Dynamic representation of fuzzy knowledge based on fuzzy petri net ...
Expert Systems with Applications 41 (2014) 1369–1376
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Dynamic representation of fuzzy knowledge based on fuzzy petri net and genetic-particle swarm optimization Wei-Ming Wang ⇑, Xun Peng, Guo-niu Zhu, Jie Hu, Ying-Hong Peng School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Keywords: Fuzzy knowledge Petri nets Knowledge representation Learning algorithms Particle swarm optimization
a b s t r a c t Information in some fields like complex product design is usually imprecise, vague and fuzzy. Therefore, it would be very useful to design knowledge representation model capable to be adjusted according to information dynamics. Aiming at this objective, a knowledge representation scheme is proposed, which is called DRFK (Dynamic Representation of Fuzzy Knowledge). This model has both the features of a fuzzy Petri net and the learning ability of evolutionary algorithms. An efficient Genetic Particle Swarm Optimization (GPSO) learning algorithm is developed to solving fuzzy knowledge representation parameters. Being trained, a DRFK model can be used for dynamic knowledge representation and inference. Finally, an example is included as an illustration. 2013 Published by Elsevier Ltd.
1. Introduction In the real world, there exist many problems people have not had a fundamental understanding of, and the information people are obtaining is uncertain information. Most human knowledge, how-ever, is typically expressed in vague and imprecisely defined concepts and the inference is mostly supported by common-sense and intuitive reasoning (Ribaric & Hrkac, 2012). Aiming to problem solving in specific areas, the knowledgebased systems depends not only on theoretical knowledge determined in specific areas, but more on experience and common sense of experts. The uncertainty of objective things or phenomena in the real world leads to the fact that people’s information and knowledge in various cognitive domains are mostly inaccurate, which requires that the knowledge representation and processing model in the expert system can reflect this uncertainty. Therefore, how to represent and process the uncertainty of knowledge has become one of the important research issues on artificial intelligence. There are important issues underlying knowledge representation that have not yet been adequately addressed. Such issues are those of the modeling and verification of the knowledge-based systems (Ashon, 1995; Mengshoel & Delab, 1993). As a case in point, the conventional techniques such as simulation methods and analytical methods do not provide tools for representing the dynamic behavior of the KBSs as well as for modeling the different aspects of fuzzy information of these systems (Garg, Ahson, & Gupta, 1991; Polat & Guvenir, 1993; Zadeh, 1989).
⇑ Corresponding author. Tel.: +86 (0) 21 34206170; fax: +86 (0) 21 34206179. E-mail addresses: [email protected], [email protected] (W.-M. Wang). 0957-4174/$ - see front matter 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.eswa.2013.08.034
As an important modeling and computational paradigm, Petri nets (PN) have been widely used (Murata, 1998). Machine learning with fuzzy AND-OR neurons and with fuzzy Petri nets have been proposed by Pedrycz (1989). In order to deal with uncertain information or knowledge, fuzzy Petri nets (FPNs) have been introduced, which can be used to represent Horn clauses or Non-Horn clauses and represent and execute the fuzzy rules (Jeffrey, Lobo, & Murata, 1996; Konar & Mandal, 1996). To improve the FPN adjusting (or learning) ability, a generalized fuzzy Petri net (GFPN) is proposed (Pedrycz & Gomide, 1994). To solve the learning problem, an adaptive fuzzy Petri net (AFPN) is proposed by adjusting weights the same as those in a neural network (NN) (Looney, 1994). But the weight adjustment is off-line. To overcome this weakness, several researchers have recently investigated FPN learning ability by using evolutionary algorithms (Huang, Yang, Wang, & Tsai, 2010). In fact, the knowledge learning above was under the framework of neural networks. Adaptive Fuzzy Petri Net (AFPN) has also the learning ability of a neural network, but it does not need to be transformed into neural networks, and Back Propagation algorithm is developed for the knowledge learning under generalized conditions (Li & Lara-Rosano, 2000). However, the learning algorithm is based on a special transition firing rule, it is necessary to know certainty factors of each consequence proposition in the system. Obviously, this restriction is too strict for a knowledge based system. Instead of using the Neural Network, this study tries swarm intelligence: particle swarm optimization (PSO). Proposed by Kennedy and Eberhart and inspired by social behavior in nature, PSO is a population-based search algorithm that is initialized with a population of random solutions, called particles (Kennedy, 1995; Kennedy, Eberhart, & Shi, 2001). Each particle in the PSO flies
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through the search space at a velocity that is dynamically adjusted according to its own and its companion’s historical behavior. Because particle swarm optimization is powerful, easy to implement, and computationally efficient, numerous researches on PSO theories or applications have been reported Recently (Chatterjee & Siarry, 2006; Jiang & Etorre, 2005; Mendes, Kennedy, & Neves, 2004; Shi, Liang, Lee, Lu, & Wang, 2005). The rest of the paper is organized as follows. The related work concerning the proposed scheme is introduced in Section 2. The proposed model of dynamic representation of fuzzy knowledge (DRFK) is covered in Section 3. The GPSO learning algorithm for solving knowledge representation parameters is presented in Section 4. An illustrative case study, fault diagnose of launch vehicle is presented in Section 5. Finally, conclusions are drawn in Section 6. 2. Fuzzy production rules and fuzzy Petri net 2.1. Fuzzy production rules Fuzzy production rules (FPRs) are widely used in knowledge based systems to represent fuzzy and uncertain concepts. FPRs are usually presented in the form of a fuzzy IF-THEN rule in which both the antecedent and the consequent are fuzzy concepts, denoted by fuzzy sets. To effectively represent both the fuzziness and the uncertainty in FPRs, several knowledge parameters such as confidence value, weight, and threshold have been incorporated into FPRs. Thus, a FPR has the following format: Definition 1. R : If AthenCðCF ¼ lÞ; Th; W where A ¼ ða1 ; a2 ; . . . ; an Þ is the antecedent portion which comprises one or more propositions connected by either ‘‘AND’’ or ‘‘OR’’. Each proposition ai ð1 6 i 6 nÞ may have a fuzzy variable of the antecedent. The parameter l is the confidence value of the rule and it represents the strength of belief of the rule. The symbol Th represents a set of threshold values specified for the propositions in the antecedent A. The set of weights assigned to the propositions a1 ; a2 ; . . . ; an , is given by W ¼ ðw1 ; w2; . . . ; wn Þ.
be powerful modeling formalisms. Modeling using FPN has many advantages compared to other modeling schemes. The graphic representation of FPN makes the models relatively simple and legible. The knowledge representation based on FPN has two basic general formulation, shown as ‘‘AND’’ rule representation model based on FPN (as Fig. 1) and ‘‘OR’’ rule representation model based on FPN (as Fig. 2). The fuzzy production rules based on the Petri Net is shown in Fig. 3, where pj (j = 1, 2, . . ., n) denotes the antecedent (premise) part of a given rule, it defines the fuzzy region in the input space; di (i = 1–3) denotes the consequent (decision, conclusion) part of that rule, it specifies the output in the fuzzy region; li (i = 1–3) is the confidence value that is associated with the conclusion being pursued. wij is the weight that weighs the importance of every antecedent to its consequent propositions. k is the threshold value. The model includes three rules:
IF p1 and p2 THEN d1 ðk1 ; l1 ; x11 ; x12 Þ IF p1 and p3 and p4 THEN d2 ðk2 ; l2 ; x21 ; x22 Þ IF p1 and p4 and p5 THEN d3 ðk3 ; l3 ; x31 ; x32 ; x33 Þ The knowledge representation parameters of the FPN is updated or modified frequently, it may be regarded as dynamic systems. Suitable models for them should be adaptable. In other words, the models must have ability to adjust themselves according to the systems’ changes. However, the lack of adjustment (learning) mechanism in FPNs cannot cope with potential changes of actual systems. Therefore, it would be very useful to develop the parameters self-learning mechanism in the FPN capable to be adjusted like human cognition and thinking, according to knowledge dynamics to achieve the dynamic representation of fuzzy knowledge. The model of DRFK is shown in Fig. 4. Aiming at the adjustment (self-leaning) of the representation parameters in the DRFK, firstly, building the physical model of DRFK based on fuzzy Petri net to organize the fuzzy knowledge representation parameters.
2.2. Fuzzy Petri net Definition 2. An FPN is a 9-tuple, given by FPN = hP, Tr, T, D, I, O, Th, n, Wi where P = (p1, p2, . . ., pn) is a finite set of places; Tr = (tr1, tr2, . . ., trm) is a finite set of transitions; T = (t1, t2, . . ., tm) is a set of fuzzy truth tokens in the interval [0, 1] associated with the transitions (tr1, tr2, . . ., trm),respectively; D = (d1, d2,. . .,dn) is a finite set of propositions, where proposition dk corresponds to place pk; P \ Tr \ D = £; cardinality of (P) = cardinality of (D); I:Tr ? P1 is the input function, representing a mapping from transitions to bags of (their input) places; O:Tr ? P1 is the output function, representing a mapping from transitions to bags of (their output) places; TH = (th1, th2, . . ., thm) represents a set of threshold values in the interval ½0; 1 associated with transitions (tr1, tr2, . . ., trm), respectively; n:P ? [0, 1] is an association function hereafter called fuzzy belief, representing a mapping from places to real values between 0 and 1; n(pi) = ni(say); W = wij is the set of weights from the jth transition to the ith place, where i and j are integers. 3. The model of dynamic representation of fuzzy knowledge Fuzzy production rule is one of the most basic researched areas of knowledge based engineering (KBE). We have chosen the fuzzy Petri net (FPN) as an alternative modeling and analysis formalism. FPN formalism is a derivative of PNs which have been demonstrated to
p1 d1
w1
p2 d 2
w2
pn d n
μ, λ d P
.. .
t
wn
Fig. 1. ‘‘AND’’ rule.
p1 d1 p2 d 2 .. .
pn d n
μ1 , λ1 t 1
μ2 , λ2 t 2
d P
μn , λn tn Fig. 2. ‘‘OR’’ rule.
p1 p2 p3 p4 pn
w11 w12 w21 w22 w23 w31 w32 w33
λ1 μ1
d1
rule1
λ1 μ1
d2
rule2
λ1 μ1
d3
rule3
Fig. 3. Fuzzy knowledge representation based on Petri net.
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Fig. 4. Dynamic representation of fuzzy knowledge (DRFK) model.
Secondly, transforming the physical model to mathematical model through the fired transition function, besides, the fuzzy-reliability curve is developed to achieve the fuzzy degree of the antecedent proposition. Finally, applying the improved particle swarm optimization to solve the parameters to realize the ability of self-learning is the most important part in the adjustment (self-learning) mechanism.
Since belief computation involves the fuzzy OR (max) operation, the places may be regarded as OR neurons. For the purpose of further research, it’s meaningful for the fuzzy knowledge representation based on FPN to build the mathematical model. Aiming at that, we introduce the fired transition function and maximum operation continuous function. (1) Fired transition function is defined as the following:
yðXÞ ¼ 1=ð1 þ ebðkÞ Þ 4. GPSO-based self-learning algorithm in the DRFK 4.1. The performance evaluation function In this paper, we suggest that the fuzzy knowledge is described by the fuzzy production rules and the knowledge modeling is realized by mapping these rules into the DFRK based on fuzzy Petri nets. For any transition t, if the certainty factors associated with the tokens of all its input places are all greater than their thresholds, then the transition is enabled and fires instantly. Definition 3. 8t 2 T; if 8pij 2 IðtÞ; the transition is enable.
Pn
j¼1 MðP ij Þ
ð3Þ
where b is a positive number, k is the corresponding threshold, X is the inputs of the transition. The function result depends on the comparison between X and k on the condition that b is massive positive number: when X exceeds k, the result is one, otherwise, it’s zero. Therefore, no matter if the transition is enabled, the result can be represented by yðXÞ X l uniformly. The maximum operation continuous function is given as the following
t ¼ maxðx1 ; x2 Þ
x1 x2 þ bðx1 x2Þ 1þe 1 þ ebðx2 x1Þ
ð4Þ
h ¼ maxðx1 ; x2 ; x3 Þ ¼ maxðmaxðx1 ; x2 Þ; x3 Þ ¼ maxðt; x3 Þ wij ef ðtÞ means that
t x3 þ 1 þ ebðtx3Þ 1 þ ebðx3 tÞ
ð5Þ
Definition 4. A transition t j is enabled if pi possesses fuzzy beliefs for "pi I(tj) An enabled transition fires by generating a fuzzy truth token (FTT) at its output arc. The value of the FTT is given by
where b is a massive positive number (3). Suppose that the DRFK model to be studied is n-layered Petri Net with b ultimate places. And r learning samples are used to train the FPN model. The performance evaluation function is defined as the following:
# 8" ^ ^ > < if fni jpi 2 Iðt i Þg thj fni jpi 2 Iðt i Þg > thj tj ðt þ 1Þ ¼ 8i > 8i : 0 otherwise
E ¼ 21
Since FTT computation at a transition involves taking fuzzy AND (min) of the beliefs of its input places, a transition may be regarded as an AND neuron. Definition 5. After the firing of ti ’s the fuzzy belief of nk, at place pk , where pk O(tj)"j, is given by
8j
ð6Þ
i¼1 j¼1
ð1Þ
" # _ _ nk ðt þ 1Þ ¼ nk ðtÞ ftj ðt þ 1Þg
r X b 2 X Mi ðpj Þ M1i ðpj Þ
ð2Þ
where Mi(pj) and M1i ðpj Þ represent the actual marking value and the expected one of the ultimate place respectively. The tokens of ultimate places depend on the value of weight, threshold, and confidence; therefore, the performance evaluation function can be replaced by
E ¼ yðw1 ; w2 ; w3 ; . . . ; wn ; lkÞ
ð7Þ
where E is the error of the model wi ði ¼ 1; 2; . . . ; nÞ is the weight l is the confidence value k is the threshold. Using Neural Network and BP algorithm, the learning process will be broken because of the discontinuity after taking the derivative of the performance evaluation function. While, like
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other evolutionary algorithms, particle swarm optimization (PSO) algorithm didn’t take the derivative of the fired transition function. So, we propose the particle swarm optimization (PSO) algorithm for self-learning of representation parameters in the DRFK. 4.2. Standard particle swarm optimization PSO is an optimization algorithm, modeled after the social behavior of flocks of birds. PSO is a population-based search process where individuals initialized with a population of random solutions, referred to as particles, are grouped into a swarm. Each particle in the swarm represents a candidate solution to the optimization problem, and if the solution is made up of a set of variables, the particle can correspondingly be a vector of variables. In a PSO system, each particle is ‘‘flown’’ through the multidimensional search space, adjusting its position in search space according to its own experience and that of neighboring particles. The particle therefore makes use of the best position encountered by itself and that of its neighbors to position itself toward an optimal solution. The performance of each particle is evaluated using a predefined fitness function, which encapsulates the characteristics of the optimization problem. In this study, the larger the value of fitness function, the better the particle is.
vkþ1 id
¼
wvkid
þ
k c1 rand1 ðpbestid
xkid Þ
þ
k c2 rand2 ðgbestd
xkid Þ
Some improved PSO algorithms have also been developed. Fan (2002) and Robinson J., S., and R. S. (2002) proposed similar hybrid PSO algorithms by introducing operations of GAs into PSO systems. After a hybrid probability is assigned to each particle, the algorithm selects a certain number of particles into a pool according to the hybrid probabilities at each stage of iteration. The particles in the pool are randomly separated into couples. Each couple reproduces two children by crossover. Then the children are used to replace their parents of the previous particles to keep the number of particles unchanged. Recently, several researchers have combined PSO and GA to form different hybrid algorithms. Shi et al. (2005) combined a variable population-size genetic algorithm (VPGA) and PSO to form a hybrid algorithm that generates initial populations for VPGA and PSO according to a certain proportion. The new population is obtained by evolution according to the corresponding algorithm rules. Gandelli et al.(2007) randomly divided the total population into two subpopulations and evolved the two sub-populations using GA and PSO operations, respectively. Kao and Zahara
Generation of initial population
Evaluation of each individual
ð8Þ
where k is the current step number, w is the inertia weight, c1 and c2 are the acceleration constants, rand1 and rand2 are two random numbers in the range [0, 1], xkid is the current position of the particle, k pbest mhboxid is the best one of the solutions this particle has reached, k gbestd is the best one of the solutions all the particles have reached. After calculating the velocity, the new position of every particle can be worked out:
xkþ1 ¼ xkid þ vkþ1 id id
Termination Criteria met?
Y
N Velocity Update
ð9Þ Position Update
The PSO algorithm performs repeated applications of the update equations above until a specified number of iteration has been exceeded, or until the velocity updates are close to zero. Whereas the PSO does not possess the crossover and mutation processes used in GAs, it finds the optimum solution by swarms following the best particle. Compared to GAs, the PSO has much more profound intelligent background and could be performed more easily. Based on its advantages, the PSO is not only suitable for science research, but also engineering applications, in the fields of evolutionary computing, optimization and many others. In recent years there have been a lot of reported works focused on the PSO which has been applied widely in the function optimization, artificial neural network training, pattern recognition, fuzzy control and some other fields.
N
Converge to local optima?
Y Crossover
Final solution Fig. 5. Flow chart of GPSO.
4.3. Genetic-PSO In standard versions of PSO, velocity consist of three parts, the first is previous velocity of the particle, the second and third parts are the terms associated with their best positions in the past. The PSO algorithm updates a population of particles on a basis of information about each particle’s previous best performance and the best particle in the population. In PSO, only best positions give out the information to others. However, when the position of a particle equals its personal best position or the global best particle, the velocity is only influenced by the inertial term. Therefore, if this particle stays on the global best position, which is also the personal best position, for a number of iterations, its velocity tends to be zero and the particle stagnates.
Fig. 6. Fitness valve’s evolution curve of three algorithms.
W.-M. Wang et al. / Expert Systems with Applications 41 (2014) 1369–1376
(2008) proposed a GA–PSO hybrid algorithm for geometric proportional populations. Further, to avoid the particle to be stuck in the local minimum, Kuo and Han (2011) integrated the mutation mechanism of GA with PSO. Valdez, Melin, and Castillo (2011) combined GA and PSO using fuzzy logic to integrate the results of both methods and for parameter tuning. With this analysis, standard versions of PSO aren’t appropriate for the DRFK that the paper proposes because of the algorithm’s prematurity. Consequently, we propose an improved Genetic Particle Swarm Algorithm (GPSO) to solve the self-learning problem of representation parameters in the DRFK. GPSO operates the particles through cross-breeding operation, when the algorithm converges to local optima. A general flow chart for the GPSO is shown in Fig. 5. Suppose that x (x1, x2, x3 ,..., xm) is the particle that maybe converge to local optima, then choose two of them (parent1(x), parent2(x)) as the parents randomly. Finally, the offspring generates as the follow formula:
childðxÞ ¼ p parent 1 ðxÞ þ ð1 pÞparent 2 ðxÞ childðv Þ ¼
parent1 ðv Þ þ parent 2 ðv Þ jparent1 ðv Þj jparent1 ðv Þ þ parent 2 ðv Þj
ð10Þ
ð11Þ
Besides, using the cross-breeding operation to overleap local optima, the improved PSO remains the satisfactory converging character. In fact, ‘‘adventure molecular’’ may fly out of the food area to find the new place when flocks of birds have found food in nature. The phenomenon is in sympathy with the idea of GPSO. For the DRFK, detail procedure for self-learning of representation parameters is as following (Pseudo-code of GPSO learning algorithm in the DRFK are in the Table 1.):
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Step 1: Build the performance evaluation function; besides, assign the value to the population size, inertia weight, acceleration constants, and maximum genetic number. Step 2: Initialize position and velocity of the particles swarm, and calculate the fitness of the particles. Besides, initialize the pbest and gbest of each sub-swarm as the standard PSO. Step 3: Update the particles swarm according to Eqs. (8) and (9), and calculate the corresponding fitness of the update swarm.
k vkþ1 ¼ wvkid þ c1 rand1 pbestid xkid id k þ c2 rand2 gbestd xkid
xkþ1 ¼ xkid þ vkþ1 id id Step 4: Compare the current function value to the former best position’s function value, then update the gbest with the litter one. Step 5: Store the particles and its function value into vector min, and count out the difference between the two elements in order during the last five in vector min. If the result is less than 0.00001, then the algorithm may converge to local optima, at that time, the particle should be dealt with crossbreeding operation as the Eqs. (10) and (11). Otherwise, turn to step 6.
childðxÞ ¼ p parent 1 ðxÞ þ ð1 pÞparent 2 ðxÞ
childðv Þ ¼
parent 1 ðv Þ þ parent2 ðv Þ jparent 1 ðv Þj jparent 1 ðv Þ þ parent2 ðv Þj
Table 1 Pseudo-code of GPSO learning algorithm in the DRFK. GPSO code: function[xm, fv] = GeneticPSO(fitness, N, c1, c2, w, M, D) format long; for i = 1:N for j = 1:D x(i,j)=rand: v(i,j)=rand: end end ...... for t = 1:m for i = 1:N v(i.:)=w⁄v + c(i.:)+cl⁄rand⁄(y(i.:)-x(ii,:))+c2⁄rand⁄(pg-x(i,:)); x(i.:)= x(i.:)+ v(i.:); if x(i,:)>1 x(i,:)=1; end if fitness}}(x(i,:)) < p(i) p(i)= fitness (x(i,;)) y(i,;)= x(i,;); end if p(i) 4 ifnorm(p(i) - max(1, L)) + norm(p(i) - max(1, L - 1)) + norm(p(i) - max(1, L - 2)) + norm(p(i) - max(1, L - 3)) + norm(p(i) - max(1, L - 4)) < 0.0005 num pool = round (N) pool x = x(1:numpool,:); pool vx = v(1:numpool,:); for i = 1:numpool seed1 = floor(rand()⁄(numpool-1))+1; seed2 = floor(rand()⁄(numpool-1))+1; pb = rand();childx1(i, :) = pb ⁄ Poolx(seed1, :) + (1 - pb) ⁄ Poolx(seed2, :); ...... end
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Table 2 Simulation result after 20 iteration. Result
GA
PSO
GPSO
(w1, w2,l; kÞ
(0.7453, 0.3066, 0.6220, 0.5723)
(0.9025, 0.3447, 0.5246, 0.6583)
(0.7020, 0.2687, 0.6741, 0.5052)
Table 3 Simulation result after 2000 iteration.
*
Result
GA
PSO
GPSO
(w1, w2, l; kÞ Time* Accuracy* Precision*
(0.7664, 0.2932, 0.6175, 0.5767) 35.2525s 85.9% 2.369 10–9
(0.7092, 0.2713, 0.6673, 0.5124) 10.7121s 39.25% 1.25 108
(0.7047, 0.2696, 0.6715, 0.5013) 15.7442s 74.2% 1.739 109
Time means the average time to get the parameters; Accuracy means the accuracy rate of algorithm; Precision means the accuracy level.
Table 4 Comparison for expected value and solving result. NO.
Expected value
GA
PSO
GPSO
1 2 3 4 5 6 7 8 9 10
0.5213 0 0.3573 0.4882 0.6987 0.3868 0.3865 0.4450 0.5647 0.6890
0.5212 0 0.3572 0.4880 0.6988 0.3867 0.3864 0.4450 0.5646 0.6890
0.5214 0 0.3573 0.4881 0.6988 0.3866 0.3867 0.4450 0.5646 0.6890
0.5214 0 0.3574 0.4883 0.6988 0.3868 0.3690 0.4450 0.5790 0.68894
4.5. Expert curve of initial place in the DRFK
Table 5 Places in the representation model. P1 P2 P3 P4 P9
Lack of feedback voltage The feedback wire broke The pressure in the chamber is too high The pressure in by-pass pipe is too high Piston rod of servo mechanism doesn’t close
which aimed at perfecting the representation of the fuzzy knowledge. (The expectation value of the fuzzy knowledge representation parameters are w1 ¼ 0:7232w2 ¼ 0:2768 l ¼ 0:6543k ¼ 0:5422). Comparison of results of PSO, GA, GPSO algorithms are shown as Fig. 6, Tables 2–4. After the optimization of the 20 iterations by three algorithms, the result in Table 2 shows that the GPSO and PSO have the quicker convergence rate than GA, while PSO may easily converge to local optima, which causes the solved fuzzy representation parameters inaccurate. Besides, through the optimization of the 2000 iterations by these algorithms, the result in the Tables 3 and 4 show the high-accuracy, high-precision and short-optimization time of the GPSO, while the PSO’s accuracy and convergence precision are lower. As a result, the GPSO is appropriate for the solver and self-learning of the fuzzy representation parameters in DRFK.
P5 P6 P7 P8
Feedback potentiometer broke The regulator of zero position broke The voltage is abnormal after the servo mechanism has electricity The cylinder hasn’t move
Step 6: The PSO algorithm performs repeated applications of the update equations above until a specified number of iteration has been exceeded, or until the velocity updates are close to zero. 4.4. Numerical results The proposed GPSO are tested using a fuzzy production rule. To examine the efficiency of the algorithms, the comparisons are made with the standard GA and PSO algorithms. Conducting simulations with solving the representation parameters of the fuzzy production rules that ‘‘IF the pressure of the pressure vessel is too high and the pressure release valve became invalid THEN the pressure vessel turn out to be abnormal’’ by the GA, PSO and GPSO. where p1 and p2 correspondingly present the proposition ‘‘ the pressure of the pressure vessel is too high ‘‘and’’ the pressure release valve became invalid ‘‘,d1 and d2 correspondingly represent the confidence value of that proposition, w1 and w2 correspondingly represent the consequence of that proposition to the conclusion, l represents the confidence value that is associated with the conclusion being pursued, k represents the threshold,
The token in the initial place is the reliability of fuzzy proposition, which is difficult to determine. The traditional method isn’t applicable for some industry’s special requirements (e.g., real-time requirement in the Fault diagnose), for the poor efficiency and high subjectivity. Aiming at the structure characteristic of DRFK based on FPN, we introduce the fuzzy statistical method to find the expert curve, which determine the token in the initial place of the DRFK model. The certain set is related to an initial place A, whose vague concept level is a. Once determined, the Aa determine the a clearly, which means the extension of a. In every operation, the value of e is certain, while the Aa is changed. If the number of experiment is n, h represents the number of e e Aa, then the reliability frequency of e, which belong to the vague proposition is w ¼ hn. With the n increasing, the reliability of e changes to stabilize, and the stabilize value is the fuzzy reliability of e. Finally, by the interpolation in the discrete point, where the fuzzy reliability is determined, the expert curve is established.
5. Example The launch vehicle is composed of the structure of the rocket, dynamical system, control system, remote-measuring system, ballistic trajectory measure and wireless security system, separation system and auxiliary system. Design parameters and principle of structure choice generally rely on the experience of designer. Besides, in order to get a satisfied scheme, it’s important to make decision with the uncertainty knowledge (e.g., the missing data, randomized information, fuzzy rule, expertise experience and so on).
W.-M. Wang et al. / Expert Systems with Applications 41 (2014) 1369–1376
3
λ1 μ1
t1 3
ω1
ω2
t3
3
μ3
μ7 λ7
μ5 λ5
t7
t5
Fault diagnose of launch vehicle is the typical example of the fuzzy knowledge representation and inference. Because of the complexity of launch structure, the traditions techniques of fault diagnose have some limitations. However, the DRFK proposed in the paper can not only represent the fuzzy knowledge in the fault diagnose area accurately, but also inference effectively, which based on the self-learning ability of DRFK model to dynamic adaptation to the change information and knowledge. 5.1. Fuzzy representation model of fault diagnose
Fig. 7. The representation model of fault diagnose based on DRFK.
1.5
Reliability
3
3
3
λ3
t6
μ4 t λ4 4
t2
μ2
1
Fig. 7 shows the fuzzy knowledge representation model of the fault diagnose of the servo mechanism in a launch vehicle. Where wi is the weight, li represents the confidence, ki represents the threshold. The value of weight without presentation is one acquiescently Places in the representation model are in Table 5. 5.2. The ultimate place function
0.5 0
2
4
6
8
According to the fired transition function, the maximum operation continuous function, Table 6 shows the mathematical expression of the token in P5–P8, that is prepared for the self-learning of representation parameters in the DRFK.
10
Pressure of Chamber (Unit: 0.1MP) 5.3. The fuzzy-reliability curve of initial places
Fig. 8. High-pressure of chamber (P3).
Applying the expect curve method, getting the fuzzy-reliability curve of P3, P4 and P7 as Figs. 8–10. Both of the token in the P1 and P2 are 1, which are the certainty propositions.
1.5
Reliability
3
λ2
1
5.4. Results and discuss
0.5 0
2
4
6
8
After representation model of fault diagnose based on DRFK has been developed, the dynamic adjustment of representation parameters is conducted on the base of the GPSO learning method. The solving results are listed in following:
10
Pressure of by-pass Pipe (0.1MP)
½w1 ; w2 ¼ ½0:5621; 0:4379
Fig. 9. The high-pressure of by-pass pipe (P4).
½l1 ; l2 ; l3; l4; l5; l6; l7 ¼ ½0:9585; 0:8542; 0:8458; 0:8639; 0:7889; 0:8845; 0:9986 1.5
½k1 ; k2 ; k3 ; k4 ; k5 ; k6 ; k7 Reliability
3
μ6 λ6
3
1375
1
¼ ½0:4589; 0:2986; 0:5898; 0:6352; 0:5286; 0:6658; 0:1478 Comparison for the solving results and expected results are in the Figs. 11–14: (where the circle represents the expectation data and the plus sign represents solving results data) From comparison, it is clear that the DRFK method can complete the dynamic adjustment of representation parameters effectively, and approach the desired data with good accuracy. From the actual application of the proposed method in this paper, compared with traditional methods such as NN, the DRFK based method can not only represent fuzzy dynamic knowledge, but also adjust the
0.5 0
0
100
200
300
400
500
600
Voltage of System (Unit: V) Fig. 10. The abnormal pressure in system (P7).
Table 6 Mathematical expression of the places. Token
Mathematical expression
M(p5)
x1 ¼ Mðp1 Þl1 =ð1þe
M(p6)
x3 = M(p3) w1 + M(p4) w2 Mðp6 Þ ¼ x3 l3 =ð1 þ ebðMðp3 Þk3 Þ
M(p8)
x4 x5 þ 1þebðx x4 ¼ Mðp4 Þl4 =ð1 þ ebðMðp4 Þk4 Þ Þx5 ¼ Mðp5 Þl5 =ð1 þ ebðMðp5 Þk5 Þ ÞMðp8 Þ ¼ max x4 x5 1þebðx 5 x4 4 x5 Þ
M(p9)
x6 x7 þ 1þebðx x6 ¼ Mðp7 Þmu6 =ð1 þ ebðMðp6 Þk6 Þ Þx7 ¼ Mðp8 Þl7 =ð1 þ ebðMðp7 Þk7 Þ ÞMðp9 Þ ¼ max x6 x7 1þebðx 7 x6 6 x7 Þ
bðMðp1 Þk1 Þ
ÞÞ
x2 ¼ Mðp2 Þl2 =ð1þe
bðMðp2 Þk2 Þ
ÞÞ
x1 x1 Mðp5 Þ ¼ max x1 x2 1þebðx þ 1þebðx 1 x2 Þ 2 x1
W.-M. Wang et al. / Expert Systems with Applications 41 (2014) 1369–1376
Fault Probability
1376
No. of Result Comparison
used NN learning algorithm. The validity of this method has been demonstrated by using it in the fault diagnoses of launch vehicle. The generic scheme of the DRFK method should find applications in many-to-many knowledge based system as well as in design of complex product. The improvement of the DRFK modeling ability will be studied in the future. Via the improvement, it is to realize the aim of modeling complex product design and overcoming complexity phenomena. For the future work, we hope that the measurable fuzzy knowledge evaluation method interrelated design problem is studied.
Fig. 11. The comparison result in P5.
Acknowledgement
Fault Probability
This work was supported by the National Science Foundation of China under Grants no. 51005148. References
No. of Result Comparison
Fault Probability
Fig. 12. The comparison result in P6.
No. of Result Comparison
Fault Probability
Fig. 13. The comparison result in P7.
No. of Result Comparison Fig. 14. The comparison result in P8.
representation parameter through learning by itself based on GPSO algorithm. So the proposed method manifests powerful advantages in dynamic representation of fuzzy knowledge.
6. Conclusion Uncertain information and dynamic knowledge are usual phenomena in the design process of complex product. Therefore, dynamic knowledge representation and inference are the critical problems to be solved in the modeling and analysis of complex product design. This paper presented a dynamic representation of fuzzy knowledge (DRFK) method. The GPSO learning algorithm is proposed to self-learning of fuzzy representation parameters in the DRFK. It has shown that the proposed learning algorithm is computationally efficient, much simpler than other commonly
Ashon, S. I. (1995). Petri net models of fuzzy neural networks. IEEE Transaction System, Man and Cybernetics, 25, 926–932. Chatterjee, A., & Siarry, P. (2006). Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization. Computers & Operations Research, 33(3), 859–871. Fan, H. Y. (2002). A modification to particle swarm optimization algorithm. Engineering Computations, 19(8), 970–989. Gandelli, A., Grimaccia, F., Mussetta, M. et al. (2007). Development and validation of different hybridization strategies between GA and PSO. In Proceedings of the IEEE congress on evolutionary computation (pp. 2782–2785). Singapore. Garg, L., Ahson, S. I., & Gupta, P. V. (1991). A fuzzy Petri net for knowledge representation and reasoning. Information Processing Letters, 39, 165–171. Huang, A. F. M., Yang, S. J. H., Wang, M., & Tsai, J. J. P. (2010). Improving fuzzy knowledge integration with particle swarm optimization. Expert Systems with Applications, 37, 8770–8783. Jeffrey, J., Lobo, J., & Murata, T. (1996). A high-level Petri net for goal directed semantics of horn clause logic. IEEE Transactions on Knowledge Data Engineering, 8, 241–259. Jiang, C.-W., & Etorre, B. (2005). A hybrid method of chaotic particle swarm optimization and linear interior for reactive power optimization. Mathematics and Computers in Simulation, 68(1), 57–65. Kao, Y., & Zahara, E. (2008). A hybrid genetic algorithm and particle swarm optimization for multimodal functions. Applied Soft Computing, 8(2), 849–857. Kennedy, J., & Eberhart, R. C. (1995). Particle swarm optimization. In Proceedings of IEEE international conference on neural networks (pp. 1942–1948). Piscataway, NJ. Kennedy, J., Eberhart, R. C., & Shi, Y. (2001). Swarm intelligence. San Francisco, CA: Morgan Kaufmann Publishers Inc. 2001. Konar, A., & Mandal, A. K. (1996). Uncertainty management in expert systems using fuzzy Petri nets. IEEE Transaction on Knowledge and Data Engineering, 8, 96–105. Kuo, R. J., & Han, Y. S. (2011). A hybrid of genetic algorithm and particle swarm optimization for solving bi-level linear programming problem-a case study on supply chain mode. Applied Mathematical Modelling, 35(8), 3905–3917. Li, X., & Lara-Rosano, F. (2000). Adaptive fuzzy petri nets for dynamic knowledge representation and inference. Expert Systems with Applications, 19, 235–241. Looney, C. G. (1994). Fuzzy reasoning in information, decision and control systems. Dordrecht: Kluwer. Mendes, R., Kennedy, J., & Neves, J. (2004). The informed particle swarm: simpler, maybe better. IEEE Transactions on Evolutionary Computation, 8(3), 204–210. Mengshoel, O. J., & Delab, S. (1993). Knowledge validation: Principles and practice. IEEE Expert, 8, 62–68. Murata, T. (1998). Petri nets: Properties, analysis and applications. Proceedings of IEEE, 77, 541–580. Pedrycz, W. (1989). A fuzzy cognitive structure for pattern recognition. Pattern Recognition Letter, 9, 305–313. Pedrycz, W., & Gomide, F. (1994). A generalized fuzzy petri net model. IEEE Transaction Fuzzy Systems, 2, 295–301. Polat, F., & Guvenir, H. (1993). UVT: A unification-based tool for knowledge base verification. IEEE Expert, 8, 69–75. Ribaric, S., & Hrkac, T. (2012). A model of fuzzy spatio-temporal knowledge representation and reasoning based on high-level Petri nets. Information Systems, 37, 238–256. Robinson J., Sinton, S., & Yahya, R. S. (2002). Particle swarm, genetic algorithm, and their hybrids: Optimization of a profiled corrugated horn antenna. In IEEE antennas and propagation society international symposium (vol. I, pp. 314–317). San Antonio. Shi, X.-H., Liang, Y.-C., Lee, H.-P., Lu, C., & Wang, L.-M. (2005). An improved GA and a novel PSO-GA-based hybrid algorithm. Information Processing Letters, 93(5), 255–261. Valdez, F., Melin, P., & Castillo, O. (2011). An improved evolutionary method with fuzzy logic for combining particle swarm optimization and genetic algorithms. Applied Soft Computing, 11(2), 2625–2632. Zadeh, L. A. (1989). Knowledge representation in fuzzy logic. IEEE Transactions on Knowledge and Data Engineering, 1, 89–100.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Dynamic representation of fuzzy knowledge based on fuzzy petri net and genetic-particle swarm optimization Wei-Ming Wang ⇑, Xun Peng, Guo-niu Zhu, Jie Hu, Ying-Hong Peng School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Keywords: Fuzzy knowledge Petri nets Knowledge representation Learning algorithms Particle swarm optimization
a b s t r a c t Information in some fields like complex product design is usually imprecise, vague and fuzzy. Therefore, it would be very useful to design knowledge representation model capable to be adjusted according to information dynamics. Aiming at this objective, a knowledge representation scheme is proposed, which is called DRFK (Dynamic Representation of Fuzzy Knowledge). This model has both the features of a fuzzy Petri net and the learning ability of evolutionary algorithms. An efficient Genetic Particle Swarm Optimization (GPSO) learning algorithm is developed to solving fuzzy knowledge representation parameters. Being trained, a DRFK model can be used for dynamic knowledge representation and inference. Finally, an example is included as an illustration. 2013 Published by Elsevier Ltd.
1. Introduction In the real world, there exist many problems people have not had a fundamental understanding of, and the information people are obtaining is uncertain information. Most human knowledge, how-ever, is typically expressed in vague and imprecisely defined concepts and the inference is mostly supported by common-sense and intuitive reasoning (Ribaric & Hrkac, 2012). Aiming to problem solving in specific areas, the knowledgebased systems depends not only on theoretical knowledge determined in specific areas, but more on experience and common sense of experts. The uncertainty of objective things or phenomena in the real world leads to the fact that people’s information and knowledge in various cognitive domains are mostly inaccurate, which requires that the knowledge representation and processing model in the expert system can reflect this uncertainty. Therefore, how to represent and process the uncertainty of knowledge has become one of the important research issues on artificial intelligence. There are important issues underlying knowledge representation that have not yet been adequately addressed. Such issues are those of the modeling and verification of the knowledge-based systems (Ashon, 1995; Mengshoel & Delab, 1993). As a case in point, the conventional techniques such as simulation methods and analytical methods do not provide tools for representing the dynamic behavior of the KBSs as well as for modeling the different aspects of fuzzy information of these systems (Garg, Ahson, & Gupta, 1991; Polat & Guvenir, 1993; Zadeh, 1989).
⇑ Corresponding author. Tel.: +86 (0) 21 34206170; fax: +86 (0) 21 34206179. E-mail addresses: [email protected], [email protected] (W.-M. Wang). 0957-4174/$ - see front matter 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.eswa.2013.08.034
As an important modeling and computational paradigm, Petri nets (PN) have been widely used (Murata, 1998). Machine learning with fuzzy AND-OR neurons and with fuzzy Petri nets have been proposed by Pedrycz (1989). In order to deal with uncertain information or knowledge, fuzzy Petri nets (FPNs) have been introduced, which can be used to represent Horn clauses or Non-Horn clauses and represent and execute the fuzzy rules (Jeffrey, Lobo, & Murata, 1996; Konar & Mandal, 1996). To improve the FPN adjusting (or learning) ability, a generalized fuzzy Petri net (GFPN) is proposed (Pedrycz & Gomide, 1994). To solve the learning problem, an adaptive fuzzy Petri net (AFPN) is proposed by adjusting weights the same as those in a neural network (NN) (Looney, 1994). But the weight adjustment is off-line. To overcome this weakness, several researchers have recently investigated FPN learning ability by using evolutionary algorithms (Huang, Yang, Wang, & Tsai, 2010). In fact, the knowledge learning above was under the framework of neural networks. Adaptive Fuzzy Petri Net (AFPN) has also the learning ability of a neural network, but it does not need to be transformed into neural networks, and Back Propagation algorithm is developed for the knowledge learning under generalized conditions (Li & Lara-Rosano, 2000). However, the learning algorithm is based on a special transition firing rule, it is necessary to know certainty factors of each consequence proposition in the system. Obviously, this restriction is too strict for a knowledge based system. Instead of using the Neural Network, this study tries swarm intelligence: particle swarm optimization (PSO). Proposed by Kennedy and Eberhart and inspired by social behavior in nature, PSO is a population-based search algorithm that is initialized with a population of random solutions, called particles (Kennedy, 1995; Kennedy, Eberhart, & Shi, 2001). Each particle in the PSO flies
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through the search space at a velocity that is dynamically adjusted according to its own and its companion’s historical behavior. Because particle swarm optimization is powerful, easy to implement, and computationally efficient, numerous researches on PSO theories or applications have been reported Recently (Chatterjee & Siarry, 2006; Jiang & Etorre, 2005; Mendes, Kennedy, & Neves, 2004; Shi, Liang, Lee, Lu, & Wang, 2005). The rest of the paper is organized as follows. The related work concerning the proposed scheme is introduced in Section 2. The proposed model of dynamic representation of fuzzy knowledge (DRFK) is covered in Section 3. The GPSO learning algorithm for solving knowledge representation parameters is presented in Section 4. An illustrative case study, fault diagnose of launch vehicle is presented in Section 5. Finally, conclusions are drawn in Section 6. 2. Fuzzy production rules and fuzzy Petri net 2.1. Fuzzy production rules Fuzzy production rules (FPRs) are widely used in knowledge based systems to represent fuzzy and uncertain concepts. FPRs are usually presented in the form of a fuzzy IF-THEN rule in which both the antecedent and the consequent are fuzzy concepts, denoted by fuzzy sets. To effectively represent both the fuzziness and the uncertainty in FPRs, several knowledge parameters such as confidence value, weight, and threshold have been incorporated into FPRs. Thus, a FPR has the following format: Definition 1. R : If AthenCðCF ¼ lÞ; Th; W where A ¼ ða1 ; a2 ; . . . ; an Þ is the antecedent portion which comprises one or more propositions connected by either ‘‘AND’’ or ‘‘OR’’. Each proposition ai ð1 6 i 6 nÞ may have a fuzzy variable of the antecedent. The parameter l is the confidence value of the rule and it represents the strength of belief of the rule. The symbol Th represents a set of threshold values specified for the propositions in the antecedent A. The set of weights assigned to the propositions a1 ; a2 ; . . . ; an , is given by W ¼ ðw1 ; w2; . . . ; wn Þ.
be powerful modeling formalisms. Modeling using FPN has many advantages compared to other modeling schemes. The graphic representation of FPN makes the models relatively simple and legible. The knowledge representation based on FPN has two basic general formulation, shown as ‘‘AND’’ rule representation model based on FPN (as Fig. 1) and ‘‘OR’’ rule representation model based on FPN (as Fig. 2). The fuzzy production rules based on the Petri Net is shown in Fig. 3, where pj (j = 1, 2, . . ., n) denotes the antecedent (premise) part of a given rule, it defines the fuzzy region in the input space; di (i = 1–3) denotes the consequent (decision, conclusion) part of that rule, it specifies the output in the fuzzy region; li (i = 1–3) is the confidence value that is associated with the conclusion being pursued. wij is the weight that weighs the importance of every antecedent to its consequent propositions. k is the threshold value. The model includes three rules:
IF p1 and p2 THEN d1 ðk1 ; l1 ; x11 ; x12 Þ IF p1 and p3 and p4 THEN d2 ðk2 ; l2 ; x21 ; x22 Þ IF p1 and p4 and p5 THEN d3 ðk3 ; l3 ; x31 ; x32 ; x33 Þ The knowledge representation parameters of the FPN is updated or modified frequently, it may be regarded as dynamic systems. Suitable models for them should be adaptable. In other words, the models must have ability to adjust themselves according to the systems’ changes. However, the lack of adjustment (learning) mechanism in FPNs cannot cope with potential changes of actual systems. Therefore, it would be very useful to develop the parameters self-learning mechanism in the FPN capable to be adjusted like human cognition and thinking, according to knowledge dynamics to achieve the dynamic representation of fuzzy knowledge. The model of DRFK is shown in Fig. 4. Aiming at the adjustment (self-leaning) of the representation parameters in the DRFK, firstly, building the physical model of DRFK based on fuzzy Petri net to organize the fuzzy knowledge representation parameters.
2.2. Fuzzy Petri net Definition 2. An FPN is a 9-tuple, given by FPN = hP, Tr, T, D, I, O, Th, n, Wi where P = (p1, p2, . . ., pn) is a finite set of places; Tr = (tr1, tr2, . . ., trm) is a finite set of transitions; T = (t1, t2, . . ., tm) is a set of fuzzy truth tokens in the interval [0, 1] associated with the transitions (tr1, tr2, . . ., trm),respectively; D = (d1, d2,. . .,dn) is a finite set of propositions, where proposition dk corresponds to place pk; P \ Tr \ D = £; cardinality of (P) = cardinality of (D); I:Tr ? P1 is the input function, representing a mapping from transitions to bags of (their input) places; O:Tr ? P1 is the output function, representing a mapping from transitions to bags of (their output) places; TH = (th1, th2, . . ., thm) represents a set of threshold values in the interval ½0; 1 associated with transitions (tr1, tr2, . . ., trm), respectively; n:P ? [0, 1] is an association function hereafter called fuzzy belief, representing a mapping from places to real values between 0 and 1; n(pi) = ni(say); W = wij is the set of weights from the jth transition to the ith place, where i and j are integers. 3. The model of dynamic representation of fuzzy knowledge Fuzzy production rule is one of the most basic researched areas of knowledge based engineering (KBE). We have chosen the fuzzy Petri net (FPN) as an alternative modeling and analysis formalism. FPN formalism is a derivative of PNs which have been demonstrated to
p1 d1
w1
p2 d 2
w2
pn d n
μ, λ d P
.. .
t
wn
Fig. 1. ‘‘AND’’ rule.
p1 d1 p2 d 2 .. .
pn d n
μ1 , λ1 t 1
μ2 , λ2 t 2
d P
μn , λn tn Fig. 2. ‘‘OR’’ rule.
p1 p2 p3 p4 pn
w11 w12 w21 w22 w23 w31 w32 w33
λ1 μ1
d1
rule1
λ1 μ1
d2
rule2
λ1 μ1
d3
rule3
Fig. 3. Fuzzy knowledge representation based on Petri net.
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Fig. 4. Dynamic representation of fuzzy knowledge (DRFK) model.
Secondly, transforming the physical model to mathematical model through the fired transition function, besides, the fuzzy-reliability curve is developed to achieve the fuzzy degree of the antecedent proposition. Finally, applying the improved particle swarm optimization to solve the parameters to realize the ability of self-learning is the most important part in the adjustment (self-learning) mechanism.
Since belief computation involves the fuzzy OR (max) operation, the places may be regarded as OR neurons. For the purpose of further research, it’s meaningful for the fuzzy knowledge representation based on FPN to build the mathematical model. Aiming at that, we introduce the fired transition function and maximum operation continuous function. (1) Fired transition function is defined as the following:
yðXÞ ¼ 1=ð1 þ ebðkÞ Þ 4. GPSO-based self-learning algorithm in the DRFK 4.1. The performance evaluation function In this paper, we suggest that the fuzzy knowledge is described by the fuzzy production rules and the knowledge modeling is realized by mapping these rules into the DFRK based on fuzzy Petri nets. For any transition t, if the certainty factors associated with the tokens of all its input places are all greater than their thresholds, then the transition is enabled and fires instantly. Definition 3. 8t 2 T; if 8pij 2 IðtÞ; the transition is enable.
Pn
j¼1 MðP ij Þ
ð3Þ
where b is a positive number, k is the corresponding threshold, X is the inputs of the transition. The function result depends on the comparison between X and k on the condition that b is massive positive number: when X exceeds k, the result is one, otherwise, it’s zero. Therefore, no matter if the transition is enabled, the result can be represented by yðXÞ X l uniformly. The maximum operation continuous function is given as the following
t ¼ maxðx1 ; x2 Þ
x1 x2 þ bðx1 x2Þ 1þe 1 þ ebðx2 x1Þ
ð4Þ
h ¼ maxðx1 ; x2 ; x3 Þ ¼ maxðmaxðx1 ; x2 Þ; x3 Þ ¼ maxðt; x3 Þ wij ef ðtÞ means that
t x3 þ 1 þ ebðtx3Þ 1 þ ebðx3 tÞ
ð5Þ
Definition 4. A transition t j is enabled if pi possesses fuzzy beliefs for "pi I(tj) An enabled transition fires by generating a fuzzy truth token (FTT) at its output arc. The value of the FTT is given by
where b is a massive positive number (3). Suppose that the DRFK model to be studied is n-layered Petri Net with b ultimate places. And r learning samples are used to train the FPN model. The performance evaluation function is defined as the following:
# 8" ^ ^ > < if fni jpi 2 Iðt i Þg thj fni jpi 2 Iðt i Þg > thj tj ðt þ 1Þ ¼ 8i > 8i : 0 otherwise
E ¼ 21
Since FTT computation at a transition involves taking fuzzy AND (min) of the beliefs of its input places, a transition may be regarded as an AND neuron. Definition 5. After the firing of ti ’s the fuzzy belief of nk, at place pk , where pk O(tj)"j, is given by
8j
ð6Þ
i¼1 j¼1
ð1Þ
" # _ _ nk ðt þ 1Þ ¼ nk ðtÞ ftj ðt þ 1Þg
r X b 2 X Mi ðpj Þ M1i ðpj Þ
ð2Þ
where Mi(pj) and M1i ðpj Þ represent the actual marking value and the expected one of the ultimate place respectively. The tokens of ultimate places depend on the value of weight, threshold, and confidence; therefore, the performance evaluation function can be replaced by
E ¼ yðw1 ; w2 ; w3 ; . . . ; wn ; lkÞ
ð7Þ
where E is the error of the model wi ði ¼ 1; 2; . . . ; nÞ is the weight l is the confidence value k is the threshold. Using Neural Network and BP algorithm, the learning process will be broken because of the discontinuity after taking the derivative of the performance evaluation function. While, like
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other evolutionary algorithms, particle swarm optimization (PSO) algorithm didn’t take the derivative of the fired transition function. So, we propose the particle swarm optimization (PSO) algorithm for self-learning of representation parameters in the DRFK. 4.2. Standard particle swarm optimization PSO is an optimization algorithm, modeled after the social behavior of flocks of birds. PSO is a population-based search process where individuals initialized with a population of random solutions, referred to as particles, are grouped into a swarm. Each particle in the swarm represents a candidate solution to the optimization problem, and if the solution is made up of a set of variables, the particle can correspondingly be a vector of variables. In a PSO system, each particle is ‘‘flown’’ through the multidimensional search space, adjusting its position in search space according to its own experience and that of neighboring particles. The particle therefore makes use of the best position encountered by itself and that of its neighbors to position itself toward an optimal solution. The performance of each particle is evaluated using a predefined fitness function, which encapsulates the characteristics of the optimization problem. In this study, the larger the value of fitness function, the better the particle is.
vkþ1 id
¼
wvkid
þ
k c1 rand1 ðpbestid
xkid Þ
þ
k c2 rand2 ðgbestd
xkid Þ
Some improved PSO algorithms have also been developed. Fan (2002) and Robinson J., S., and R. S. (2002) proposed similar hybrid PSO algorithms by introducing operations of GAs into PSO systems. After a hybrid probability is assigned to each particle, the algorithm selects a certain number of particles into a pool according to the hybrid probabilities at each stage of iteration. The particles in the pool are randomly separated into couples. Each couple reproduces two children by crossover. Then the children are used to replace their parents of the previous particles to keep the number of particles unchanged. Recently, several researchers have combined PSO and GA to form different hybrid algorithms. Shi et al. (2005) combined a variable population-size genetic algorithm (VPGA) and PSO to form a hybrid algorithm that generates initial populations for VPGA and PSO according to a certain proportion. The new population is obtained by evolution according to the corresponding algorithm rules. Gandelli et al.(2007) randomly divided the total population into two subpopulations and evolved the two sub-populations using GA and PSO operations, respectively. Kao and Zahara
Generation of initial population
Evaluation of each individual
ð8Þ
where k is the current step number, w is the inertia weight, c1 and c2 are the acceleration constants, rand1 and rand2 are two random numbers in the range [0, 1], xkid is the current position of the particle, k pbest mhboxid is the best one of the solutions this particle has reached, k gbestd is the best one of the solutions all the particles have reached. After calculating the velocity, the new position of every particle can be worked out:
xkþ1 ¼ xkid þ vkþ1 id id
Termination Criteria met?
Y
N Velocity Update
ð9Þ Position Update
The PSO algorithm performs repeated applications of the update equations above until a specified number of iteration has been exceeded, or until the velocity updates are close to zero. Whereas the PSO does not possess the crossover and mutation processes used in GAs, it finds the optimum solution by swarms following the best particle. Compared to GAs, the PSO has much more profound intelligent background and could be performed more easily. Based on its advantages, the PSO is not only suitable for science research, but also engineering applications, in the fields of evolutionary computing, optimization and many others. In recent years there have been a lot of reported works focused on the PSO which has been applied widely in the function optimization, artificial neural network training, pattern recognition, fuzzy control and some other fields.
N
Converge to local optima?
Y Crossover
Final solution Fig. 5. Flow chart of GPSO.
4.3. Genetic-PSO In standard versions of PSO, velocity consist of three parts, the first is previous velocity of the particle, the second and third parts are the terms associated with their best positions in the past. The PSO algorithm updates a population of particles on a basis of information about each particle’s previous best performance and the best particle in the population. In PSO, only best positions give out the information to others. However, when the position of a particle equals its personal best position or the global best particle, the velocity is only influenced by the inertial term. Therefore, if this particle stays on the global best position, which is also the personal best position, for a number of iterations, its velocity tends to be zero and the particle stagnates.
Fig. 6. Fitness valve’s evolution curve of three algorithms.
W.-M. Wang et al. / Expert Systems with Applications 41 (2014) 1369–1376
(2008) proposed a GA–PSO hybrid algorithm for geometric proportional populations. Further, to avoid the particle to be stuck in the local minimum, Kuo and Han (2011) integrated the mutation mechanism of GA with PSO. Valdez, Melin, and Castillo (2011) combined GA and PSO using fuzzy logic to integrate the results of both methods and for parameter tuning. With this analysis, standard versions of PSO aren’t appropriate for the DRFK that the paper proposes because of the algorithm’s prematurity. Consequently, we propose an improved Genetic Particle Swarm Algorithm (GPSO) to solve the self-learning problem of representation parameters in the DRFK. GPSO operates the particles through cross-breeding operation, when the algorithm converges to local optima. A general flow chart for the GPSO is shown in Fig. 5. Suppose that x (x1, x2, x3 ,..., xm) is the particle that maybe converge to local optima, then choose two of them (parent1(x), parent2(x)) as the parents randomly. Finally, the offspring generates as the follow formula:
childðxÞ ¼ p parent 1 ðxÞ þ ð1 pÞparent 2 ðxÞ childðv Þ ¼
parent1 ðv Þ þ parent 2 ðv Þ jparent1 ðv Þj jparent1 ðv Þ þ parent 2 ðv Þj
ð10Þ
ð11Þ
Besides, using the cross-breeding operation to overleap local optima, the improved PSO remains the satisfactory converging character. In fact, ‘‘adventure molecular’’ may fly out of the food area to find the new place when flocks of birds have found food in nature. The phenomenon is in sympathy with the idea of GPSO. For the DRFK, detail procedure for self-learning of representation parameters is as following (Pseudo-code of GPSO learning algorithm in the DRFK are in the Table 1.):
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Step 1: Build the performance evaluation function; besides, assign the value to the population size, inertia weight, acceleration constants, and maximum genetic number. Step 2: Initialize position and velocity of the particles swarm, and calculate the fitness of the particles. Besides, initialize the pbest and gbest of each sub-swarm as the standard PSO. Step 3: Update the particles swarm according to Eqs. (8) and (9), and calculate the corresponding fitness of the update swarm.
k vkþ1 ¼ wvkid þ c1 rand1 pbestid xkid id k þ c2 rand2 gbestd xkid
xkþ1 ¼ xkid þ vkþ1 id id Step 4: Compare the current function value to the former best position’s function value, then update the gbest with the litter one. Step 5: Store the particles and its function value into vector min, and count out the difference between the two elements in order during the last five in vector min. If the result is less than 0.00001, then the algorithm may converge to local optima, at that time, the particle should be dealt with crossbreeding operation as the Eqs. (10) and (11). Otherwise, turn to step 6.
childðxÞ ¼ p parent 1 ðxÞ þ ð1 pÞparent 2 ðxÞ
childðv Þ ¼
parent 1 ðv Þ þ parent2 ðv Þ jparent 1 ðv Þj jparent 1 ðv Þ þ parent2 ðv Þj
Table 1 Pseudo-code of GPSO learning algorithm in the DRFK. GPSO code: function[xm, fv] = GeneticPSO(fitness, N, c1, c2, w, M, D) format long; for i = 1:N for j = 1:D x(i,j)=rand: v(i,j)=rand: end end ...... for t = 1:m for i = 1:N v(i.:)=w⁄v + c(i.:)+cl⁄rand⁄(y(i.:)-x(ii,:))+c2⁄rand⁄(pg-x(i,:)); x(i.:)= x(i.:)+ v(i.:); if x(i,:)>1 x(i,:)=1; end if fitness}}(x(i,:)) < p(i) p(i)= fitness (x(i,;)) y(i,;)= x(i,;); end if p(i) 4 ifnorm(p(i) - max(1, L)) + norm(p(i) - max(1, L - 1)) + norm(p(i) - max(1, L - 2)) + norm(p(i) - max(1, L - 3)) + norm(p(i) - max(1, L - 4)) < 0.0005 num pool = round (N) pool x = x(1:numpool,:); pool vx = v(1:numpool,:); for i = 1:numpool seed1 = floor(rand()⁄(numpool-1))+1; seed2 = floor(rand()⁄(numpool-1))+1; pb = rand();childx1(i, :) = pb ⁄ Poolx(seed1, :) + (1 - pb) ⁄ Poolx(seed2, :); ...... end
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Table 2 Simulation result after 20 iteration. Result
GA
PSO
GPSO
(w1, w2,l; kÞ
(0.7453, 0.3066, 0.6220, 0.5723)
(0.9025, 0.3447, 0.5246, 0.6583)
(0.7020, 0.2687, 0.6741, 0.5052)
Table 3 Simulation result after 2000 iteration.
*
Result
GA
PSO
GPSO
(w1, w2, l; kÞ Time* Accuracy* Precision*
(0.7664, 0.2932, 0.6175, 0.5767) 35.2525s 85.9% 2.369 10–9
(0.7092, 0.2713, 0.6673, 0.5124) 10.7121s 39.25% 1.25 108
(0.7047, 0.2696, 0.6715, 0.5013) 15.7442s 74.2% 1.739 109
Time means the average time to get the parameters; Accuracy means the accuracy rate of algorithm; Precision means the accuracy level.
Table 4 Comparison for expected value and solving result. NO.
Expected value
GA
PSO
GPSO
1 2 3 4 5 6 7 8 9 10
0.5213 0 0.3573 0.4882 0.6987 0.3868 0.3865 0.4450 0.5647 0.6890
0.5212 0 0.3572 0.4880 0.6988 0.3867 0.3864 0.4450 0.5646 0.6890
0.5214 0 0.3573 0.4881 0.6988 0.3866 0.3867 0.4450 0.5646 0.6890
0.5214 0 0.3574 0.4883 0.6988 0.3868 0.3690 0.4450 0.5790 0.68894
4.5. Expert curve of initial place in the DRFK
Table 5 Places in the representation model. P1 P2 P3 P4 P9
Lack of feedback voltage The feedback wire broke The pressure in the chamber is too high The pressure in by-pass pipe is too high Piston rod of servo mechanism doesn’t close
which aimed at perfecting the representation of the fuzzy knowledge. (The expectation value of the fuzzy knowledge representation parameters are w1 ¼ 0:7232w2 ¼ 0:2768 l ¼ 0:6543k ¼ 0:5422). Comparison of results of PSO, GA, GPSO algorithms are shown as Fig. 6, Tables 2–4. After the optimization of the 20 iterations by three algorithms, the result in Table 2 shows that the GPSO and PSO have the quicker convergence rate than GA, while PSO may easily converge to local optima, which causes the solved fuzzy representation parameters inaccurate. Besides, through the optimization of the 2000 iterations by these algorithms, the result in the Tables 3 and 4 show the high-accuracy, high-precision and short-optimization time of the GPSO, while the PSO’s accuracy and convergence precision are lower. As a result, the GPSO is appropriate for the solver and self-learning of the fuzzy representation parameters in DRFK.
P5 P6 P7 P8
Feedback potentiometer broke The regulator of zero position broke The voltage is abnormal after the servo mechanism has electricity The cylinder hasn’t move
Step 6: The PSO algorithm performs repeated applications of the update equations above until a specified number of iteration has been exceeded, or until the velocity updates are close to zero. 4.4. Numerical results The proposed GPSO are tested using a fuzzy production rule. To examine the efficiency of the algorithms, the comparisons are made with the standard GA and PSO algorithms. Conducting simulations with solving the representation parameters of the fuzzy production rules that ‘‘IF the pressure of the pressure vessel is too high and the pressure release valve became invalid THEN the pressure vessel turn out to be abnormal’’ by the GA, PSO and GPSO. where p1 and p2 correspondingly present the proposition ‘‘ the pressure of the pressure vessel is too high ‘‘and’’ the pressure release valve became invalid ‘‘,d1 and d2 correspondingly represent the confidence value of that proposition, w1 and w2 correspondingly represent the consequence of that proposition to the conclusion, l represents the confidence value that is associated with the conclusion being pursued, k represents the threshold,
The token in the initial place is the reliability of fuzzy proposition, which is difficult to determine. The traditional method isn’t applicable for some industry’s special requirements (e.g., real-time requirement in the Fault diagnose), for the poor efficiency and high subjectivity. Aiming at the structure characteristic of DRFK based on FPN, we introduce the fuzzy statistical method to find the expert curve, which determine the token in the initial place of the DRFK model. The certain set is related to an initial place A, whose vague concept level is a. Once determined, the Aa determine the a clearly, which means the extension of a. In every operation, the value of e is certain, while the Aa is changed. If the number of experiment is n, h represents the number of e e Aa, then the reliability frequency of e, which belong to the vague proposition is w ¼ hn. With the n increasing, the reliability of e changes to stabilize, and the stabilize value is the fuzzy reliability of e. Finally, by the interpolation in the discrete point, where the fuzzy reliability is determined, the expert curve is established.
5. Example The launch vehicle is composed of the structure of the rocket, dynamical system, control system, remote-measuring system, ballistic trajectory measure and wireless security system, separation system and auxiliary system. Design parameters and principle of structure choice generally rely on the experience of designer. Besides, in order to get a satisfied scheme, it’s important to make decision with the uncertainty knowledge (e.g., the missing data, randomized information, fuzzy rule, expertise experience and so on).
W.-M. Wang et al. / Expert Systems with Applications 41 (2014) 1369–1376
3
λ1 μ1
t1 3
ω1
ω2
t3
3
μ3
μ7 λ7
μ5 λ5
t7
t5
Fault diagnose of launch vehicle is the typical example of the fuzzy knowledge representation and inference. Because of the complexity of launch structure, the traditions techniques of fault diagnose have some limitations. However, the DRFK proposed in the paper can not only represent the fuzzy knowledge in the fault diagnose area accurately, but also inference effectively, which based on the self-learning ability of DRFK model to dynamic adaptation to the change information and knowledge. 5.1. Fuzzy representation model of fault diagnose
Fig. 7. The representation model of fault diagnose based on DRFK.
1.5
Reliability
3
3
3
λ3
t6
μ4 t λ4 4
t2
μ2
1
Fig. 7 shows the fuzzy knowledge representation model of the fault diagnose of the servo mechanism in a launch vehicle. Where wi is the weight, li represents the confidence, ki represents the threshold. The value of weight without presentation is one acquiescently Places in the representation model are in Table 5. 5.2. The ultimate place function
0.5 0
2
4
6
8
According to the fired transition function, the maximum operation continuous function, Table 6 shows the mathematical expression of the token in P5–P8, that is prepared for the self-learning of representation parameters in the DRFK.
10
Pressure of Chamber (Unit: 0.1MP) 5.3. The fuzzy-reliability curve of initial places
Fig. 8. High-pressure of chamber (P3).
Applying the expect curve method, getting the fuzzy-reliability curve of P3, P4 and P7 as Figs. 8–10. Both of the token in the P1 and P2 are 1, which are the certainty propositions.
1.5
Reliability
3
λ2
1
5.4. Results and discuss
0.5 0
2
4
6
8
After representation model of fault diagnose based on DRFK has been developed, the dynamic adjustment of representation parameters is conducted on the base of the GPSO learning method. The solving results are listed in following:
10
Pressure of by-pass Pipe (0.1MP)
½w1 ; w2 ¼ ½0:5621; 0:4379
Fig. 9. The high-pressure of by-pass pipe (P4).
½l1 ; l2 ; l3; l4; l5; l6; l7 ¼ ½0:9585; 0:8542; 0:8458; 0:8639; 0:7889; 0:8845; 0:9986 1.5
½k1 ; k2 ; k3 ; k4 ; k5 ; k6 ; k7 Reliability
3
μ6 λ6
3
1375
1
¼ ½0:4589; 0:2986; 0:5898; 0:6352; 0:5286; 0:6658; 0:1478 Comparison for the solving results and expected results are in the Figs. 11–14: (where the circle represents the expectation data and the plus sign represents solving results data) From comparison, it is clear that the DRFK method can complete the dynamic adjustment of representation parameters effectively, and approach the desired data with good accuracy. From the actual application of the proposed method in this paper, compared with traditional methods such as NN, the DRFK based method can not only represent fuzzy dynamic knowledge, but also adjust the
0.5 0
0
100
200
300
400
500
600
Voltage of System (Unit: V) Fig. 10. The abnormal pressure in system (P7).
Table 6 Mathematical expression of the places. Token
Mathematical expression
M(p5)
x1 ¼ Mðp1 Þl1 =ð1þe
M(p6)
x3 = M(p3) w1 + M(p4) w2 Mðp6 Þ ¼ x3 l3 =ð1 þ ebðMðp3 Þk3 Þ
M(p8)
x4 x5 þ 1þebðx x4 ¼ Mðp4 Þl4 =ð1 þ ebðMðp4 Þk4 Þ Þx5 ¼ Mðp5 Þl5 =ð1 þ ebðMðp5 Þk5 Þ ÞMðp8 Þ ¼ max x4 x5 1þebðx 5 x4 4 x5 Þ
M(p9)
x6 x7 þ 1þebðx x6 ¼ Mðp7 Þmu6 =ð1 þ ebðMðp6 Þk6 Þ Þx7 ¼ Mðp8 Þl7 =ð1 þ ebðMðp7 Þk7 Þ ÞMðp9 Þ ¼ max x6 x7 1þebðx 7 x6 6 x7 Þ
bðMðp1 Þk1 Þ
ÞÞ
x2 ¼ Mðp2 Þl2 =ð1þe
bðMðp2 Þk2 Þ
ÞÞ
x1 x1 Mðp5 Þ ¼ max x1 x2 1þebðx þ 1þebðx 1 x2 Þ 2 x1
W.-M. Wang et al. / Expert Systems with Applications 41 (2014) 1369–1376
Fault Probability
1376
No. of Result Comparison
used NN learning algorithm. The validity of this method has been demonstrated by using it in the fault diagnoses of launch vehicle. The generic scheme of the DRFK method should find applications in many-to-many knowledge based system as well as in design of complex product. The improvement of the DRFK modeling ability will be studied in the future. Via the improvement, it is to realize the aim of modeling complex product design and overcoming complexity phenomena. For the future work, we hope that the measurable fuzzy knowledge evaluation method interrelated design problem is studied.
Fig. 11. The comparison result in P5.
Acknowledgement
Fault Probability
This work was supported by the National Science Foundation of China under Grants no. 51005148. References
No. of Result Comparison
Fault Probability
Fig. 12. The comparison result in P6.
No. of Result Comparison
Fault Probability
Fig. 13. The comparison result in P7.
No. of Result Comparison Fig. 14. The comparison result in P8.
representation parameter through learning by itself based on GPSO algorithm. So the proposed method manifests powerful advantages in dynamic representation of fuzzy knowledge.
6. Conclusion Uncertain information and dynamic knowledge are usual phenomena in the design process of complex product. Therefore, dynamic knowledge representation and inference are the critical problems to be solved in the modeling and analysis of complex product design. This paper presented a dynamic representation of fuzzy knowledge (DRFK) method. The GPSO learning algorithm is proposed to self-learning of fuzzy representation parameters in the DRFK. It has shown that the proposed learning algorithm is computationally efficient, much simpler than other commonly
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