IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

4443

Extended Kalman Filter Based Learning Algorithm for Type-2 Fuzzy Logic Systems and Its Experimental Evaluation Mojtaba Ahmadieh Khanesar, Student Member, IEEE, Erdal Kayacan, Student Member, IEEE, Mohammad Teshnehlab, and Okyay Kaynak, Fellow, IEEE

Abstract—In this paper, the use of extended Kalman filter for the optimization of the parameters of type-2 fuzzy logic systems is proposed. The type-2 fuzzy logic system considered in this study benefits from a novel type-2 fuzzy membership function which has certain values on both ends of the support and the kernel, and uncertain values on other parts of the support. To have a comparison of the extended Kalman filter with other existing methods in the literature, particle swarm optimization and gradient descent-based methods are used. The proposed type-2 fuzzy neuro structure is tested on different noisy input–output data sets, and it is shown that extended Kalman filter has a better performance as compared to the gradient descent-based methods. Although the performance of the proposed method is comparable with the particle swarm optimization method, it is faster and more efficient than the particle swarm optimization method. Moreover, the simulation results show that the proposed novel type-2 fuzzy membership function with the extended Kalman filter has noise rejection property. Kalman filter is also used to train the parameters of type-2 fuzzy logic system in a feedback error learning scheme. Then, it is used to control a real-time laboratory setup ABS and satisfactory results are obtained. Index Terms—Antilock braking system (ABS), extended Kalman filter (EKF), feedback error learning (FEL), identification, type-2 fuzzy logic systems (T2FLSs), type-2 fuzzy neural network.

I. I NTRODUCTION

E

VEN WHEN an industrial control system has an accurate model, numerous uncertainties may arise due to the precision of the sensors, noise produced by the sensors, environmental conditions of the sensors and the nonlinear characteristics of the actuators and sensors. In cases when the designer does not have an accurate model and/or the system has some uncertainties, model-free approaches are generally preferred. The most common approaches to model-free design in the literature are Artificial Neural Networks (ANNs) and Fuzzy Logic Systems

Manuscript received November 8, 2010; revised March 8, 2011; accepted April 28, 2011. Date of publication May 5, 2011; date of current version June 19, 2012. This work was supported by the Bogazici University Research Fund under Project 09HA203D. M. A. Khanesar and M. Teshnehlab are with the Faculty of Electrical Engineering, Control Department, K. N. Toosi University of Technology, Tehran 19697, Iran (e-mail: [email protected]; [email protected]). E. Kayacan and O. Kaynak are with the Department of Electrical and Electronics Engineering, Bogazici University, Istanbul 34342, Turkey (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIE.2011.2151822

(FLSs). It is to be noted that “fuzzy logic is a precise conceptual system of reasoning, deduction and computation in which the objects of discourse and analysis are, or are allowed to be, associated with imperfect information. Imperfect information is information which in one or more respects is imprecise, uncertain, incomplete, unreliable, vague or partially true” [1]. The combination of fuzzy systems and neural networks, Fuzzy Neural Networks (FNNs), have become a popular approach in engineering fields to solve control, identification, prediction, pattern recognition, etc. [2], [3] problems, because FNNs carry the advantages of both techniques simultaneously. Type-2 fuzzy sets have been introduced by Zadeh as the extension of type-1 fuzzy sets [4]. In type-1 fuzzy sets membership functions are totally certain, whereas in type-2 fuzzy sets membership functions are themselves fuzzy. The latter case results in that the antecedents and the consequents of the rules are uncertain. Mendel and Karnik have further developed the theory of type-2 fuzzy sets [5]. Type-2 fuzzy logic systems (T2FLSs) appear to be a more promising method than their type-1 counterparts for handling uncertainties such as noisy data and changing environments [6], [7]. In [8] and [9] the effects of the measurement noise in type-1 and type-2 fuzzy logic controllers and identifiers are simulated to perform a comparative analysis. It is concluded that the use of type-2 fuzzy logic controllers in real world applications [10] which exhibit measurement noise and modeling uncertainties can be a better option. Many different optimization methods are used over the years to estimate the parameters of type-1 and type-2 fuzzy models. These optimization methods can basically be put in two categories: derivative-based and derivative-free optimization methods. Genetic algorithm [11], particle swarm optimization (PSO) [12], singular value QR decomposition [13] and cell mapping [14] can be cited as some examples of derivative-free methods. On the other hand, examples of derivative-based optimization methods are gradient descent [15], simplex method [16], least square [17] and Extended Kalman Filter (EKF) [12]. Derivative-free methods are less likely to get entrapped in local minima. They are also easier to implement because they do not need derivatives which may be hard to calculate. Furthermore, they generally converge faster. Among derivativebased methods, the entrapment possibility in a local minima of second-order methods like Kalman filter (KF) is less than the first order methods.

0278-0046/$26.00 © 2011 IEEE

4444

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

EKF has previously been used to optimize the parameters of ANNs [18] and Type-1 Fuzzy Logic Systems (T1FLSs) [12], [19]. KF as a second-order gradient-based method can be a powerful tool to optimize the parameters of T2FLSs. It is known to be an optimal estimator for linear dynamical systems with white noise. Moreover, by linearizing the system around the current parameter estimate, it can be extended for nonlinear dynamical systems with colored noise [18]. In this paper, decoupled EKF is used to estimate the parameters of both the premise and the consequent parts of T2FLSs. Decoupled EKF is a variant of EKF which assumes that certain parameters interact with each other only at a second-order level. In this way, instead of having one group of parameters, we have some groups of parameters with small interaction between groups and the computational cost of EKF is greatly reduced. The results of applying EKF to the training of the T2FLSs are then compared with the first order gradient-based method of back-propagation and population based method of PSO for the premise part. The performance of all methods in the presence of measurement noise is also investigated. It is also shown that the performance of T2FLSs is better than the performance of type-1 counterparts in presence of higher levels of noise and uncertainty. The novelties of this paper are in the following aspects: First of all, a new type-2 fuzzy membership function is proposed. Second, EKF is used to optimize the parameters of both the premise and the consequent parts of T2FLSs for the first time. In the paper, KF is also used in a Feedback Error Learning (FEL) structure to control a real-time laboratory setup Antilock Braking System (ABS). This structure comprises a fixed controller to ensure the stability of the system and an adaptive intelligent feedforward controller in parallel to the fixed controller which improves the performance of the controller. The outputs of the fixed feedback controller is regarded as the error signal and it is used to train the inverse model of the plant [20]. From a control theoretic viewpoint FEL can be conceived of as an adaptive control technique. To date several implementations of FEL in industrial plants have been reported. In [20]–[22] FEL is used to train a neural network and is simulated on an inverted pendulum. In addition, to achieve a better performance, a variable learning rate is used to tune the parameters of neural networks. In [23] an enhanced FEL is used to control an n-degree of freedom robotic manipulator. In [24] FEL approach is used for load frequency control in interconnected power system. In [21], [22] KF is used to tune the parameters of neural networks in a FEL scheme. Stability analysis of FEL is considered in several papers. In [25] the stability of FEL for stable and stably invertible linear systems is proved. In [26] the stability property of FEL for a class of nonlinear dynamical systems is considered. The main body of the paper contains five sections: In Section II, the structure of a T2FLS with different type-2 fuzzy membership functions is discussed, and a novel type-2 fuzzy membership function is introduced. Section III introduces the parameter update rules of Gradient Descent (GD), KF and EKF methods used in this paper. To illustrate the applicability and the efficacy of the proposed EKF method, the prediction of the chaotic Mackey–Glass system and the identification of a ABS

Fig. 1. Type-2 fuzzy set with (a) uncertain standard deviation and (b) uncertain mean.

Fig. 2.

Type-2 fuzzy set with (a) uncertain width and (b) uncertain center.

are studied in Section IV and the simulation studies obtained are presented. KF is used to train the parameters of a T2FLS in a FEL scheme. Finally, in Section VI, the conclusions are given. II. S TRUCTURE OF T2FLS W ITH D IFFERENT F UZZY T YPE -2 M EMBERSHIP F UNCTIONS A. Existing Type-2 Fuzzy Membership Functions in the Literature There exist a number of type-2 fuzzy membership functions in the literature, i.e., triangular, Gaussian, trapezoidal, sigmoidal, pi-shaped, etc. Gaussian type membership functions are widely used in which uncertainties can be associated to the mean and the standard deviation. In Fig. 1(a) and (b), Gaussian type-2 fuzzy sets with uncertain standard deviation and uncertain mean are shown. The mathematical expression for the type-2 membership function is expressed as   1 (x − c)2 μ ˜(x) = exp − (1) 2 σ2 where c and σ are the center and the variance of the membership function, x is the input vector. In Fig. 2(a) and (b), triangular type-2 fuzzy sets with uncertain width and uncertain center are shown. The mathematical expression for the triangular type-2 membership function is expressed as   x−c   If |x − c| < d  (2) μ ˜(x) = 1 − d 0 otherwise

KHANESAR et al.: KALMAN FILTER BASED LEARNING ALGORITHM FOR TYPE-2 FUZZY LOGIC SYSTEMS

4445

Fig. 4. Structure of T2FLS with the proposed novel fuzzy type-2 membership function.

C. Structure of T2FLS Used in This Paper Fig. 3. Geometries of the proposed type-2 fuzzy membership function with different values for (a) a1 and a2 . a1 = a2 = 1, (b) a1 = 1.2 and a2 = 0.8, (c) a1 = 1.4 and a2 = 0.6.

where c and d are the center and the width of the membership function and they can be some uncertain values, x is the input vector. B. Novel Type-2 Fuzzy Membership Function: Ellipsoidal Membership Function Motivated by the fact that the biggest uncertainty in the membership value is usually in the mid range, a novel type-2 fuzzy membership function is introduced in this study. It has certain values on both ends of the support and the kernel, and some uncertain values on other values of the support. The mathematical expression for the novel type-2 membership function is expressed as  a  a1   , a2 < a < a1 If |x − c| < d (3) 1 −  x−c μ ˜(x) = d 0 otherwise where c and d are the center and the width of the membership function, x is the input vector. The parameters a1 and a2 determine the width of the uncertainty of the proposed membership function, to be selected as

The interval T2FLS considered used in this paper benefits from type-2 membership functions in the premise part and crisp numbers in the consequent part. This structure is called A2C0 fuzzy system [27] and it is shown in Fig. 4 for a 2-input, single output system with 3 memberships functions for each input (and therefore 9 rules). The rules are such that the consequent parts are TSK type. The ith rule of A2-C0 fuzzy system can be written as IF x1 is A˜j1 and x2 is A˜j2 and . . . and xn is A˜jn n  THEN uj = wij xi + bj

(5)

i=1

where x1 , x2 , . . . , xn are the input variables, uj (j = 1, . . . , M ) are the output variables, A˜ij represents interval type-2 fuzzy sets for jth rule of the ith input. wij and bj (i = 1, . . . , n, j = 1, . . . , M ) are the parameters in the consequent part of the rules. In Fig. 4, the function u(.) includes wij as the multiplication of the input values and bj as the bias values. The final output of the system can be written as [27]

M j j=1 f uj uT SK/A2−C0 = ... 1/ M (6) j j=1 f 1

f 1 ∈[f 1 ,f ]

f M ∈[f M ,f

M

]

j

a1 > 1

and

0 < a2 < 1.

(4)

Fig. 3(a)–(c) shows the geometries of the proposed membership function for a1 = a2 = 1, a1 = 1.2, a2 = 0.8 and a1 = 1.4, a2 = 0.6, respectively. As can be seen from Fig. 3(a), the shape of the proposed type-2 membership function is transformed into to a type-1 triangular membership function when its parameters are selected as a1 = a2 = 1. As the parameters responsible for the width of uncertainty (a1 and a2 ) and the parameters responsible for the center and the support of the proposed membership function (c and d, respectively) are decoupled from each other in the proposed type-2 membership function (3), it is possible to train them independently.

where f j and f are given by f j (x) = μA˜ (x1 ) ∗ . . . ∗ μA˜ (xn ) j1

j

jn

f (x) = μA˜j1 (x1 ) ∗ . . . ∗ μA˜jn (xn )

(7)

in which ∗ represents the t-norm which is the product operator in this study. The output of the fuzzy system in closed form is obtained by [27]

M j j=1 f uj uT SK/A2−C0 = M

M j j j=1 f + j=1 f

M j j=1 f uj + M (8)

M j j j=1 f + j=1 f

4446

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

M

uT SK/A2−C0 =

j j j=1 (f + f )uj

M j M j j=1 f + j=1 f

.

(9)

In this way, the firing of each rule is defined as follows: j

fj + f

M j . j j=1 f + j=1 f

rj = M

(10)

 ∂E ∂u ∂f j ∂μij ∂E = ∂a1ij ∂u ∂f j ∂μij ∂a1ij j

III. PARAMETER U PDATE RULES The design of T2FLS includes the determination of the unknown parameters that are the parameters of the antecedent and the consequent parts of the fuzzy If–Then rules. In this paper, EKF and KF are applied to the premise and the consequent parts, respectively. To show the effectiveness of Kalman filtering for T2FLS, back-propagation and PSO are also used to optimize the parameters of T2FLS, and their performances are compared. Four different learning methods that are used in this paper are briefly explained in the following subsections.

In this method, the cost function is defined as 1 (ud − u)2 2

(11)

where ud and u are the desired and the current output values of the network, respectively. The parameters wij , bj and a1ij , a2ij , cij and dij are adjusted using the gradient descent method as follows: wij (t + 1) = wij (t) − γ

∂E ∂wij

∂E bj (t + 1) = bj (t) − γ ∂bj cij (t + 1) = cij (t) − γ

∂E ∂cij

(12)

(13)

(14)

(15)

∂E a2ij (t + 1) = a2ij (t) − γ ∂a2ij

(16)

dij (t + 1) = dij (t) − γ

∂E ∂dij

(17)

where γ is the learning rate. The derivatives in (12) and (13) are determined by the following formulas: ∂E ∂u ∂uj ∂E = ∂wij ∂u ∂uj ∂wij ∂E ∂E ∂u ∂uj = . ∂bj ∂u ∂uj ∂bj

(21)

The parameters of the T2FLS can thus be updated using (12)–(17) together with (19)–(21). The derivatives in (18) are calculated as ∂E ∂u ∂uj ∂u = u(t) − ud (t), = 1, = rj xi and = rj . ∂u ∂uj ∂wij ∂bj (22)

The derivations of the governing equations of EKF are commonly available in the literature [12], [19], [28], [29]. In this section, the optimization of the parameters of T2FLS using EKF is briefly studied. Consider the output of a FLS in the form of u = f (X, θ) in which X ∈ Rn is the input of the system and θ is the vector of all unknown parameters of T2FLS including d’s, m’s, a1 ’s and a2 ’s of type-2 membership function and wij , bj , the weights of the consequent part θk+1 = f (θk ) + ωk yk = h(θk ) + νk

∂E ∂a1ij

a1ij (t + 1) = a1ij (t) − γ

 ∂E ∂u ∂f j ∂μij ∂E = . ∂a2ij ∂u ∂f j ∂μij ∂a2ij j

B. EKF-Based Learning Method

A. Gradient Descent-Based Learning Method

E=

The derivatives in (14)–(17) are determined by the following formulas:   ∂E ∂u ∂f j ∂μij ∂u ∂f j ∂μij ∂E = + (19) ∂cij ∂u ∂f j ∂μij ∂cij ∂f j ∂μij ∂cij j   ∂E ∂u ∂f j ∂μij ∂E ∂u ∂f j ∂μij = + (20) ∂dij ∂u ∂f j ∂μij ∂dij ∂f j ∂μij ∂dij j

(23)

where the vector θk is the states of the system at time k, ωk is the process noise, vk is the observation noise f (.) is the nonlinear state equation and h(.) is nonlinear output function. It is assumed that θ0 is the initial estimate of θ and that noise sequences ωk and νk are Gaussian and independent of each other, and E(θ0 ) = θ0 

E (θ0 − θ0 )(θ0 − θ0 )T = P0 E(ωk ) = 0   E ωk ωlT = Qδkl E(νk ) = 0   E νk νlT = Rδkl

(24) (25) (26) (27) (28) (29)

where E(.) is the expectation operator and δkl is the Kronecker delta. Using KF, it is possible to estimate the states (θk ) of the system f (θk ) = f (θˆk ) + Fk .(θk − θˆk ) + H.O.T.

(18)

h(θk ) = h(θˆk ) + Hk .(θk − θˆk ) + H.O.T.

(30)

KHANESAR et al.: KALMAN FILTER BASED LEARNING ALGORITHM FOR TYPE-2 FUZZY LOGIC SYSTEMS

where

 ∂f (θ)  Fk = ∂θ θ=θˆk  ∂h(θ)  HkT = . ∂θ θ=θˆk

(31) (32)

Neglecting the higher-order terms (HOT) the system in (30), (23) can be rewritten as θk+1 = Fk θk + ωk + φk yk = HkT θk + νk + ϕk

(33)

where φk and ϕk are defined by φk = f (θˆk ) − Fk θˆk

(34)

ϕk = h(θˆk ) − Hk θˆk .

(35)

The Kalman estimation of the parameters in (33) is as [12], [19], [28], [29]   θˆk = f (θˆk−1 ) + Kk yk − h(θˆk−1 ) 

there are four parameters for each type-2 membership function. The parameters of the premise part are put in one group. The parameters of the consequent part are put in one vector, and they make the other group of parameters. In this way, two groups of parameters are considered. The parameter adaptation rule for the parameters in the ith group is as shown below    i i + Kki yki − hi θˆk−1 θˆki = θˆk−1  −1  T Kki = Pki Hki Ri + Hki Pki Hki  T i Pk+1 = Pki − Kki Hki Pki + Qi .

(36)

In above, Kk is known as the Kalman gain. and Fk is considered as the identity matrix. In the case of linear systems, it can be shown that Pk is the covariance matrix of the state estimation error, and that the state estimate θˆk+1 is optimal in the sense that it approaches E[θk+1 |y0 , y1 , . . . , yk ] for large k. However, for nonlinear systems the filter is not optimal and the estimates are only approximately conditional means [12], [19], [28], [29]. In applying EKF to T2FLS, all the unknown parameters of T2FLS are gathered in a vector. The computational cost of EKF is of the order of AB 2 where A is the dimension of the output of dynamical system and B is the number of the parameters. The number of parameters for a T2FLS depends on the number of inputs and the number of membership functions considered for each input. For a T2FLS with n inputs, c membership function for all of the inputs and M number of rules, the number of training parameters is 4oc + (n + 1)oM . Here, M is a large number, being the multiplication of the number of membership function of the inputs. If the number of training samples are considered as o, the computational cost of KF when applied to T2FLS will be 16oc2 + o(n + 1)2 M 2 + 8oc(n + 1)M which is a very big number. To reduce the computational cost of KF, decoupled Kalman filter is introduced in the next subsection. C. Decoupled Kalman Filter To achieve a faster and easy to implement Kalman filter, it is used in a decoupled form in this study, based on the assumption that certain parameters interact with each other only at a second-order level. The decoupled KF has been previously used in [18], [30]. For the T2FLS considered in this paper,

(37)

In above, the function h is the feedforward equation of T2FLS defined as in (9). It is to be noted that for the two groups of parameters that have been considered, and each group has its own Kalman parameters K, P, R, Q, H. The unknown parameter of the first group θ1 consists of the parameters of the premise part as a1 ’s, a2 ’s, c’s and d’s of the membership functions such that θ1 = [a111 , . . . , a1nM , a211 , . . . , a2nM , c11 , . . . , cnM , d11 , . . . , dnM ]T . In this way H 1 is defined as H1 =

−1 HkT Pk Hk

Kk = P k H k R +   Pk+1 = Fk Pk − Kk HkT Pk FkT + Q.

4447

∂h . ∂θ1

(38)

The update rule for the parameter θ1 and its corresponding covariance matrix P 1 is calculated as in (37). The second group of the parameters θ2 consists of the parameters of the consequent part of T2FLS. The parameters of the consequent part are put in a single vector format as θ2 = [w11 , . . . , wn1 , . . . , wM 1 , . . . , wM n , b1 , . . . , bM ]T (39) and the corresponding H matrix for the second group is defined as in (37) H2 =

∂h . ∂θ2

(40)

The corresponding update rule for the parameters of the second group θ2 and its covariance matrix P 2 are obtained as in (41)–(43). With the use of decoupled EKF, the computational cost of estimation is greatly reduced and it is now of the order of 16oc2 + o(n + 1)2 M 2 . The ratio of the computational cost of decoupled EKF to standard EKF is shown in (41) 16c2 + (n + 1)2 M 2 Decoupled EKF = . Standard EKF 16c2 + (n + 1)2 M 2 + 8c(n + 1)M (41) D. PSO-Based Learning Method In this paper, to show the efficiency of the proposed training method based-on EKF, it is compared with one of the well known population-based training methods, namely PSO. PSO was originally designed and introduced by Eberhart and Kennedy [31], [32] in 1995. It is a population-based search algorithm, based on the simulation of the social behavior of birds, bees or a school of fish. This vector has also one assigned

4448

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

vector which determines the next movement of the particle and is called the velocity vector. Assume that our search space is d-dimensional, and the ith particle of the swarm can be represented by a d-dimensional position vector Xi = (xi1 , xi2 , . . . , xid ). The velocity of the particle is denoted by Vi = (vi1 , vi2 , . . . , vid ). Also consider best visited position for the particle is Pibest = (pi1 , pi2 , . . . , pid ) and also the best position explored so far is Pgbest = (pg1 , pg2 , . . . , pgd ). So the position of the particle and its velocity is being updated using following equations: Vi (k + 1) = ωVi (k) + c1 φ1 (Pibest − Xi (k)) + c2 φ2 (Pgbest − Xi (k)) Xi (k + 1) = Xi (k) + Vi (k + 1)

(42) (43)

where c1 and c2 are positive constants, and φ1 and φ2 are two random variables with uniform distribution between 0 and 1. In this equation, ω is the inertia weight which shows the effect of the previous velocity vector on the new vector. An upper bound is placed on the velocity in all dimensions Vmax . This limitation prevents the particle from moving too rapidly from one region in the search space to another. The algorithm for the PSO can be summarized as follows. 1) Initialize the swarm Xi , the positions of the particles are randomly initialized within the hypercube of the feasible space. 2) Evaluate the performance F of each particle, using its current position Xi (k). 3) Compare the performance of each individual to its best performance so far: if F (Xi (k)) < F (Pibest ) F (Pibest ) = F (Xi (k)) and Pibest = Xi (k).

Fig. 5.

Hybrid algorithm based on the combination of PSO and KF.

TABLE I P REDICTION ACCURACIES OF THE T2FLS W ITH THE P ROPOSED T YPE -2 M EMBERSHIP F UNCTION AND T1FLS U SING EKF

(44)

4) Compare the performance of each particle to the global best particle: if F (Xi (k)) < F (Pgbest ) F (Pgbest ) = F (Xi (k)) and Pgbest = Xi (k).

(45) A. Prediction of Chaotic Mackey-Glass Time Series

5) Change the velocity of the particle according to (42). 6) Move each particle to a new position using (43). 7) Go to step 2, and repeat until convergence. In this paper, the combination of PSO and KF is used for the training of T2FLS. Fig. 5 shows the hybrid algorithm based on the combination of PSO and KF.

In this section, the EKF is used to estimate the parameters of a T2FLS, used to predict the noisy chaotic Mackey-Glass time series. This chaotic system is a well-known benchmark problem in the literature described by the following dynamic equation [33]: x(t) ˙ = 0.2

IV. S IMULATION S TUDIES T2FLS trained by EKF is tested on three different data sets. To show the effectiveness of this learning structure, three other learning algorithms are designed and tested on the same data sets, which are gradient descent for both premise and consequent parts, PSO for premise part and KF for consequent part and GD for premise part and KF for consequent part. These methods are briefly called GD plus GD, PSO plus KF and GD plus KF, respectively.

x(t − τ ) − 0.1x(t). 1 + x10 (t − τ )

(46)

The numerical values selected for the chaotic system above are τ = 17, x(0) = 1.2 in this study. The predictor goal is to predict x(t + 1) using the inputs x(t − 3), x(t − 2), x(t − 1) and x(t). For each input two membership functions are used. The number of the rules of the system is therefore equal to 16. The number of training data is selected as 750, and the number of test data is 335. To study the effect of noise in the proposed system, six experiences using three different optimization methods are performed in

KHANESAR et al.: KALMAN FILTER BASED LEARNING ALGORITHM FOR TYPE-2 FUZZY LOGIC SYSTEMS

4449

TABLE II P REDICTION ACCURACIES OF THE T2FLS W ITH THE P ROPOSED T YPE -2 M EMBERSHIP F UNCTION U SING D IFFERENT L EARNING M ETHODS FOR M ACKEY-G LASS DATA S ET

Fig. 6. Convergence graph of different learning algorithms for Mackey-Glass time series.

Fig. 7. Actual Mackey-Glass time series and the output of the T2FLS.

which the data are corrupted with different levels noise. As the measure of noise level, signal to noise ratio (SNR) is used. Table I shows the Root Mean Square Error (RMSE) values of the T2FLS with the proposed type-2 membership function and T1FLS using EKF over ten times of simulations. To be able to make a fair comparison, triangular type membership functions are used in the T1FLS. Although the prediction performance of the T2FLS is better than the T1FLS especially for higher levels of noise on the data, their accuracies are similar for the lower levels of noise on the data. These results are expected results from a T2FLS, as similar conclusions are reported in the literature [34]. Encouraged by the conclusion stated, only the results for T2FLSs are given in the rest of the simulations to keep the manuscript at a reasonable size. Table II shows the RMSE values of the T2FLS with the proposed type-2 membership function using EKF, gradient descent (GD) method plus KF, GD plus GD method and PSO plus KF learning methods in the presence of different levels of noise on the data sets used both for training and testing over ten times of simulations. The results show that the performance of the PSO plus KF learning method is better than the other three methods mentioned above for almost all levels of noise on the data set. While the performance of EKF is comparable with the results obtained by PSO plus KF, the computation time for EKF (around 9 seconds for 10 iterations) is almost 5 times less than PSO plus KF (around 100 seconds for 10 iterations). Moreover, the performance of EKF is better than the performance of GD plus KF and GD plus GD, and their computation times are

very close. The computation time for GD plus KF is around 8 seconds for 10 iterations. Fig. 6 shows the convergence graph of different learning algorithms for Mackey-Glass time series in the presence of white noise (SNR = 10 dB). As can be seen from that figure, the convergence performance of GD method is the most poorly performing algorithm used in this paper. On the other hand, PSO+KF outperforms the other two algorithms. Besides, the convergence performance of EKF is comparable with the PSO+KF algorithm. Fig. 6 also shows that EKF converges in the first few epochs. Fig. 7 shows the real data and the results of the T2FLS using EKF. As can be seen from Fig. 7, the prediction accuracy of T2FLS using EKF is quite satisfactory for such a noisy data. Fig. 8 shows the proposed novel type-2 fuzzy membership functions before (solid line) and after (dotted line) the training using EKF. As can be seen from it, the parameters of the type-2 membership functions are initially set to a1 = a2 = 1 which means that they are type-1. Once the training is finished, it is seen that the widths of the proposed membership functions are grown which means that they take the

type-2 form. Fig. 9 shows the mean value of |a1 − a2 | for T2FLS trained by EKF plus KF as a measurement of the width of the proposed membership function over ten times of simulations. It can be seen that the width of the proposed membership function is larger for higher levels of noise which is an expected behavior from a T2FLS. To compare the computational costs of EKF and PSO plus KF, the time durations for the error to reduce to a predefined

4450

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

TABLE III C OMPARISON OF THE C OMPUTATION T IME B ETWEEN PSO+KF AND EKF W ITH D IFFERENT N UMBER OF RULES FOR M ACKEY-G LASS DATA S ET (SNR = 10 dB)

TABLE IV C OMPARISON OF THE C OMPUTATION T IME B ETWEEN GD AND EKF W ITH D IFFERENT N UMBER OF RULES FOR M ACKEY-G LASS DATA S ET (SNR = 10 dB)

Fig. 8. Proposed novel type-2 fuzzy membership functions before (solid line) and after (dotted line) the training using EKF.

Fig. 10.

ABS laboratory setup.

Fig. 11.

(a) The brake torque (input) of ABS and (b) slip (output) of ABS.

Fig. 9. Width of the proposed novel type-2 fuzzy membership function after training using EKF wrt. the noise level.

RMSE level are measured for ten times and different number of rules for both methods for SNR = 10 dB noise, and the results are shown in Table III. As can be seen from the table, these results show that although PSO gives more precise results, it has higher computational burden. Similar to Table III, Table IV shows the result of comparison of the computation time for between EKF and GD. Although one epoch for GD-based method takes less time, it takes much more epochs for it to converge. B. Identification of Antilock Braking System (ABS) Data Set In this section the T2FLS with the proposed novel membership function using EKF is used to identify the data set of an ABS. ABS is an electronically controlled system that helps the driver to maintain control of the vehicle during emergency braking by preventing the wheels from lock up.

In this paper, the ABS laboratory setup [35] manufactured by Inteco Ltd. is used (Fig. 10). The input–output data set is collected in a closed loop fashion using a PD controller. Fig. 11 shows the input–output data set in which input is the braking torque and the output is the current slip value of ABS. In the identification process, u(t), y(t − 1) are chosen as the inputs of the T2FLS proposed, and y(t) is the target value. Three membership functions are considered for u(t) and two membership functions are used for y(t − 1). In this way, the

KHANESAR et al.: KALMAN FILTER BASED LEARNING ALGORITHM FOR TYPE-2 FUZZY LOGIC SYSTEMS

4451

Fig. 13. Structure of FEL using KF to train the parameters of T2FLS.

Fig. 12. Actual ABS data set and the output of the T2FLS. TABLE V P REDICTION ACCURACIES OF THE T2FLS AND T1FLS W ITH D IFFERENT T RAINING M ETHODS FOR THE I DENTIFICATION OF ABS DATA S ET

resulting fuzzy system will have 6 rules. Fig. 12 shows the actual ABS data and the output of the T2FLS using EKF. The training performance of different algorithms on T2FLS are summarized in Table V over ten times of simulations. As can be seen from the table, training T2FLS using PSO plus KF has the best training results while the best test result belongs to EKF plus KF training of T2FLS. Therefore, for this data set, EKF has the best generalization performance over other training methods. V. F EEDBACK E RROR L EARNING C ONTROL OF A L ABORATORY S ETUP ABS U SING K ALMAN F ILTER In this section, Kalman filter is used to train the parameters of T2FLS in a FEL scheme. The controller is implemented on a real-time ABS. The concept of FEL is first introduced by Kawato in an effort to establish a stable controller which can learn the inverse dynamic of the system under control [36]. The block diagram of the proposed controller can be seen in Fig. 13. As can be seen from this figure, the control signal which is the braking-torque applied to the system is the summation of the output values of PID controller and T2FLS. The PID controller coefficients KP , KD and KI are set to 500, 2, and 40, respectively. The T2FLS is trained in such a way that the output of the PID controller becomes zero, and then T2FLS takes the responsibility of controlling the plant. The output of PID controller is considered as the error signal to train the parameters of T2FLS. Since the inputs of the T2FLS are the desired signal and the error signal and the output of the T2FLS is the control signal, the T2FLS learns the inverse dynamics of the ABS. ABS is an electronically controlled system that helps the driver to maintain control of the vehicle during emergency brak-

Fig. 14. Output of PID controller and T2FLS.

ing by preventing the wheels from lock up. During accelerating or braking, friction forces that are generated between the wheel and the road surface are proportional to the normal load of the vehicle. The coefficient of this proportion is called the road adhesion coefficient and it is denoted by μ. Studies show that μ is a nonlinear function of the wheel slip, which is defined as the measure of relative difference between the vehicle and the wheel velocities [37]. The mathematical formula for the wheel slip can be represented as λ=

V − Rω V

(47)

where V is the forward velocity of the vehicle, ω is the angular velocity of the wheel, and R is the effective radius of the corresponding wheel. While a wheel slip of zero indicates that the wheel velocity and the vehicle velocity are the same, a ratio of one indicates that the tire is not rotating and the wheels are skidding on the road surface; i.e., the vehicle is no longer steerable. The typical μ − λ curve is obtained from the data of numerous experiments. Most of the ABS controllers are expected to keep the vehicle slip at a particular level, where the corresponding friction force (i.e., road adhesion coefficient) reaches its maximum value. Zanten states in [38] that the wheel slip should be kept between 0.08 and 0.3 to achieve optimal performance. Furthermore, some research papers show that the reference wheel slip does not have to be a constant value [39]. In this

4452

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

TABLE VI P ERFORMANCE OF THE D IFFERENT C ONTROL A LGORITHMS ON N OISY AND N OISE -F REE E NVIRONMENTS

paper, the reference slip value has been selected as both a constant value and a function of the vehicle velocity. The ABS laboratory setup used in this study is shown in Fig. 10. A. Simulation Studies In this section, several control algorithms are simulated on the ABS model. The first control algorithm is a conventional PID controller working alone. To compare the performance of T2FLS with KF, this control algorithm is compared with T2FLS which is trained by GD, and also T1FLS which is trained by KF. These controllers are simulated on noisy and noise-free environments. In addition, the performance of these controllers is tested when the slip reference is a constant signal value and when it is a velocity dependent value. Fig. 14 shows the output of PID and T2FLS which is trained by KF in noise-free environment when the reference slip value is a constant signal. It can be observed that the output of PID controller tends to go to zero after the transient response. Meanwhile, the T2FLS learns the inverse dynamics of the system and takes the responsibility of the control of the system. As a result, the output of T2FLS increases and the output of PID controller decreases simultaneously. Table VI shows the performance of the different control algorithms on noisy and noise-free environments both for constant and velocity dependent slip reference values. It can be seen that FEL structure has much better performance than the conventional PID controller. It can also be observed that T2FLS trained by KF gives better performance than T2FLS trained by GD. While the performances of T2FLS and T1FLS are similar in noise-free environment, T2FLS outperforms its type-1 counterpart in noisy environments. The results given in Table VI can be visualized in Figs. 15–18. B. Real-Time Experiments Since the sampling time is restricted to 1ms for the laboratory setup available, the type-zero TSK fuzzy model is used and only the parameters of the consequent part have been trained. As can be seen from Fig. 13 the inputs of T2FLS are e(i), e(i) and desired value. The number of membership functions considered for the inputs of the system are two, three

Fig. 15.

Response of four different controllers applied to the model of ABS.

Fig. 16. Response of four different controllers applied to the model of ABS in the presence of noise.

and two for e(i), e(i) and desired value, respectively, and the resulting fuzzy system has 12 rules. Because of the fact that the laboratory setup available is highly nonlinear and it has uncertainties, ten different experiments for each case are done, and the mean values are given. All results below are for a car with the initial longitudinal velocity of V = 70 km/h maneuvering on a straight line. Two approaches are followed to determine the reference wheel slip value: In the first case,

KHANESAR et al.: KALMAN FILTER BASED LEARNING ALGORITHM FOR TYPE-2 FUZZY LOGIC SYSTEMS

Fig. 17. Response of four different controllers applied to the model of ABS with velocity dependent slip reference.

Fig. 18. Response of four different controllers applied to the model of ABS with velocity dependent slip reference in the presence of noise.

the reference wheel slip is taken to be constant (λ = 0.2). As mentioned earlier, the desired range for slip value is between 0.08 and 0.3 so the choice of 0.2 as the desired value for slip is reasonable. In the second case, to check the performance of this controller under varying slip conditions, the reference slip is considered to vary as a function of the vehicle velocity. For this purpose, a pseudo-static curve is used to calculate the reference wheel slip and the corresponding tire friction coefficient. These values are used to construct a table, which relates the vehicle speed to the peak values of tire road friction coefficient and to the reference wheel slip. Next, the reference wheel slip is made available to the proposed controller at each step of the control loop [39]. There is a velocity threshold which states the minimum velocity level for applying ABS control algorithms. Under this minimum value of the velocity, the system becomes unstable if ABS algorithm is applied. Under such a circumstance, the maximum braking torque should be applied to the wheels without considering the target value of slip. As a result, the error is computed when the system is operating above the minimum velocity (15 km/h) mentioned above. The mean value of ten experiments in terms of the sum of the squared error obtained for T1FLS trained with KF, T2FLS trained with GD and T2FLS

4453

Fig. 19. Real-time response of the PID and T2FLS for constant slip reference.

Fig. 20. Real-time response of the T1FLS and T2FLS for constant slip reference.

trained with KF are 7.92 ± 0.26, 8.04 ± 0.13 and 7.67 ± 0.15, respectively. The parameters of the EKF for the implementation part are selected as follows: R = 1000, Q = 0.1I12 and P = 10I12 where I represents the identity matrix. The learning rate for the GD algorithm is taken as 0.02. The sum of squared error for the conventional PID controller is seen to be much higher than any of the FEL schemes, being typically bigger than 15. These quantitative results can be seen in Figs. 19–21. Similar to the simulation section in this study, T2FLS trained with KF gives the best performance for the real-time experiments. As mentioned earlier, to check the performance of the controller proposed in this study under varying reference slip conditions, some experiments are carried out when the reference slip is considered to vary as a function of the vehicle velocity. Encouraged by all the simulation and real-time experiments given in this study, this is done only for the T2FLS trained with KF and the result is given in Fig. 22. It can be observed that the control algorithm proposed has the ability to control such a nonlinear and uncertain system with a satisfactory performance. VI. C ONCLUSION In this paper, EKF as a powerful second-order gradient method is used to train the parameters of T2FLS. To show

4454

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 11, NOVEMBER 2012

KF is used to train the parameters of the T2FLS in a FEL scheme too and it is implemented on a real-time laboratory setup ABS. Two different types of slip reference signal (constant and velocity dependent) are used. It is shown that for constant slip reference, T2FLS trained by KF gives the best results when compared to T2FLS trained by GD and T1FLS trained by KF. T2FLS trained by KF in FEL scheme is also implemented on the ABS system when the reference slip value is a function of the velocity of the system and satisfactory results are obtained. ACKNOWLEDGMENT

Fig. 21. Real-time response of the T2FLS trained by GD and KF for constant slip reference.

The authors would like to acknowledge the financial support of the Bogazici University Research Fund with the project number 09HA203D. R EFERENCES

Fig. 22. Real-time response of the T2FLS trained with KF for velocity dependent slip reference.

the effectiveness of the use of EKF for the training of the parameters of T2FLSs, other gradient-based and gradient-free methods are used. The results show that EKF outperforms first order gradient-based methods, while its computation time does not increase dramatically. Although the results obtained by EKF are comparable with the results of PSO, which is a gradient-free method, but EKF is a seen to be a computationally more effective method. In the paper a comparison is also presented between a T1FLS and a T2FLS trained by EKF. They used for the prediction of Mackey-Glass chaotic system and for the identification of ABS data set in the presence of measurement noise. It is seen that the performance of T2FLS is better than those of type-1 counterparts in the presence of higher levels of noise and uncertainties. The simulation results also show that the width of the proposed membership function increases as more noise power is injected to the system which means that the performance improvement of T2FLS over type-1 counterpart is much higher for the higher levels of noise. This fact indicates that T2FLSs should be preferred over T1FLS in the presence of noise and uncertainties in the system.

[1] L. Zadeh, “Toward extended fuzzy logic-a first step,” Fuzzy Sets Syst., vol. 160, no. 21, pp. 3175–3181, Nov. 2009. [2] F.-J. Lin and P.-H. Chou, “Adaptive control of two-axis motion control system using interval type-2 fuzzy neural network,” IEEE Trans. Ind. Electron., vol. 56, no. 1, pp. 178–193, Jan. 2009. [3] R. H. Abiyev and O. Kaynak, “Type 2 fuzzy neural structure for identification and control of time-varying plants,” IEEE Trans. Ind. Electron., vol. 57, no. 12, pp. 4147–4159, Dec. 2010. [4] L. A. Zadeh, “The concept of linguistic variable and its application to approximate reasoning,” Inf. Sci., vol. 8, no. 3, pp. 199–249, 1975. [5] N. N. Karnik, J. M. Mendel, and Q. Liang, “Type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 7, no. 6, pp. 643–658, Dec. 1999. [6] C.-F. Juang and C.-H. Hsu, “Reinforcement interval type-2 fuzzy controller design by online rule generation and q-value-aided ant colony optimization,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 6, pp. 1528–1542, Dec. 2009. [7] C.-F. Juang and C.-H. Hsu, “Reinforcement ant optimized fuzzy controller for mobile-robot wall-following control,” IEEE Trans. Ind. Electron., vol. 56, no. 10, pp. 3931–3940, Oct. 2009. [8] R. Sepulveda, P. Melin, A. Rodriguez, A. Mancilla, and O. Montiel, “Analyzing the effects of the footprint of uncertainty in type-2 fuzzy logic controllers,” Eng. Lett., vol. 13, no. 2, pp. 138–147, 2006. [9] M. A. Khanesar, M. Teshnehlab, E. Kayacan, and O. Kaynak, “A novel type-2 fuzzy membership function: Application to the prediction of noisy data,” in Proc. Comput. Intell. Meas. Syst. Appl., Sep. 2010, pp. 128–133. [10] M. Biglarbegian, W. Melek, and M. Mendel, “Design of novel interval type-2 fuzzy controllers for modular and reconfigurable robots: Theory and experiments,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1371– 1384, Apr. 2011. [11] F. Leung, H. Lam, S. Ling, and P. Tam, “Optimal and stable fuzzy controllers for nonlinear systems based on an improved genetic algorithm,” IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 172–182, Feb. 2004. [12] M. A. Shoorehdeli, M. Teshnehlab, and A. K. Sedigh, “Training ANFIS as an identifier with intelligent hybrid stable learning algorithm based on particle swarm optimization and extended Kalman filter,” Fuzzy Sets Syst., vol. 160, no. 7, pp. 922–948, Apr. 2009. [13] G. Mouzouris and J. M. Mendel, “A singular-value-QR decomposition based method for training fuzzy systems in uncertain environments,” J. Intell. Fuzzy Syst., vol. 5, pp. 367–374, 1997. [14] S. Smith and D. Comer, “Automated calibration of a fuzzy logic controller using a cell state space algorithm,” IEEE Control Syst. Mag., vol. 11, no. 5, pp. 18–28, Aug. 1991. [15] J. Mendel, “Computing derivatives in interval type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 12, no. 1, pp. 84–98, Feb. 2004. [16] S. Sakai and T. Takahama, “Learning fuzzy control rules by vector simplex method,” in Proc. Joint 9th IFSA World Congr. and 20th NAFIPS Int. Conf., Jul. 2001, vol. 5, pp. 2541–2546. [17] M. A. Shoorehdeli, M. Teshnehlab, A. K. Sedigh, and M. A. Khanesar, “Identification using ANFIS with intelligent hybrid stable learning algorithm approaches and stability analysis of training methods,” Appl. Soft Comput., vol. 9, no. 2, pp. 833–850, Mar. 2009. [18] D. Simon, “Training radial basis neural networks with the extended Kalman filter,” Neurocomputing, vol. 48, no. 1–4, pp. 455–475, Oct. 2002.

KHANESAR et al.: KALMAN FILTER BASED LEARNING ALGORITHM FOR TYPE-2 FUZZY LOGIC SYSTEMS

[19] D. Simon, “Training fuzzy systems with the extended Kalman filter,” Fuzzy Sets Syst., vol. 132, no. 2, pp. 189–199, Dec. 2002. [20] X. Ruan, M. Dinga, D. Gong, and J. Qiaoa, “On-line adaptive control for inverted pendulum balancing based on feedback-error-learning,” Neurocomputing, vol. 70, no. 4–6, pp. 770–776, Jan. 2007. [21] X. Ruan, L. Liu, N. Yu, and M. Ding, “A model of feedback error learning based on Kalman estimator,” in Proc. 6th WCICA, 2006, vol. 1, pp. 4190–4194. [22] Z. Shaobai, R. Xiaogang, and C. XieFeng, “A new cerebellar learning scheme of feedback error based on Kalman estimator,” in Proc. CCDC, Jul. 2008, pp. 4064–4069. [23] M. J. Er and K. C. Liew, “Control of Adept One SCARA robot using neural networks,” IEEE Trans. Ind. Electron., vol. 44, no. 6, pp. 762–768, Dec. 1997. [24] K. Sabahi, M. Teshnehlab, and M. A. Shoorhedeli, “Recurrent fuzzy neural network by using feedback error learning approaches for LFC in interconnected power system,” Energy Conver. Manag., vol. 50, no. 4, pp. 938–946, Apr. 2009. [25] A. Miyamura and H. Kimura, “Stability of feedback error learning scheme,” Syst. Control Lett., vol. 45, no. 4, pp. 303–316, Apr. 2002. [26] J. Nakanishi and S. Schaal, “Feedback error learning and nonlinear adaptive control,” Neural Netw., vol. 17, no. 10, pp. 1453–1465, Dec. 2004. [27] M. Begian, W. Melek, and J. Mendel, “Stability analysis of type-2 fuzzy systems,” in Proc. Int. Conf. FUZZ-IEEE, 2008, pp. 947–953. [28] B. Anderson and J. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [29] A. Gelb, Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974. [30] G. Puskorius and L. Feldkamp, “Neurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks,” IEEE Trans. Neural Netw., vol. 5, no. 2, pp. 279–297, Mar. 1994. [31] R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proc. 6th Int. Symp. MHS, Oct. 1995, pp. 39–43. [32] J. Kennedy and R. Eberhart, Swarm Intelligence. San Francisco, CA: Morgan Kaufmann, 2001. [33] M. Mackey and L. Glass, “Oscillation and chaos in physical control system,” Science, vol. 197, pp. 287–289, Jul. 1977. [34] R. Sepulveda, O. Castillo, P. Melin, O. Montiel, and L. Aguilar, “Evolutionary optimization of interval type-2 membership functions using the human evolutionary model,” in Proc. Int. FUZZ-IEEE, London, U.K., 2007, pp. 1–6. [35] User’s Manual, The Laboratory Antilock Braking System Controlled From PC, Inteco Ltd., Krakow, Poland, 2006. [36] M. Kawato and H. Gomi, “A computational model of four regions of the cerebellum based on feedback-error learning,” Biol. Cybern., vol. 68, no. 2, pp. 95–103, Dec. 1992. [37] A. Harifi, A. Aghagolzadeh, G. Alizadeh, and M. Sadeghi, “Designing a sliding mode controller for antilock brake system,” in Proc. Int. Conf. Comput. Tool EUROCON, Nov. 2005, vol. 1, pp. 258–261. [38] A. Zanten, R. Erhardt, and A. Lutz, “Measurement and simulation of transients in longitudinal and lateral tire forces,” SAE Trans., vol. 99, no. 6, pp. 300–318, 1990. [39] Y. Oniz, E. Kayacan, and O. Kaynak, “A dynamic method to forecast the wheel slip for antilock braking system and its experimental evaluation,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 2, pp. 551–560, Apr. 2009.

Mojtaba Ahmadieh Khanesar (S’07) was born in Esfahan, Iran, in 1982. He received the B.Sc. and M.Sc. degrees in control engineering from K. N. Toosi University of Technology, Tehran, Iran, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree in control engineering. His research interests include soft computing, fuzzy sets and systems, intelligent control, adaptive control systems, particle swarm optimization.

4455

Erdal Kayacan (S’06) was born in Istanbul, Turkey, on January 7, 1980. He received the B.Sc. degree in electrical engineering from Istanbul Technical University, Istanbul, Turkey, in 2003 and the M.Sc. degree in systems and control engineering from Bogazici University, Istanbul, Turkey, in 2006, where he is currently working toward the Ph.D. degree in electrical and electronic engineering, and he is a research assistant at the same department. His research interests include soft computing, intelligent control, fuzzy theory, grey system theory.

Mohammad Teshnehlab received the B.Sc. degree in electrical engineering from Stony Brook University, Stony Brook, NY, in 1980, the Master degree in electrical engineering from Oita University, Oita, Japan, in 1991 and the Ph.D. degree in computational intelligence from Saga University, Saga, Japan, in 1994. He is currently an Associate Professor of control systems with the Department of Electrical and Computer Engineering, K.N. Toosi University of Technology, Tehran, Iran. He is the author or a coauthor of more than 130 papers that have appeared in various journals and conference proceedings. His main research interests are neural networks, fuzzy systems, evolutionary algorithms, swarm intelligent, fuzzy-neural networks, pattern recognition, meta heuristic algorithms and intelligent identification, prediction and control.

Okyay Kaynak (M’80–SM’90–F’03) received the B.Sc. (first-class honors) and Ph.D. degrees in electronic and electrical engineering from the University of Birmingham, Birmingham, U.K., in 1969 and 1972, respectively. From 1972 to 1979, he held various positions in industry. In 1979, he joined the Department of Electrical and Electronics Engineering, Bogazici University, Istanbul, Turkey, where he is currently a Full Professor, holding the UNESCO Chair on Mechatronics. He has held long-term (near to or more than a year) Visiting Professor/Scholar positions at various institutions in Japan, Germany, the U.S., and Singapore. His current research interests include intelligent control and mechatronics. He is the author of three and the editor of five books. Additionally, he is the author or a coauthor of more than 300 papers that have appeared in various journals and conference proceedings. Dr. Kaynak is active in international organizations, has served on many committees of the IEEE, and was the President of the IEEE Industrial Electronics Society during 2002–2003.