IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013 171
Fuzzy Nash Equilibriums in Crisp and Fuzzy Games
Alireza Chakeri and Farid Sheikholeslam
Abstract—In this paper, we introduce fuzzy Nash equilibrium to de-termine a graded representation of Nash equilibriums in crisp and fuzzygames. This interpretation shows the distribution of equilibriums in thematrix form of a game and handles uncertainties in payoffs. In addition, anew method to rank fuzzy values with the user’s viewpoint is investigated.By this means, the definition of satisfaction function, which provides theresult of comparison in the form of real value, is developed when users havepreferences regarding the payoffs.
Index Terms—Fuzzy game, fuzzy Nash equilibrium, fuzzy preferencerelation, fuzzy value, satisfaction function (SF).
I. INTRODUCTION
A game is a decision-making system that involves more than onedecision maker each having profits that conflict with each other. Astrategic game first defines each player’s actions (strategy). The com-bination of all the players’ strategies will determine an outcome tothe game and the payoffs to all players in which each player tries tomaximize his own payoff. The traditional game theory assumes thatall data of a game are known exactly by players. However, in realgames, the players are often not able to evaluate exactly the game dueto lack of information, imprecision in the available information of theenvironment, or the behavior of the other players. Initially, fuzzy setswere used by Butnariu [1] in noncooperative game theory. He usedfuzzy sets to represent the belief of each player for strategies of otherplayers. There have been several approaches to extend fuzzy coop-erative games; Butnariu [2] introduced core and stable sets in fuzzycoalition games where a degree of participation of players in a coali-tion is assigned. Moreover, Mares [3] considered fuzzy core in fuzzycooperative game where possibility of each fuzzy coalition is fuzzyinterval as an extension of core in classic TU games. In addition, hediscussed Shapely value in cooperative game with deterministic char-acteristic and fuzzy coalition. Fuzzy game theory has been applied tomany competitive decision-making situations [4]–[18]. Vijay et al. [4]considered a game with fuzzy goals and fuzzy parameters and provedthat such a game is equivalent to a primal–dual pair of certain fuzzylinear programming (FLP) problems with fuzzy goals and parameters.Chen and Larbani [6] discussed multiple attribute decision making witha two-person zero-sum game and achieved simpler criteria to solve thecorresponding FLP. Liu and Kao [7] defined the value of games infuzzy form. Nishizaki and Sakawa [8] discussed fuzzy bimatrix game,and by using a nonlinear programming, the equilibriums were searched.Maeda [9] studied zero-sum bimatrix games with fuzzy payoffs. In [10],a fuzzy differential game approach was proposed to solve the N-personquadratic differential noncooperative and cooperative game. Kima andLeeb [13] considered fuzzy constraints, as well as fuzzy preference,and proved some theorems on the existence of equilibrium. Song andKandel [14] used a multigoal problem, where the degree of satisfactionfor each goal was a fuzzy one and the overall payoff is a weightedsum of the satisfaction of all goals. They assumed that each player
Manuscript received April 11, 2010; revised November 14, 2010 andNovember 13, 2011; accepted April 13, 2012. Date of publication June 6,2012; date of current version January 30, 2013.
The authors are with the Department of Electrical and ComputerEngineering, Isfahan University of Technology, Isfahan, Iran (e-mail:[email protected]; [email protected]).
Digital Object Identifier 10.1109/TFUZZ.2012.2203308
has a fuzzy knowledge about his opponents mixed strategies, i.e., eachplayer assigns a membership function to the probability distribution ofhis opponents’ strategies. Garagic and Cruz [15] transformed a gamewith fuzzy strategies and fuzzy payoffs to a crisp game using fuzzyIF–THEN rules. Subsequently, they discussed the Nash equilibriums inthe equivalent crisp game; they proved that this crisp game has at leastone pure strategy Nash equilibrium. Li et al. [16] employed two fuzzyapproaches, including the fuzzy multicriteria decision-making methodand the theory of fuzzy moves to investigate the game of chicken.Their model incorporates the player’s subjective manner and impre-cise knowledge to the game model. In [17] and [18] a mathematicalprogramming approach of fuzzy matrix games with intuitionistic fuzzypayoffs and interval-valued intuitionistic fuzzy (IVIF) payoffs [19] wasdeveloped. It is proven that each matrix game with IVIF payoffs has asolution.
In this paper, we develop a new approach to N-person crisp andfuzzy noncooperative games to obtain Nash equilibriums for thesekinds of problems. The most significant advantages of using the pro-posed method are the range of game-theoretic problems that can beanalyzed and the information about equilibriums that can be obtain-able to players. In the proposed approach, the definition of equilibriumin crisp and fuzzy games has been generalized to show distribution ofNash equilibriums in matrix games as well as present the amount of op-timality of the players’ strategies by a degree. In this regard, we do notneed to determine whether a pure strategy is Nash equilibrium. Instead,we assign a graded membership to any pure strategy that describes towhat possibility it is Nash equilibrium. Hence, we can consider strate-gies with high degrees of equilibrium which are not necessarily theequilibrium points. In fuzzy games, the fuzzy Nash equilibrium ap-proach is more appropriate for real-world problems which are modeledby game theory. The proposed approach avoids loss of any informationthat happens by the defuzzification method in games and handles un-certainty of payoffs through all steps of finding the Nash equilibrium.It shall be noted that in this approach, the existence of theorem forequilibrium was not established, since the focus is not on the existenceof equilibrium but in the degree of equilibrium.
This paper is structured as follows. Section II introduces the conceptof the degree of being the Nash equilibrium in games with crisp pay-offs using the fuzzy preference relation. In Section III, a new functionmodeling the SF is defined. In this definition, the weights of the domainin the fuzzy values are considered directly in the formula. In addition,Section III explains a new approach in fuzzy games using the satisfac-tion function (SF). Moreover, the consequence of player’s viewpointsin Nash degrees of cells is discussed. The conclusion highlights themain findings of this paper.
II. GAMES WITH CRISP PAYOFFS
A. Noncooperative N-Person Games
This section contains the background on game theory, which areneeded to develop the fuzzy Nash equilibrium.
Games have been classified by the number of players, the numberof strategies, the nature of the payoffs function, and cooperativeness.A normal game consists of a set of players, their strategies, and thepayoffs available for all combinations of players’ strategies.
A noncooperative N-person strategic game can be formulated asfollows [20].
1) There are N players to be denoted by P1 , P2 , ..., PN .2) There are a finite number of alternatives for each player to choose
from. Let si denote the number of alternatives available to Pi , andfurther denote the index set {1, 2, ..., si} by Xi , with a typicalelement of Xi , designated as xi .
172 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013
TABLE ISAMPLE GAME WITH CRISP PAYOFFS
TABLE IIRESULTS OF THE GAME OF TABLE I
3) If ∀i, Pi chooses a strategy xi ∈ Xi , then the payoff forPi is a single number Πi (x1 , x2 , ...., xN ). In addition,{Πi (xi , x−i )|∀xi ∈ Xi , ∀x−i ∈ X1 × ... × Xi−1 × Xi+1 × ...×XN } is the set of all payoffs for Pi , where x−i denotes thestrategies chosen by other players.
4) Players play with a policy of maximizing their payoff, they takeinto account the possible rational choices of the other players,and they also make decision independently.
Unlike one-player decision making, where optimality has an explicitmeaning, in multiperson decision making, the optimality is in the formof Nash equilibrium. A pure strategy Nash equilibrium is a strategyset wherein, if a player knows his opponent’s strategy, he is totallysatisfied with his equilibrium strategy and is reluctant to change hisstrategy [21].
Definition 1: Classical game theory determines a cell as a Nashequilibrium if and only if it maximizes all players’ payoffs when otherplayers refrain from changing their strategies; (x∗
1 , . . . , x∗N ) is the pure
strategy Nash equilibrium if and only if [22]
Πi (x∗1 , . . . , x
∗i , . . . , x
∗N )
= Πi (x∗i , x
∗−i ) ≥ Πi (xi , x
∗−i )
= Πi (x∗1 , . . . , xi , . . . , x
∗N )
∀i ∈ {1, . . . , N} , ∀xi �= x∗i ∈ Xi . (1)
One can rephrase the classical game theory solution to a simplealgorithm in two steps.
1) Replace Πi (xi , x−i ) with 1 if Πi (x, x−i ) is maximized by x =xi ; otherwise, replace it with 0.
2) Find the minimum of the elements of each cell: if this valueequals 1, the cell is Nash equilibrium, and if this value gets zero,the cell is not Nash equilibrium.
For instance, consider the simple game of Table I where each cellincludes two crisp payoffs: the first for player 1 and the second forplayer 2. Players 1 and 2 have three strategies, namely, J1 , J2 , J3 andK1 , K2 , K3 , respectively. Table II shows the Nash equilibrium cells.
B. Fuzzy Nash Equilibriums in Games With Crisp Payoffs
According to classical game theory, if a player knowing others’strategies chooses a strategy whereby he cannot get maximum pay-off, he will completely regret his choice. However, just as the Table Idepicts, in real-world problems, there may be situations where the dif-ference between payoffs are negligible, e.g., payoffs 0.323 and 0.324
Fig. 1. Algorithm for finding the degree of belonging to the Nash Equilibriumfuzzy set for each cell of the crisp payoffs game.
in Table I. If a player chooses 0.323, he/she will be quite satisfied withhis/her choice. Therefore, one can consider 0.322, 0.323, and 0.324to have approximately the same value in Table I. This approximationprompted the employment of fuzzy logic to make a soft measurementbetween payoffs. In this paper, instead of the logic “greater than” rela-tion, a new measure is defined, i.e., that of the amount being greater,for instance, the potential amount of being greater between 1000 and 1is greater than that between 2 and 1.The amount of being greater maybe perceived differently by each particular player. In other words, itdepends on the mental state and beliefs of that player. The more metic-ulous and greedy a player is, the greater a difference he/she perceivesbetween slightly different payoffs.
A new term is defined in this paper to determine the amount ofbeing greater between two payoffs in (2); this value is interpretedas the preference between two payoffs. Then, using fuzzy preferencerelation matrix [23], priorities are calculated by the least deviationmethod [24], in which the priority vector is a vector which determinesthe degree of importance of alternatives. Here, the grades of being Nashequilibriums are modeled according to the priority that players feel fortheir strategies. This definition for the grade of being Nash equilibriumseems meaningful because, if a player knows the opponent’s strategy,he is satisfied with his strategy to the degree that this strategy haspriority for him. The greater priority the players feel for each cell, thegreater the possibility that the cell is the game’s equilibrium.
Algorithm: First, it is necessary to normalize all payoffs and makethe range between 0 and 1. Let a and b be the payoffs of selectingstrategies i, j, respectively, for a player if the opponents’ strategies arefixed. The amount of being greater between a and b can be calculatedas follows:
pij =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
min{
δ(a − b) + (1 − δ)(
a − b
b
)
+ .5, 1}
, a > b
0.5, a = b
1 − min{
δ(b − a) + (1 − δ)(
b − a
a
)
+ .5, 1}
, a < b.
(2)The algorithm that is shown in Fig. 1 is designed to determine towhat degree a cell belongs to the fuzzy set of Nash equilibriums. Thisalgorithm assigns each cell the minimum priority of players as the
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013 173
TABLE IIIPRIORITY MATRIX OF THE GAME OF TABLE I
TABLE IVDEGREE OF BEING NASH EQUILIBRIUM FOR THE GAME OF TABLE I USING
MINIMUM AS T-NORM OPERATOR
Fig. 2. Priority of payoffs (0.97, 0.3, 0.81) versus δ.
degree of being Nash equilibrium. However, the minimum operatorcan be replaced by any other T-norm.
Example 1: The algorithm that is described in Fig. 1 is implementedin the game of Table I assuming δ = 0.5. The results of the first twosteps are shown in Table III.
In the first row of the game (see Table I), player 2 has approximatelythe same payoffs and player 1 has relatively high payoffs for all cells.Therefore, one may expect they have approximately the same degrees ofbeing the Nash equilibrium. This is exactly what the proposed algorithmhas calculated and is obvious in the first row of Table IV. In otherwords, a relatively small difference in player 2’s payoff has resulted inthe distribution of an equilibrium degree in the first row.
The effect of δ on the priorities of payoffs (0.97, 0.3, 0.81) and (0.85,0.29, 0.58) is shown in Figs. 2 and 3, respectively. The only correlationthat can be identified in these figures is that as δ increases, the priorityof the highest payoff decreases. As the figures shows, there is no linearrelationship between δ and the priority.
Fig. 3. Priority of payoffs (0.85, 0.29, 0.58) versus δ.
III. GAMES WITH FUZZY PAYOFFS
A. Possibility of Being Greater Between Fuzzy Values
In the fuzzy decision making and fuzzy game theory with fuzzypayoffs, ranking the fuzzy value is a necessary procedure. Variousmethods to rank fuzzy subsets have been planned [25]–[32]. Althoughmost methods can only rank fuzzy values, in [25], the credibility mea-sure as the summation of possibility and necessity measure is used toshow the degree of greatness. However, their method cannot considerthe possibility distribution of fuzzy values. In [26]–[28], the SF as thetruth value of an arithmetic comparison between fuzzy values was in-troduced. However, the method in [28] can only rank the fuzzy valueswhen there are viewpoints but cannot show the degree of being greater.In addition, it is unclear how a fuzzy number with indefinite substancecan compare with viewpoint. In other words, the fuzzy sets have natureof possibility, but viewpoints are constructed as user’s preferences andinterests.
This paper introduces a new method to calculate SF when usershave a viewpoint. The user’s viewpoint is incorporated in the domainof value. By this means, each element x is extended to V (x) · x, whereV (x)is the user viewpoint. Hence, the membership function of fuzzyvalues is modified dependent on viewpoint as follows:
μA ′(x) = maxh (x )
μA (h(x)) (3)
where h is a multivalued mapping, i.e., h(V (x) · x) = x.In the proposed method, the length of α-cuts of the fuzzy value is
increased or decreased according to the user viewpoint. We suggestthe following formula for comparison between fuzzy values A and B,where their modified membership function is used:
SFV (A < B) =
∫ ∞−∞
∫ y
−∞μA ′(x) Θ μB ′(y) dx dy
∫ ∞−∞
∫ ∞−∞ μA ′(x) Θ μB ′(y) dx dy
(4)
SFV (B < A) =
∫ ∞−∞
∫ ∞y
μA ′(x) Θ μB ′(y) dx dy∫ ∞−∞
∫ ∞−∞
μA ′(x) Θ μB ′(y) dx dy(5)
where the operator Θ is a T-norm, e.g., it can be min or the multiplica-tion operator.
SFV (A < B)and SFV (A > B) determine the possibility of truthof the A < B and A > B, i.e., they represent the possibility that fuzzy
174 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013
TABLE VRANKING FUZZY VALUES A AND B USING METHOD IN [28]
TABLE VIRANKING FUZZY VALUES A AND B USING (4) AND (5)
Fig. 4. Two simple fuzzy values A and B.
Fig. 5. Three viewpoints V1 , V2 , and V3 show pessimistic, neutral, andoptimistic player, respectively.
number A is smaller than B and A is larger than B, respectively. It isobvious that SFV (A < B) + SFV (A > B) = 1.
Since A′ and B ′ in definition (3) are fuzzy values, all propertieswhich were confirmed in [28] hold for definitions (4) and (5) in thispaper.
Example 2: The following example is presented to show how theuser viewpoint affects the value of SF. It consists of the categories ofoptimism, pessimism, and neutral user. Consider two fuzzy values Aand B and three viewpoints V1 , V2 , and V3 shown in Figs. 4 and 5,respectively. Table V shows a comparison between fuzzy values A andB using the approach in [28], and Table VI shows the results of acomparison between fuzzy values A and B using (4) and (5).
As shown in Table V, the method to rank fuzzy values in [28]is insensitive to the user’s viewpoints, and it is unable to determinethe effect of viewpoints since it can only rank fuzzy values whenuser interests exist. However, it seems that the result for pessimisticplayers V1 gives A larger than B, the result of proposed method iscongruous as expected. If the user is a pessimistic one, he is satisfiedwith the lower value. Hence, the low value has great importance to theuser, and SFV (A < B) is less than 0.5, because A is closer to zero.Additionally, if a user is optimistic, he/she prefers to choose the highvalue, and the high value is more important to him/her than a low value.Table VI shows that SFV (A < B) for an optimistic user is bigger thanSFV (A < B) for a neutral user, since in the optimistic one, the high
TABLE VIISIMPLE GAME WITH FUZZY PAYOFFS
value has more weights and fuzzy valueB has value of domain in highvalue.
B. Fuzzy Nash Equilibriums in Games With Fuzzy Payoffs
There have been many studies to define a game in fuzzy parameters.As discussed earlier, a game has four main components: a set of players,a set of strategies for each player, a set of payoffs, and preferencerelationship. Defining each of these components as a fuzzy componentwould lead to a fuzzy game. Most of the previous works on fuzzy gamesis concerned with defining fuzzy payoffs and, as a result, defining apreference on these fuzzy payoffs. An example is in an election, wherethe candidates may select different campaign issues on which to focus.Different issues may bring different votes, and the number of votescan only be estimated. For instance, candidates may think that if theyconcentrate on a specific issue for each of the number of votes, there isa possibility. Fuzzy sets theory is shown to be an appropriate means tomodel these uncertainties.
In this paper, a new method is proposed for finding degree of beingNash equilibrium of each cell. This explanation determines the distri-bution of the degree of being Nash equilibrium in the matrix game.In fact, the algorithm in classical game theory, which is mentioned inSection II, is modified in the case of having uncertainty in payoffs,i.e., it is a generalization of the classical game theory algorithm. Inclassical game theory, a crisp payoff is clearly greater than another oneor not, but in the case of a fuzzy payoff, there is uncertainty in rankingfuzzy values. These uncertainties are shown by the degree of truth ofarithmetic comparison, i.e., the SF.
Definition 2: if each player has viewpoint Vi , every N-tuple strategy(x1 , . . . , xN ) has a possibility of being pure strategy Nash equilibriumwith the degree of
μNash (x1 , . . . , xN )
= min∀i∈{1 , . . . ,N }
(
min∀x �=xi ∈X i
(
SFV i
(Πi (x1 , . . . , xi , . . . , xN )
≥ Πi (x1 , . . . , x, . . . , xN )
)))
. (6)
In addition, in matrix games, it can be stated in two steps.1) Replace Πi (xi , x−i ) with min
∀x �=xi ∈X i
(SFV i(Πi (xi , x−i ) >
Πi (x, x−i ))).2) Find the minimum of the elements of each cell. This value is the
degree of being Nash equilibrium for that cell.Example 3: Regarding crisp games, two person games are discussed
because they are easier to be considered, but they can be generalized tomore than two players. For instance, consider the fuzzy game describedin Table VII where T (a, b) denotes a fuzzy triangular number with acenter on a and boundaries on a ± b such as Fig. 6. The results of theproposed algorithm on the game are shown in Tables VIII and IX forneutral players.
To analyze the effect of user viewpoint in the distribution of Nashdegrees, consider V1 and V3 in Fig. 5 for both players, in which V1
shows a pessimistic player, and V3 shows an optimistic one. Tables X
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 1, FEBRUARY 2013 175
Fig. 6. Simple fuzzy triangular number A (T(5,1)).
TABLE VIIIFINDING NASH EQUILIBRIUM DEGREE FOR EACH NEUTRAL PLAYERS
TABLE IXFINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR NEUTRAL PLAYERS
TABLE XFINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR PESSIMISM PLAYERS
TABLE XIFINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR OPTIMISM PLAYERS
and XI determine the graded Nash of each cell for pessimism andoptimism players, respectively.
Comparing the results of Tables X and XI, one can conclude that asplayers become less greedy, i.e., experiencing higher degrees of satis-faction from lower payoffs, the degrees of being the Nash equilibriumbecome more widely distributed in the games matrix, and the degreesgrow closer to each other. This effect occurs because when playersbecome less greedy, the priority of different payoffs will increase andbecome more similar to others. In the first row of the game, whenplayers are optimistic, they prefer to choose T (3, 2), and hence, thispayoff yields more SF than when players are neutral. Moreover, whenplayers are pessimistic, cells with a high Nash degree in neutral and
optimistic cases change to cells with a low Nash degree because of theinversion of the user viewpoint.
IV. CONCLUSION
In this paper, a new approach has been introduced to analyze gamesmore realistically than previous models. In the first part, only the pref-erence relationship is generalized to a fuzzy one, i.e., the relationshipof “greater than or equal” is extended to a fuzzy one, which describeshow much a crisp number is greater than or equal to another number. Incrisp games, a fuzzy preference relation was employed to compare pay-offs and calculate the priority of each payoff using the least deviationmethod. Using this priority, a value of being equilibrium is computed,and it is shown that this value yields more realistic results.
In the case of having fuzzy payoffs, the definition of SF when playershave viewpoints is improved. The proposed method incorporates playerviewpoints in the domain of fuzzy value and transforms it to anotherfuzzy value. The algorithm for finding the Nash degree of each cell isproposed. Finally, the effect of different viewpoints on the result of thegame is studied. Comparing the results with the fuzzy Nash equilibrium,the results obtained through this strategy were more sensitive to thepayoffs.
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A Three-Domain Fuzzy Wavelet System for SimultaneousProcessing of Time-Frequency Information and Fuzziness
Zhi Liu, C. L. Philip Chen, Yun Zhang, Han-Xiong Li,and Yaonan Wang
Abstract—Traditional wavelet system is a two-domain (time and fre-quency domains) wavelet system (2DWS), which works only in time andfrequency domains. The 2DWS is not able to treat time-frequency informa-tion and fuzziness simultaneously. For this reason, a three-domain (fuzzy,time, and frequency domains) fuzzy wavelet system (3DFWS) is proposed,where the three-domain mechanism provides a solution to handle fuzzy un-certainties and time-frequency information together. The major advantageof 3DFWS is able to use the prior knowledge via the novel fuzzy domainto analyze uncertain data and signals, which will enhance the potentials of2DWS. Experimental and simulation studies show that the performance ofthe proposed 3DFWS is superior to the traditional one for simultaneousprocessing of time-frequency and fuzziness.
Index Terms—Fuzzy system, fuzzy wavelet network (FWN), three-domain fuzzy wavelet system (3DFWS), uncertainty modeling.
I. INTRODUCTION
Wavelet transforms have provided a significant technique in sig-nal processing, because it offers multiresolution analysis in the time-frequency domain and is able to extract more information at differentfrequency bands or in different time intervals [1]–[3]. The wavelet sys-tems have been used to analyze the complicated time-varying signalsuccessfully [37], [39], [41], [42]. Recently, the wavelet systems havebeen integrated with numerous soft computing methods to enhance theadaptive and learning capability in complicated engineering situations[4]–[6], [35], [36]. The soft computing paradigms of wavelet networks(WNs), fuzzy systems, and Bayesian classifiers have been studied tofuse the advantages among these intelligent strategies together [7]–[13], [45], [46]. Wavelet neural networks have been studied to combinethe advantages of radial basis function (RBF) neural networks withwavelet systems for nonparametric estimation [10]. The fuzzy waveletnetwork (FWN) has been studied for the approximation of arbitrarynonlinear functions [11], dynamic modeling, and control [12], [13],where each fuzzy rule is employed a subwavelet neural network asthe consequent part. However, all the aforementioned methods stillstay at the stage of two-domain wavelet system (2DWS), where the2DWS would analyze the signals only via time and frequency domain.
Manuscript received June 28, 2011; revised March 6, 2012; accepted May15, 2012. Date of publication June 11, 2012; date of current version January 30,2013. This work was supported by the National Natural Science Foundation ofChina under Project 60974047 and Project U1134004, the Science and Technol-ogy Plan of Guangdong Province under Project 2009B010900051, the Scienceand Technology Plan of Guangzhou (2010Y1-C591), the 2011 Zhujiang NewStar, the FOK Ying Tung Education Foundation of China under Project 121061,the High-Level Professionals Project of Guangdong Province, the 973 Programof China under Projects 2011CB302801 and 2011CB013104, and the MacauScience and Technology Development Fund under Grant 008/2010/A1.
Z. Liu and Y. Zhang are with the Department of Automation, GuangdongUniversity of Technology, Guangzhou 510006, China (e-mail: [email protected]).
C. L. P. Chen is with the Faculty of Science and Technology, University ofMacau, Taipa, Macau (e-mail: [email protected]).
H.-X. Li is with the Department of System Engineering and EngineeringManagement, City University of Hong Kong, Hong Kong and also with theSchool of Mechanical and Electrical Engineering, Central South University,China (e-mail: [email protected]).
Y. Wang is with the Department of Electrical Engineering, Hunan University,Changsha 410079, China (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TFUZZ.2012.2204265
