).

(b) In case wU,ω(t) is constant over some interval [t t + τ )(and not immediately after t + τ ) then, F [·] is backlash andt + τ ∈

tk + τd, tk +

πω

+ τa; k ∈ N.

(c) In case wU,ω(t) is discontinuous at t then, F [·] is backlash-inverse and τa = τd = 0.

(d) Whatever the operator type, one of the following equalitiesholds for some k ∈ N:t + τ t +

π

ω

=

tk + τd tk +

π

ω

or

t + τ t +π

ω

=

tk +

π

ω+ τa tk +

2πω

where τ = τa = τd = 0 is null in the case of backlash-inverse.

(2) If wU,ω(t) exhibits no useful characteristic then, F [·] may bememory or memoryless. Furthermore:

fa (U|G(jω)|) = fd (U|G(jω)|) ,fa (−U|G(jω)|) = fd (−U|G(jω)|) andτa = τd = 0. �

Proof (Outline). In the case of backlash-inverse operator, condition(9) entails a hysteresis cycle like in (Fig. 5a). Accordingly, theworking point (xU,ω(t), wU,ω(t)) moves (in steady-state) only onthe lateral borders, jumping periodically from one border to theother (Fig. 5a). Consequently, the (undisturbed) output wU,ω(t)exhibits discontinuities at (and only at) the instants tk and/or tk+ π

ω

(modulo 2πω). It turns out that, if wU,ω(t) is discontinuous at some

time t then, it should be concluded that the involved nonlinearityis a backlash-inverse type and ϕ(ω) = ω t (modulo π ), due to(7). Similar argument can be developed with backlash operators.Then, under condition (9), the hysteresis cycle looks like in Fig. 4a.

Fig. 7. The figure illustrates the situationwhere a backlash operator is excitedwitha sine input xU,ω(t) whose amplitude/frequency couple (U, ω) is noninformative.Then, the resulting output wU,ω(t) is constant and the obtained hysteresis cycleturns out to be a horizontal segment.

Accordingly, the working point (xU,ω(t), wU,ω(t)) moves along apiece of each of the bordering lines, (x, fa(x)) and (x, fd(x)), andon horizontal segments connecting these lines. Consequently, theoutput wU,ω(t) features constant stages starting at the instantstk and/or tk +

πω

(modulo 2πω). It turns out that, if wU,ω(t)

exhibits a constant stage starting at some time t then, it should beconcluded that the involved nonlinear operator is a backlash typeand ϕ(ω) = ω t (modulo π ), due to (7). Whatever the operatortype (being backlash or backlash-inverse), the outputwU,ω(t)willexhibit no useful output characteristics if the couple (U, ω) is suchthat (9) does not hold i.e. if fa(U|G(jω)|) = fd(U|G(jω)|) andfa(−U|G(jω)|) = fd(−U|G(jω)|). This is illustrated by Figs. 6a and6b which also shows that, in such a situation, one has τa = τd = 0even for backlash operators. �

Remark 2. (1) In addition to the possible recognition of thenonlinearity type, Proposition 1 (Part 1) shows that the usefuloutput characteristics make easier the determination of thephase ϕ(ω) (modulo π ). In turn, this will make it easier toidentify the nonlinearity bordering lines.

(2) It follows from Proposition 1 (Part 2) that, the absence of usefuloutput characteristics does not entail thememoryless nature ofthe system nonlinearity. Nevertheless, due to the polynomialnature of the nonlinearity borders, one may conclude thatthe system nonlinearity is actually memoryless if the systemoutput exhibits no useful characteristics for more than ndifferent informative couples of the form (Uh, ω) (h =

1 . . . n). Indeed, the number of crossing points of two nthdegree polynomials cannot exceed n i.e. fa (Uh|G(jω)|) =

fd (Uh|G(jω)|) (h = 1 . . . n) ⇒ fa = fd.(3) Another difference between backlash and backlash-inverse

operators is that the latter leads to a varying output wU,ω(t)whatever U|G(jω)| = 0. This may not be the case in presenceof backlash operators. Indeed, it is easily checked (e.g. Fig. 7)that, if the internal signal amplitudeU|G(jω)| is not sufficientlylarge then, the point (xU,ω(t), wU,ω(t)) only moves along ahorizontal segment, leading to a constant output. This situationis easily coped with letting U be large enough.

3.3. Data acquisition

Proposition 1 shows that,when a sine input u(t) = U cos(ω t) isapplied to the system, one must make sure that the correspondingcouple (U, ω) is informative in the sense of Definition 1. Thepoint is that the definition of input informativeness involvesthe undisturbed output wU,ω(t) which is not accessible tomeasurements. Fortunately, an accurate estimator of wU,ω exists,thanks to the (steady-state) periodicity of wU,ω and the ergodicityof the noise ξ . The estimator, denoted wU,ω , is obtained by

F. Giri et al. / Automatica 50 (2014) 768–783 773

Table 1Input–output data acquisition.

Choose an initial set of amplitude/frequency couples, (Uh, ωh; h = 1 . . .m), with different frequencies ω1 . . . ωm , choose a threshold 0 < ε ≪ 1, and repeat thefollowing steps for h = 1 . . .m:1. Apply the input u(t) = Uh cos(ωht) to the Wiener system (1)–(5) and collect the steady-state output yUh,ωh (t) over a sufficiently large interval, say 0 ≤ t ≤ 2Nπ/ωh(with N large enough).2. Generate the periodic signals wUh,ωh,N−1(t) and wUh,ωh,N (t), using (10a)–(10b), and compute the following relative errors, using the power norm defined by (11)letting there T = 2π/ωh:

e1(h,N) =

wUh ,ωh ,N (t)−wUh ,ωh ,N−1(t)wUh ,ωh ,N (t)

and

e2(h,N) =

wUh ,ωh ,N (t)−wUh ,ωh ,N (t−2πωh)

wUh ,ωh ,N (t) .

If e1(h,N) < ε and e2(h,N) < ε then, go to Step 3. Otherwise, increase N and repeat Step 2.3. If wUh,ωh,N (t) is constant, increase the amplitude Uh and go back to Step 1. If wUh,ωh,N (t) cannot be made time varying by taking Uh larger (a situation that occurswhen G(jωh) = 0) then, change ωh and go back to Step 1. When a varying wUh,ωh,N (t) is obtained, retain the input–output data obtained with the couple (Uh, ωh).

performing a T -periodic averagingwith T = 2π/ω (Ljung, 1999, p.232):

wU,ω,N(t) =1N

Nk=1

yU,ω

t + k

2πω

for t ∈

0

2πω

(10a)

wU,ω,N

t + k

2πω

= wU,ω,N(t,N) for k = 1, 2, 3 . . . (10b)

whereN is any sufficiently large integer. The estimator (10a)–(10b)was shown in Giri et al. (2009) to be uniformly consistent i.e.wU,ω,N(t) converges (w.p.1 as N → ∞) to wU,ω(t), whatevert . Making use of this estimator and of Definition 1 (Part 1), acollection of suitable sine inputs, and corresponding outputs, canbe selected based on Proposition 1 and subsequent observations.This leads to the practical procedure of Table 1where the followingpower norm is defined, for any T -periodic signal s(t):

∥s(t)∥ def=

1T

T

0s(t)2dt

1/2

. (11)

The integer N depends on the couple (Uh, ωh). Since a finitenumber of couples are involved, one only retain the greatest N .The collection of data sets thus obtained constructively satisfy thefollowing properties:

Proposition 2. The procedure of Table 1, when applied to theWiener system (1)–(5), yields m data sets, {Uh cos(ωh t), yUh,ωh(t),wUh,ωh,N(t); 0 ≤ t ≤ N 2π

ωh} (h = 1 . . .m), with the following fea-

tures:

(1) w.p.1 as N → ∞, the signal wUh,ωh,N(t) is periodic with period2π/ωh (h = 1 . . .m).

(2) The frequency gain G(jωh) are nonzero (h = 1 . . .m). �

Proof. Recall that wUh,ωh,N(t) converges (w.p.1 as N → ∞) towUh,ωh(t), whatever t and (U, ω). Then, it readily follows using(10a), (10b) and (11) that, limN→∞ e1(h,N) = 0 (w.p.1 as N →

∞). On the other hand, we know that wU,ω(t) is periodic withperiod 2π/ωh. Since wUh,ωh,N(t) converges to wUh,ωh(t), it followsthat limN→∞ e2(h,N) = 0 (w.p.1 as N → ∞). Since both relativeerrors converge to zero as N → ∞, it only takes a finite numberof iterations with the iterative search in Step 2 of Table 1 to finda suitable value of the integer N . The fact that e2(h,N) vanishesmeans that wUh,ωh,N(t) becomes periodic, with period 2π/ωh. But,this does not guarantee that wUh,ωh,N(t) is time-varying. Then, Step3 involves a change of the amplitude Uh until a varying wUh,ωh,N(t)is obtained. In this respect, note that a time-varying wUh,ωh,N(t)implies that G(jωh) = 0 (because if G(jωh) = 0 then wUh,ωh(t)would be time-invariant due to (6a)–(6b)). �

4. Nonlinearity parameter estimation

4.1. Fundamental system parameterizations

Throughout this subsection, the Wiener system (1)–(5) isexcited by a sine input u(t) = U cos(ω t) for some informativecouple (U, ω), belonging to the set {(Uh, ωh); h = 1 . . .m} foundwith Table 1. Using (6a), (6b) and (3), it readily follows from (8b)and (8d) that:

yU,ω(t) =

ni=0

ai (U|G(jω)|)i (cos(ωt − ϕ(ω))) i + ξ(t)

=

ni=0

α∗

i (ω) (cos(ωt − ϕ(ω))) i + ξ(t),

for all t ∈

tk +

π

ω+ τa tk +

2πω

, k ∈ N (12a)

and

yU,ω(t) =

ni=0

di (U|G(jω)|)i (cos(ωt − ϕ(ω))) i + ξ(t)

=

ni=0

δ∗

i (ω) (cos(ωt − ϕ(ω))) i + ξ(t),

for all t ∈

tk + τd tk +

π

ω

, k ∈ N (12b)

where α∗

i (U, ω) and δ∗

i (U, ω) are lumped parameters defined by:

α∗

i (U, ω) = ai(U|G(jω)|)i,

δ∗

i (U, ω) = di(U|G(jω)|)i, (i = 0 . . . n).(13)

Introduce the following, (ω,ψ)-parameterized, time function:

σω,ψ (t) =1 + sgn (sin(ωt − ψ))

2. (14)

Recall that, in view of (7), the tk’s are periodic instants that satisfythe property:

sin(ωt − ϕ(ω)) > 0 ontk tk +

π

ω

and

sin(ωt − ϕ(ω)) < 0 ontk +

π

ωtk +

2πω

. (15a)

Then, it follows from (14) that:

σω,ϕ(ω)(t) = 1 ontk tk +

π

ω

and

σω,ϕ(ω)(t) = 0 ontk +

π

ωtk +

2πω

. (15b)

774 F. Giri et al. / Automatica 50 (2014) 768–783

Then, Eqs. (12a)–(12b) can be recast in the following singlecompact form, for all t ∈

tk + τd tk +

πω

∪

tk +

πω

+ τa tk +2πω

and all k ∈ N:

yU,ω(t) =

ni=0

α∗

i (U, ω)1 − σω,ϕ(ω)(t)

(cos(ωt − ϕ(ω))) i

+

ni=0

δ∗

i (U, ω)σω,ϕ(ω)(t)

× (cos(ωt − ϕ(ω))) i + ξ(t) (16a)

= XTω,ϕ(ω)(t)θ

∗

U,ω + ξ(t), (16b)

with:

θ∗

U,ω =α∗

0(U, ω) · · ·α∗

n(U, ω) δ∗

0(U, ω) · · · δ∗

n(U, ω)T (17a)

XTω,ϕ(ω)(t) = [(1 − σω,ϕ(ω)(t)) (1 − σω,ϕ(ω)(t))

cos(ωt − ϕ(ω)) · · · (1 − σω,ϕ(ω)(t))(cos(ωt − ϕ(ω)))nσω,ϕ(ω)(t) σω,ϕ(ω)(t)cos(ωt − ϕ(ω)) · · · σω,ϕ(ω)(t)(cos(ωt − ϕ(ω)))n]. (17b)

At this point, it is important to emphasize the plurality of thecouple (G(s), F [x]) which defines the Wiener model (1)–(2). Ineffect, any couple of the form (G(s)/K , F [K x]) is also a model,whatever K = 0. Then, one should get benefit of this modelplurality to reduce, to some extent, the complexity of the systemidentification problem. A judicious way to reach this goal is tofocus on the particularmodel

G(s), F [ x]

def= (G(s)/K , F [K x])with

K = U1|G(jω1)|sgn (sin(− G(jω1))), where (U1, ω1) is the firstinformative amplitude/frequency couple determined in Table 1.By Proposition 2 (Part 2), one has K = 0. Furthermore, it isreadily checked that, the couple

G(s), F [ x]

is the onlymodel that

features the two properties, U1|G(jω1)| = 1 and ϕ(ω1) ∈ [0 π)(modulo 2π ) with ϕ(ω1)

def= − G(jω1). Now, to avoid additional

notations, the particular model (G(s), F [ x]) will continue to bedenoted (G(s), F [x]). Accordingly, this model is the only one thatenjoys the following couple of properties:

U1|G(jω1)| = 1 and ϕ(ω1) ∈ [0 π) (modulo 2π) (18)

where ϕ(ω1) = − G(jω1) (see (6a)). An immediate benefit onegets from these properties is that Eqs. (13) and (17a) boil down to:

α∗

i (U1, ω1) = ai, δ∗

i (U1, ω1) = di, (i = 0 . . . n) (19a)

θ∗

U1,ω1=

a0 . . . an d0 . . . dn

T. (19b)

It turns out that, the parameter vector θ∗

U1,ω1is only composed

of the nonlinearity parameters (i.e. the ai’s and the di’s) and so isindependent on (U1, ω1). All the other vectors, θ∗

Uh,ωh(h = 2 . . .m),

depend on the quantity Uh|G(jωh)|, as pointed out by (13). On theother hand, using Eqs. (19a), (19b) and (13), one gets the followingadditional properties which in turn will prove to be useful:

sgnα∗

i (Uh, ωh)

= sgn(ai),

sgnδ∗

i (Uh, ωh)

= sgn(di); (i = 0 . . . n; h = 2 . . .m)(20a)

diα∗

i (Uh, ωh) = aiδ∗

i (Uh, ωh), (i = 0 . . . n; h = 2 . . .m). (20b)

For convenience, we introduce the set S ⊂ R2(n+1) including allvectors satisfying (20a)–(20b). More specifically, one has:

θ =α0 . . . αn δ0 . . . δn

T∈ S

⇔ sgn(αi) = sgn(ai), sgn(δi) = sgn(di), anddiαi = aiδi, (i = 0 . . . n; h = 2 . . .m). (21)

In the next subsection, a class of prediction error cost functionswillbe introduced which, combined with properties (18), (19a), (19b)and (21), will yield two constrained optimization problems. Thefirst problem will be shown to have a unique solution, namely thecouple (ϕ(ω1), θ

∗

U1,ω1). In turn, the second problem will generally

have a unique solution i.e. (ϕ(ωh), θ∗

Uh,ωh) (h = 2 . . .m). The

situation where the second problem has a not unique solution iswhen theWiener system involves a symmetric nonlinear operatorF [·] (Giri et al., 2013). By definition, a symmetric operator is onewhich satisfies F [x] = F [−x], for any signal x. This is the casewhenthe operator borders are symmetricalwith respect to the axis x = 0i.e. fd(x) = fa(−x), for all real numbers x. Presently, one gets from(3) that, F [·] is symmetric if:

ai = (−1)idi (or, equivalently, di = (−1)iai). (22)In this case, it follows from (6b) that wU,ω = F [xU,ω] = F [−xU,ω]and, from (6a), that xU,ω(t) = U|G(jω)| cos(ωt − ϕ(ω)) and−xU,ω(t) = U|G(jω)| cos (ωt − ϕ(ω)+ π). These remarks showthat, in presence of symmetric operators, the two phase valuesϕ(ω) and ϕ(ω) + π are indistinguishable based on the systeminput/output signals. Additional general comments on symmetricoperators can be found in Giri et al. (2013).

4.2. Prediction error optimization problems

To get benefit of the systemparameterizations (12a)–(12b), oneneeds a triplet (t, τa, τd), next called optimization-domain indices,such that:t + τd t +

π

ω

∪

t +

π

ω+ τa t +

2πω

=

tk + τd tk +

π

ω

∪

tk +

π

ω+ τa tk +

2πω

, (23a)

for some k ∈ N. This holds if, one has for some k ∈ N:t = tk, τd = τd, τa = τa or

t = tk + π/ω, τd = τa, τa = τd. (23b)Given a triplet (t, τa, τd) satisfying (23a) or (23b), the following

cost function can be defined:

J∗U,ω(ψ, θ) =

I(U,ω)

EyU,ω(t)− XT

ω,ψ (t)θ2

dt,

for ψ ∈ [0 2π), θ ∈ R2(n+1) (24)with,

I(U, ω) =

t + τd t +

π

ω

∪

t +

π

ω+ τa t +

2πω

, (25)

where E(·) denotes the mean operator (related to the noiseξ probability law in (2)). At this point, one may notice thatthe optimization-domain indices (t, τa, τd) depend on (U, ω) ∈

{(Uh, ωh); h = 1 . . .m}, because so do the quantities tk, τa, τd.To alleviate the text, such a dependence is solely emphasizedthrough the notation I(U, ω) in (25). Based on (24)–(25), two keyoptimization problems can now be stated as follows:Optimization problem 1 (related to (ϕ(ω1), θ

∗

U1,ω1)): minimize

J∗U1,ω1(ψ, θ), under the constraint 0 ≤ ψ < π (due to (18)). The

solution of this problem is denoted, argmin 0≤ψ<πθ∈R2(n+1)

J∗U1,ω1(ψ, θ).

Optimization problem 2 (related to (ϕ(ωh), θ∗

Uh,ωh; h = 2 . . .m)):

minimize J∗Uh,ωh(ψ, θ), under the constraint θ ∈ S, using (21).

The solution of this problem is simply denoted, argmin 0≤ψ<2πθ∈S

J∗Uh,ωh(ψ, θ).

The above problems are investigated in the following proposi-tion, the proof of which is placed in Appendix A. This propositionconstitutes a key step in the identification method design.

F. Giri et al. / Automatica 50 (2014) 768–783 775

Table 2Selection of the optimization-domain indices.

Consider the m (estimated) outputswUh,ωh,N (t); 0 ≤ t ≤ N 2π

ωh; h = 1 . . .m

obtained using Table 1. For each h = 1 . . .m, an estimate (tN , τa,N , τd,N ) of (t, τa, τd) is

obtained following the next steps:(1) If wUh,ωh,N (t) exhibits no useful characteristic then, let τa,N = τd,N = 0 and tN be any time.(2) If wUh,ωh,N (t) exhibits (in steady-state) periodic discontinuities then, let τa,N = τd,N = 0 and tN be any time where wUh,ωh,N (t) is discontinuous.(3) (a) If wUh,ωh,N (t) exhibits periodic constant stages then, let tN be any time where wU,ω begins a constant stage and let tN + τa,N be the time when that stage ends.

(b) If further wUh,ωh,N (t) begins its next constant stage (after tN ) at time tN + π/ω then, the time when that stage ends equals tN + π/ω + τd,N . Otherwise, letτd,N = 0.

Proposition 3. Let theWiener system (1)–(5) be excited with u(t) =

U cos(ω t), for some informative couple (U, ω) ∈(Uh, ωh); h =

1 . . .m. Let the corresponding system data (obtained using Table 1)

be used in the cost function J∗U,ω(ψ, θ). Then, one has the followingproperties:

(1) Optimization Problem 1 has a unique solution, namely:

arg min0≤ψ<πθ∈R2(n+1)

J∗U1,ω1(ψ, θ) = (ϕ(ω1), θ

∗

U1,ω1), modulo 2π (26)

where the mention ‘modulo 2π ’ concerns the phase only.(2) Optimization Problems 2 has generally a unique solution, namely:

arg min0≤ψ<2πθ∈S

J∗Uh,ωh(ψ, θ) = (ϕ(ωh), θ

∗

Uh,ωh),

modulo 2π, (h = 2 . . .m). (27)

It is only in the particular case of a symmetric operator F [·] that,the problem has a second solution equal to (ϕ(ωh)+ π, θ∗

Uh,ωh),

modulo 2π . Then, the phase can only be determined modulo π .(3) Whatever (U, ω) ∈ {(Uh, ωh); h = 1 . . .m}, the parameter

vector θ∗(ω) is related to the system data and the phase ϕ(ω)by the following least-squares (LS) expression:

θ∗

U,ω =

I(U,ω)

Xω,ϕ(ω)(t)XTω,ϕ(ω)(t)dt

−1

×

I(U,ω)

wU,ω(t) Xω,ϕ(ω)(t) dt. (28)

Remark 3. (1) It is shown in the proof of Proposition 3 (Part 2)that, each cost function J∗Uh,ωh

(ψ, θ) (h = 2 . . .m) has twominima, namely (ϕ(ωh), θ

∗

Uh,ωh) and (ϕ(ωh)+ π, θ−

Uh,ωh)with

θ−

Uh,ωh=

α−

0 (U, ω) · · ·α−

n (U, ω) δ−

0 (U, ω) · · · δ−

n (U, ω)T

where α−

i (U, ω) = (−1)iδ∗

i (U, ω) and δ−

i (U, ω) = (−1)iα∗

i(U, ω). Except for the particular case of symmetric operators,one has θ−

Uh,ωh∈ S. This makes clear that, it is the constraint

θ ∈ S (in Optimization Problem2) that rules out the unsuitableminimum (ϕ(ωh)+ π, θ−(ωh)).

(2) In the case of a symmetric operator F [·], one has θ−

Uh,ωh=

θ∗

Uh,ωhdue to (22) and so θ−

Uh,ωh∈ S, making impossible

to distinguish between the two solutions (ϕ(ωh), θ∗

Uh,ωh) and

(ϕ(ωh) + π, θ−

Uh,ωh). Accordingly, the phase function ϕ(ω) =

− G(jω) can only be determined modulo π . It is worthpointing out that this fact is independent of the method usedto determine the phase. It is a structural limitation resultingfrom the series system structure and the inaccessibility ofthe intermediary signal x to measurements. Indeed, it is seenfrom (1)–(2) that: w(t) = F [G(s)u](t) = F [−(−G(s)u)] (t),whatever the input signal u(t). That is, the couples (G(s), F [x])and (−G(s), F [−x]) are both representative of the Wienersystem i.e. are indistinguishable from the input and outputsignals. In the case of symmetric operators, i.e. when F [−x] =

F [x], one gets w(t) = F [G(s)u](t) = F [−G(s)u] (t) whatever

the input signal u(t). This implies that the Wiener system isdescribed by the two couples (G(s), F [x]) and (−G(s), F [x]).It turns out that the operator F [x] can be exactly determinedbecause it is common to both couples. Also, the frequencyresponse function modulus |G(jω)| can be exactly determinedsince | − G(jω)| = |G(jω)|. But, the phase function G(jω) canonly be determinedmoduloπ because −G(jω) = G(jω)+π(modulo 2π ).

To get benefit of Proposition 3, one needs an estimate (tN ,τa,N , τd,N) of some triplet (t, τa, τd) satisfying (23a). To this end, thepractical algorithm of Table 2 is designed based on Proposition 1.

Since wUh,ωh,N(t) is a consistent estimator of wU,ω(t), it readilyfollows that (tN , τa,N , τd,N) converges (w.p.1 as N → ∞) to sometriplet (t, τa, τd) satisfying (23a). In other words, one has w.p.1:tN + τd,N tN +

π

ω

∪

tN +

π

ω+ τa,N tN +

2πω

−−−→N→∞

tk + τd tk +

π

ω

∪

tk +

π

ω+ τa tk +

2πω

(29)

for some k ∈ N. Then, it is suggested by Eqs. (24)–(25) that thefollowing practical prediction error cost function is defined foreach informative couple (U, ω) ∈ {(Uh, ωh); h = 1 . . .m}:

JU,ω,N(ψ, θ) =

I(U,ω,N)

EyU,ω(t)− XT

ω,ψ (t)θ2

dt (30)

with:

I(U, ω,N) =

tN + τd,N tN +

π

ω

∪

tN +

π

ω+ τa,N tN +

2πω

. (31)

It readily follows, comparing (30)–(31) and (24)–(25) that, onehas for all (ψ, θ), JU,ω,N(ψ, θ) −−−→

N→∞

J∗U,ω(ψ, θ) (w.p.1). Conse-

quently:

arg min0≤ψ<πθ∈R2(n+1)

JU1,ω1,N(ψ, θ)

−−−→N→∞

arg min0≤ψ<πθ∈R2(n+1)

J∗U1,ω1(ψ, θ) (w.p.1) (32)

arg min0≤ψ<2πθ∈S

JUh,ωh,N(ψ, θ)

−−−→N→∞

arg min0≤ψ<2πθ∈S

J∗Uh,ωh(ψ, θ) (w.p.1) (h = 2 . . .m). (33)

4.3. Estimation of the nonlinearity parameters (ai, di; i = 0 . . . n)

In this subsection, the focus is made on the optimizationproblem min 0≤ψ<π

θ∈R2(n+1)JU1,ω1,N(ψ, θ) which is a practical version

of the ideal problem min 0≤ψ<πθ∈R2(n+1)

J∗U1,ω1(ψ, θ) (i.e. Optimization

776 F. Giri et al. / Automatica 50 (2014) 768–783

Table 3Practical algorithm for nonlinearity parameter estimation.

Given the data,U1 cos(ω1 t), yU1,ω1 (t), wU1,ω1,N (t); 0 ≤ t ≤ N 2π

ω1

obtained by Table 1 and the associated indices (tN , τa,N , τd,N ) provided by Table 2. Compute (ϕ(ω1,N), θU1,ω1 (N)) as follows:(1) Case where wU1,ω1,N (t) exhibits a discontinuity, or starts a constant-stage, at tN :

ϕ(ω1,N) =

ω1 tN if ω1 tN ∈ [0π) (modulo 2π)ω1 tN + π otherwise (38)

θU1,ω1 (N) =

I(U1,ω1,N)

Xω1,ϕ(ω1,N)(t)XTω1,ϕ(ω1,N)

(t)dt−1

×I(U1,ω1,N)

wU1,ω1,N (t) Xω1,ϕ(ω1,N)(t) dt (39)(2) Case where wU1,ω1,N (t) exhibits no useful characteristic:ϕ(ω1,N) = argmin0≤ψ≤π JU1,ω1,N (ψ, θ(ψ,N)) (40)θU1,ω1 (N) obtained using (39) (41)where:

θ(ψ,N) =

I(U1,ω1,N)

Xω1,ψ (t)XTω1,ψ

(t)dt−1

×I(U1,ω1,N)

wU1,ω1,N (t) Xω1,ψ (t) dt .

Problem 1). One knows, by (26) that this has a unique solution,namely (ϕ(ω1), θ

∗

U1,ω1). Furthermore, we know by (19b) that the

components of θ∗

U1,ω1are nothing other than (ai, di; i = 0 . . . n) i.e.

the parameters of the system nonlinearity. These considerationssuggest the following estimator for (ϕ(ω1), θ

∗

U1,ω1):

ϕ(ω1,N), θU1,ω1(N)

= arg min0≤ψ<πθ∈R2(n+1)

JU1,ω1,N(ψ, θ). (34)

The following proposition is an immediate consequence of (34),(32) and (26):

Proposition 4. Let the Wiener system (1)–(5) be excited by u(t) =

U1 cos(ω1 t), where (U1, ω1) being the first informative ampli-tude/frequency couple obtained by Table 1. Then, the estima-tor (ϕ(ω1,N), θU1,ω1(N)) converges to (ϕ(ω1), θ

∗

U1,ω1), w.p.1, as

N → ∞.

The above result is a major achievement. But, one needs apractical algorithm to solve the theoretical problem (34). Again,let us first investigate this issue for the ideal optimization problemmin 0≤ψ<π

θ∈R2(n+1)J∗U1,ω1

(ψ, θ). Clearly, themain difficulty lies in the fact

that J∗U1,ω1(ψ, θ) is non-quadratic in the variable ψ . Nevertheless,

there is a situation where the optimal value of ψ can easily bedetermined. Specifically, if the steady-state (undisturbed) outputwU1,ω1(t) exhibits a useful characteristic starting at t (Definition 1,Part 2) then, one has by Proposition 1 (Part 1) that, ϕ(ω1) = ω1 t(modulo π ). Besides, we know that ϕ(ω1) ∈ [0 π) (modulo 2π ).Then, one has the following relation between ϕ(ω1) and t:

ϕ(ω1) =

ω1 t if ω1 t ∈ [0π) (modulo 2π)ω1 t + π otherwise. (35)

Once ϕ(ω1) is determined, one makes use of the LS expression(28) to get θ∗

U1,ω1. The above discussion leads to Part 1 of the

algorithm of Table 3 (hereafter). Now, let us consider the casewherewU1,ω1(t) exhibits no useful characteristics. Then, θ∗

U1,ω1and

ϕ(ω1) will conjunctly be determined using the separable least-squares algorithm. Specifically, one gets from (28):

θ(ψ) =

I(U1,ω1)

Xω1,ψ (t)XTω1,ψ

(t)dt−1

×

I(U1,ω1)

wU1,ω1(t) Xω1,ψ (t) dt. (36)

The inverted matrix on the right side of (37) is actually invertible(see the proof of Proposition 3, Part 3). Now, substituting the rightsides of (36) to θ in J∗U1,ω1

(ψ, θ) one gets a nonlinear functioninvolving a single unknown variable i.e. ψ . By Proposition 3, thisproblem has a unique solution which is precisely equal to ϕ(ω1). It

turns out that:

ϕ(ω1) = arg min0≤ψ≤π

J∗U1,ω1(ψ, θ(ψ)). (37)

The optimization task (37) is tractable since it is a one-dimensionsearch problem: the minimum search could be performed byplotting J∗U1,ω1

(ψ, θ(ψ)) in function of ψ over [0 π). The searchresult is then used in (36) to get the parameter vector θ∗

U1,ω1, due

to Proposition 3. This approach is commonly referred to separableleast-squares and is a formof relaxation. These remarks lead to Part2 of Table 3.

5. Linear subsystem estimation

In this subsection, the aim is to get accurate estimates of thefrequency gains G(jωh) where the frequencies ωh (h = 1 . . .m)are those selected in Table 1. First, notice that by (18) one has|G(jω1)| = 1/U1. Also, a consistent estimate ϕ(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 = 2 . . .m. This problem is presently dealtwith by considering Optimization Problem 2 and the correspond-ing analysis in Proposition 3 (Part 2). The latter suggests the fol-lowing family of estimators:ϕ(ωh,N), θUh,ωh(N)

= arg min

0≤ψ<2πθ∈S

JUh,ωh,N(ψ, θ)

(h = 2 . . .m). (42)

The following proposition is an immediate consequence of (42),(33) and Proposition 3 (Part 2):

Proposition 5. Let the Wiener system (1)–(5) be excited by u(t) =

Uh cos(ωh t) with (Uh, ωh; 2, . . . ,m) being the informative ampli-tude/frequency couples obtained by Table 1. Then, one has:

(1) If the system nonlinear operator F [·] is not symmetric then,w.p.1as N → ∞, the estimator (ϕ(ωh,N), θUh,ωh(N)) converges to(ϕ(ωh), θ

∗

Uh,ωh), modulo 2π (with respect to the first argument),

for all h = 2 . . .m.(2) If the operator F [·] is symmetric then, w.p.1 as N → ∞, the

estimator (ϕ(ωh,N), θUh,ωh(N)) converges to (ϕ(ωh), θ∗

Uh,ωh),

modulo π , for all h = 2 . . .m.

The above result is a major achievement. But, it needs tobe completed with a practical algorithm solving the theoreticalproblem (42). Inspired by Table 3, the algorithm of Table 4 isproposed.

Note that the corrections in Steps 1b and 2b are justified byProposition 3 (Part 2) and Remark 3. Also, in coherence with (17a)

F. Giri et al. / Automatica 50 (2014) 768–783 777

Table 4Practical algorithm of the estimator (ϕ(ωh,N), θUh,ωh (N)) defined by (42).

Repeat the following process for h = 2 . . .m, where the system dataUh cos(ωh t), yUh,ωh (t), wUh,ωh,N (t); 0 ≤ t ≤ N 2π

ωh

are those generated by Table 1 and the

corresponding optimization-domain indices, denoted (tN , τa,N , τd,N ), are those provided by Table 2. Also, S denotes the estimate of the set S defined by (21), replacingthere the coefficients (ai, di) by their estimates previously obtained using Table 3.(1) If wUh,ωh,N (t) exhibits a discontinuity, or a constant-stage starting, at tN then, compute the quantities ϕ = ωh tN and

θ =

I(Uh,ωh,N)

Xωh,ϕ(t)XTωh,ϕ

(t)dt−1

×I(Uh,ωh,N)

wUh,ωh,N (t) Xωh,ϕ(t) dt

In coherence with (21), θ is of the form θ =α0 . . . αn δ0 . . . δn

T.

(a) If θ ∈ S then, set ϕ(ω1,N) = ϕ and θUh,ωh (N) = θ .(b) Otherwise, set ϕ(ωh,N) = ϕ + π and θ (ωh,N) =

α′

0 . . . α′

n δ′

0 . . . δ′

n

Twith α′

i = (−1)i δi and δ′

i = (−1)iαi (using Remark 3).(2) If wUh,ωh,N (t) exhibits no useful characteristic then, compute the quantities:ϕ = argmin0≤ψ≤π JUh,ωh,N (ψ, θ(ψ,N)) (43a)

θ =

I(Uh,ωh,N)

Xωh,ϕ(t)XTω1,ϕ

(t) dt−1

×I(Uh,ωh,N)

wUh,ωh,N (t) Xωh,ϕ(t) dt (43b)with:

θ(ψ,N) =

I(Uh,ωh,N)

Xωh,ψ (t)XTωh,ψ

(t)dt−1

×I(Uh,ωh,N)

wUh,ωh,N (t) Xωh,ψ (t) dt

(a) If θ ∈ S then, set ϕ(ωh,N) = ϕ and θUh,ωh (N) = θ .(b) Otherwise, set ϕ(ωh,N) = ϕ + π and θ (ωh,N) =

α′

0 . . . α′

n δ′

0 . . . δ′

n

Twith α′

i = (−1)i δi and δ′

i = (−1)iαi .

and (19b), the following notations are introduced, where h =

2 . . .m:

θUh,ωh(N) =

α0(Uh, ωh,N) · · · αn(Uh, ωh,N)

δ0(Uh, ωh,N) · · · δn(Uh, ωh,N)T

(44)

θU1,ω1(N) =

a0(N) · · · an(N) d0(N) · · · dn(N)

T. (45)

On the other hand, it readily follows from the first equation in(13) (of course, the second equation in (13) could as well beconsidered):

|α∗

i (Uh, ωh)|n/i

= |ai|n/i(Uh|G(jωh)|)n;

(i = 1 . . . n; h = 2 . . .m).

Adding both sides of this equality over i = 1 . . .m, yields:n

i=1

|α∗

i (Uh, ωh)|n/i

=

ni=1

|ai|n/i(Uh|G(j(ωh))|)n

(h = 2 . . .m). (46)

Solving this with respect to |G(jωh)| yields:

|G(jωh)| =1Uh

n

i=1|α∗

i (Uh, ωh)|n/i

ni=1

|ai|n/i

1/n

(h = 2 . . .m). (47)

Note that (47) involves no singularity because the vector[a1 . . . an]T was supposed not to be null. The expression (47),suggests the following family of estimators:

|G(jωh,N)| =1Uh

n

i=1|αi(Uh, ωh,N)|n/i

ni=1

|ai(N)|n/i

1/n

(h = 2 . . .m). (48)

Proposition 6. Let the Wiener system (1)–(5) be considered in thesame conditions as in Proposition 5. Consider the family of estimatorsG(jωh,N) = |G(jωh,N)|e−jϕ(ωh,N) (h = 2 . . .m) where ϕ(ωh,N)is defined by (42), and computed using the practical algorithmof Table 4, and |G(jωh,N)| is computed by (48). Then, one has thefollowing properties, for all h ∈ {2, . . . ,m}:

(1) If the operator F [·] is nonsymmetric then, G(jωh,N) −−−→N→∞

G(jωh), w.p.1.(2) If F [·] is symmetric then,w.p.1 as N → ∞, G(jωh,N) converges

either to G(jωh) or to −G(jωh).

Proof. From Propositions 4 and 5, one gets that αi(Uh, ωh,N) andai(N) respectively converge to α∗

i (Uh, ωh) and ai (i = 0 . . . n, h =

2 . . .m). Then, it follows comparing (47) and (48) that |G(jωh,N)|converges w.p.1 to |G(jωh)| (h = 2 . . .m). On the other hand,Proposition 5 (Part 1) stipulates that, in the case of nonsymmetricnonlinearities, ϕ(ωh,N) converges w.p.1 to ϕ(ωh), for all h ∈

{1, . . . ,m}. This proves Part 1 of Proposition 6. Similarly, Part 2follows from Part 2 of Proposition 5 (Part 2). �

Remark 4. (1) The identification method thus constructed isrecapitulated by Fig. 8. This illustrates well (left row of Fig. 8)that the nonlinearity parameters are determined first usingonly the data associated with the input u(t) = U1 cos(ω1t).In a second stage, each individual gain G(jωh,N) is separatelycomputed (parallel processing) using the input u(t) =

Uh cos(ωht). Also, it is seen that the quality of the estimatesdepends on the accuracy of the indices (tN , τa,N , τd,N).Interestingly, this accuracy is made controllable through theparameter ε in Table 1: the smaller ε the better the estimates.

(2) In the case were the linear subsystem transfer function G(s)assumes a known parametric structure, its parameters canbe estimated using the previously obtained frequency gainsG(jωh,N). Basically, this is a least-squares problem. A recentdiscussion of the optimal approximation of general measuredfrequency response functions using parametric models can befound in Pintelon and Schoukens (2012).

6. Simulation

6.1. Wiener system with backlash nonlinear operator

Presently, the system (1)–(5) is characterized by G(s) =

0.5/(s2 + 0.7s+ 0.1) and a backlash operator F [·] bordered by thefunctions:

fa(x) = −1.05 + 1.15x − 0.15x2 + 0.05x3 andfd(x) = 1.05 + 1.15x + 0.15x2 + 0.05x3.

At this stage, the structure of G(s) (being a second order) and thenature of F [·] (being a backlash) are not known to the user. The

778 F. Giri et al. / Automatica 50 (2014) 768–783

Fig. 8. Schematic recapitulation of the identification method.

noise ξ(t) is a sequence of normally distributed (pseudo) randomnumbers, with zero-mean and standard deviation σξ = 0.35.According to the identification method described in Sections 4and 5, the first step is input–output data acquisition followingthe procedure of Table 1. Doing so, a number of sine inputs sig-nals u(t) = Uh cos(ωht), and associated output signals, are ob-tained and used in the next identification steps. The first couple(U1, ω1) thus selected is U1 = 1, ω1 = 0.01π (rad/s). Ac-cording to Section 4.1, the system model plurality is coped withby focusing on the model

G(s), F [ x]

def= (G(s)/K , F [K x]) with

K = U1|G(jω1)|sgn (sin(− G(jω1))). Presently, one has K = 4.93and G(s) =

0.1014s2+0.7s+0.1

. This model is the only to feature the twoproperties, U1|G(jω1)| = 1 and ϕ(ω1) ∈ [0 π) (modulo π ) withϕ(ω1)

def= − G(jω1). A direct consequence of this choice is prop-

erty (10a)–(10b) i.e. the components of the lumped parameter vec-tor θ∗

U1,ω1are nothing other than the parameters of the nonlin-

earity F [K x], denoted (ai, di; i = 0 . . . 3). The vector θ∗

U1,ω1is

estimated using the algorithm of Table 3. To this end, the sys-tem output yU1,ω1(t) is collected on a sufficiently large interval0 ≤ t ≤ N 2π/ω1. Presently, N = 75 and a sample of the (steady-state) output is shown by Fig. 9a. The undisturbed output estimate,wU1,ω1(t,N), generated using (10a)–(10b), is shown by Fig. 9b andit clearly contains a constant stage. Then, one concludes (by Propo-sition 1, Part 1b) that the nonlinearity F [·] is a backlash and, sincethe constant stage starts at time tN = 6.7 s, one concludes using(38) that ϕ(ω1,N) = tN × ω1 = 0.21 (rad) and θU1,ω1(N) is com-puted using (39). By definition, the components of θU1,ω1(N) areestimates of the nonlinearity parameters (ai, di; i = 0 . . . 3).

A number of couples (Uh, ωh) have been selected as explainedabove and the corresponding input–output data generated byTable 1 have been based upon to get the estimates G(jωh,N) usingTable 4 and the estimator (48). For space limitation, the results arepresently provided only for twomore couples (Uh, ωh), see Table 5.The latter shows the phase estimates ϕ(ωh,N) and the lumpedparameter vector estimates θUh,ωh(N), the component of which areαi(Uh, ωh,N), δi(Uh, ωh,N). These, together with the previouslyobtained estimates

ai(N), di(N); i = 0 . . . 3

, are used in (48)

to get estimates |G(jωh,N)| of the frequency gain modulus. Allestimates are shown, together with their true values, in Table 5.Clearly, the estimates are quite close to their true values. To furthervalidate the results, the true system and the estimated model are

Fig. 9a. Steady-state output y(t) obtained with (U1, ω1) over two periods of time.

Fig. 9b. Estimated undisturbed output wU1,ω1 (t,N) over one period of time.

both submitted to the following input signal:

u(t) = U1 cos(ω1t)+ U2 cos(ω2t)+ U2 cos(ω2t) (49)

where the Uh’s are those of Table 5. Note that the input is a sumof sinusoids whose frequencies are those for which an estimateG(jωh,N) is available. This is coherent with the fact that, so far,the structure of the transfer function G(s) has not been supposed

F. Giri et al. / Automatica 50 (2014) 768–783 779

Table 5Phase and lumped parameter estimates—Backlash case.

h 1 2 3

Uh 1 1 1ωh (rad) 0.01π 0.05π 0.1π

ϕ(ωh) (rad) 0.218 0.97 1.56ϕ(ωh,N) (rad) 0.21 0.96 1.59

θ∗

Uh,ωh(N) [−1.05 5.67 −3.65 5.99 1.05 5.67 3.65 5.99] [−1.05 4.31 −2.11 2.64 1.05 4.31 2.11 2.64] [−1.05 2.61 −0.77 0.59 1.05 2.61 0.77 0.59]

θUh,ωh (N) [−1.06 5.71 −3.62 6.03 1.04 5.62 3.68 5.95] [−1.07 4.28 −2.14 2.61 1.03 4.34 2.15 2.61] [−1.03 2.58−0.75 0.56 1.03 2.64 0.775 0.61]

|G(jωh)| 1 0.76 0.46|G(jωh,N)| 1 0.747 0.447

Fig. 10a. Samples of system and model responses to the input signal (49).

Fig. 10b. Samples of system and model responses to a uniformly distributed(pseudo) random input.

to be known making impossible to compute the model output forgeneral input signals. Again, the noise ξ(t) is a normally distributedsequence, with zero-mean and standard deviation σξ = 1.15.The resulting true system output y(t) and the estimated modeloutput, denoted y(t), are plotted in Fig. 10a. Clearly, a quitesatisfactory system-model matching is observed. This is furtheremphasized using the output estimation error variance estimatedby the expression:

σ 2e =

19000

9000

0e2(t)dt, with e(t) = y(t)− y(t). (50)

Presently, one gets σe = 1.75 which is comparable to the noisestandard deviation.

Now, if the structure G(s) = λ/(s2 +γ1s+γ0) is supposed to bea priori known then, one can get estimates of the coefficient vec-tor Θ∗

= [γ1 γ0 λ] T = [0.7 0.1 0.1014] T using the available fre-quency response estimates G(jωh,N). As pointed out in Remark 4(Part 2), this fundamentally is a least squares problem. Doing so,one gets ΘN =

0.709 0.105 0.105

T . The identified modelcan now be validated using a general shape input signal. Here, thetrueWiener system and the identifiedmodel are both excitedwitha (pseudo) random input signal uniformly distributed in the in-terval [−25, 25]. Again, the noise ξ(t) is a normally distributedsequence, with zero-mean and standard deviation σξ = 0.1. Theresulting system and model outputs are shown in Fig. 10b whichshows a satisfactory system-model matching. The correspondingoutput estimation error has a standard variation, computed using(50), of σe = 0.201 which is comparable to the noise standard de-viation.

6.2. Wiener system with backlash-inverse nonlinear operator

The Wiener system (1)–(5) is now characterized by G(s) =

0.1/(s+0.2)(s+1) and a backlash-inverse operator F [·]with cross-ing borders defined by fa(x) = 1 + x + x2 and fd(x) = 3 + x − x2.Again the structure of G(s) and the nature of F [·] (being backlash-inverse) are not a priori known. The noise ξ(t) is a sequence ofnormally distributed random variables with zero-mean and stan-dard deviation σξ = 0.25. According to Section 4.1, the systemmodel plurality is copedwith by focusing on themodel

G(s), F [ x]

def= (G(s)/K , F [K x]) with K = U1|G(jω1)|sgn (sin(− G(jω1))).Presently, one has K = 0.74 and G(s) =

0.135s2+1.2s+0.2

. This model isthe only to feature the two properties,U1|G(jω1)| = 1 and ϕ(ω1) ∈

[0 π). Following the identification method of Sections 4 and 5, thefirst step is input–output data acquisition following the procedureof Table 1. Doing so, a number of informative couples (Uh, ωh) aredetermined the values of which are shown in Table 6. Let us il-lustrate how the procedure of Table 1 operates by considering thecouple (U2, ω2). This particular couple is focused on because theresulting system output signal, yU2,ω2(t), and the undisturbed ver-sion, wU2,ω2(t,N), exhibit no useful characteristic (Figs. 11a and11b). Then, Table 2 (Part 1) stipulates that τa,N = τd,N = 0 and tN isarbitrarily chosen. Letting tN = 0, one gets from (31) the optimiza-tion interval I(U2, ω2,N) = [0 2π/ω]. Then, Table 3 (Part 2) sug-gests that the function JU2,ω2,N(ψ, θ(ψ,N)), defined by (36)–(37),is plotted in function of ψ on the interval 0 ≤ ψ ≤ π . Fig. 12shows that the only global minimum of JU2,ω2,N(ψ, θ(ψ,N)) is lo-cated atψopt = 0.8. Then, one concludes using Table 4 (Part 2) thatϕ = 0.8 (rad). This allows the computation of θ using the least-squares expression in Table 4 (Part 2). It is checked that θ ∈ Swhich, according to Table 4 (Part 2a), means that one must setϕ(ω2,N) = 0.8 (rad) and θU2,ω2(N) = θ .

780 F. Giri et al. / Automatica 50 (2014) 768–783

Table 6Phase and lumped parameter estimates—Backlash-inverse case.

h 1 2 3

Uh 1.5 2.57 2.57ωh (rad/s) 0.01π 0.05π 0.1π

ϕ(ωh) (rad) 0.19 0.82 1.31ϕ(ωh,N) (rad) 0.21 0.8 1.27

θ∗(ωh) [1.00 0.74 0.55 3.00 0.74 − 0.55] [1.00 1.00 0.997 3.00 1.00 − 0.997] [1.00 0.66 0.43 3.00 0.66 − 0.43]θ (ωh,N) [1.02 0.72 0.53 3.03 0.73 − 0.58] [1.02 1.03 0.970 2.97 1.04 − 0.960] [1.04 0.63 0.48 2.95 0.62 − 0.45]

|G(jωh)| 0.67 0.52 0.340|G(jωh,N)| 0.67 0.54 0.356

Fig. 11a. Steady-state output y(t) obtainedwith (U2, ω2), over two periods of time.

Fig. 11b. Estimated undisturbed output signal wU2,ω2 (t,N)over one period of time.

Fig. 12. Plot of JU2,ω2,N (ψ, θ(ψ,N)) in function of ψ .

7. Conclusion

The problem of system identification is addressed for Wienersystems where the linear subsystem, described by (1)–(2), maybe parametric or not, finite order or not. The nonlinear ele-ment is any memory operator of backlash or backlash-inversetype, that is bordered by polynomial lines. The latter are allowedto be noninvertible and crossing so that memory and memorylesspolynomial nonlinearities can be handled within a unified theo-retical framework. The identification problem is dealt with usinga two-stage approach combining frequency tools and prediction-error optimization. In the first stage, the nonlinearity parametersare estimated applying the estimator of Table 3. In the secondstage, the transfer function response is identified using the algo-rithm of Table 4 and the estimator (48). All involved estimators areconsistent and the identificationmethod indifferently applies to alltypes of nonlinear operators. Furthermore, the prior knowledge ofthe type of the underlying operator (being memoryless or mem-ory backlash or backlash-inverse) is not required. To the author’sknowledge no previous study has solved the identification prob-lem for a so large class of Wiener systems.

Appendix A. Proof of Proposition 3

Proof of Part 1. From (16b) and (24), one gets for all (U, ω) ∈

{(Uh, ωh); h = 1 . . .m}:

J∗U,ω(ψ, θ) =

I(U,ω)

wU,ω(t)− XT

ω,ψ (t)θ2

dt

+

I(U,ω)

E(ξ(t))2dt (A.1)

where the cross product term has been ignored because ξ(t) is un-correlated with wU,ω(t) and Xω,ψ (t). Indeed, ξ(t) is uncorrelatedwith wU,ω(t) and Xω,ψ (t) because, on one hand, wU,ω(t) only de-pends on the control input u(t) (which is supposed to be uncor-related with the noise input) and, on the other hand, Xω,ψ (t) is adeterministic function due to (17b). Clearly, the second termon theright side of (A.1) is nonnegative and independent on θ . Then, theminimum of J∗U,ω(ψ, θ) is reached when the first part on the leftside is null. This is the case if, and only if:

wU,ω(t) = XTω,ψ (t)θ, for (almost) all t ∈ I(U, ω). (A.2)

Recall that θ is of the following form (see (21)):

θ =α0 . . . αn δ0 . . . δn

T. (A.3)

Then, it follows from (A.2) using (17b) that:

wU,ω(t) =

ni=0

αi1 − σω,ψ (t)

(cos(ωt − ψ)) i

+ δiσω,ψ (t) (cos(ωt − ψ)) i , (A.4)

F. Giri et al. / Automatica 50 (2014) 768–783 781

for (almost) all t ∈ I(U, ω). As sin(ωt −ψ) is periodic with period2π/ω, there is an interval (b1 c1) ⊂ I(U, ω) (resp. (b2 c2) ⊂

I(U, ω)) such that sin(ωt−ψ) is negative on (b1 c1) (resp. positiveon (b2 c2)). Then, it follows from (15a)–(15b) that σω,ψ (t) = 0 on(b1 c1) and σω,ψ (t) = 1 on (b2 c2). This together with (A.2) yield:

wU,ω(t) =

nj=0

αi (cos(ωt − ψ)) idef= g2(cos(ωt − ψ)),

on (b1 c1) (A.5)

wU,ω(t) =

nj=0

δi (cos(ωt − ψ)) idef= g1(cos(ωt − ψ)),

on (b2 c2). (A.6)

Clearly, the functions gi (i = 1, 2) are polynomials with nonzerodegrees. Then, applying the technical lemma of Appendix B, itfollows from (A.5)–(A.6) thatψ = ϕ(ω) (modulo π ). We have thusproved the following result:

(ψ, θ) is a minimum of J∗U,ω(ψ, θ) ⇒ ψ

= ϕ(ω) (modulo π ). (A.7)

In Problem Optimization 1, the function J∗U1,ω1(ψ, θ) is mini-

mized under the constraint 0 ≤ ψ < π . Then, if (ψ, θ) is a solu-tion of this problem, it follows from (A.7) thatψ = ϕ(ω1) (modulo2π ). Then, by definition of (b1 c1) and (b2 c2), one has (b1 c1) ⊂t + τd t +

πω

and (b2 c2) ⊂

t +

πω

+ τa t +2πω

. Then, it fol-

lows comparing Eqs. (A.5) and (12a) that, αi = α∗

i (U1, ω1) (i =

0 . . . n). Similarly, comparing (A.6) with (12b) one gets that, δi =

δ∗

i (U1, ω1) (i = 0 . . . n). Then, by (A.3) and (17a), one has θ =

θ∗(ω1). Eq. (26) is proved.

Proof of Part 2. Optimization Problem 2 amounts to minimizingJ∗U,ω(ψ, θ), with (U, ω) = (Uh, ωh) (h = 2 . . .m), under theconstraint θ ∈ S. By definition of S (see (22)), θ enjoys the followingproperties:

sgn(αi) = sgn(ai), sgn(δi) = sgn(ai);(i = 0 . . . n; h = 2 . . .m) (A.8)

diαi = aiδi, (i = 0 . . . n; h = 2 . . .m). (A.9)

Comparing (A.8) and (20a), one gets:

sgn(αi) = sgn(α∗

i (ω1) ) and

sgn(δi) = sgn(δ∗

i (ω1) ), for i = 0 . . . n. (A.10)

Now, suppose (ψ, θ) is a solution of the Optimization Problem2, for some h = 2 . . .m. One immediately gets from (A.7) thatψ =

ϕ(ωh) (modulo π ). Let us prove that, except for the case wherethe nonlinear operator F [·] is symmetric, one has ψ = ϕ(ωh)(modulo 2π ). To this end, suppose that, for some h = 2 . . .m, ψ =

ϕ(ωh)+ π (modulo 2π ). Then, (A.5) rewrites using (A.10):

wUh,ωh(t) =

nj=0

(−1)iαi sgn(α∗

i (ωh))

× (cos(ωht − ϕ(ωh)))i , on (b1 c1). (A.11)

By definition, (b1 c1) is an interval over which sin(ωht − ψ)is negative. Then, ψ = ϕ(ωh) + π implies that (b1 c1) is also an

interval overwhich sin(ωht−ϕ(ωh)) is positive. Thenone gets from(15a) that (b1 c1) ⊂

tk tk +

πω

. Then, replacing in (12b) (U, ω)

by (Uh, ωh) and comparing the obtained equation with (A.11), onegets:

δ∗

i (Uh, ωh) = (−1)iαi (i = 0 . . . n). (A.12)

Similarly, one gets using (A.6) and (12a):

α∗

i (Uh, ωh) = (−1)iδi (i = 0 . . . n). (A.13)

Using (13) and (A.10), Eq. (A.12) implies:

sgn(di) = (−1)isgn(ai) (i = 0 . . . n). (A.14)

Furthermore, multiplying (A.12) by di and (A.13) by ai, com-paring the obtained equalities, and using (A.9), one gets di δ∗

i(Uh, ωh) = ai α∗

i (Uh, ωh). Multiplying both sides by di, one obtains:

δ∗

i (Uh, ωh)d2i = ai α∗

i (Uh, ωh)di = δ∗

i (Uh, ωh)a2i (A.15)

where the second equality is obtained using (20b). From (A.15) onegets d2i = a2i which, due to (A.14) yields:

di = (−1)iai (i = 0 . . . n). (A.16)

But, this only holds when the operator F [·] is symmetric (see(22)). Hence, in the general case of nonsymmetric operators, thestarting assumption ψ = ϕ(ωh) + π (modulo 2π ), cannot holdand so one has ψ = ϕ(ωh) (modulo 2π ). In turn, this impliesthat θ = θ∗(ωh) and the proof is quite similar to the case h = 1,developed in Part 1 leading to (26).

Proof of Part 3. Minimizing J∗U,ω(ψ, θ)with respect to θ amountsto minimize the first term on the right side of (A.1). Deriving thatterm with respect to θ and letting the derivative be zero, one getsthe least-squares expression (28) in Part 3. It remains to showthat the matrix

I(U,ω) Xω,ϕ(ω)(t)X

Tω,ϕ(ω)(t)dt is actually invertible.

Clearly, thismatrix is nonnegative definite. Then, if is sufficient (forthat matrix to be invertible) to show that it is positive definite. Tothis end suppose that:

ZT

IXω,ϕ(ω)(t)XT

ω,ϕ(ω)(t) dtZ = 0, for some Z ∈ R2n+2

or, equivalently: cr

br

ZTXω,ϕ(ω)(t)

2dt = 0 (r = 1, 2),

for some Z ∈ R2n+2 (A.17)

with [b1 c1) =t + τd t +

πω

and [b2 c2) =

t +

πω

+ τa t +2πω

.

Recall that, by (15b) and (17b), Xω,ϕ(ω)(t) boils down to:

Xω,ϕ(ω)(t)

=0 · · · 0 1 cos(ωt − ϕ(ω)) · · · (cos(ωt − ϕ(ω))) n

Ton

t + τd t +

π

ω

(A.18)

Xω,ϕ(ω)(t)

=1 cos(ωt − ϕ(ω)) · · · (cos(ωt − ϕ(ω))) n 0 · · · 0

Ton

t +

π

ω+ τa t +

2πω

. (A.19)

The structure of Xω,ϕ(ω)(t) suggests for Z the partition Z =ZT1 ZT

2

Twith dim Z1 = dim Z2 = n + 1. Let the components

782 F. Giri et al. / Automatica 50 (2014) 768–783

of Z1 (resp. Z2) be denoted z, i (resp. z2,i), i = 1 . . . n + 1. Based onthe above observations, it follows from (A.17), that:

z1,1 + z1,2 cos(ωt − ϕ(ω))+ · · · + z1,n+1(cos(ωt − ϕ(ω)))n

= 0 ont +

π

ω+ τa t +

2πω

(A.20)

z2,1 + z2,2 cos(ωt − ϕ(ω))+ · · · + z2,n+1(cos(ωt − ϕ(ω)))n

= 0 ont + τd t +

π

ω

. (A.21)

By definition of the triplet (t, τa, τd) (see (23a)–(23b)) andthe time sequence { tk} (see (7)), the function cos(ωt − ϕ(ω)) ismonotone on both time intervals in (A.20) and (A.21). On the otherhand, both polynomials in (A.20) and (A.21) are nth degree and soassumes at most n different real roots. Therefore, one gets from(A.20)–(A.21) that all polynomial coefficients zi,j (i = 1, 2; j =

0 . . . n) are null, which implies that Z = 0. We have thus shownthat, if the statement (17) holds then Z = 0. This proves that thematrix

I Xω,ϕ(ω)(t)X

Tω,ϕ(ω)(t)dt is invertible. This ends the proof of

Proposition 3. �

Appendix B. Technical lemma

Consider the Wiener system (1)–(5) being excited with u(t) =

U cos(ω t) for some informative couple (U, ω). Suppose there isa finite-degree polynomial function g , a time interval (b c), anda real number ψ such that: wU,ω(t) = g(χψ (t)) on (b c), with

χψ (t)def= cos(ωt − ψ). Then, one has ψ = ϕ(ω), modulo π .

Proof. As wU,ω(t) is not constant on (b c), it follows from (8a)–(8d) that there is a subinterval (b c) of (b c) such that (b c) ⊂tk + τd tk +

πω

or (b c) ⊂

tk +

πω

+ τa tk +2πω

for some k ∈ N.

Suppose the first case holds (the proof is similar if the second caseholds). Then, one has from (8b):

wU,ω(t) = fd(xU,ω(t)) = fdU|G(jω)|χϕ(ω)(t)

,

∀t ∈ (b c). (B.1)

On the other hand, by definition of χψ (t) one has:

χψ (t) = cos (ωt − ϕ(ω)+ ϕ(ω)− ψ)

= ϑ χϕ(ω)(t)±

1 − χ2

ϕ(ω)(t)1 − ϑ2

with ϑ = cos(ϕ(ω)−ψ). AswU,ω(t) = g(χψ (t)), for all t ∈ (b c),it follows that:

wU,ω(t) = gϑχϕ(ω)(t)±

1 − χ2

ϕ(ω)(t)1 − ϑ2

,

for all t ∈ (b c). (B.2)

Comparing (B.1) and (B.2), one gets:

fd (U|G(jω)|z) = gϑ z ±

1 − z2

1 − ϑ2

,

for all z ∈ (z1 z2) (B.3)

where (z1 z2) denotes the image of the interval (b c) by thefunction cos(ωt − ψ). Note that (z1 z2) is nonempty because theabove function is monotone on (b c). Now, since fd is a nth-degreepolynomial, it follows from (B.3) that:

dn+1

dzn+1g

ϑ z ±

1 − z2

1 − ϑ2

=

dn+1

dzn+1fd(U|G(jω)|z) = 0, for all z ∈ (z1 z2). (B.4)

But, this is impossible if g is a finite degree polynomial unlessϑ2

= 1 or, equivalently, ϕ(ω)−ψ = 0 modulo π . This proves thelemma. �

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Fouad Giri received a Ph.D. from the Institut NationalPolytechnique of Grenoble, France, in 1988 and is nowDistinguished Professor at the University of Caen Basse-Normandie, Caen, France. He is currently serving asthe General Chair of the IFAC Int. Workshops ALCOSP2013 and PSYCO 2013. He is the Vice-Chair of theIFAC TC ‘Adaptive and Learning Systems’ and is holdingmembership positions in the IFAC TCs on ‘Modelling,Identification and Signal Processing’ and ‘Power Plants andPower Systems’, and the IEEE CSS TC ‘System Identificationand Adaptive Control’. He is Associate Editor of the IFAC

Journal Control Engineering Practice, of the IEEE Transactions on Control SystemsTechnology and of the Editorial Board of IEEE CSS conferences. His research interestsinclude Nonlinear System Identification, Adaptive Nonlinear Control, ConstrainedControl, and control applications to Power Converters, ElectricMachines and PowerSystems. He has supervised 20 Ph.D.s and has co-authored three books and over 250journal and conference papers on these topics.

Abdelhadi Radouane was born in 1969. He receivedthe Agrégation degree in Electrical Engineering fromthe ENSET of Rabat-Morocco in 1996. He is currentlycompleting his Ph.D. in Automatic Control under thesupervision of Professor F. Giri. His research focuses onnonlinear system identification based on multi-modelrepresentations.

Adil Brouri was born in 1973. He received the Agrégationdegree in Electrical Engineering from the ENSET of Rabat-Morocco and completed a Ph.D. in Automatic Controlunder the supervision of Professor F. Giri. He is conductinga research activity on nonlinear system identification andhas published several journal and conference papers onthis topic.

Fatima-Zahra Chaoui obtained a Ph.D. degree in Auto-matic Control from the Institut National Polytechnique ofGrenoble-France, in 2000. She has spent long-term visitsat the Laboratoire d’Automatique de Grenoble and the GR-EYC, University of Caen, both in France. Since 1995, shehas been successively Assistant Professor and Professor atthe Ecole Normale Supérieure d’Enseignement Technique(ENSET) of Rabat-Morocco. Her research interest includesnonlinear system identification and constrained control.She published over 100 journal and conference papers onthese topics and has supervised 9 Ph.D.s.