Decoupling Static Nonlinearities in a ParallelWiener-Hammerstein System: A First-order

ApproachPhilippe Dreesen∗, Maarten Schoukens∗, Koen Tiels∗ and Johan Schoukens∗

∗Vrije Universiteit Brussel, Dept. VUB-ELECPleinlaan 2, B-1050 BRUSSELS, Belgium.philippe.dreesen@vub.ac.be

Abstract—We present a method to decompose a static MIMO(multiple-input-multiple-output) nonlinearity into a set of SISO(single-input-single-output) polynomials acting on internal vari-ables that are related to the inputs and outputs of the MIMOnonlinearity by linear transformations. The method is inspiredon the small-signal analysis of nonlinear circuits and proceedsby collecting first-order information of the MIMO functioninto a set of Jacobian matrices. A simultaneous diagonalizationof the set of Jacobian matrices is computed using a tensordecomposition, providing the required linear transformations,after which also the coefficients of the internal SISO polynomialscan be computed. The method is validated on measurements ofa parallel two-branch Wiener-Hammerstein identification setup.

I. INTRODUCTION

Modern measurement instruments frequently make use ofadvanced signal processing and control algorithms that aredesigned and implemented using nonlinear models. Sensorlinearization and nonlinear pre-compensation of actuators aretypical examples. Therefore, aspects regarding nonlinear sys-tem identification and modeling pose important questions forthe instrumentation and measurement community.

Recent years have witnessed a shift in research interest fromlinear system identification [1], [2], [3] to nonlinear identifi-cation methods. Modeling non-linear systems is covered indifferent fields like statistical learning and machine learning[4], [5], [6], [7], [8], but most of these methods are typicallynot specifically developed to deal with dynamics and oftenhave limited means for dealing with noise.

In the system identification community, non-linear exten-sions of well-known methods have been studied [9], [10], suchas non-linear ARX and non-linear ARMAX models [11], butthese methods often lack the ability to provide the user withan understandable model. Moreover, the resulting models donot match with the underlying physical nature of the systems,in which the nonlinearities are often localized at a certainplace in the system or they appear in certain signal paths.These shortcomings suggest the use of block-oriented systemidentification methods [12], [13], [14], which consist of theinterconnection of linear time-invariant (LTI) system blocksand static nonlinear functions (SNL).

In this paper we will consider the case of block-orientedidentification of a parallel Wiener-Hammerstein system (see

bottom figure in Figure 1). The identification procedure pro-posed in [15] leads to a static multiple-input-multiple-output(MIMO) nonlinearity, rather than static single-input-single-output nonlinearities for each parallel branch (see top figurein Figure 1).

We address the problem of unraveling the static MIMOnonlinearity into a set of static SISO nonlinearities, againproviding the user with an intuition into the identified system(Figure 1). This continues the work initiated in [15], [16], [17],[18].

LTI

LTI

SNL

SNL

LTI

LTI

+upWH(t) ypWH(t)

LTI

LTI

SNL

LTI

LTI

+upWH(t) ypWH(t)

Fig. 1: Decoupling in a parallel Wiener-Hammerstein struc-ture. The figure on the top represents the result of the identi-fication procedure [15], where the LTI blocks represent lineartime-invariant dynamical systems and the SNL blocks repre-sent coupled static nonlinearities. The figure on the bottomrepresents a decoupled structure, where the static nonlinearblock is decoupled into two single-branch nonlinearities.

This paper introduces a new way to decompose a staticMIMO function into SISO polynomials. In the general sense,a decoupling procedure provides insight into the nonlinearfunction, but may also reduce the number of parameters, orserve as a simplification or an approximation. We will describea decoupling procedure that uses first-order information ofthe MIMO mapping and relies on a tensor decompositionmethod. The decoupling method will then be integrated into

This full text paper was peer-reviewed at the direction of IEEE Instrumentation and Measurement Society prior to the acceptance and publication.

978-1-4799-6144-6/15/$31.00 ©2015 IEEE

the parallel Wiener-Hammerstein identification procedure of[15] and validated on measurements of a parallel Wiener-Hammerstein test system.

II. DECOUPLING PROCEDURE

A. Problem Statement

The decoupling task we tackle in this paper is visualizedin Figure 2. Consider a multivariate mapping f : Rm →Rn, defined as f = (f1(u), . . . , fn(u)), in the variablesu = (u1, . . . , um). Each fi(u) depends on all inputs ui. Forinstance, when fi(u) are polynomials of total degree d, eachfi(u) contains

(m+dm

)coefficients, of which many correspond

to ‘mixed’ monomials consisting of several variables ui, e.g.,u1u2, u21u3, u2u33, etc.

A decoupled representation of f(u) is defined as

f(u) = Wg(VTu), (1)

where V ∈ Rm×r and W ∈ Rn×r are linear transformationmatrices that relate the input variables u and the outputvariables y to the internal variables x ∈ Rr and z ∈ Rr bythe relations x = VTu and y = Wz, respectively. The vectorfunction g : Rr → Rr represents a set of univariate (SISO)functions gi(xi), defined as

g(x1, . . . , xr) = (g1(x1), . . . , gr(xr)),

with gk : R → R. The number r corresponds to the numberof internal univariate functions gi(xi) and may exceed m andn.

B. A Simultaneous Matrix Diagonalization Problem

The idea behind the proposed method is to capture thebehavior of f(u) by means of its first-order derivatives in aset of operating points u(k), with k = 1, . . . , N . The first-order information of f(u) is contained in the Jacobian matrixof f(u), denoted by J(u) and defined as

J(u) =

∂f1∂u1

(u) . . . ∂f1∂um

(u)...

. . ....

∂fn∂u1

(u) . . . ∂fn∂um

(u)

.By evaluating the Jacobian matrix in the operating pointsu(k), k = 1, . . . , N , we will show that the decoupling taskis solved by a simultaneous diagonalization1 of the set ofJacobian matrices J(u(k)). After the transformations V andW are determined, also an estimation of the internal univariategi(xi) can be reconstructed.

The first-order derivatives of the parameterization (1) aregiven by

J(u) = W diag (g′i(vT

i u))VT , (2)

1In the classical literature, the simultaneous (or joint) diagonalization refersto the simultaneous congruence transformation Ak = VDkV

T , where theAk are square matrices (see [19] for a recent survey paper). In this paper,simultaneous diagonalization concerns the nonsymmetrical problem Ak =WDkV

T , where, in addition to having different linear transformations onthe left and on the right, the matrices Ak are not necessarily square.

where g′i(xi) = dgi(xi)/ dxi, which follows from applyingthe chain rule to f(u) = Wg(VTu). This implies that thefirst-order derivatives of the parameterization (1), evaluated atthe points u(k), lead to the simultaneous diagonalization ofthe set of matrices

J(u(k)) = W diag(g′i(vT

i u(k)))VT ,

in which the matrix factors W and V do not depend on thechoice of the operating points u(k).

Simultaneous matrix diagonalization can be computed bytensor methods. Consider therefore the n ×m × N Jacobiantensor J that is constructed by stacking the Jacobian matricesJ(u(k)) behind each other. The canonical polyadic decomposi-tion (CP decomposition2) [20], [21], [22] expresses the tensorJ as a sum of rank-1 terms. Tensor J is thus written as

J =

r∑i=1

wi vi hi, (3)

where r is a positive integer and denotes the outer product(i.e., for a tensor X = abc, the element in position (i, j, k)is given by ai · bj · ck). We have that

W =[w1 . . . wr

],

V =[v1 . . . vr

]and

H =[h1 . . . hr

],

where H contains the evaluations of the g′i(vTi u) in the N

operating points as

hki = g′i(vT

i u(k)).

We have now that J(u(k)) = W diag(hk1, . . . , hkN )VT .Figure 3 summarizes how the tensor of Jacobian matricesleads to a simultaneous matrix diagonalization problem thatis solved by the CP decomposition.

The CP decomposition has an attractive property that guar-antees ‘essential uniqueness’ [23], [24]. This means that thefactors V, W and H are defined uniquely up to column-wisescaling and permutation invariances.

In [25] the condition for essential uniqueness was shown toboil down to min(m, r)+min(n, r) ≥ r+2 for the decouplingtask. It is important to keep in mind that the CP decompositionrequires the solution of a nonconvex optimization problem.Note that the uniqueness property does not imply that theoptimizaton routine to compute the CP decomposition is notharmed by the problem of local minima. It only states that inan (exact) CP decomposition the retrieved factors V, W andH are unique (up to column-wise scaling and permutation).Software to compute the CP decomposition is widely available,see e.g., tensorlab for MATLAB [26].

A suitable value for r needs to be fixed when computing theCP decomposition. Determining r is a part of the decouplingtask and is in practice based on the decomposition error forthe CP decomposition for several candidate values for r.

2The CP decomposition is sometimes called CanDecomp or PARAFAC.

u1

...

um

f(u)

y1

...

yn

↔

u1

...

um

VT

g1(x1)x1

...

gr(xr)xr

W

z1

zr

y1

...

yn

Fig. 2: Decomposing a static nonlinear function f(u) into a decoupled structure: Find from y = f(u) the transformations Vand W and the functions gi(xi) that constitute the parallel structure f(u) = Wg(VTu).

(a)

=

=

=

=

W V

H

WH V

W V

w1v1

h1

+ . . . +

wrvr

hr

(b)

Fig. 3: From a set of Jacobian matrices to the CP decom-position. The first-order information of f(u) is collected in aset of operating points u(k), with k = 1, . . . , N (indicatedby the colored patches on the surface shown in (a)). Thecorresponding Jacobian matrices J(u(k)) are placed in a three-way tensor J (b). Each Jacobian matrix J(u(k)) can bewritten as J(u(k)) = W diag(g′i(v

Ti u

(k)))VT . This resultsin a simultaneous matrix diagonalization problem, which iscomputed by the CP decomposition.

C. Reconstruction gi(xi) CoefficientsThe CP decomposition returns the factors V, W and

H, but we still need to determine the functions gi(xi).

We present a method for reconstructing the coefficients ofthe univariate polynomials gi(xi) using the results of theCP decomposition and input-output samples (u(k),y(k)). Wemake use of the fact that the columns of H contain theevaluations of the differentiated gi(xi) in the N operatingpoints as hki = g′i(x

(k)i ). The internal functions gi(xi) can be

obtained from the (symbolic) integration of the r (univariate)interpolation polynomials obtained from hki = g′i(x

(k)i ). After

the integration step, the constants of integration remain to bedetermined, which is achieved through the solution of a linearsystem of equations. The complete approach is described inthe following paragraphs.

A polynomial fitting problem using the data (x(k)i , hki) pro-

vides a (symbolic) representation of g′i(xi), for i = 1, . . . , r.We solve a system of linear equations involving the N × dVandermonde matrix containing the powers of the x

(k)i for

i = 1, . . . , r. For branch i we have the Vandermonde system

1 x(1)i . . .

(x(1)i

)d−11 x

(2)i . . .

(x(2)i

)d−1...

......

1 x(N)i . . .

(x(N)i

)d−1

c′i,0c′i,1

...c′i,d−1

=

h1ih2i

...hNi

,

where c′i,δ denotes the coefficient corresponding to degree δof g′i(xi).

A symbolic integration of g′i(xi) provides the gi(xi) up tothe correct value of the constant of integration. We have nowthat

y(k) = Wg(VTu(k)),

in which we separate the unknown constant terms, denoted bygc, from the calculated g0(VTu) as

g(VTu) = g0(VTu) + gc.

Finding the constants gc amounts to solving the system

Wgc = y(k) −Wg0(VTu(k)), (4)

which completes the computation of the functions gi(xi).Remarks. If n ≥ r, solving (4) is possible in the exactsense, and has a unique solution. If n < r, the system (4) isunderdetermined; in this case an exact solution of (4) exists,but it is not unique. Consequently, the reconstruction of theinternal gi(xi) is not uniquely possible. However, using any

of the valid reconstructions, the input-output reconstruction off(u) still holds (in the exact sense).

III. DECOUPLING IN PARALLEL WIENER-HAMMERSTEINIDENTIFICATION

We briefly review the parallel Wiener-Hammerstein iden-tification and decoupling procedure introduced in [15], [18]and improved recently in [17]. In this paper we will focuson the decoupling of the identified static nonlinear MIMOmapping. The identification procedure itself is not of maininterest in this paper, and therefore we refer the interestedreader to abovementioned manuscripts for details.

We will apply the proposed decoupling procedure (Sec-tion II) to a two-branch parallel Wiener-Hammerstein systemidentification setup and discuss the results.

A. Review Measurement Setup [17]

The front and back LTI blocks of the parallel Wiener-Hammerstein system (Figure 1) are third order IIR filters.The static nonlinearity of each branch is realized with adiode-resistor network. The excitation signals are generatedwith the Agilent/HP E1445A arbitrary waveform generator(AWG) at a sampling frequency of 625 kHz. An internal low-pass filter with a cut-off frequency of 250 kHz is used asreconstruction filter for the input signal. In- and output signalsof the system are measured by the alias-protected acquisitionchannels (Agilent/HP E1430A) at a sampling frequency of78 kHz. The AWG and acquisition cards are synchronized toavoid leakage errors.

Buffers with high input impedance, added between theacquisition cards and the in- and output of the system avoidthat the circuit is loaded by the 50 Ω input impedance of theacquisition card. The buffers are very linear (≈ 85 dBc at fullscale and 1 MHz) up to 10 V peak to peak, and have an inputimpedance of 1 MΩ and a 50 Ω output impedance.

The input signal upWH is a random phase multisine (see [2]),containing M = 131072 samples with a flat amplitudespectrum. The excited frequency band ranges from fs/M tofs/2, such that upWH(t) = A

∑M/2n=1 cos(2πnkfs/M + φn),

where the phases φn are independent uniformly distributedrandom variables ranging from 0 to 2π. Twenty realizationsof the multisines are used. The input signal is applied at fivedifferent RMS values that are linearly distributed between100 mV and 1 V.

The signals are measured at a sampling frequency of78 kHz, which is eight times slower than the samplingfrequency at the generator side, resulting in 16384 samplesper period.

B. Review Identification Procedure [17]

The identification procedure of [17] can be summarized inthe following steps:

1) The best linear approximation (BLA), defined in [2], ofthe system under test is determined at different operatingpoints. BLAs are determined using excitation signals with

different power spectra, slightly changing the operatingpoint of the system under test for each BLA.

2) It can be shown that GBLA(s) =∑ri=1 αiH

(i)(s)S(i)(s),where H(i)(s) and S(i)(s) denote the LTI systems inbranch i at the front and back of the parallel Wiener-Hammerstein structure, respectively. The dynamics (polesand zeros) of the BLAs are estimated using a commondenominator approach (each branch has the same poles).The zeros of the BLAs are distributed over the differentbranches using a singular value decomposition (SVD) ofthe numerator coefficients [15].

3) The poles and zeros in each branch are assigned to thefront and back LTI systems H(i)(s) and S(i)(s). Thisis achieved by considering all possible partitionings ofpoles and zeros, while estimating a multivariate low-degree (e.g., degree three) MIMO polynomial to modelthe static nonlinearity. The model resulting in the leastsimulation error is selected as the final model.

Remarks. In step 3, a large amount of candidate models needsto be considered, each of which involves the estimation of anonlinear MIMO function f(u). For this reason, the proceduretypically employs third-order MIMO polynomials to modelthe nonlinearity, as the corresponding parameter estimationproblem is linear-in-the-parameters. It is argued that a third-degree function is able to represent dominant even and oddnonlinearities of f(u).

In [17] the identified static MIMO nonlinearity is decoupledby computing CP decompositions of the coefficient tensors ofthe third-order polynomials (obtained in the last step of theidentification procedure), from which a decoupled representa-tion is obtained. Finally the decoupled representation is usedas an initialization to estimate a high-degree (e.g., d = 15)decoupled polynomial in a final nonlinear optimization step.

The method presented in this paper takes a differentapproach. After performing the identification procedure, a(coupled) neural network model is estimated that accuratelyrepresents the nonlinear MIMO mapping f(u). The procedureof Section II is then applied to decouple this neural networkmodel into a set of univariate polynomials. The first-orderdecoupling approach is indeed able to cope with the neuralnetwork model class, provided that the Jacobian matrices canbe computed, implying that the activation functions in thenodes of the neural network need to be differentiable.

C. Results Measurement ExampleThe decoupling procedure of Section II is validated on the

identification of a two-branch parallel Wiener-Hammersteinstructure as shown in Figure 1. The system under test isthe custom-built measurement setup introduced in [17]. Themeasurement setup, signal generation and data acquisitionof [17] are summarized in Section III-A. The identificationprocedure summarized in Section III-B is performed, andafterwards an accurate model of the static nonlinearity is fittedusing an artificial neural network model [6] with ten neuronsin the hidden layer, with a tansig activation function, and alinear activation function in the output nodes.

The results of the algorithm of Section II are compared forfive different input amplitude levels. The Jacobian tensor iscomputed in N = 8000 points that were obtained from inputsamples that are filtered with the front branch LTI filters. TheJacobian matrices of the neural network model are computedusing numerical differentiations. The 2 × 2 × 8000 Jacobiantensor is decomposed using a rank-two CP decompositionusing tensorlab [26], resulting in the transformation matricesV and W, as well as the factor H. The coefficients ofthe internal mappings gi(xi) are finally obtained using themethod described in Section II-C. After obtaining the decou-pled descriptions y = Wg(VTu), the decoupled model isthen plugged into the parallel Wiener-Hammerstein structureand the performance is assessed on the basis of independentvalidation data sets.

Observations: Table I summarizes the results of the ex-periments. We show the root-mean-square value rms(e), thestandard deviation σe and the mean value µe of the simulationerror, defined as:

rms(e) =√

1N

∑Nk=1 e

2(k),

σe =√

1N

∑Nk=1 (e(k)− µe)2,

µe = 1N

∑Nk=1 e(k),

where the simulation error e(k) = ypWH(k) − ypWH(k) is thedifference between the measured output ypWH and the modeledoutput ypWH of the parallel Wiener-Hammerstein system atsample k.

The two top rows in Table I contain the results obtainedin [17] for the sake of comparison. The three bottom rowscontain the figures of interest for this paper. A coupled neuralnetwork is used as the starting point for the presented method,and hence serves as a ‘ground truth’ model. The results forthe decoupled 3rd and 7th degree polynomials obtained withthe presented method are shown in the last two rows.

Interpretation: The algorithm of Section II assumes thata decoupled representation exists, however, this is not thecase in the identification context due to modeling errors andmeasurement noise. It should therefore be expected that theresults of the decoupling step are suboptimal. Nevertheless,the procedure of Section II obtains decoupled degree three rep-resentations that show significantly better performance whencomparing to the degree three decoupling of [17]. This maynot come as a surprise, since the decoupling obtained in [17]is obtained using a degree three coupled representation that isnot able to capture accurately the nonlinear saturation effectsin the system.

We conclude that one of the attractive aspects of the pre-sented method is its applicability to non-polynomial nonlinear-ities, which results in a more accurate modeling. An additionalsignificant improvement is achieved by considering a degreeseven decoupling. It should be mentioned that further improve-ments are possible by using higher degrees, however they may

0 2 4 6 8 10 12 14 16 18 20−100

−80

−60

−40

−20

0

20

mag

nitu

de(d

B)

frequency (kHz)

Fig. 4: Measured output spectrum and spectra of the simulationerrors of the system on a validation dataset with rms(upWH) =550mV. The measured spectrum is shown in full black line.The red plus signs represent the results using the degree threedecoupling of [17]. The results for the decoupling obtainedwith the procedure of Section II are shown with blue crossesfor degree three and cyan stars for degree seven. The resultsusing the coupled neural network model are shown with greendiamond signs.

−8 −6 −4 −2 0 2 4 6 8 ×10−3−0.1858

−0.1858

−0.1857

−0.1857

−0.1856

−0.1856

xi

g i(xi)

−8 −6 −4 −2 0 2 4 6 ×10−3

0.1961

0.1962

0.1962

0.1962

0.1962

0.1962

Fig. 5: The estimated static nonlinear functions of the firstand second branch of the parallel Wiener-Hammerstein modelare shown in blue crosses for g1(x1) and black plus signs forg2(x2).

require a careful consideration of interpolation aspects, e.g.,using a regularization mechanism to avoid overmodeling.

In general it is observed that as the excitation amplituderms(upWH) increases, the performance decreases, which is tobe expected, as the contribution of the nonlinearity becomesmore significant. Figure 4 shows the measured and modeledoutput spectra on the validation set (rms(u) = 550 mV).Figure 5 shows the identified nonlinear functions g1(x1) andg2(x2).

TABLE I: Results on validation data (mV)

Excitation amplitude rms(upWH) (mV) 100 325 550 775 1000

coupled 3rd degree polynomial [17]rms(e) 4.70 7.74 13.42 15.07 22.40

µe -3.43 1.83 6.61 8.96 5.91σe 3.22 7.52 11.68 12.12 21.61

decoupled 3rd degree polynomial [17]rms(e) 12.08 22.82 31.05 35.95 49.02

µe 8.39 3.95 2.45 -7.33 -10.06σe 8.69 22.47 30.95 35.20 48.00

coupled neural networkrms(e) 0.30 0.51 0.37 0.60 1.08

µe -0.01 0.38 -0.06 -0.07 -0.33σe 0.30 0.33 0.37 0.60 1.03

decoupled 3rd degree polynomial (Algo. Sec. II)rms(e) 0.31 2.52 8.61 19.63 42.49

µe -0.02 2.37 6.44 16.58 -38.52σe 0.31 0.86 5.71 10.52 17.93

decoupled 7th degree polynomial (Algo. Sec. II)rms(e) 0.31 1.00 2.80 2.32 8.20

µe -0.01 0.93 2.59 0.87 -6.86σe 0.31 0.38 1.05 2.14 4.50

ACKNOWLEDGMENT

This work was supported in part by the Fund for ScientificResearch (FWO-Vlaanderen), by the Flemish Government(Methusalem), the Belgian Government through the Interuniversity Poles of Attraction (IAP VII) Program, and bythe ERC advanced grant SNLSID, under contract 320378.M. Schoukens is currently an FWO Aspirant, supported byFWO-Vlaanderen.

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