INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 4, NO. 3, SEPTEMBER 2006

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Hysteresis Modeling using Fuzzy Subtractive Clustering Siamak Tafazoli, Mathieu Leduc and Xuehong Sun

Abstract— This paper summarizes work undertaken in the area of modeling Shape Memory Alloy (SMA) and airfoil hysteresis using a Sugeno-type fuzzy modeling approach based on subtractive clustering. Two alternative approaches to develop a fuzzy model for hysteresis are proposed and evaluated. The first consists in building a mirror image of the lower curve in order to model both curves concurrently and the second consists in modeling the upper and lower curves separately. In each case linear and quadratic Sugeno models were tested. Simulation results show that those models perform better than different types of interpolations and neural networks in term of root mean c 2006 Yang’s Scientific Research square (rms) error. Copyright ° Institute, LLC. All rights reserved. Index Terms— Hysteresis, shape memory alloys (SMA), airfoil, subtractive clustering, Takagi-Sugeno fuzzy modeling, neural network .

I. I NTRODUCTION

T

HE subject of Fuzzy Logic and Fuzzy modeling has received much attention in the past years due to its ability to model non-linear dynamics of complex systems [5]. Multi-valued functions such as hysteresis are considered hard non-linearities since they are complex to model and are considered a failure to linear analysis. Hysteresis is a nonlinear phenomenon that happens in a whole range of areas, namely in mechanical and magnetic systems. It is most usually associated with a magnetization curve whose path depends on whether the magnetizing force is increasing or decreasing [19]. Different types of modeling have been used to capture the non-linearity of hysteresis [10], [15]. Salvini et al. [17], [18] have used neural networks, trained with genetic algorithms, to generalize the Jiles-Atherton (JA) static hysteresis model for dynamic loops. The neural networks are used for the JA parameter estimation rather than modeling the hysteresis model itself. Dickinson, Hughes and Wen [7] have modelled Shape Memory Alloy (SMA) hysteresis using the Preisach model and used a feedback control approach. Gorbet, Wang and Morris [8] have also modelled SMA hysteresis using the Preisach model. Richter, Kubica and Gorbet [16] have once again modelled SMA hysteresis with the Preisach model, but with a notable difference: they replaced the traditional modeling of the wire heating using a first-order differential equation with constant Manuscript received December 13, 2005; revised July 21, 2006. Siamak Tafazoli, Mathieu Leduc and Xuehong Sun, Canadian Space Agency, 6767 route de l’Aeroport St-Hubert, QC, J3Y 8Y9, Canada. Email: [email protected](S. Tafazoli) Publisher Item Identifier S 1542-5908(06)10303-6/$20.00 c Copyright °2006 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on August 27, 2006 at http://www.YangSky.com/ijcc/ijcc43.htm

coefficients by a fuzzy heating model, which is tuned using a genetic algorithm. The Preisach model however, generally relies on a double integral resolution, which is computationally intensive. Closer to the approach presented in this paper, Azzerboni et al.[1] have used fuzzy logic to model hysteresis loops in magnetic materials by dividing the M-H plane into a given number of quadrants (each described by a membership function) and using a linguistic structure to describe the phenomenon. In this paper, we deal specifically with the subject of airfoil and SMA hysteresis modeling using Takagi-Sugeno fuzzy systems based on a subtractive clustering procedure. Actually, much like in the case of conventional hysteresis, we are dealing with two curves or a set of two outputs, each of which depends on whether the angle of attack of the airfoil is increasing or decreasing (in the case of airfoil hysteresis) or on whether temperature is increasing or decreasing (in the case of SMA hysteresis). The approaches described in this paper can be similarly applied to other hysteresis non-linearities. In order to better introduce the problem, we start, in Section II, with an overview of hysteresis. Fuzzy modeling and the modeling used for hysteresis are then explained in Section III. We particularly focus on the Sugeno-type modeling as one of the popular fuzzy modeling approaches. We illustrate how the model reduction for hysteresis takes place and we review two different approaches, namely (1) an optimized Sugeno model based on a transformation approach [19]; and (2) two separate optimized Sugeno models for the hysteresis curves. We consider, in the case of airfoils, the angle of attack as input and the normal force coefficient as output while in the case of SMAs, temperature is considered the input and the dimensional change the output. In all the above cases, we consider both first and second order Sugeno models. The main idea here is to find a structure of if-then rules of the reduced model that agrees well enough with the dynamics of the original model.

In order to make a comparison, we also model SMA and airfoil hysteresis using neural networks. Different from [17], [18], our neural network models are obtained from input and output data; while in [17], [18], the neural networks are trained to estimate parameters of JA static hysteresis model. For this purpose, Sections IV gives a brief introduction of neural networks and the method to find an optimal architecture. Finally, in Sections V and VI, results are presented for fuzzy logic and neural networks respectively. Sections VII draws some conclusions with respect to the merits of these fuzzy models.

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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 4, NO. 3, SEPTEMBER 2006

Lift

F = Resulting Lifting Force

Cho

rd

Drag

V Fig. 2. Angle of attack is measured between the chord line and the incoming air velocity vector.

Fig. 1.

Hysteresis non-linearity in the case of a SMA.

II. H YSTERESIS A. Overview Hysteresis can be described as a lagging or retardation of the effect, when the forces acting upon a body are changed, as if from velocity or internal friction. We can also see it as a temporary resistance to change from a condition previously induced, observed in magnetism, thermoelectricity, etc., on reversal of polarity. The phenomenon is best described visually in Fig. 1. This is an example of SMA hysteresis. The upper curve represents the evolution of the dimensional change when temperature is increasing and the lower curve represents its evolution when temperature is decreasing. B. SMA Hysteresis The shape memory effect refers to the way in which some materials can be deformed in a low temperature martensitic crystalline state yet always return to the same unique shape in a high temperature austenitic (parent) phase [7]. At higher temperatures, all the material is in the austenite phase. As the temperature is lowered, areas of martensite appear and grow until at sufficiently low temperatures all the material is in the martensite phase. Stress and strain can also affect such a transformation, and can control the exact nature of the martensite phase through the process of twinning. Here the temperature input is analogous to the electric field in ferroelectrics, the magnetic field in ferromagnetics or the angle of attack in the case of airfoils. The two main sources of hysteresis in SMAs are the static frictional effects associated with twinning and de-twinning processes (which can be considered as a from of domain wall pinning by “defects” with the twinning interfaces as domain walls and crystal cells along the twinning interfaces as defects), and those due to lattice defects [7]. C. Airfoil Hysteresis An airfoil is a device, which gets a useful reaction from air moving over its surface. When an airfoil is moved through the air, it is capable of producing lift. Wings, horizontal and vertical tail surfaces and propellers are all examples of airfoils. A reference line often used in discussing airfoils is the chord,

an imaginary straight line joining the extremities of the leading and trailing edges. The angle of attack (α) is the angle between the chord line and the incoming air velocity vector (Fig. 2). The airfoil is designed to increase the velocity of the airflow above its surface, thereby decreasing pressure above the airfoil. Simultaneously, the impact of the air on the lower surface of the airfoil increases the pressure below. This combination of pressure differential produces lift force upward, perpendicular to the velocity vector (Fig. 2). Hysteresis behavior is often observed in the curve of the normal force coefficient versus the angle of attack. Perhaps the most common occurrence is hysteresis around classical stall, where the lift loss at stall is not recovered until the angle of attack is significantly reduced. Low Reynolds number airfoils frequently exhibit separation bubble induced hysteresis, usually in the positive sense, that is, similar to stall hysteresis, but occasionally reversed. Changes in model geometry can induce hysteretic behavior. For example slot lip spoilers deployed in conjunction with Fowler flaps have been shown to exhibit significant aerodynamic hysteresis in the curve of the normal force coefficient versus the angle of attack for large flap deflections where none may occur for lower flap deflection angles [4], [21]. III. F UZZY MODELING A. Overview Fuzzy theory finds its application in dealing with problems in which there is imprecision caused by the absence of sharp criteria. A good example is the human language. Linguistic terms can be defined by fuzzy sets and we can formulate fuzzy if-then rules. Operators like AND and OR and the implication IF: THEN are defined and so we can calculate with statements given in this form. The application of fuzzy methods in modeling of non-linear systems is becoming more and more popular. It is our intention to follow this approach in an attempt at investigating the applicability of these methods to a hard non-linearity problem such as hysteresis. When employing fuzzy sets to cover the range of variation of the state variables, the resulting number of rules becomes large. The amount of rules is proportional to both the number of membership functions in the fuzzy sets and the number of input variables. Thus the size of the rule base constituting the model increases with the dimension of the problem. There are

TAFAZOLI, LEDUC & SUN, HYSTERESIS MODELING USING FUZZY SUBTRACTIVE CLUSTERING

various kinds of fuzzy modeling. One is to describe the inputoutput relation of data with a fuzzy relation. Another approach is to divide the input-output space into clusters. Fuzzy rules are then generated by projection of these clusters onto the input space. Each rule represents one or several clusters, which can be interpreted as local models. A method to avoid having to deal with rule bases whose size increase with the dimensionality of the problem, is to classify the training set of data points in a pre-determined number of fuzzy groups or clusters, based upon a similarity criterion. By representing each cluster by its center or prototype data point, a rule base of relatively small size is obtained, containing one rule per cluster. A variety of fuzzy clustering techniques are available and will be described in the next subsection.

does not involve any iterative non-linear optimization and the computation time grows only linearly with the dimension of the problem. After finding the fuzzy clusters to establish the number of fuzzy rules and the rule premises, we then optimize the rule consequents by applying the Least Square Estimation method (LSE) as explained below. The first task here is to assume the order of the Sugeno model. A first-order model is initially considered and is given by (for N rules): R1 : If X is A1 , then w1 (u) = p10 + p11 u .. . Ri : If X is Ai , then wi (u) = pi0 + pi1 u

B. modeling Hysteresis This subsection deals with the approximation of SMA and airfoil hysteresis by Sugeno-type fuzzy modeling [20]. Since hysteresis is a hard non-linearity, it is complex to model. To avoid complications due to the hard non-linearity of multivalued functions and to save processing time as a result, we want to investigate whether fuzzy system modeling in general, and Sugeno-type modeling specifically, is capable of dealing with this kind of problems. By using the very same input data and comparing the results from the fuzzy model with the output provided by the experimental model, we can determine to what extent, Sugeno-type modeling could be successful in simulating the experimental output. The first step in our procedure was to perform the clustering of the numerical data. The purpose of clustering is to distill natural groupings of data from a large data set and to obtain a concise representation of a system’s behavior. Different clustering algorithms have been developed. One that has been widely used is the fuzzy C-means (FCM) [2]. Its weakness is that, as with most non-linear optimization problems, its solution depends largely on an initial guess. In fact, starting with a given number of clusters and an initial guess for each cluster center, this algorithm will converge to a solution which could be just a local minimum or a saddle point of a cost function. Another algorithm, proposed by Yager and Filev [21], is called the Mountain Method. It estimates the number and the initial locations of cluster centers. More specifically, this method consists of making a grid of the data space and computing a potential value for each grid point based on its distances to the actual data points. Although this algorithm is simple and powerful, the computation time grows exponentially with the dimension of the problem. Due to its automatic generation capability of rules, we chose the subtractive clustering method by Chiu [4]. This algorithm presents a number of other advantages. First, each data point (not a grid point) is considered as a potential cluster center. As a result, the number of points to be evaluated is simply equal to the number of data points and independent of the dimension of the problem. Second, it is no longer necessary to specify a grid resolution. Third, we use the cluster estimation method as the basis of a fuzzy model identification algorithm. The latter

17

RN

.. . : If X is AN , then wN (u) = pN 0 + pN 1 u

where the parameters pi0 and pi1 are optimized by LSE. For a given input uj , the fuzzy model output w(uj ) is computed as N X

w(uj )

=

µAi (uj )wi (uj )

i=1 N X

µAi (uj )

i=1

=

N X

βij wi (uj ) =

i=1

N X

βij (pi0 + pi1 uj )

(1)

i=1

where βij =

µAi (uj ) N X

(2)

µAi (uj )

i=1

and the membership function µAi is given by “

µAi (uj ) = e

−4

uj −ci ri ·range

”2

(3)

where ri is the cluster radius, ci is the cluster center and the input range is defined as: range = max(input values) − min(input values)

(4)

Now considering all inputs uj , j = 1, 2, . . . , M w(u1 ) =

N X i=1

w(uM ) =

N X

βi1 (pi0 + pi1 u1 ) .. .

(5)

βiM (pi0 + pi1 uM )

i=1

or, expressed in matrix notation, − → − → AΠ = W

(6)

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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 4, NO. 3, SEPTEMBER 2006

Fig. 4.

Mirror Image approach.

Fig. 5.

Separation approach.

Fig. 3. Linear Sugeno models for clustered data (upper part) and membership functions (lower part).

where



β11 u1  .. A= . β1M uM

β11 .. .

... .. .

βN 1 u1 .. .

β1M

...

βN M uM

 βN 1 ..  .  βN M

→ − Π = ( p11 p10 . . . pN 1 pN 0 )T − → W = ( w1 . . . wM )T

(7)

(8) (9)

This is a least-squares estimation problem where A is a − → known constant matrix, W is a known vector of output values, → − and Π is a vector of parameters to be estimated. The wellknown pseudo-inverse solution that minimizes − → − → kA Π − W k2 (the squared L2-norm) is given by − → − → Π = ((AT A)−1 AT )W

(10)

(11)

Higher-order Sugeno models can also be considered and can be derived similarly. It is to be expected that the quadratic model can identify a system with a lower root-mean-squared (rms) errors for the same number of rules or could achieve equal performance with a smaller number of rules, when compared to the linear Sugeno model. Fig. 3 illustrates the approach in the case of linear models. Two different approaches were selected to transform the input-output data: Method 1: Producing a continuous curve by adding the “mirror” image of the lower curve to the end of the upper curve (the “Mirror Image” approach). The lower curve is mirrored around the rightmost input value and the input range is then increased accordingly. Fig. 4 illustrates this approach. Method 2: Separating the two curves into an upper curve and a lower curve (the “Separation” approach). It is noted that a single data point in both curves is shared to allow continuity of the models. Fig. 5 illustrates this approach. By using 67% of the data (2 out of each 3 data points), we identified an optimized Sugeno model for the mirror approach

(in each of the linear and quadratic cases) and two optimized models for the separation method (in each of the linear and quadratic cases). The minimization of the least square error on the training data was based on different clustering parameters such as accept ratio (²), reject ratio (²), squash factor (η) and cluster radius (ra ). The ranges of those parameters were [0.0− 1.0], [0.0 − 0.9], [0.1 − 3.0] and [0.1 − 1.0], respectively. The step sizes for accept ratio, reject ratio, squash factor and cluster radius were selected as 0.1, 0.1, 0.1 and 0.05, respectively. The number of clusters in each case was also chosen to be optimal after finding optimized models for a range of clusters (1 to 15). After developing the models, the remaining 33% of the data was used to test their performance. IV. N EURAL N ETWORK A. Overview An artificial neural network is a computational device to model the mapping between a set of input data and output data. Its basic computation unit is called a neuron. Typically, a neuron has several inputs and one output. It performs two steps computation. In the first step, each input is multiplied by its weight and the results are summed up. In the second step, the result from the first step is passed through an activation function, which can be a threshold function or a sigmoid

TAFAZOLI, LEDUC & SUN, HYSTERESIS MODELING USING FUZZY SUBTRACTIVE CLUSTERING

f (Σ) W 1 1,1

x1

W 1 1,2

f (Σ)

W 1 2,2

Fig. 6.

W 2 1,1

W 1 2,1 W 1 3,1

x2

19

W 1 3,2

W 2 1,2

f (Σ)

y1

W 2 1,3

f (Σ)

A feed forward neural network.

function, etc. Neurons are connected in various manners to form different neural networks. Fig. 6 is an example of a feed forward neural network. It has two neuron layers. Sometime, the inputs are accounted as one layer. x1 and x2 are the input; i y1 is the neural network output; The wj,k are weights which will be trained according to the training data. The objective of training is to find the adequate weights such that the output of the neural network will be as close as possible to the target output. For this purpose, many training algorithms are developed for updating the weights. One of the algorithms we use in the paper is Levenberg-Marquardt algorithm [11], [14]. The application of Levenberg-Marquardt to neural network training is described in [9]. This algorithm appears to be the fastest method for training moderate-sized feed forward neural networks (up to several hundred weights) [6]. It trains neural networks at a rate 10 to 100 times faster than the usual gradient descent back propagation method. The readers are referred to [3] for an introduction to the field of Neural Networks.

Fig. 7.

Performance for mirror image data.

Fig. 8.

Performance for upper curve data.

B. Performance Versus Number of Hidden Nodes for SMA Hysteresis We decided to use the feed forward neural networks to model the Hysteresis. In order to use the neural network to solve the problem at hand, the first problem need to be solved is to define the architecture of the neural network, i.e., how many layers and how many nodes in each layer. Essentially, this is a system identification problem [12]. We try to find a model to fit the training data well; at the same time, we are expecting this model to fit the test data well in the future. This is the generalization property of a model. Given a set of training data, there are many models that fit the training data well. The problem is how to find the one with good capacity of generalization. There is no systematic way to do this. The rule of thumb is to follow Occam’s razor principle [13], which states a preference for simple models. Thus, we started out by using simple architecture to conduct the experiments and find the good one. We have conducted extensive experiments to find optimal network architecture. We found that the architecture with one hidden layer and around five hidden nodes can achieve better results than other choices. Some of the detailed results are presented as follows. We used SMA Hysteresis data to conduct the experiments to decide the number of hidden nodes. The result plots are presented in Fig. 7, Fig. 8 and Fig. 9. From these plots, we can see when the number of hidden nodes gets large, the RMS errors for checking data gets large, though the RMS errors

for training data keeps small. This indicates the network gets overfitted when the number of hidden nodes is large. We found the generation capability is good around five hidden nodes. We also conducted experiments using Airfoil Hysteresis data. Similar results are produced. Relevant plots are presented in Fig. 10, Fig. 11 and Fig. 12. V. R ESULTS FOR F UZZY MODELING A. “Mirror Image” Approach for SMA Hysteresis The linear model has a rms error of 2.18E − 03 for the training data and has the following clustering parameters: ra = 0.15, ² = 0.1, ² = 0.1 and η = 1.6. The rms error for the checking data is 5.69E − 03. The quadratic model has a rms error of 2.47E − 03 for the training data and has the following clustering parameters: ra = 0.1, ² = 0.2, ² = 0.2 and η = 3. The rms error for the checking data is 5.76E − 03. The linear model required 12 rules whereas the quadratic one required only 7. The rules of these models as well as their cluster centers are presented in Table I and Table II. Relevant plots

20

Fig. 9.

INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 4, NO. 3, SEPTEMBER 2006

Performance for lower curve data.

Fig. 10.

Performance for mirror image data.

Fig. 11.

Performance for upper curve data.

Fig. 12.

Performance for lower curve data.

Fig. 13. Actual data and fuzzy model for training data (linear model: 12 rules and quadratic model: 7 rules).

Fig. 14. Actual data and fuzzy model for checking data (linear model: 12 rules and quadratic model: 7 rules).

TAFAZOLI, LEDUC & SUN, HYSTERESIS MODELING USING FUZZY SUBTRACTIVE CLUSTERING

21

TABLE I 12-RULE LINEAR SUGENO MODEL Rule i

Cluster center

1

-67.00

0.0119

2

-37.94

0.0049

3

-8.38

4

21.87

5

TABLE III 6-RULE LINEAR SUGENO MODEL

Consequents Pi1 Pi0

Rule i

Cluster center

1.2628

1

-67.00

-173.6

-64556

0.5824

2

-27.93

-14449.2

-4865405

-0.0055

0.1811

3

-8.38

-42642.7

1215591

-0.0046

0.2940

4

21.87

-28946.5

2131173

61.81

-0.0044

0.3584

5

52.62

-9582.7

1708993

6

99.85

-0.0086

0.8351

6

81.98

-789.6

303638

7

131.89

-0.0126

1.5772

8

161.76

-0.0259

3.9382 8.8859

TABLE IV 4-RULE QUADRATIC SUGENO MODEL

9

188.78

-0.0480

10

218.47

-0.1442

28.203

11

235.38

-0.1768

44.5571

12

255.43

-0.0341

10.5016

TABLE II 7-RULE QUADRATIC SUGENO MODEL Rule i

Pi2

Consequents Pi1

Pi0

-37.94

-0.0002

-0.0174

0.0551

2

-8.38

0.0001

-0.0058

0.2438

3

52.62

0

-0.0027

0.2299

4

131.89

0

-0.0062

0.4172

5

188.78

0

0.0041

-0.4989

6

224.74

0.0001

-0.0446

4.4912

7

255.43

-0.0001

0.0699

-9.2796

1

Cluster center

Consequents Pi1 Pi0

Rule i

Cluster center Pi2

Consequents Pi1 Pi0

1

-57.63

-0.5

-199.7

2

-8.38

-3.3

-1067.0

-19987 -84897

3

52.62

18.3

-226.0

326119

4

95.64

-4.7

3029.7

-551268

for each linear and quadratic cases are presented in Fig. 13 and Fig. 14. B. “Separation” Method for SMA Hysteresis: Upper Curve (Increasing Temperature) The linear model has a rms error of 8.58E − 06 for the training data and has the following clustering parameters: ra = 0.85, ² = 0.6, ² = 0.1 and η = 0.3. The rms error for the checking data is 1.01E − 02. The quadratic model has a rms error of 8.36E − 06 for the training data and has the following clustering parameters: ra = 0.9, ² = 0.1, ² = 0.1 and η = 0.7. The rms error for the checking data is 1.01E − 02. The linear model required 6 rules whereas the quadratic one required 4. The rules of these models as well as their cluster centers are presented in Table III and Table IV. Relevant plots for each linear and quadratic cases are presented in Fig. 15 and Fig. 16.

Fig. 15. Actual data and fuzzy model for training data (linear model: 6 rules and quadratic model: 4 rules).

C. “Separation” Method for SMA Hysteresis: Lower Curve (Decreasing Temperature) The linear model has a rms error of 2.26E − 03 for the training data and has the following clustering parameters: ra = 0.25, ² = 0.6, ² = 0.6 and η = 0.4. The rms error for the checking data is 2.37E − 03. The quadratic model has a rms error of 3.08E − 03 for the training data and has the following clustering parameters: ra = 0.45, ² = 0.1, ² = 0.1 and η = 1.9. The rms error for the checking data

Fig. 16. Actual data and fuzzy model for checking data (linear model: 6 rules and quadratic model: 4 rules).

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TABLE V 6-RULE LINEAR SUGENO MODEL Rule i

Cluster center

Consequents Pi1 Pi0

1

15.71

-0.0036

2

29.52

0.0917

0.8977

3

44.51

0.6680

-18.1152

4

59.39

1.5045

-80.9794

5

73.62

1.3168

-109.0370

6

87.19

0.2949

-37.1231

0.1344

TABLE VI 2-RULE QUADRATIC SUGENO MODEL Rule i

Cluster center Pi2

Consequents Pi1

Pi0

1

-59.17

-1.148E-05

-0.0061

0.0122

2

49.51

2.437E-05

-0.0027

0.0614

is 2.47E − 03. The linear model required 6 rules whereas the quadratic one required only 2. The rules of these models as well as their cluster centers are presented in Table V and Table VI. Relevant plots for each linear and quadratic cases are presented in Fig. 17 and Fig. 18. D. “Mirror Image” Approach for Airfoil Hysteresis The linear model has a rms error of 8.43E − 03 for the training data and has the following clustering parameters:ra = 0.15, ² = 0.2, ² = 0.1 and η = 1.4. The rms error for the checking data is 1.79E − 02. The quadratic model has a rms error of 7.32E − 03 for the training data and has the following clustering parameters: ra = 0.15, ² = 0.7, ² = 0.4 and η = 0.3. The rms error for the checking data is 2.09E−02. The linear model required 12 rules whereas the quadratic one required only 7. The rules and the plots are not presented here due to space limitations.

Fig. 18. Actual data and fuzzy model for checking data (linear model: 6 rules and quadratic model: 2 rules).

E. “Separation” Method for Airfoil Hysteresis: Upper Curve (Increasing Angle) The linear model has a rms error of 2.33E − 02 for the training data and has the following clustering parameters: ra = 0.5, ² = 0.6, ² = 0.6 and η = 0.5. The rms error for the checking data is 1.67E − 02. The quadratic model has a rms error of 1.79E − 02 for the training data and has the following clustering parameters: ra = 0.55, ² = 0.1, ² = 0.1 and η = 1.8. The rms error for the checking data is 1.65E−02. The linear model required 3 rules whereas the quadratic one required only 2. The rules and the plots are not presented here due to space limitations. F. “Separation” Method for Airfoil Hysteresis: Lower Curve (Decreasing Angle) The linear model has a rms error of 5.31E − 03 for the training data and has the following clustering parameters: ra = 0.35, ² = 0.8, ² = 0.7 and η = 0.3. The rms error for the checking data is 9.06E − 03. The quadratic model has a rms error of 4.30E − 03 for the training data and has the following clustering parameters: ra = 0.35, ² = 0.8, ² = 0.8 and η = 0.5. The rms error for the checking data is 6.95E−03. The linear model required 6 rules whereas the quadratic one required only 4. The rules and the plots are not presented here due to space limitations. G. Performance Versus Number of Clusters

Fig. 17. Actual data and fuzzy model for training data (linear model: 6 rules and quadratic model: 2 rules).

The quadratic Sugeno models, when compared to the linear ones, achieve comparable or better performance with a smaller number of rules (clusters). Fig. 19 to Fig. 24 illustrate the relationship between the number of clusters and the rms errors for both the training data and the checking data. Note that each model (for each number of clusters) was optimized based on the same procedure as before. It can also be seen that in both linear and quadratic cases, models with too many clusters tend to overfit the training data and perform poorly (with large rms errors for the checking data). It is also to be noted that the

TAFAZOLI, LEDUC & SUN, HYSTERESIS MODELING USING FUZZY SUBTRACTIVE CLUSTERING

Fig. 19.

SMA mirror curve model performance.

Fig. 20.

SMA upper curve model performance.

performance of the quadratic models starts to decrease (rms errors for checking data increase) with a smaller number of clusters than in the linear cases; the quadratic models therefore overfit the training data more than the linear ones due to their higher number of parameters.

Fig. 21.

SMA lower curve model performance.

Fig. 22.

Airfoil mirror curve model performance.

Fig. 23.

Airfoil upper curve model performance.

VI. R ESULTS FOR N EURAL N ETWORKS The detailed results for neural networks are presented as follows. A. “Mirror Image” Approach for SMA Hysteresis These are the results using SMA hysteresis data. The neural network has a RMS error of 0.0047793 for the training data and 0.0091891 for checking data. Relevant plots are presented in Fig. 25 and Fig. 26. B. “Separation” Method for SMA Hysteresis: Upper Curve (Increasing Temperature) For upper curve data, the neural network has a RMS error of 0.0003138 for the training data and 0.0040532 for checking

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Fig. 24.

Fig. 25.

INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 4, NO. 3, SEPTEMBER 2006

Airfoil lower curve model performance.

Fig. 27.

Actual data and Neural Network prediction for training data.

Fig. 28.

Actual data and Neural Network prediction for checking data.

Actual data and Neural Network prediction for training data.

data. Relevant plots are presented in Fig. 27 and Fig. 28. C. “Separation” Method for SMA Hysteresis: Lower Curve (Decreasing Temperature) For lower curve data, the neural network has a RMS error of 0.0021043 for the training data and 0.0046909 for checking data. Relevant plots are presented in Fig. 29 and Fig. 30. D. “Mirror Image” Approach for Airfoil Hysteresis The neural network has a RMS error of 0.0270411 for the training data and 0.0313319 for checking data. Relevant plots are presented in Fig. 31 and Fig. 32. E. “Separation” Method for Airfoil Hysteresis: Upper Curve (Increasing Angle) Fig. 26.

Actual data and Neural Network prediction for checking data.

The neural network has a RMS error of 0.0118454 for the training data and 0.0345624 for checking data. Relevant plots are presented in Fig. 33 and Fig. 34.

TAFAZOLI, LEDUC & SUN, HYSTERESIS MODELING USING FUZZY SUBTRACTIVE CLUSTERING

Fig. 29.

Actual data and Neural Network prediction for training data.

Fig. 32.

Actual data and Neural Network prediction for checking data.

Fig. 30.

Actual data and Neural Network prediction for checking data.

Fig. 33.

Actual data and Neural Network prediction for training data.

Fig. 31.

Actual data and Neural Network prediction for training data.

Fig. 34.

Actual data and Neural Network prediction for checking data.

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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 4, NO. 3, SEPTEMBER 2006

Fig. 35.

Actual data and Neural Network prediction for training data.

Fig. 36.

Actual data and Neural Network prediction for checking data.

F. “Separation” Method for Airfoil Hysteresis: Lower Curve (decreasing Angle) The neural network has a RMS error of 0.0061308 for the training data and 0.0171966 for checking data. Relevant plots are presented in Fig. 35 and Fig. 36. G. Summary Statistics and Performance Table VII and Table VIII compare the RMS among the linear and quadratic fuzzy models (for an optimal number of clusters and optimized model parameters), linear and cubic spline interpolations and neural networks. It can be seen that the fuzzy models perform better than both types of interpolation and neural networks in almost all cases. It is however unclear which fuzzy model order (linear or quadratic) performs better for an optimal number of clusters. VII. C ONCLUSIONS The problem of hysteresis for SMAs, or any other physical system for that matter, can be solved in different ways. In

this paper we are considering two different fuzzy methods of solving the problem or developing the model for the hysteresis curve: • By transformation of the input data, i.e., increasing the range of the input, creating a mirror image of the lower curve and then developing a single model for the newly found function. • By separating the data into two parts, one for the upper part of the curve and the other for the lower part. In this case, two different models are developed for two different input-output systems, which have a common data point in order to ensure continuity of the model. Both models were generated by taking 67% of the initial data as training data and the remaining 33% were used as checking data. Both models generated were found to agree well with the original data within the given rms errors. It is not clear which approach (mirror image or separation) yields better results. Both approaches are therefore recommendable for tackling hysteresis modeling. The quadratic fuzzy models generally achieve better performance (for a given number of rules) than the linear ones with both the training and checking data. However, as the number of rules is increased, both the quadratic and linear fuzzy models tend to overfit the training data and perform poorly (and the quadratic models have a tendency to overfit more quickly). The fuzzy logic modeling approaches presented in this paper perform better than linear and cubic spline interpolation techniques, and better than neural networks modeling approaches, as was shown. Additional work could also include a further optimization of the membership functions and the consequents parameters considered (e.g. ANFIS). This could prevent a possible overfitting of the training data by using an additional data set (different from the validation data set). ACKNOWLEDGMENT The authors would like to acknowledge the contributions of Dr. D. Nikanpour and Dr. A. Mohanad for providing the experimental SMA and airfoil data, respectively. R EFERENCES [1] B. Azzerboni, M. Carpentieri, G. Finocchio, M. Ipsale, and F. La Foresta, “Fuzzy Approach to Modelling Scalar Hysteresis,” Proceedings of the IEEE International Magnetics Conference, Boston, MA, USA, March 30-April 3, 2003. [2] J.C. Bezdek, “Cluster validity with fuzzy sets,” Journal of Cybernetics, vol. 3, pp. 58-71, 1974. [3] M. Caudill, Neural Networks Primer, San Francisco, CA: Miller Freeman Publications, 1989. [4] S. L. Chiu, “Fuzzy Model Identification Based on Cluster Estimation,” Journal of Intelligent and Fuzzy systems, vol.2, pp. 267-278, 1994. [5] K. Demirli and P. Muthukumaran, “Fuzzy System Identification with Higher Order clustering,” Journal. of Intelligent and Fuzzy Systems, Vol. 9, No. 3-4, 2000, pp. 129-158. [6] Howard Demuth, Mark Beale, Neural Network Toolbox, User’s Guide, Version 4, 2004, The MathWorks, Inc. [7] C. A. Dickinson, D. C. Hughes, and J.T. Wen, “Hysteresis in Shape Memory Alloy Actuators: the Control Issues,” Proceedings of SPIE, The International Society for Optical Engineering, vol. 2715, pp. 494-506, 1996. [8] R.B. Gorbet, D.W.L. Wang, and K.A. Morris, “Preisach Model Identification of a Two-Wire SMA Actuator,” Proceedings of the IEEE International Conference on Robotics & Automation, Leuven, Belgium, May 1998.

TAFAZOLI, LEDUC & SUN, HYSTERESIS MODELING USING FUZZY SUBTRACTIVE CLUSTERING

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TABLE VII RMS ERRORS FOR SMA CHECKING DATA USING THE FUZZY MODELS, NEURAL NETWORK AND TWO TYPES OF INTERPOLATION (NUMBER OF CLUSTERS FOR FUZZY MODELS APPEAR IN PARENTHESES) Interpolation Linear Cubic Spline

Fuzzy Models Linear Quadratic

NN

Mirror

8.06E-03

6.85E-03

5.69E-03 (12)

5.76E-03 (7)

9.19 E-03

Upper

1.27E-02

9.46E-03

1.01E-02 (6)

1.01E-02 (4)

4.05 E-03

Lower

4.20E-03

3.56E-03

2.37E-03 (6)

2.47E-03 (2)

4.69 E-03

TABLE VIII RMS ERRORS FOR AIRFOIL CHECKING DATA USING THE FUZZY MODELS, NEURAL NETWORK AND TWO TYPES OF INTERPOLATION (NUMBER OF CLUSTERS FOR FUZZY MODELS APPEAR IN PARENTHESES) Interpolation Linear Cubic Spline

Fuzzy Models Linear Quadratic

NN

Mirror

3.38E-02

2.86E-02

1.79E-02 (12)

2.09E-02 (7)

3.13 E-02

Upper

4.50E-02

4.95E-02

1.67E-02 (3)

1.65E-02 (2)

3.46 E-02

Lower

1.51E-02

1.40E-02

9.06E-03 (6)

6.95E-03 (4)

1.72 E-02

[9] Hagan, M. T., and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Transactions on Neural Networks, vol. 5, no. 6, pp. 989-993, 1994. [10] D. Landman, and C. P. Britcher, “Experimental Investigation of Multielement Airfoil Lift Hysteresis due to Flap Rigging,” Journal of Aircraft, vol. 38, no. 4, pp. 703-708, July-August 2001. [11] K. Levenberg, “A method for the solution of certain problems in least squares,” Quart. Appl. Math., 1944, Vol. 2, pp. 164-168. [12] Lennart Ljung, System Identification - Theory For the User, 2nd ed, PTR Prentice Hall, Upper Saddle River, N.J., 1999. [13] D.J.C. MacKay, “Bayesian Methods for Adaptive Models,” Ph. D thesis, California Institute of Technology, 1991. [14] D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math., 1963, Vol. 11, pp. 431-441. [15] T. J. Mueller, “The Influence of Laminar Separation and Transition on Low Reynolds Number Airfoil Hysteresis,” Journal of Aircraft, vol. 22, no. 9, pp. 763-770, September 1985. [16] J. Richter, E. Kubica, and R. Gorbet, “Fuzzy Modelling of a Shape Memory Alloy Actuator,” Proceedings of 2nd Canada-US CanSmart Workshop on Smart Materials and Structures, Montreal, Quebec, Canada, 10-11 October 2002. [17] A. Salvini, and F.R. Fulginei, “Genetic Algorithms and Neural Networks Generalizing the Jiles-Atherton Model of Static Hysteresis for Dynamic Loops,” IEEE Transactions on Magnetics, vol. 38, no. 2, pp. 873-876, March 2002. [18] A. Salvini, F.R. Fulginei, and C. Coltelli, “A Neuro-Genetic and TimeFrequency Approach to Macromodelling Dynamic Hysteresis in the Harmonic Regime,” IEEE Transactions on Magnetics, vol. 39, no. 3, pp. 1401-1404, May 2003. [19] S. Tafazoli and K. Demirli, “Fuzzy Modelling of Hysteresis from InputOutput Data,” Joint 9th IFSA and 20th NAFIPS International Conference Proc., pp. 3009-3014, July 25-28, 2001, Vancouver, Canada. [20] T. Takagi, and M. Sugeno, “Fuzzy Identification of Systems and its applications to Modelling and Control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116-132, 1985. [21] R.R. Yager, and D.P. Filev, “Approximate Clustering Via the Mountain Method,” IEEE Transactions on Systems, Man, and Cybernetics, vol 24, no. 8, pp. 1279-1284, August 1994.