Frequency Domain Identification of Multiple InputMultiple Output Nonlinear Systems

Akshya K Swain Cheng-Shun LinDepartment of Electrical & Computer Engineering

University of Auckland,Private Bag-92019,New Zealand, Email: a.swain@auckland.ac.nz

E.M.A.M.MendesDepartment of Electronic EngineeringUniversidade Federal de Minas Gerais,

Av. Antonio Carlos 6627, BH-MG,31270-901,BrazilEmail: emmendes@cpdee.ufmg.br

Abstract— The proposed study introduces a total least squareswith structure selection (TLSS) algorithm to identify continuoustime differential equation models from generalized frequencyresponse function matrix (GFRFM) of multiple-input multiple-output(MIMO) nonlinear system. The estimation procedure isprogressive where the parameters of each degree of nonlinearityof each subsystem is estimated beginning with the estimationof linear terms and then adding higher order nonlinear terms.The algorithm combines the advantages of both the total leastsquares and orthogonal least squares with structure selection(OLSSS). The error reduction ratio (ERR) feature of OLSSSare exploited to provide an effective way of detecting the correctmodel structure or which terms to include into the model andthe total least squares algorithm provides accurate estimatesof the parameters when the data is corrupted with noise. Theperformance of the algorithm has been compared with theweighted complex orthogonal estimator and has been shown tobe superior.

I. INTRODUCTION

Identification of continuous time differential equation mod-els have attracted the attention of researchers from variety ofdisciplines during last few decades [1]. The obvious reasonsfor working with continuous time models is that most physicalsystems are inherently continuous in time and parametersof such models can have simple physical interpretation. Amajor difficulty of identification of continuous time modelsis associated with the numerical errors induced while comput-ing derivatives of input-output from raw (noisy)data. Severaltechniques using Poisson moment functional, piecewise con-stant block functions, Hermite and Legendre polynomials, anddifferent types of modulating functions have been proposed byresearchers in the past to minimize and possibly to avoid theseerrors during estimation [2].

Another approach to avoid the errors arising due to dif-ferentiation is to utilize the frequency domain informationto fit continuous time models. Several techniques have beendeveloped to estimate continuous time models for linear sys-tems using frequency response data which has been compre-hensively reviewed in [3]. Extending these techniques intononlinear systems are not without difficulties. In [4] [5],several algorithms have been proposed to estimate parametersof single input single output (SISO) nonlinear differentialequation models exploiting the invariant feature of generalizedfrequency response functions . The great advantage of thisapproach is that direct calculation of derivatives of the input-

output signals is avoided. However, in earlier works, emphasishas been given to noise free case and the the accuracy of theestimates is heavily dependent on the choice of the frequencyrange and the weighting parameter [5] [6]. Li and Billings[7] suggested a method based around instrumental variables toproduce unbiased estimates from the noisy frequency responsedata.

The present study introduces a new algorithm based ontotal least squares to obtain accurate estimates of the para-meters under noisy circumstances. This algorithm combinesthe feature of total least squares with the error reductionration test of Swain and Billings [5] to include significantterms into the model. The estimation procedure is progressivewhere the terms associated with each order of nonlinearity ineach subsystem is estimated beginning with the estimation oflinear terms , followed by estimation of quadratic nonlinearterms and so on. The performance of the algorithm has beencompared with the weighted orthogonal estimator [5] atdifferent levels of noise considering an example of a coupledDuffing’s oscillator and has been shown to be superior.

The paper is organized as follows. Section-II introducesthe total least squares with structure selection algorithm.In section-III, the Volterra modelling of MIMO system isdiscussed. The estimation procedure to fit continuous timemodels for each subsystem is described in section-IV. Resultsof simulation are included in section-V to demonstrate theeffectiveness of the proposed algorithm.

II. TOTAL LEAST SQUARES WITH STRUCTURE SELECTION

Since the structure selection of the TLSSS algorithm is doneusing an error reduction ratio (ERR) test which is a byproductof orthogonal least squares algorithm [8] [5], it is appropriateto introduce briefly about the ERR test and orthogonal leastsquares.

A. Orthogonal Least Squares with Structure Selection

Consider a system which can be modeled as

z(jω) =M∑i=1

θipi(jω) + ξ(jω) (1)

where θi,i = 1, ......M are the real unknown deterministic pa-rameters of the system associated with the complex regressorspi(jω),i = 1, ....M . z(jω) is a complex dependent variable or

1–4244–0342–1/06/$20.00 c© 2006 IEEE ICARCV 2006

the term to regress upon and ξ(jω) represents the modelingerror. If ‘N’ measurements of z(jω) and pi(jω) are availableat ωi, i = 1, .....N , the complex system of equation (1) canbe represented in matrix form (after partitioning into real andimaginary parts) as

Z = Pθ + Ξ (2)

where P is an 2N × M matrix of known regressors and Ξ =2N × 1 real vector.. The matrix ‘P’ will be referred to as theinformation matrix. By using QR factorization, matrix P canbe decomposed into

P = QR (3)

where R is an M × M triangular matrix with 1’s on thediagonal and 0’s below the diagonal and Q is an 2N × Mmatrix with orthogonal columns qi such that, QT Q = D andthe diagonal elements of D are given as

di = qTi qi =

2N∑j=1

qi(ωj)T qi(ωj), 1 ≤ i ≤ M (4)

Since the space spanned by the set of orthogonal basis vectorsqi is the same space spanned by the original set of vectors pi,itis possible to represent equation (2) by the auxiliary equation

Z = Qg + Ξ (5)

The orthogonal least squares solution g can be calculated from

g = D−1QT Z or,

gi =qTi Z

qTi qi

, 1 ≤ i ≤ M (6)

The original parameters θ can be recovered by solving thetriangular system

Rθ = g (7)

Multiplying equation (5) by itself and taking the time averagegives

1N

ZT Z =1N

M∑i=1

g2i qT

i qi +1N

ΞT Ξ (8)

The output energy consists of two terms; the first term is thepart of output energy explained by the regressors whereas thesecond term accounts for the unexplained energy. Due to theorthogonal nature of qis, the contribution that each term makesto the output energy can be computed independently as g2

i qTi qi.

Expressing this as a fraction of the overall output energy givesthe Error Reduction Ratio (ERRi)

[ERR]i =giq

Ti qi

ZT Z, 1 ≤ i ≤ M (9)

Thus the value of ERR indicates the significance of a candidateterm and is used to include relevant terms into the modelin accordance with their contribution to the overall outputvariance (energy). Normally at the beginning all availablecandidate terms are examined and the term which contributesthe maximum ERR is included in the model. This is repeateduntil all candidate terms have been exhausted or the sum ofERR approaches 100 percent.

B. Total Least Squares Solution

Consider the system of equation (2). Most of the standardleast squares based algorithm assume that the output (depen-dent) variable Z contains error and the regressor matrix Pis error free. However, as it will be shown in the followingsection, the regressors in the frequency domain formulationare also corrupted by noise. The total least squares algorithm[9] estimates the parameters in such cases by minimizing thefollowing Frobenius norm :

minimize||∆P ; ∆Z||F (10)

where ∆P and ∆Z represent errors in P and Z respectively.The algorithm uses singular value decomposition and is com-bined with the ERR test to yield TLSSS [10]:

C. TLS solution with Structure Selection

Given P ∈ RN×M and Z ∈ RN , the following algorithmcomputes the total least squares estimate of θ denoted as θTLS

such that equation (10) is satisfied. The relevant terms areselected using the ERR test.

1) Apply the orthogonal least squares or weighted orthogo-nal least squares algorithm to select the significant terms.

2) Apply SVD algorithm to the upper triangular part of thematrix [P (1 : M, 1 : M); Z(1 : M ] resulting from theQR factorization.

3) Save the matrix V where V is obtained from the SVDalgorithm

4) Determine the rank η5) Compute a Householder matrix Γ such that V = V Γ,

then V (M + 1, M − η + 1 : M) = 0if vM+1,M+1 �= 0for i = 1 : Mθi=−vi,M+1./vM+1,M+1

endend

III. VOLTERRA MODELLING OF MIMO SYSTEMS

Before formulating the estimation problem in the frequencydomain, the concept of self , cross and average kernel trans-form are introduced following [11]. The traditional methodof representing the input-output behaviour of a wide class ofsingle input single output (SISO) nonlinear systems is givenby the Volterra series [12].

y(t) =N∑

n=1

y(n)(t) (11)

where the n-th order output of the system y(n)(t) is given by

y(n)(t) =∫ ∞

−∞. . .

∫ ∞

−∞hn(τ1, . . . , τn)

n∏i=1

u(t − τi)dτi, n > 0

(12)

where hn(τ1, . . . , τn) is called the n-th order Volterra kernelor generalized impulse response function of order n. Themultidimensional (n > 1) Fourier transform of the n-th order

impulse response yields the n-th order transfer function orgeneralized frequency response function(GFRF)

Hn(jω1, jωn)

=∫ ∞

−∞

∫ ∞

−∞hn(τ1, ., τn)e(−j(ω1τ1+.+ωnτn))dτ1, .dτn

(13)

Note that when n=1, equation (13) gives the linear transferfunction. When the system is excited with more than one input,the analysis becomes more complex.

The output of the ‘j1-th ’ subsystem of an r-input m-output system possessing nonlinearity up to degree Nl maybe expressed as

yj1(t) =Nl∑

n=1

y(n)j1

(t) (14)

where y(n)j1

(t) is the n-th order component of the output yj1(t).When the r-inputs are denoted as uβ1(t), . . . uβr(t) eqn(14)can be expressed as

yj1(t) =Nl∑

n=1

r∑β1=1

r∑β2=β1

. . .

r∑βn=βn−1

∫ ∞

−∞

. . .

∫ ∞

−∞h(j1:β1,...βn)

n (τ1, ..τn)

× uβ1(t − τ1)..uβn(t − τn)dτ1, ..dτn (15)

where h(j1:β1,...βn)n (τ1, . . . τn) is the n-th order Volterra kernel

of the j1th subsystem. The superscripts β1, ...βn in the kernelcorrespond to the inputs uβ1(t), . . . uβn(t) that take part inthe n-dimensional convolution with h(j1:β1,...βn)

n (τ1, . . . τn).The kernels are called self-kernels when all the superscriptsβ1, . . . . . . βn in h(j1:β1,...βn)

n (τ1, . . . τn) are equal; for exampleh(j1:11...1)

n (τ1, . . . τn) is the n-th order self kernel of the systemcorresponding to the inputs u1(t), otherwise they are calledcross-kernels [11].

Consider the example of a two-input system. The output ofthe j1th subsystem of this system which includes only kernelsup to degree 2 is given by

yj1(t) =∫ ∞

−∞h

(j1:1)1 (τ1)u1(t − τ1)dτ1

+∫ ∞

−∞h

(j1:2)1 (τ1)u2(t − τ1)dτ1

+∫ ∞

−∞

∫ ∞

−∞h

(j1:11)2 (τ1, τ2)

× u1(t − τ1)u1(t − τ2)dτ1, dτ2

+∫ ∞

−∞

∫ ∞

−∞[h(j1:12)

2 (τ1, τ2) + h(j1:21)2 (τ2, τ1)]

× u1(t − τ1)u2(t − τ2)dτ1, dτ2

+∫ ∞

−∞

∫ ∞

−∞h

(j1:22)2 (τ1, τ2)

× u2(t − τ1)u2(t − τ2)dτ1, dτ2 (16)

The self-kernels of MIMO system is symmetric i.e.h

(j1:11)2 (τ1, τ2) = h

(j1:11)2 (τ2, τ1) and the symmetric self-

kernel is defined as

2!h(j1:11)2 (τ1, τ2) =

[h

(j1:11)2 (τ1, τ2) + h

(j1:11)2 (τ2, τ1)

](17)

However, the cross-kernels of MIMO system is not symmetrici.e h

(j1:12)2 (τ1, τ2) �= h

(j1:12)2 (τ2, τ1). Analogous to symmetric

self-kernel, an average cross-kernel is defined as

2!h(j1:12)2avg

(τ1, τ2) =[h

(j1:12)2 (τ1, τ2) + h

(j1:21)2 (τ2, τ1)

](18)

The multi-dimensional Fourier transform of these kernels givescorresponding transforms.

IV. PROBLEM FORMULATION

The procedure of computing the generalized frequencyresponse function matrix (GFRFM) has been detailed in [11].This procedure is recursive where each lower order GFRFMcontains no effects from higher-order terms. This featureis exploited for parameter estimation where the parameterscorresponding to each degree of nonlinearity can be estimatedindependently beginning with estimation of parameters oflinear terms and then the parameters of quadratic nonlinearterms and so on. The estimation process proceeds as per thefollowing steps :

1) Obtain the GFRFM of the system2) Estimate the parameters associated with linear terms

by utilizing information in the first order GFRFM i.eGFRFM (1)

3) Estimate the parameters of quadratic nonlinear termsutilizing information in the second order GFRFM(GFRFM (2)) and so on

The procedure is demonstrated considering the example ofa two-input two-output TITO quadratically nonlinear systemdescribed by

a1y1(t) + a2y1(t) + y1(t) + b1y2(t) + b2y2(t)+c1y

21(t) + c2y

22(t) + c3y1(t)y2(t) = d1u1(t) (19)

α1y2(t) + α2y2(t) + y2(t) + β1y1(t) + β2y1(t)+γ1y

21(t) + γ2y

22(t) + γ3y1(t)y2(t) = δ1u2(t) (20)

(21)

A. Estimation of First and Second Order GFRFM

The first and second order GFRFM of this system consistsof

GFRFM (1) =

[H

(1:1)1 (jω1) H

(1:2)1 (jω1)

H(2:1)1 (jω1) H

(2:2)1 (jω1)

](22)

GFRFM (2)

=

[H

(1:11)2 (jω1, jω2) H

(1:12)2 (jω1, jω2) H

(1:22)2 (..)

H(2:11)2 (jω1, jω2) H

(2:12)2 (jω1, jω2) H

(2:22)2 (..)

](23)

The first order GFRFM (1) is computed following [13] [11]which gives[

H(1:1)1 (jω1) H

(1:2)1 (jω1)

H(2:1)1 (jω1) H

(2:2)1 (jω1)

]

=[

a1(jω1)2 + a2(jω1) + 1 b1(jω1) + b2

β1(jω1) + β2 α1(jω1)2 + α2(jω1) + 1

]−1

(24)

B. Estimation of Cross-Kernel Transform or Average CrossKernel Transform

From GFRFM(2) in equation(23),the first and thirdcolumns are the direct kernel transforms and the calculation ofthese are similar to the calculation of the GFRF of a SISO sys-tem [13]. But the estimation of second column of GFRFM(2)

which consists of H(1:12)2 (jω1, jω2) and H(2:12)

2 (jω1, jω2) isslightly different. This is done as followsSplit the two-tone input ejω1t + ejω2t as

u1(t) = ejω1t and u2(t) = ejω2t (25)

The output of the first and second subsystem then becomes

y1(t) = H(1:1)1 (jω1)ejω1t + H

(1:2)1 (jω2)ejω2t

+ [H(1:12)2 (jω1, jω2) + H

(1:21)2 (jω2, jω1)]︸ ︷︷ ︸

2!H(1:12)2avg

(jω1,jω2)

ej(ω1+ω2)t

+terms of repetitious combinations of frequencies

y2(t) = H(2:1)1 (jω1)ejω1t + H

(2:2)1 (jω2)ejω2t

+ [H(2:12)2 (jω1, jω2) + H

(2:21)2 (jω2, jω1)]︸ ︷︷ ︸

2!H(2:12)2avg

(jω1,jω2)

ej(ω1+ω2)t

+terms of repetitious combinations of frequencies(26)

Substituting the values of y1(t), y2(t), u1(t) and u2(t)in the system equation and extracting the coefficients ofej(ω1+ω2)t yields[

d11 d12

d21 d22

] [2!H(1:12)

2avg(jω1, jω2)

2!H(2:12)2avg

(jω1, jω2)

]=

[n11

n22

](27)

where

d11 = a1(jω1 + jω2)2 + a2(jω1 + jω2) + 1,

d12 = b1(jω1 + jω2) + b2

d21 = β1(jω1 + jω2) + β2 andd22 = α1(jω1 + jω2)2 + α2(jω1 + jω2) + 1

n11 = −2c1

[H

(1:1)1 (jω1)H

(1:2)1 (jω2)

]−2c2

[H

(2:1)1 (jω1)H

(2:2)1 (jω2)

]−c3

[H

(1:1)1 (jω1)H

(2:2)1 (jω2)

+H(1:2)1 (jω2)H

(2:1)1 (jω1)

](28)

n22 = −2γ1

[H

(1:1)1 (jω1)H

(1:2)1 (jω2)

]−2γ2

[H

(2:1)1 (jω1)H

(2:2)1 (jω2)

]−γ3

[H

(1:1)1 (jω1)H

(2:2)1 (jω2) + H

(1:2)1 (jω2)H

(2:1)1 (jω1)

](29)

After computing the GFRFM, the next issue is to find if it ispossible to relate the unknown parameters of the model to theelements of GFRFM. In the following it will be shown thatthe relation between the unknown parameters and elements ofGFRFM can be put in a linear-in-parameters framework suchthat these can be estimated using standard least squares basedtechniques.

C. Estimation of the Linear Terms

Step-1.1The frequency domain equivalent of the system of eqn(19)corresponding to the first column of GFRFM(1) is given by

[a1(jω1)2 + a2(jω1)1 + 1]H(1:1)1 (jω1)

+[b1(jω1) + b2]H(2:1)1 (jω1) = d1 (30)

This can be written as

−H(1:1)1 (jω1) = a1(jω1)2H

(1:1)1 (jω1) − a2(jω1)1H

(1:1)1 (jω1)

−b1(jω1)1H(2:1)1 (jω1) − b2H

(2:1)1 (jω1) + d1

(31)

The parameters a1, a2, b1, b2 and d1 can be solved using leastsquares based techniques by inserting values for the frequencyresponse function H(1:1)

1 (jω1) and H(2:1)1 (jω1)in eqn(31)

Step-1.2 :Similarly the parameters α1, α2, β1, β2 and δ1 of subsystem-2 can be estimated from the frequency domain equivalentcorresponding to the second column of the GFRFM(1)

−H(2:2)1 (jω1) = α1(jω1)2H

(2:2)1 (jω1) − α2(jω1)1H

(2:2)1 (jω1)

−β1(jω1)1H(1:2)1 (jω1) − β2H

(1:2)1 (jω1) + δ1

(32)

D. Estimation of the Second Order Nonlinear Terms

Note that different terms of the differential equation modelcan be divided into three parts

1) Pure Output Nonlinear terms.2) Pure Input Nonlinear terms.3) Input-Output Cross Product terms

The pure output nonlinear terms contribute to all the self andcross kernel transforms. However, pure input nonlinear termsand input-output cross product terms do not contribute to allkernel transforms. As an example for TITO quadratic nonlinearsystem, the input-output cross product term (y1(t)u1(t)) donot contribute to last column of GFRFM (2). Similarly theterm (y1(t)u2(t)) do not contribute to the first column ofGFRFM (2). The details of contribution of different typesof terms can be found in [11]. In the present examplewhich contains only pure output nonlinear terms, any column

of GFRM (2) can be utilized for estimating the nonlinearparameters.

Considering the frequency domain equivalent correspondingto second column of GFRFM (2) the procedure of estimatingparameters of nonlinear terms are illustrated.

The frequency domain equivalent of the sub system-1corresponding to the second column of GFRFM(2) is givenby

[a1(jω1 + jω2)2 + a2(jω1 + jω2) + 1]

×2!H(1:12)2avg

(jω1, jω2)

+ [b1(jω1 + jω2) + b2]2!H(2:12)2avg

(jω1, jω2)

= −2c1

[H

(1:1)1 (jω1)H

(1:2)1 (jω2)

]−2c2

[H

(2:1)1 (jω1)H

(2:2)1 (jω2)

]−c3

[H

(1:1)1 (jω1)H

(2:2)1 (jω2)

+H(1:2)1 (jω2)H

(2:1)1 (jω1)

](33)

The parameters c1, c2 and c3 can be estimated from eqn(33)once the values of the H(.)(.) have been inserted.Note that theparameters a1, a2, b1, b2 associated with linear output termshave been estimated in step-1 and therefore their contributionto the cross kernel transform are brought to left hand side ofthe equation. Similarly the parameters of second subsystemwill be estimated from the equation

[α1(jω1 + jω2)2 + α2(jω1 + jω2) + 1]

×2!H(2:12)2avg

(jω1, jω2)

+ [β1(jω1 + jω2) + β2]2!H(1:12)2avg

(jω1, jω2)

= −2γ1

[H

(1:1)1 (jω1)H

(1:2)1 (jω2)

]−2γ2

[H

(2:1)1 (jω1)H

(2:2)1 (jω2)

]−γ3

[H

(1:1)1 (jω1)H

(2:2)1 (jω2)

+H(1:2)1 (jω2)H

(2:1)1 (jω1)

](34)

V. SIMULATED EXAMPLE

The effectiveness of the new estimator is demonstratedconsidering an example of multi degree freedom Duffing’soscillator whose dynamics are governed by [14].

y1 + 40y1 + 2 × 104y1 − 20y2 − 104y2 + (y1 − y2)2 = u1

y2 + 40y2 + 2 × 104y2 − 20y1 − 104y1 + (y1 − y2)2 = u2 (35)

The first order generalised frequency response function matrixof this system are obtained using the procedure detailed insection-IV. In order to reconstruct the linear part of system’sdifferential equation model corresponding to subsystem-1 and2, 100 equally spaced frequency response data were generatedfrom systems first order GFRFM in the frequency range of0-35 Hz. To demonstrate the structure selection feature of thealgorithm, an overparameterized (8th order) transfer functionwas initially specified. The algorithm, which follows closely

Terms θ θ θ θ θSNR=40 SNR=30 SNR=20 SNR=10

y1 1 0.9988 0.9984 1.0161 1.1773y1 40 40.3029 40.9343 42.6361 44.5635y2 -20 -20.3582 -21.0325 -22.1893 -18.4773y2 -1e4 -9993.5 -9956.5 -9644 -7611.2u1 1 1.0091 1.0262 1.0066 1.0091

TABLE I

PARAMETER ESTIMATES USING WEIGHTED ORTHOGONAL LEAST

SQUARES FOR SUBSYSTEM-1(WITH LAMDA=10000)

Terms θ θ θ θ θSNR=40 SNR=30 SNR=20 SNR=10

y1 1.0 0.9997 0.9986 0.9910 0.9385y1 40.0 39.9746 39.8457 38.7688 30.3664y2 -20.0 -20.0585 -20.2455 -20.5047 -19.2001y2 -1e04 -9.9963e03 -9.9931e03 -1.003e04 -1.004e04u1 1.0 1.0007 1.0016 0.9995 0.9563

TABLE II

PARAMETER ESTIMATES USING TOTAL LEAST SQUARES FOR

SUBSYSTEM-1

the structure selection feature of the weighted complex orthog-onal estimator [5], selected the correct model structure. Sincein practice, the frequency response data could be corruptedby noise, the performance of the estimator was studied andcompared with the orthogonal estimator by adding complexwhite Gaussian noise of varying SNR to the clean frequencyresponse data. The parameters associated with the linear termswere estimated using both total least squares and orthogonalleast squares. The error in the estimates are calculated as

ε =|θ − θ|

θ× 100 (36)

where θ and θ represents the true and estimated value of theparameters respectively. Due to limitations of space only theresults of the parameter estimates and associated errors forsubsystem-1 are summarized in Table-I,II, III and IV.From the tables it is obvious that as the noise level increases,performance of the orthogonal algorithm degrades where asthe total least squares algorithm performs better under noisyconditions. However,estimates of one of the parameter in TLSgives more error at SNR=10 compared to weighted orthogonalleast squares (WOLS). Note that performance of WOLS is

Terms ε ε ε εSNR=40 SNR=30 SNR=20 SNR=10

y1 0.12 0.16 1.61 17.73y1 0.7573 2.3358 6.5903 11.4088y2 1.791 5.1625 10.9465 7.6135y2 0.065 0.435 3.56 23.888u1 0.19 0.66 2.62 11.39

TABLE III

PERCENTAGE OF ERRORS IN PARAMETER ESTIMATES USING ORTHOGONAL

LEAST SQUARES FOR SUBSYSTEM-1 AT DIFFERENT LEVELS OF NOISE.

Terms ε ε ε εSNR=40 SNR=30 SNR=20 SNR=10

y1 0.0304 0.1422 0.9 6.1495y1 0.0636 0.3858 3.0779 24.084y2 0.4291 1.2277 2.5237 3.9993u1 0.0665 0.1564 0.0476 4.37y2 0.037 0.0688 0.2692 4.4308

TABLE IV

PERCENTAGE OF ERRORS IN PARAMETER ESTIMATES USING TOTAL LEAST

SQUARES FOR SUBSYSTEM-1 AT DIFFERENT LEVELS OF NOISE.

critically dependent on the choice of tuning parameter λ andthis has been carefully selected by trial and error to get thebest possible results under noisy conditions. The selection ofproper tuning parameter is not without difficulties. The TLSalgorithm, on the other hand is not dependent on any tuningparameter and therefore easy to implement. In order to improvefurther the performance of TLS algorithm, a weighted versionof TLS may be used which will be reported in future. Toreconstruct the nonlinear parts of the system the average kerneltransform was computed following the procedure in section-IV. 100 equally spaced frequency response data were used inthe frequency range of 0-30 Hz. An over parameterized modelof second order nonlinearity with 20 model terms were initiallyspecified. The proposed algorithm detects the correct modelstructure. The errors in the parameter estimates for subsystem-1 is shown in Table-V and VI. The results for subsystem-2 issimilar and therefore not presented here.

Terms ε ε ε εSNR=40 SNR=30 SNR=20 SNR=10

y21 4.94 4.66 5.59 24.2

y1y2 0.4759 0.13 0.75 15.765y22 5.89 6.28 6.08 8.74

TABLE V

PERCENTAGE OF ERRORS IN PARAMETER ESTIMATES OF NONLINEAR

TERMS OF SUBSYSTEM-1 AT DIFFERENT LEVELS OF NOISE USING

ORTHOGONAL LEAST SQUARES.

Terms ε ε ε εSNR=40 SNR=30 SNR=20 SNR=10

y21 2.7676 2.4379 0.0367 22.5082

y1y2 0.6923 1.0058 3.139 10.7275y22 5.4575 5.7523 5.75778 6.5168

TABLE VI

PERCENTAGE OF ERRORS IN PARAMETER ESTIMATES OF NONLINEAR

TERMS OF SUBSYSTEM-1 AT DIFFERENT LEVELS OF NOISE USING TOTAL

LEAST SQUARES.

VI. CONCLUSIONS

A new algorithm for estimating continuous time nonlin-ear differential equation models using data from generalizedfrequency response function matrix have been introduced. The

structure selection property of the algorithm helps to determinethe correct model structure i.e. which terms to include intothe model. The method does not require differentiation of theinput-output data. Moreover, nonlinear models are fitted foreach subsystem sequentially by building in the linear modelterms followed by quadratic terms and so on. The performanceof the algorithm has been compared with the weighted complexorthogonal estimator considering an example of multi degreefreedom Duffing’s oscillator and has been shown to give betterparameter estimates under the effect of significant noise.

ACKNOWLEDGEMENTS

Akshya Swain acknowledges that part of this work wassupported by University of Auckland grant ref 3605678/9273. E. Mendes acknowledges the support of CNPq under thegrant 301313/96-2.

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