Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing

Modeling Belt-Servomechanism by Chebyshev Functional Recurrent Neuro-Fuzzy Network*

Yuan-Ruey HUANG**, Yuan KANG***, Ming-Hui CHU**** and Yeon-Pun CHANG*** **Department of , Nanya Institute of Technology, Chung Li 32023, Taiwan, R.O.C. ***Department of Mechanical Engineering, Chung Yuan Christian University, Chung Li 32023, Taiwan, R.O.C. E-mail: [email protected] ****Department of Mechatronic Engineering, Tung Nan Institute of Technology, Taipei, 22203, Taiwan, R.O.C.

Abstract A novel Chebyshev functional recurrent neuro-fuzzy (CFRNF) network is developed from a combination of the Takagi-Sugeno-Kang (TSK) fuzzy model and the Chebyshev recurrent neural network (CRNN). The CFRNF network can emulate the nonlinear dynamics of a servomechanism system. The system nonlinearity is addressed by enhancing the input dimensions of the consequent parts in the fuzzy rules due to functional expansion of a Chebyshev polynomial. The back propagation algorithm is used to adjust the parameters of the antecedent membership functions as well as those of consequent functions. To verify the performance of the proposed CFRNF, the experiment of the belt servomechanism is presented in this paper. Both of identification methods of adaptive neural fuzzy inference system (ANFIS) and recurrent neural network (RNN) are also studied for modeling of the belt servomechanism. The analysis and comparison results indicate that CFRNF makes identification of complex nonlinear dynamic systems easier. It is verified that the accuracy and convergence of the CFRNF are superior to those of ANFIS and RNN by the identification results of a belt servomechanism.

Key words: Neuro-Fuzzy Model, Chebyshev Functional Recurrent Neural Network, Back Propagation Algorithm, Belt Servomechanism, Adaptive Neuro Fuzzy Inference System

1. Introduction

For decades, it has been proven that the neural emulators can provide the dynamical relations between the inputs and outputs for a nonlinear plant. The well-trained neural emulator is useful for servo controller design in simulation phase. Recent years, many studies about the nonlinear neural emulators were proposed. Narendra and Parthasarthy (1) used nonlinear functional mapping of neural networks to emulate a plant dynamics in relation to their output and input sensitivity. They utilized forward multiple layer neural networks which provide static mapping functions only. Additionally, in several other studies (2)−(3) , it has been shown that a fuzzy model is a universal approximator, however, it is difficult to quantify the fuzzy terms. The neuro-fuzzy networks combine the advantages of the fuzzy model (i.e., transparency), and neural networks (i.e., learning capabilities), and thus can provide a more complete explanations of the results than do other black-box *Received 3 Sep., 2007 (No. 07-0481) (4) [DOI: 10.1299/jamdsm.2.949] models such as neural networks alone .

949 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing A recurrent neuro-fuzzy network was used by Zhang and Morris (5) for multi-step ahead prediction. In their method the outputs of the neuro-fuzzy model were used as input to the network, through one or several time delay units. Panchariya et al. (6) and Zhang (7) also used neuro-fuzzy networks with good system identification results. In additional, Jang (8) proposed using an adaptive neuro fuzzy inference system (ANFIS) as a neuro fuzzy method for the adjustment and identification of the parameters of a TSK fuzzy model. In the ANFIS, fuzzy decision rules are implemented via the decision criterion of the neural network (9) ; however, this type of forward neural network with multiple layers provides static mapping functions only. Wu and Er (10) integrated the fuzzy model and neural network approach to obtain an approximation of the nonlinear model of a dynamic system. Chebyshev series are frequently used for approximations to functions and are much more efficient than power series of the same degree (11) . Moreover, Chebyshev polynomials for the nonlinear expansion are easier to compute than trigonometric functions (12) . It has been proved that Chebyshev neural network (CNN) has powerful representation capabilities whose input is generated by using a subset of Chebyshev polynomials (13) . Patra et al. (14) showed how to improve the identification accuracy of Chebyshev functional link artificial neural networks by increasing the dimensions of the input vector. Hence, we consider the Chebyshev polynomials as for functional approximation. The proposed CFRNF uses Chebyshev recurrent neural network (CRNN ) as the consequent of fuzzy rule, which makes it to approximate a complex nonlinear system becomes easier. Flat belt servomechanisms, which transmit motion and power via friction between the belt and pulleys, are widely used in industrial applications, because of their low vibration, low noise and low cost. However slippage between the belt and the pulley, and belt elasticity can induce high order non-linear dynamics and uncertainty into the servomechanism systems. In addition the dynamic characteristics of the belt can be changed by variations in temperature and humidity. Due to their high order non-linearity, and time variation, it is difficult to establish an accurate belt-driven servomechanism model with the method of conventional parameters identification. Many controls of belt servomechanisms with the use of fuzzy control algorithms (15) , and sliding mode control schemes (16) are not so succeed due to the incomplete identification of nonlinear dynamical systems. Karimi et al. (17) proposed the multi-model adaptive control for a flexible system with low damped vibration modes and large parameter variations, but this type of system is very difficult to control. This paper aims to use the CFRNF for the identification of a flat-belt servomechanism. The back propagation algorithm is used to adjust the parameters of the antecedent membership functions as well as those of consequent functions. On the basis of the fuzzy model approach, the nonlinear dynamics of the flat-belt servomechanism can be divided into several local models. A CRNN is used as an approximator for each one of the local models in a nonlinear system. The expansion of the Chebyshev functions increases the input dimensions of the consequent parts in the fuzzy rules, because the hyper planes of the CRNN provide greater discrimination capability in the input space. The manner to qualify a neural network model by using training data to learn and evaluating its output by different sets of test data is described in Nellles (18) . The performance of the CFRNF, in comparison to the ANFIS and RNN methods, in a flat-belt servomechanism case study is done.

Nomenclature e : error between output of vmr : center of the rth membership plant and model X : input vector of plant

950 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing E : cost function yp : output of plant Fh : activation function of yˆ r : output of the rth CRNN hidden-layer neurons Y : output vector of the plant F : activation function of o Yˆ : output vector, composed of

output-layer neurons yˆ(n), yˆ(n −1), yˆ(n − 2),", of h : output of hidden-layer j CFRNF

neurons η : learning rate k : elasticity coefficient of belt α : momentum factor net : net input to the jth j δo : error propagated through the output hidden-layer neurons layer of CRNN netk : net input to the output-layer ρ : density of input m vm neurons φi : the ith Chebyshev polynomial Nφ : order of the Chebyshev σ mr : standard deviation of the rth polynomials membership R : total number of rules µmr : the rth membership function V, vm : input vector and µr : degree of input matches to the rth rule the mth input of CFRNF ∏ : “And” operator w ji : weights between the hidden ψ r : normalized strength of the rth rule layer and the input layer ζ i : the ith Chebyshev coefficient w kj : weights between the hidden layer and the output layer

2. Design of the Chebyshev Functional Recurrent Neuro-Fuzzy Network

Figure 1 shows the architecture of a six-layer Chebyshev functional recurrent neuro-fuzzy network (CFRNF) for the identification of a nonlinear system. The input vector V = (X,Y)T of the CFRNF consists of X and Y which are input and output of the plant,

respectively. The output of the plant includes yp (n) and a bank of unit

delays, yp (n −1), yp (n − 2), yp (n − 3)," , etc. In the first layer, Gaussian membership

functions represented by µm1(vm ),", µmr (vm ),",µmR (vm ) for the mth input are utilized

to fuzzify all inputs v,m m = 1, 2,", M, of CFRNF. In the second layer, the products of membership function values of all inputs are the outputs of inference layer, which are M denoted by ∏ µmr (Vm ) , r = 1, 2,", R . In the third layer, µr (V) is normalized due to m=1

Fig. 1 Schematic diagram of the proposed CFRNF for a nonlinear plant

951 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing dividing by the sum of all products and denoted by ψ r . In the fourth layer, the Chebyshev recurrent neural network (CRNN) is used to generate the consequent parameters of rules, which increases the dimensions of consequent parameters because of increasing the input dimensions via the Chebyschev polynomials. Multipling the output of the Rth CRNN

yˆ R and the normalized strength of the Rth rule ψ R gives the Rth rule output shown

as ψ R yˆ R in the Figure 1. In the fifth layer, the estimated output of the plant yˆ(n ) can be R ˆ ˆ obtained by defuzzified as y = ∑ψ r yr (n) . In the sixth layer, the difference between the r=1 plant output Y and the estimated output Yˆ is utilized to modify the parameters of CRNN by using the back propagation algorithm. The details of the identification as shown in Fig. 1 will be discussed in the following paragraphs.

2.1 Fuzzification The purpose of clustering is to divide the initial data set into homogenous groups based on the similarity of properties. A clustering technique is used to automatically generate rules, and each cluster represents a group of associated data in a data space. Gaussian function is chosen as membership function in this study due to its continuous differentiable characteristics. The membership functions of X and Y are shown in Fig. 2. The centers (19) of these Gaussian functions, µmr , can be obtained by using the subtractive clustering . When applying subtractive clustering to a set of inputs, V = (X,Y)T , each data is considered

as the candidates for cluster centers. A density measure associated to the input vm , is defined as 2 v m − v j − n r ( c ) 2 2 ρ m = ∑ e (1) j=1

where rc is a positive constant and used to define the neighborhood radius of each data point. A data point means a datum in a set. A data point scored the highest density measure represents a cluster center. Therefore, the data point with the highest density measure is selected as the cluster center. After computing the density measure for each point, the data

point with the highest density measure denoted by ρc1 is selected as the first cluster

center vm1 . When the first cluster center has been selected, the density measure of all the

remaining data point is reduced. The density measure for each data point vm is revised as follows 2 v −v − m m1 (rb )2 2 ρm = ρm − ρc1e (2)

1 where the rb is a positive constant and 0.9

defines the neighborhood radius with 0.8

sensitive reduction in its density measure. 0.7

Therefore, the density measures of those 0.6

data points close to the selected center are 0.5 significantly reduced. The data point with 0.4 Membership Degree Membership the highest revised density measure is 0.3 selected as the next cluster center and all of 0.2 the density measures for data points are 0.1 0 revised again. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Each node in the first layer corresponds Fig. 2 Gaussian function used as membership functions for fuzzification to an input variable vm , and transmits the

952 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing input variable to the next layer directly. A Gaussian membership function is adopted

for µmr , r = 1,2,",R . Each node in the fuzzification layer is determined by the Gaussian membership function that specifies the degree of an input value which belongs to a fuzzy set. The Gaussian membership function is defined by. 2  1  v − v   µ (v ) = exp−  m mr   r = 1,2,..., R (3) mr m  2  σ     mr  

where vmr is the center and determined from Eq. (2), σ mr is the standard deviation of the

Gaussian membership function; and vm represents the input to the node. The Gaussian membership is obtained from Eq. (3). The goal in adjusting the centers and widths is to find the optimal locations of Gaussian memberships and achieve a certain amount of response overlap between each Gaussian membership and its neighbors so that they form a smooth and contiguous interpolation over those regions of the input space.

The characteristic parameters of Gaussian membership functions of vm can be updated by using the back propagation algorithm to minimize the cost function as defined by q q 1 1 ˆ 2 2 E = ∑(yp (n) − y(n)) = ∑e (n) (4) 2 n=1 2 n=1

where yp (n) is the desired output and yˆ(n ) is the output of the CFRNF at the nth time step. The identification error, e , is defined as the difference between the output of the plant,

yp , and the output of the model yˆ .

The center vmr and width σ mr of Gaussian membership shown in Eq. (3) can be

updated by back propagation method. The updated antecedent parameters, vandmr σ mr

for vm , can be determined by ∂E vmr (n +1) = vmr (n) −ηv , (5) ∂vmr ∂E where ηv denotes the learning rate; and is ∂vmr

∂E ∂E ∂yˆ(n) ∂µ ∂µ R (v − v ) = r mr = −e(n)ψ [yˆ − ψ yˆ ][ m mr ] (6) ∂v ∂yˆ(n) ∂µ ∂µ ∂v r r ∑ l l 2 mr r mr mr l=1 σ mr and ∂E σ mr (n +1) = σ mr (n) −ησ , (7) ∂σ mr

where ησ stands for the learning rate and ∂E ∂E ∂yˆ(n) ∂µ ∂µ R (v − v )2 = r mr = −e(n)ψ [yˆ − ψ yˆ ][ m mr ] (8) ∂σ ∂yˆ(n) ∂µ ∂µ ∂σ r r ∑ l l 3 mr r mr mr l=1 σ mr ∂E ∂E The training will be terminated as cost function E and its gradients and ∂vmr ∂σ mr become less than an acceptable small value.

2.2 Rule inference In the second layer, the aggregated value of the antecedent for the rth rule can be obtained by the firing strength of the rule, which applies the ‘And’ operator and denoted by ∏ . The multiplication of the input membership values is determined by M µr (V) = Π µmr (vm ) (9) m=1

where µr (V) is the strength or degree of the input vm which is designated to be the

953 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing rth rule. Thus, the normalized membership function can be obtained by

µr (V) ψ r = R (10) ∑µr (V) r=1

where ψ r ≥ 0 . In the fouth layer and the fifth layer, the model output can be obtained by R ˆ ˆ y(n) = ∑ψ r yr (n) (11) r=1

where yˆ r (n) is the output at the nth time step of the rth CRNN. The gradient descent method is adopted to tune the antecedent and consequent parameters of rules in order to

improve the accuracy of model. The resultant differences between yandyp ˆ are utilized as feedback to update parameters.

3. Chebyshev Recurrent Neural Network of Consequent

3.1 Design of CRNN A generalized IF-THEN form of the rth fuzzy CRNN rule of CFRNF can be defined

by IF v1(n ) is µ1r AND v2 (n ) is µ2r AND …AND vm (n ) is µmr , … AND

vM (n) is µMr (n ) THEN yˆ r (n) = Fo (X, Y, h j(n −1 )) , r = 1,", R, where vandm µmr designated for the input and output of the rth membership node, respectively; R is the

total number of rules; Fo stands for the activation function of CRNN which describes the

consequent part of fuzzy rule, and provides the rule with its output, yˆ r (n ) , at the nth time step. The architecture of a Chebyshev recurrent neural network is shown in Fig. 3. The input vector V = (X,Y)T of the CRNN consists of X , the input of the plant, and Y , the output

vector of the plant, which includes yp (n ) and a bank of unit delays

yp (n −1), yp (n − 2)," . Chebyshev polynomials are chosen for functional expansion of the input patterns of CRNN. Hence, each input pattern passing through a functional expansion creates corresponding elements of the N-dimensional expanded vector. A function f (v) defined in a closed interval [−1, 1] can be approximated by

the Nφ th order Chebyshev polynomials as

Nφ f(v) = ∑ζ iφi (v) (12) i=0

z −1 w h1 h1

v1 " vM

w Fh (net1 ) X 11 w k1 φ1 (X) w J1 # ψ ψ ψ w r " 1 " R 1N yˆ (n) r yˆ 1 yˆ w kj × × " × R Fh (net J ) Fo (net) φN (Y) w JN Y ψ r yˆ r ψ 1 yˆ 1 " ψ R yˆ R

h J w hJ z −1 yˆ ∂E − w (n +1) = w (n) −η + α∆w (n −1) ji ji ∂w (n) ji ji + y p Fig. 3 Architecture of a typical CRNN

954 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing

where ζi is the ith Chebyshev coefficient, and φi (v) denotes the ith Chebyshev polynomial. A general form of the Chebyshev polynomials defined in the closed interval [−1, 1] can be expressed by

φ0 (vm ) = 1, φ1(vm ) = vm , "

φi (vm ) = 2vmφi−1(vm ) − φi−2 (vm ), i ≥ 2 (13) The operation of the jth hidden neuron at the nth step in the time history can be obtained by

h j (n) = Fh (net j(n)) (14a) and N net j (n) = w hjh j(n −1) + ∑w jiφi (vm ) (14b) i=1

where Fh (⋅) is the activation function of the hidden layer and taken as the hyperbolic 1− exp(−2net) tangent function, F (net) = tanh(net) = . φ represents the ith function of h 1+ exp(−2net) i

the Chebyshev polynomials; w jiφi (vm ) and h j(n) are the net input and output at the

jth neuron of the hidden layer, respectively; h j(n −1) represents the unit delay of hidden

layer; and N are the number of input-layer neurons. The weight wji represents the connection coefficient between the jth node at the hidden layer and the ith node, one of the coefficients of the Chebyshev polynomials, at the input layer of the CRNN. Thus, the hyper plane, i.e. functional expansion, consists of N basis functions, with the relationship between the input and output of the hidden layer, as shown in Eq. (14b). The output of the hidden layer acts as feedback to the input layer via a bank of unit delays, a process referred

to as self-recurrence. The weights whj connect the self-recurrence of the hidden layer. The output of the rth CRNN is the value of neural consequent parameter of the rth fuzzy rule which can be determined by J yˆ (n) = F (net ) = F ( w h (n)) (15) r o k o ∑ kj j j=1

where Fo (⋅) is the hyperbolic tangent function of the output layer and J are the number of hidden-layer neurons. The subscript k denotes the single node of output layer and k is equal

to one. wkj represents the connective weights between the jth node at the hidden layer and the node of the output layer.

3.2 Learning Algorithm of CRNN The weighting coefficients of the CRNN can be updated by using the back propagation method for the minimization of the cost function. The error to be propagated through the output layer of the CRNN is

∂E ∂E ∂yˆ(n) ∂yˆ r (n) ∂Fo (netk ) δo = = = −e(n)ψ r (16) ∂netk ∂yˆ(n) ∂yˆ r (n) ∂netk ∂netk where E has been defined in Eq. (4). Therefore, the steepest descent equations of the error gradient with respect to the weights of the output layer are determined by

∂E ∂E ∂yˆ(n) ∂yˆ r (n) ∂netk = = δ0h j(n) (17) ∂wkj ∂yˆ(n) ∂yˆ r (n) ∂netk ∂wkj Similarly, the error gradient with respect to the weights of the hidden layer is determined by ∂E ∂E ∂yˆ(n) ∂yˆ (n) ∂net ∂h (n) ∂net = r k j j ∂w ∂yˆ(n) ∂yˆ (n) ∂net ∂h (n) ∂net ∂w ji r k j j ji (18) ∂Fh (netj) =δowkj (φi (vm) + hj(n −1)) ∂netj

955 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing The updated CRNN weights can now be expressed as ∂E wkj(n +1) = wkj(n) −η +α∆wkj(n −1) (19) ∂wkj(n) for the output layer, and ∂E w ji (n +1) = w ji(n) −η +α∆w ji (n −1) (20) ∂w ji(n) for the hidden layer, where η and α represent the learning and momentum factor,

respectively; and ∆w ji (n −1 ) and ∆w kj(n −1 ) denote the weight updates at the (n −1)th training step.

4. Case study

An actual flat-belt servomechanism as shown in Fig. 4 was set up. The mechanism is composed of a PM DC , an encoder, a fiber optic sensor (PW-PH02) with a measuring accuracy of 3555 rpm and a servo . The physical parameters of the servomotor are listed in Table 1. The encoder built into the PM DC motor is capable of 500 pulses per revolution. A personal computer with the 12 bit D/A converter interface (MPC-6A-C40) and a LabVIEW AT-MIO16E-1 data acquisition interface are used to generate control signals and collect the measured data. The sampling time was50 ms . The stroboscopic speed test disk with reflecting patches on the driven pulley is shown in Fig. 4(c). The angular velocity of driven pulley can be measured by a fiber optic sensor which counts the number of pulses within the sampling time. For example, the angular velocity of n 1 driven pulley is rps = 2π ( ) , where n denotes the number of measured pulses within 0.05 180 the sampling time. There are 180 reflecting patches in the out ring of stroboscopic speed test disk. Recurrent neural networks have been shown to be particularly appropriate for task such as dynamic system modeling. However, most learning algorithms of Recurrent neural networks are slow. One of limitations of ANFIS is its restricted application in dynamic mapping domains due to its feed forward structure, or lack of efficient learning procedures for feedback connections. For example, the ANFIS architecture with two fuzzy if-then rules based on a first order Sugeno model can be expressed as:

Rule 1: If ( x is A)1 and (y is B)1 then (f1 = p1x + q1y + r1 )

Rule2: If ( x is A)2 and (y is B)2 then (f2 = p2x + q2y + r2 ) (21)

where p,i qandi ri are linear parameter, and A,i Bi are nonlinear parameters. The two CFRNF inputs, the input voltage to the motor and the speed of the driven pulley, are denoted by X and Y , respectively. A Gaussian membership function is assigned to each cluster. Because the Chebyshev polynomials is used as the input vector of CRNN which is the consequent of the fuzzy rule, It should be noted to choose for the CRNN a suitable functional expansion plane, comprised of a subset of Chebyshev polynomials to represent the input pattern as a fuzzy inference system.

Table1. Physical parameters of the PM DC motor (Maxon motor, 2332966)

resistance Ra 3.18 Ω

induction La 0.53 mH moment of inertia J 24.3 gcm2

torque constant KT 23Nm A back emf constant 0.908 Vsec/ rad max, permissible speed 9200 rpm max. continuous torque 22.66 m Nm

956 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing The coefficients of the Chebyschev polynomials of the CRNN are shown as: 2 φ1(x(n)) = x(n); φ2 (x(n)) = 2x (n) −1; 2 φ1(yp (n −1)) = yp (n −1); φ2 (yp (n −1)) = 2yp (n −1) −1; 2 (22) φ1(yp (n − 2)) = yp (n − 2); φ2 (yp (n − 2)) = 2yp (n − 2) −1; # # 2 φ1(yp (n − q)) = yp (n − q); φ2 (yp (n − q)) = 2yp (n − q) −1;

The inputs of CFRNF used in this paper are x(n) and yp (n −1) . The antecedent of CFRNF consists of six Gaussian membership functions. Each membership function has center parameter and standard deviation parameter that change the shapes of the Gaussian membership function. Therefore, the number of weights of antecedent of CFRNF are

2×6=12. After Chebyshev functional expansion, there are four neurons, φ1(x(n )) ,

φ2 (x(n)) , φ1(yp (n −1)) and φ2 (yp (n −1 )) , at the first layer. There is one neuron at hidden layer and output layer, respectively. Hence, the number of weights of nine CRNNs are ((4+1)×1+1)×9=54. Therefore, number of weights of CFRNF are 66 totally.

After try and error, the inputs of RNN are x(n) , x(n −1) , yp (n −1) ， yp (n − 2) and there are 6 neurons at the hidden layer. There is one neuron at the output layer. Hence, the number of weights of RNN are (4+6)×6+1=61.

The inputs of ANFIS are x(n) andyp (n −1 ) . There are six Gaussian membership functions used as the antecedent of ANFIS and each membership function has center parameter and standard deviation parameter. Therefore, the number of weights of antecedent of ANFIS are 2×6=12. There are nine rules in ANFIS. The number of parameters of the consequent parameters, (pi,qi,ri) , of ANFIS are 3×9=27, three parameters for each rule and nine rules in total. Therefore, the number of weights of ANFIS are 39 totally. Mean Square Error (MSE) is chosen as the performance index which is defined as 1 N 1 N 2 ˆ 2 (23) MSE = ∑e (n) = ∑(yp (n) − y(n)) N n=1 N n=1 where N denotes the number of data. The comparison of computational complexity and performances of these methods are tabulated in table 2.

Through trial and error, the learning rates η , ηv and ησ of the CFRNF, and ANFIS are set to be 0.5, 0.00001 and 0.00001, respectively. The momentum factors α of all CFRNF, ANFIS and RNN are set to be 0.7. Accordingly, the learning rateη of the RNN is set to be 0.5. Under the conditions of the operation of the belt servomechanism, the motor’s input voltage is under 0.8V. During the training phrase, the training data sets should more fully represent the domain of interest. Here the training data sets are 0.29V, 0.48V, 0.76V, where each data set with its corresponding responses comprises 1000 data pairs, and the validation

Table 2. Comparison of computational complexity and performances of CFRNF、ANFIS and RNN training phase validation phase CFRNF ANFIS RNN CFRNF ANFIS RNN MSE 3.9153×10−6 1.0675×10−5 1.6333×10−5 6.595 × 10 −6 8.1133 × 10 −5 5.9173 × 10 −3 Max. 0.01821 0.02125 0.02653 0.01909 0.04852 0.02987 error Min. -0.02004 -0.02532 -0.12968 -0.04128 -0.01016 -0.20331 error number of 66 39 61 66 39 61 weights

957 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing data sets comprise 3000 data pairs, which are composed of three different motor input voltages, for example 0.38V, 0.57V, 0.67V. The learning of the CFRNF was completed within the first 200 epochs, as shown in Fig.5. However, the MSE curves of the ANFIS and RNN leveled off after 500 and 600 epochs, respectively. It can be seen that the (a) errors predicted by the CFRNF are smaller than those predicted by the ANFIS or RNN. The MSE of the identification results, obtained during the training phase, for CFRNF, ANFIS, and RNN are 3.9153×10−6 , 1.0675×10−5 and 1.6333×10−5 , respectively. Three other data sets, different from those used in the training phrase, are used to validate the identification (b) models. The motor input voltages during the validation phase are 0.38v, 0.57v and 0.67v, and the corresponding driven-pulley velocities are 0.4rps, 0.75rps and 0.9rps, respectively. All test errors are averaged in order to obtain a reliable estimate of the model performance. The validation gives a good estimate of the expected model performance given fresh data. The validation results and validation errors

for CFRNF, ANFIS, and RNN are shown (c) in Fig. 6 and Fig. 7, respectively. The

MSE of the validation results for Fig. 4 Belt-driven servomechanism (a) actual CFRNF, ANFIS, and RNN physical model (b) illustration (c) stroboscopic −6 −5 are 6.595×10 , 8.1133×10 , and speed test disk 5.9173×10−3 , respectively. It is clear that the identification results of the CFRNF are superior to those of the ANFIS and the RNN networks. Therefore, CFRNF can be regarded as a general adaptive model for identification of nonlinear systems. The Chebyshev functional expansion increases the dimension of the input patterns of the fuzzy inference system and creates nonlinear decision boundaries in the multidimensional input space, which enhances the dynamic mapping capabilities.

5. Conclusions

CFRNF can be regarded as a general adaptive model for identification of nonlinear systems. The Chebyshev functional expansion increases the dimension of the input pattern of the fuzzy inference system and creates nonlinear decision boundaries in the multidimensional input space. If the input space is expanded appropriately with the Chebyshev polynomial, the result is fast convergence to good parameter values that capture

958 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing the underlying dynamics. The CFRNF makes identification of complex nonlinear dynamic systems easier. The identification results of a flat-belt servomechanism verify that the accuracy and convergence of the CFRNF are superior to those of ANFIS and RNN.

Acknowledgments

The authors would like to acknowledge the support of this work by grants from the National Science Council (NSC) and Center of Excellence of Research and Development Center for Membrane Technology of the Ministry of Education.

Fig. 5 Training errors (MSE) plots for the ANFIS (dashed-dotted line • ), RNN (dashed line ) and CFRNF (solid line )

1

0.5

0 0 5 10 15 (a) 20 25 30 1

0.5

0 0 5 10 15 (b) 20 25 30 1

velocity (rps) (rps) velocity 0.5

0 0 5 10 15 (c) 20 25 30 1.5 1

0.5

0 0 5 10 15 20 25 30 (d) time (second) Fig. 6 validation results (a) output of plant (b) output of CFRNF (c) output of ANFIS (d) output of RNN

0.05

0

-0.05 0 5 10 15 20 25 30 (a) 0.05

0

-0.05 0 5 10 15 20 25 30 (b) velocity error(rps) 0.1

0

-0.1

-0.2

0 5 10 15 20 25 30 time (second) (c) Fig. 7 validation errors (a) validation error of CFRNF (b) validation error of ANFIS (c) validation error of RNN

959 Journal of Advanced Mechanical Design, Vol. 2, No. 5, 2008 Systems, and Manufacturing References

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