Fuzzy c-regression models based on the BELS method for nonlinear system identification

Borhen Aissaoui(1), Moêz Soltani(1)

(1) Research Unit on Control, Monitoring and Safety of Systems (C3S)

(1) High School of Sciences and Engineering of Tunis (ESSTT), Tunisia

{borhen.issaoui, soltani_c3s@yahoo.fr}

Dorsaf Elleuch(2), Abdelkader Chaari(1) (2) Laboratory of Sciences and Techniques of Automatic

control & computer engineering (Lab-STA) (2) National Engineering School of Sfax (ENIS)

Sfax, Tunisia dorsafelleuch@yahoo.fr, assil.chaari@esstt.rnu.tn

Abstract—A fuzzy c-regression model clustering algorithm based on Bias-Eliminated Least Squares method (BELS) is presented. This method is designed to develop an identification procedure for noisy nonlinear systems. The BELS method is used to identify consequent parameters and eliminate the bias. The proposed approach has been applied to benchmark modeling problem which proved a good performance.

Keywords—Takagi-Sugeno fuzzy model; fuzzy c-regression models; Bias-Eliminated Least-Squares.

I. INTRODUCTION Fuzzy modeling algorithms are studied in many research

areas. It is known as one as better describing nonlinear systems. Takagi and Sugeno (T-S) system is a most fuzzy model studied in literature [19]. In [6], the authors present a modified version of Fuzzy C-Means (FCM) clustering algorithm called fuzzy C-Regression Models (FCRM). The FCRM algorithm develops hyper-plane-shaped clusters, while FCM algorithm develops hyper-spherical-shaped clusters. However, in [11], the authors prove that the FCRM using the Weighted Regressive Least Square (WLRS) algorithm made it sensitive to initialization. Hence, different initializations may lead, easily, to different results. To overcome this problem, the gradient descent algorithm was used to adjust the parameters of fuzzy models [9]. Recently, the orthogonal least squares (OLS) is used to identify the consequent parameters [11- 12].

The affine T-S fuzzy model structure architecture is frequently applied to analysis and synthesis of nonlinear systems ([18], [8]). The (T-S) model has been widely studied without constant parameter ([20], [22]). In [25], the authors shown that the model is assumed as a quasi-linear model and the stability analysis ensured, if the scalar term is eliminated.

In this paper, a nonlinear system modeling with T-S fuzzy models without taking into account the constant term in consequent parameter is considered. The BELS method is used to identify the consequent parameters of local linear model. The BELS method has a robust capability against noise and is quite successful in improving the robustness of a variety of modeling problem. In other word, the BELS method is applicable to arbitrary colored disturbances acting on the plant ([14-15], [26]).

This paper is organized as follows. In Section 2, fuzzy c-regression models clustering algorithm is briefly introduced. In Section 3, the proposed method is given. Some example is given to illustrate the proposed approach. Finally, the conclusion is presented in Section 5.

II. FUZZY C-REGRESSION MODELS CLUSTERING ALGORITHM

The affine T-S fuzzy model based on FCRM belongs to the range of clustering algorithms with linear prototype.

Let S={(x1, y1), …, (xN, yN)}={(xk, yk), k=1, …,N} be a set of input-output sample data pairs. Assume that the data pairs in S are drawn from c different fuzzy regression models, the hyper-plane-shaped cluster representative will be adopted has the following structure:

01 1 2 2

1

( , ) ( )..

( )

[ . ( ), 1, 2,....,]

k i k i ik i

i k i k iM kM

ik

i

iT

k i ik i

y f Ea x a x a

E

i c

bx

E

θ θ

θ

θ θ

= += + + +

= + =

++

x

x

(1)

where 1,...., Mk k kMx x ∈ℜ⎡ ⎤⎣ ⎦=x is the input vector, c is number

of rules and 11,..., , 0

Mi i iM ia a bθ +⎡ ⎤= ∈ℜ⎣ ⎦ is the parameter

vector of the corresponding local linear model. The distances (Eik(θi)) are weighted with the membership

values µik in the objective function that is minimized by the clustering algorithm and formulated as:

2

1 1( , ) ( ) ( ( ))

N cm

ik ik ik i

J Eμ= =

=∑∑U θ θ (2)

with

( ) | . |Tik i k k iyE = − xθ θ (3)

where m is the weighting exponent and µik is the membership degree of xk to ith cluster. The membership values µik have to satisfy the following conditions:

U.S. Government work not protected by U.S. copyright

1

1

[0 1]; 1, 2,...., ; 1, 2,....,

0 ; 1, 2,....,

1; 1, 2,....,

ikN

ikk

c

iki

i c k N

N i c

k N

μ

μ

μ

=

=

∈ = =

< < =

= =

∑

∑

(4)

The identification procedure of the FCRM algorithm is summarized as follows [6]. Given data S, set m>1 and specify regression models Equation (1), choose a measure of error Equation (3). Pick a termination threshold ε�0 and initialize U(0) (e.g. random). Repeat for l=1, 2, ...

Step 1. Calculate values for c model parameters θi(l) in

Equation (1) that globally minimize the restricted function Equation (2).

Step 2. Update U(l) with Eik(θi(l)), to satisfy:

21( )

1

1 0 1

0

ik

mcl ikikj jk

if E for i c

EE

otherwise

−

=

⎧ > ≤ ≤⎪⎪ ⎛ ⎞⎪= ⎨ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎪⎪⎩

∑U (5)

Until ( ) ( )1l l ε−− ≤U U then stop; otherwise set l=l+1 and

return to step 1.

III. THE NEW FCRM CLUSTERING ALGORITHM Many authors have shown that the clustering performance

severely distorted when the noisy data is presented ([24], [13], [3], [10]). To overcome this problem, noise clustering (NC) algorithm is widely used ([11], [12]). In this approach, noise is considered as a separate class. It is represented by a fictitious prototype that has a constant distance δ from all the data points. The membership µik of point xk in the noise cluster is given by:

*1

1c

k iki

μ μ=

= −∑ (6)

Thus, the membership constraint for the good clusters is effectively relaxed to:

1

1c

iki

μ=

<∑

Dave's objective function is given by :

( ) ( ) ( ) ( )2 2*

1 1 1

,c N N

m mNC ik ik k

i k k

J U V Dμ δ μ= = =

= +∑∑ ∑ (7)

for any input xk in subspace i denoted by center vi, ik ikD = −x v . The combination of noise clustering algorithm with FCRM

algorithm can lead to New FCRM (NFCRM) objective function as follows [17] :

( ) ( ) ( )

( )

2

1 1

2*

1

; , ( )N c

mnew ik ik i

k iN

mk

k

J S U Eθ μ

δ μ

= =

=

=

+

∑∑

∑

θ (8)

Where δ is a scale parameter and may be used on the idea presented in [4] as:

2 2

1 1

1 ( ( )).

N c

ik ik i

Ec N

δ γ= =

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠∑∑ θ (9)

where γ is a user-defined parameter depending on the example’s type.

To solve constrained problem Jnew with respect to µik, we introduce N Lagrange multipliers λk,,k=1,…,N. The minimization of Jnew is as follows:

( ) *1 1

, 1N c

ik k new k ik kk i

F Jμ λ λ μ μ= =

⎛ ⎞= − + −⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑ (10)

By differentiating the Lagrangian with respect to the µik, µ*k and λk and setting the derivates to zero, we obtain:

( ) ( )1 2/ 0

m

ik ik ik kF m Eμ μ λ−

∂ ∂ = − = (11)

( ) 12* */ 0mk k kF mμ δ μ λ−∂ ∂ = − = (12)

*1

/ 1 0c

k ik ki

F λ μ μ=

∂ ∂ = + − =∑ (13)

From Equations (11-12), we get:

( )

1/( 1)1/( 1)

21

mmk

ikik

m E

λμ−− ⎡ ⎤⎡ ⎤

⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(14)

and 1/( 1) 1/( 1)

* 21

m mk

k mλμ

δ

− −⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦ (15)

Using Equations (13-15), we get:

( ) ( )

1/( 1)

2/( 1) 2/( 1)

1

1

1/ 1/

mk

c m mjk

j

mE

λ

δ

−

− −

=

⎡ ⎤ =⎢ ⎥⎣ ⎦

+∑ (16)

And then substituting it into Equation (16), the following equation can be obtained:

2 21 1

1

1ik

c m mik ik

jkj

E EE

μ

δ

− −

=

=⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

∑

(17)

From Equations (1-8), the objective function of NFCRM is defined as:

2

1 1

2

1 1

1

1

( ) ( )

(

ˆ

1 )

N cm T

new ik k ik i

M

kj

N cm

ikk i

k

xJ yμ

δ μ

= =

= =

+

=

−

= −

+

∑∑ ∑

∑ ∑

θ (18)

where ,ˆ 1k kx = ⎡ ⎤⎣ ⎦x . The partial derivation of the object Equation (18) is:

1

1 1

ˆ ˆ2 ( ) 0N M

mnewik k it kt kj

ij k t

Jy x xμ θ

θ

+

= =

⎛ ⎞∂= − − =⎜ ⎟⎜ ⎟∂ ⎝ ⎠∑ ∑ (19)

and then,

1

21

ˆ ˆ( ) ( )

ˆ( )

Nm

ik k it kt kjk t j

ij N mik kjk

y x x

x

μ θ

θμ

= ≠

=

−

=∑ ∑

∑ (20)

1,2,...., ; 1,2,...., 1i c j M= = + As mentioned in [23], the non-Euclidean distance is more

robust than the Euclidean distance. By transforming Equation (3), the measure of error is defined as:

E ( ) 1 ( | [ 1]. |)Tik i k k iexp yρ= − − − xθ θ (21)

where ρ is a positive constant. Then the NFCRM objective function Equation (8) is rewritten as follows:

2

1 1

2*

1 1

J ( ; , ) ( ) (E ( ))

(1 )

N cm

new ik ik ik i

N cm

ikk i

S U θ μ

δ μ

= =

= =

=

+ −

∑∑

∑ ∑

θ (22)

Using Equations (9) and (17) can be respectively rewritten as:

2 21 1

1

1

E E( ) ( )E

ik cik ikm m

jkj

μ

δ− −

=

=

+∑ (23)

and 2 2*

1 1

1 (E ( )).

N c

ik ik ic N

δ γ= =

= ∑∑ θ (24)

NFCRM Algorithm [17]: Given data S, set m>1. Fix γ>0, ρ>0 and the parameter

vectors θi at random. Pick a termination threshold ε>0 and initialize U(0).

Repeat for l=1,2,... Step 1. Compute measure of error

Eik(θi

l) via Equation (21) Step 2. Calculate 2

*δ via Equation (24).

Step 3. Compute ( )likμ and ( )l

ijθ via Equations. (23) and (20), respectively.

Step 4. Compute ( ) ( 1)|| ||l lerr U U −= − . Until err ε≤ then stop; otherwise set l=l+1 and return to

step 1.

A. Estimation of consequent parameters by the BELS method The novel fuzzy c-regression models for decomposition of

the input-output space into multiple linear structures is used. Gaussian membership functions are usually chosen to represent the fuzzy sets in the premise part of each fuzzy rule. The premise parameters can be easily obtained using µik. The

fuzzy sets centers vik and the standard derivation σik are calculated as follows:

1

1

, 1,2,..., ; 1,2,...,

Nik kjk

ij Nikk

xi c j M

μν

μ=

=

= = =∑∑

(25)

and 2

1

1

2 ( )N

ik kj ijkij N

ikk

xμ νσ

μ=

=

−=

∑∑

(26)

Once the antecedent parameters have been fixed, the BELS method can be applied to estimate the consequent parameters for each rule. Using BELS algorithm, the consequent parameters are estimated by transforming model Equation (1) into an equivalent auxiliary model:

0( ) ( ) ( )Ty k k kφ ω= Θ + (27)

with 1[ ,...., ]TMθ θΘ = .

We introduce a (2 ) 1m n+ × auxiliary parameters vector Θ and a (2 ) 1m n+ × auxiliary regression vector as follows:

θα⎡ ⎤Θ = ⎢ ⎥⎣ ⎦

, ( )

( )( )k

kk

ϕφ

ρ⎡ ⎤

= ⎢ ⎥⎣ ⎦

(28)

where [ ]( ) ( 1) . . . ( 2 )T k y k m y k mρ = − − − and

[ ]( ) 0 . . . 0 mkα = ∈ℜ Then, it is easy to verify the following equality:

( ) ( )T Tk kφ ϕ θΘ = (29) Thus, the model can be represented by an auxiliary model equivalent to Equation (27).

The auxiliary parameter vector can be expressed by: 1ˆ ( )LS yN R Rφφ φ

−Θ = (30)

with the auto-correlation matrix Rφφ and cross-correlation matrix yRφ are:

( ) ( )TR E k kφφ φ φ⎡ ⎤= ⎣ ⎦ (31)

[ ]( ) ( )yR E k y kφ φ= (32)

Using Equation (30) ˆLSΘ is rewritten as follows:

0

1ˆ ( )LS N R Rφφ φω−Θ = Θ +

(33) With

0

1Bias R Rϕϕ ϕω−= −

The noise covariance matrix 0

Rφω(2 ) 1m n+ ×∈ℜ can be

expressed by:

0 0yR DRφω ω= (34)

with (2 ) 1

0m n

yR ω+ ×∈ ℜ is the matrix of cross-correlation

between the output and the noise and (2 )0T m m n

mD I × +⎡ ⎤= ∈ ℜ⎣ ⎦ .

By using the BELS method, the solution of Equation (27) is given by the following expression:

0

1( ) ( ) ( )BELS LS yN N R DR Nϕϕ ωθ θ −= − (35)

with [ ]0 0( ) ( )yR E y k kω ω= : noise covariance vector.

Under condition which the noise estimates covariance vector

0yR ω is attainable in some manner. The BELS algorithm is described as follows:

step 1. Initialize the parameters vector 0BELSθ =

step 2. From the sampled input-output data { }( ), ( ), 1u k y k N ≥ calculate the estimates of the covariance matrices and vectors:

1

1ˆ ( ) ( ) ( )N

T

kR N k k

Nϕϕ ϕ ϕ=

= ∑

1

1ˆ ( ) ( ) ( )N

T

k

R N k kNϕρ ϕ ρ

=

= ∑ (36)

1

1ˆ ( ) ( ) ( )N

yk

R N k y kNϕ ϕ

== ∑

1

1ˆ ( ) ( ) ( )N

yk

R N k y kNρ ρ

== ∑

step 3. Estimate the parameter vector θ by LS method using

Equation (30) step 4. Compute the estimate of the noise covariance

0yR ω

( )( )

0

11ˆ ˆ ˆ( ) ( ) ( )

ˆˆ ˆ. ( ) ( ) ( )

Ty

TLS y

R N R N R N D

R N N R N

ω ϕρ ϕϕ

ϕρ ρθ

−−=

− (37)

step 5. Calculate the BELS estimate of the system parameter vector θ

10

( ) ( ) ( )BELS LS yN N R DR Nϕϕ ωθ θ −= − (38)

IV. EXPERIMENTAL RESULTS In this section, the performance algorithm is tested. Mean square error MSE was used as a performances index, which is defined as:

( )2

1

1 ˆN

k kk

MSE y yN =

= −∑ (39)

A. Benchmark problem

Considering a nonlinear system given by equation Equation (40) [1]

11 2 2

1

( 2) ( 2.5)v

8.5k k k

k k kk k

y y yy u

y y−

+−

+ += + +

+ + (40)

where yk is system output, uk is input which is uniformly distributed in the interval [-1 1] and vk is white noise with zero mean and variance σ2, which is added to the output system at different Signal-to-Noise Ratio (SNR) levels.

Choosing SNR as follow: SNR=1 dB, 5 dB, 10 dB, 15 dB and 20 dB.

The simulations results are given for two experimental cases. The training data set contains 500 samples for input-output pairs while for the testing 1000 data pairs are generated by the following input signal:

2sin 500,250

2 20.3sin 0.1sin .250 25

k

k if ku

k k otherwise

π

π π

⎧ ⎛ ⎞ <=⎜ ⎟⎪⎪ ⎝ ⎠= ⎨⎛ ⎞ ⎛ ⎞⎪ +⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎩

(41)

Tables I-VI present results obtained with different algorithms such as: Gustafson-Kessel (GK) [5], NFCRM algorithm (NFCRMA) [11], FCM [7], fuzzy model identification (FMI) clustering algorithm [2] and NFCRM1 [16], et NFCRM2 [17].

Choosing {y(k -1), y(k - 2), u(k), u(k - 1)} as output-input variables, and the number of fuzzy rules is four. The parameter settings are: γ=0.1 and ρ= 0.1 for NFCRM_BELS. MSETr and MSETs are the MSE for training and testing data, respectively.

In Case 1, Comparing results with those cited above with regard to the noisy data. Table I shows that the best MSE is obtained by our method during training and test phases. In addition, the consequent parameters are obtained without constant terms unlike to other approaches mentioned in these tables. This shows that our model is better than others techniques reported in literature taking into account the few number parameters for each rule.

TABLE I. COMPARISON RESULTS (CASE 1)

Algorithms MSETr MSETs

FCM 0.0090 0.2220

GK 0.0046 0.1347

FMI 0.0013 0.0181

NFCRMA 5.20 e-4 0.0096

NFCRM1 4.76 e-4 0.0052

NFCRM2 3.94e-4 0.0045

NFCRM_BELS 2.65e-4 0.0028

In Case 2, the noise influence is analyzed with different SNR levels (SNR=1, 5, 10, 15 and 20db). The parameter settings are: γ=0.1 and ρ=1 for the NFCRM_BELS algorithm. As shown in Tables II–VI, we can note that, whatever the noise level is, only the proposed algorithm retained good performance. Consequently, our algorithm is more resistant to noise compared with other existing algorithms in the literature.

TABLE II.

COMPARISON RESULTS WITH SNR=20DB

Algorithms MSETr MSETs

FCM 0.0285 0.2417

GK 0.0258 0.1867

FMI 0.0134 0.0802

NFCRMA 0.0133 0.0621

NFCRM1 0.0126 0.0473

NFCRM2 0.0116 0.0442

NFCRM_BELS 0.0093 0.0276

TABLE III. COMPARISON RESULTS WITH SNR=15DB

Algorithms MSETr MSETs

FCM 0.0533 0.3164

GK 0.0471 0.2212

FMI 0.0373 0.1110

NFCRMA 0.0363 0.0806

NFCRM1 0.0355 0.0786

NFCRM2 0.0342 0.0770

NFCRM_BELS 0.0256 0.0574

TABLE IV. COMPARISON RESULTS WITH SNR=10DB

Algorithms MSETr MSETs

FCM 0.1155 0.5505

GK 0.1100 0.3042

FMI 0.1068 0.2136

NFCRMA 0.1039 0.1926

NFCRM1 0.0963 0.1836

NFCRM2 0.0947 0.1762

NFCRM_BELS 0.0688 0.1130

TABLE V. COMPARISON RESULTS WITH SNR=5DB

Algorithms MSETr MSETs

FCM 0.4577 0.8167

GK 0.3364 0.8094

FMI 0.3356 0.4859

NFCRM 0.3309 0.4465

NFCRM1 0.3158 0.4276

NFCRM2 0.3094 0.4261

NFCRM_BELS 0.2605 0.3273

TABLE VI. COMPARISON RESULTS WITH SNR=1DB

Algorithms MSETr MSETs

FCM 2.1292 2.1640

GK 1.0079 1.3765

FMI 0.9171 1.1649

NFCRMA 0.9046 1.1395

NFCRM1 0.8505 0.9491

NFCRM2 0.8092 0.9194

NFCRM_BELS 0.8041 0.8934

V. CONCLUSION In this paper, a new fuzzy c-regression model clustering algorithm based on the BELS (NFCRM_BELS) method has been presented. The application of BELS method improves the robustness of the FCRM method in noisy data. The proposed fuzzy modeling approach is applied to benchmark modeling problem. The experimental results show that the NFCRM-BELS algorithm has better performance than other methods in terms of the accuracy and the compactness.

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