Vulnerability Evaluation of Aircraft Guarantee System by Improved Fuzzy Petri Net

FENG Lin-han,YAO Xiong-liang,ZHANG A-man

(1 Naval Academy of Armament,Beijing 100161,China;2 Dept,of Naval Architecture and Ocean Eng,Harbin Engineering University,Harbin 150011,China)

1 Introduction

The sole purpose of an aircraft carrier is to provide a means of launching a strike against an enemy anywhere in the world.After the aircraft complete their mission,the carrier must provide a means of safely recovering them.The operational guidance of aircrafts on aircraft carrier is rather complicated.For the aircraft guarantee system,only to launch and recovery aircraft comprises aircraft elevation,guidance,arrangement,oil injection,ammunition reloading,trailing,launch,arresting,etc.The aircraft carrier mainly fights by aircrafts,hence the vulnerability assessment of aircraft guarantee system in battle environment is of essential importance.Recent research about ship system survivability mainly use graphic analysis method,Boolean calculation method,Monte-Carlo stochastic simulation,analytic hierarchy process,damage tree analysis,weighted fuzzy evaluation method,etc[1].However,in the analysis of aircraft guarantee system,the analysts must take into account the complex logical relationships and various weights of so many equipments or components in this system,which is a challenge for conventional expert systems to determine so many unknown weight factors.

The fuzzy Petri net(FPN)[2-14]has been used for knowledge representation and reasoning in the presence of inexact data and knowledge bases,which is an effective method to evaluate the performance of complex systems.The FPN model with learning ability[15-19]offers the benefits of both inexact reasoning and machine learning on a common platform to make unspecific factors in system get rid of experience,and the trained parameters agree better with practical situation as well.

Based on traditional FPN model,corresponding fuzzy deductive reasoning algorithm and training algorithm are presented to evaluate complex system performance.The improved particle swarm optimization algorithm with global search capability is introduced into learning algorithm to improve the training performance[20-22].

According to the damage characteristics of aircraft guarantee system,a vulnerability evaluation model is constructed based on the proposed FPN model,with considerations of complex logic relationships and various weights.Finally,the algorithm for adaptation of equipments unknown weights of this evaluation model from given sets of training instances is presented.

2 Damage characteristics of aircraft guarantee system

The aircraft guarantee system comprises varied equipments or components to accomplish various functions needed to serve aircrafts on board.According to the amount of function lost,the system vulnerability can be divided into four levels:(1)Totally loss of aircraft guarantee ability;(2)Mostly loss of aircraft guarantee ability;(3)Essential maintenance of aircraft guarantee ability;(4)Complete maintenance aircraft guarantee ability.

Adopting the idea of reliability,when the aircraft guarantee system is under attack,the damage critical equation of equipments or components with one damage mode is given by:

where R is the damage resistance of equipments,S is damage effect by the attack.Z＜0 means the equipment has been damaged already,and Z＞0 means equipment is still in good condition.

The damage state of equipments associates with various random factors.For example,the damage effect subjects to some ‘exterior’ factors,such as detonation location,charge weight,ship weight,ship type,material,etc.Similarly,the damage resistance of equipments is effected by some ‘interior’ factors,such as the material randomness,processing techniques,storage conditions,working conditions,etc.In general,the damage state of ship equipments is of random,with the damage probability presented as follows:

where fR（r） and fs（s）denote probability density function of equipment damage resistance R and damage effect S,respectively.

Moreover,the equipment damage resistance R is a ambiguous and mutual magnitude involved fuzzy graded domain.Hence,the fuzzy math is applied to describe the damage proba-bility of equipments.

Considering the types of function,the aircraft guarantee system can be divided into several subsystems,which require many equipments working together.However,different subsystems,different types of equipments,even same types of equipments in the different positions contribute to the upper system damage or function loss with diverse effect.

In general,the damage state of aircraft guarantee system is of randomness and fuzziness.And the different weights of equipments or subsystems in the system vulnerability evaluation must be concerned.How to reflect the damage characteristics in the vulnerability assessment is emphasis of this paper.In the following part,this paper introduces an improved FPN model to describe all the characteristics conveniently.

3 Improved fuzzy Petri net with learning ability

3.1 Fuzzy deductive rules based on weights

The damage states of equipments and systems cannot be accurately described.Fuzzy deductive rules can model these fuzzy relationships among multiple propositions.The assessment system of aircraft guarantee system has multiple inputs but one output.This kind of system is mostly described by the fuzzy rules OR and AND[16].Taking damage characteristics of this system into account,the fuzzy deductive rules based on weights are introduced.

(1)AND Rule

Ri:IF d1and d2and…and dnTHEN

d（ CF=μ）,λ,w1,w2…,wn

(2)OR Rule

Ri:IF d1or d2or…or dnTHEN

d（ CF=μ）,λ,w1,w2…,wn

where dj（j=1,2,…,n）is the premises proposition,d is the concluding proposition,μ is the rule belief parameter,λ is the rule threshold,between 0～1,which denotes thresholds of all preconditions support the concluding proposition.wjis weight parameter,representing various effect of different premises proposition djcontributes to concluding proposition d,and w1+w2+…+wn=1,0＜wj＜1（ j=1,2,…,n）.

The definition of weights in OR rule,which is altered in this paper,can adequately concern the various contribution of equipments or subsystems to upper system,which accord with work principle and complicated logic relationships of this presented system better.

3.2 Proposed model of fuzzy Petri net(FPN)

Fuzzy Petri net(FPN)has been proposed for knowledge representation and reasoning in the presence of inexact data and knowledge bases in 1990 by Chen et al[20].The model of FPN comprises fuzzy deductive rules,which make the descriptions of propositions clearly and simply.And FPN is capable of fuzzy reasoning ability.The assessment model for aircraft guaran-tee system in this paper has been developed based on the following model of FPN.Definition1.A FPN is a 10-tuple,given by

where P={p1,p2,…,pn}is a finite set of places,representing the damage events of equipments,subsystems,and the whole aircraft guarantee system;

T={t1,t2,…,t m }is a finite set of transitions,representing the determination of whether the failure of equipments or subsystems will result in the damage of upper system;

D={d1,d2,…,d n }is a finite set of propositions,where proposition dk corresponds to place pk;|P|=|D|,and P∩T∩D=Ø;

I（O）:T→P,is the input function,representing a mapping from transitions to bags of(their input)places;

M:P→[0，1],is the marked value of this place,true level of proposition,representing the damage probability of equipments or subsystems in the assessment system presented in this paper;

Th:T→[0，1]:Th represents a set of threshold values in the interval[0,1]associated with transitions respectively.Th（t）=μ is a failure criteria of upper system;

W={w1,w2,…,wr }is the set of weights from the jth transition to the ith place,where i and j are integers.In this paper,the sum of weights from varied transitions to one transition is 1,representing the importance of failure of equipments or subsystem to the upper system.

f:T→[0，1]is an association function hereafter called fuzzy belief,representing the assurance level of the fuzzy rule endowed by transitions.f（t）=λ is the probability of failure event;

β:P→D represents a mapping from transitions to places.

A few more definitions,which will be referred to while presenting the algorithm,are in order.

Definition 2.If pi∈I（）t ，t∈T,then the directed arc from place pito transition t is the input arc of transition t.piis one of input places of transition t.If pi∈O（）t,t∈T,then the directed arc from transition t to place piis the output arc of transition t.piis one of output places of transition t.

Normally,each fuzzy deductive rule corresponds to one or a set of transition in FPN,and propositions in fuzzy rules correspond to places in FPN one to one.The membership grade of fuzzy proposition in fuzzy deductive rules is the marked value of places.The rule belief parameters and thresholds comprise a mapping function of transitions.The weights of fuzzy rules are assigned to the relevant input arcs of transitions.

In the vulnerability evaluation model every set of damage mode of equipments or components(input places)varies,the fuzzy deductive rule is set to judge the damage status of upper system(output places)by the mapping function defined by fuzzy belief parameters and thresholds.

For arbitrary transition,if the sum of products of marked values of its all input places and weights on all of its input arcs is greater than the threshold of this transition,then this transition t is enabled.

Definition 4.The enabled transitions are fired by generating a new marked value at its output place.The value is given by:f（t） ×ΣM （ pIj）×wIj,where pIj∈I（t）,wIjis weight on corresponding input arc.If place p is an output place comprised of multi-transitions,the marked value M（P）will take the max of transferred value.

For the AND rule,the marked value is given by:

where pIj∈I）,wIjdenotes the weight on the corresponding input arc of transition t.

For the improved OR rule,the marked value is given by:

where pij∈I ti（）,i=1,2,…n.Place p corresponds to input transition ti,f ti（）presents rule fuzzy belief of transition ti,widenotes weight on corresponding input arc of transition ti,woiis weight of transition ti,wokis corresponding weight of transition tkwho transmit the maximum input value.

Definition 5.If a transition t1for a place p∈I t1（）exists,but no transition t2exist for p∈O t2（）,then the place p is called concluding places.The concluding place corresponds to outcome proposition of fuzzy rules,viz.damage state of aircraft guarantee system in this paper.

Normally,a set of transitions followed by a set of places constitutes a layer.A l-layered FPN thus contains l-1 layers of transitions followed by places,and an additional input layer consisting of places only.This model can represent inexact knowledge,such as damage status of equipments,and train uncertain parameters with sets of input-output patterns,such as weights of equipments.

The FPN model without loops describes the damage status of aircraft guarantee system by the failure assembly from top to bottom,making the logical reasoning clearly and smoothly.The definitions of parameters,such as W,Th,etc,which clearly describe the complex weight relationships and the damage conditions of each layer system,agree well with our understanding of system damage.The abundance information embedded in the FPN model for aircraft guarantee system presented in this paper is an predominance,which lacks in conventional analysis method,for example,Damage Tree Analysis.

3.3 Algorithm for training and learning for FPN

The thresholds,weights and fuzzy belief parameters are generally specified by expert knowledge,with great subjectivity.Moreover,the parameters are often difficult to specify.For the large number of unknown parameters in evaluation model of aircraft guarantee system,it is a tremendous work to determine them.Particle swarm optimization(PSO)is a population-based optimization tool,which can effectively search for optima in complex space[20-21].The PSO has been found to be robust and fast in solving nonlinear,non-differentiable,multi-objective problems.The improved PSO is applied to set multitudinous weights in the evaluation model.

3.3.1 Improved PSO

PSO is a population-based optimization tool,where the system is initialized with a population of random particles and the algorithm searches for optima by updating generations and is originally developed by Kennedy and Eberhart[22-23]and inspired by the paradigm of birds flocking.PSO consists of a swarm of particles and each particle flies through the multi-dimensional search space with a velocity,which is constantly updated by the particle’s previous best performance and by the previous best performance of the particle’s neighbors.

In a n-dimensional search space,let the position and velocity of the ith individual be represented as vectors xi=（xi1,…,xid,…,xin）and vi=（vi1,…,vid,…,vin）,respectively.The best previous experience of the ith particle is recorded and represented by Pbesti=（Pbesti1,…,Pbestid,…,Pbestin）.The global best position of the swarm found so far is denoted by Gbesti=（Gbesti1,…,Gbestid,…,Gbestin）.Then the velocities and positions will be updated[24].

To search for weights of equipments and subsystems of aircraft guarantee system by PSO algorithm,the constraint condition is introduced to ensure the sum of weights on input arcs of arbitrary transition is 1:

where wijdenotes weight on input arc of transition ti,representing the contribution of equipments or subsystem to upper system;m is total number of input places corresponding to transition ti.

With the constraint condition shown by Eq.(5),the velocities and positions of particle swarm are adjusted by the following equations,until reach the predetermined terminal rule and output Gbest as the optimum solution.

where i=1,2,…n is the index of each particle,k is the iteration number.anddenote velocity and position of d-dimension of ith particle in kth iteration,respectively.vmaxrepresents maximum velocity,xmin=0 and xmax=1 denote the minimum position and maximum position of particles.rand1（）and rand2（）are the random numbers between 0 and 1.The learning factors c1and c2are the weighting factors of the stochastic acceleration terms,set to 2.0 according to early experiences[25].

m is inertia weight.The appropriate selection of inertia weight m in Eq.(6)provides a balance between global and local explorations,requiring less iteration on average to find a sufficiently optimal solution.In this paper,m is set by the following improved equation:

where mk,mmax,mminare current inertia weight,maximum weight and minimum weight,respectively.Joptand Jmeanare best value and mean value of all the current objective function values.k is current iteration number.Hence,the inertia weight is capable of self adapting to guarantee the convergence of search procedure.

3.3.2 Selection of objective function

In order to search for adequate weights by PSO algorithm,the difference between true value and predicted value of vulnerability of aircraft guarantee system is selected,demonstrated as follows:

where Mi（pj）and（pj）denote true marked value and expected marked value of concluding place pjin the training instances.

3.4 Algorithm flow of FPN with training ability

In this section,the training algorithm by improved PSO algorithm enables the FPN model with parameter learning ability.To apply the proposed FPN model,the following steps have to be taken:

(1)The FPN model is constructed based on fuzzy deductive rules,modeling the relationships of places by defining transitions.

(2）Initiate all the marked values of initial places,and set the other places with zero.Specify thresholds and fuzzy belief parameters of transitions.

(3)Take all the parameters requiring learning as the search zone of particles,specify the search scope and maximum velocity.

(4)The initial position and velocity for each particle should be generated randomly with consideration of constraint condition Eq.(5).

(5)Update velocity and position with Eqs.(6)-(11)of each particle.

(6)Evaluate objective function values for all particles with r sets of training data,and determine whether or not to update Pbest and Gbest.

(7)If the current status reaches predetermined terminal rule,then the search procedure is stopped and output Gbest as solution,otherwise go to step 5).

3.5 Numerical illustration for the improved FPN

To verify the improved FPN model in this paper,we apply the algorithm presented in the above section to analyze a FPN illustration from Ref.[26].The FPN model is shown in Fig.1.The initial places are p1,p2,p3,p5,p7,and concluding places are p6,p10.Suppose the ideal parameters are w11=0.3,w12=0.5,w13=0.2,μ1=0.87,λ1=0.3,μ2=0.7,λ2=0.3,w41=0.62,w42=0.38,μ4=0.9,λ4=0.3,w51=0.35,w52=0.65,μ5=0.8,λ5=0.3.The weights for OR rule are set to 1.

Select 100 sets of sample data to train the FPN model shown in Fig.1.The number of particles is 20,c1=c2=2.After k=150 times running of the algorithm,the parameters are determined as w11=0.319 0,w12=0.477 3,w13=0.203 7,w41=0.604 8,w42=0.395 2,w51=0.350 0,w52=0.650 0.After the training of the network is over,we can use the network for predicting 10 sets of random data.The comparison results between true values and predicted values by our model(see Tab.1)demonstrate the good learning performance of the proposed FPN model.

Fig.1 The fuzzy Petri net in Ref.[26]

Tab.1 Simulation results comparisons

4 Vulnerability assessment model of aircraft guarantee system based on improved FPN

4.1 Vulnerability assessment model of aircraft guarantee system

According to the function characteristics,aircraft guarantee system is divided into four subsystems:command system,landing system,launching system and schedule system.The critical positions on flight deck of typical aircraft carrier[27]is shown in Fig.2.

Fig.2 Critical positions on flight deck of typical aircraft carrier

With consideration of damage characteristics and logical relationships of aircraft guarantee system,the quantitative vulnerability assessment system based on improved FPN is constructed by fuzzy deductive rules based on function diagram.The accomplished FPN model is shown in Fig.3.The main equipments,which denotes initial input places in the presented FPN model:p1,p3,p5,p7are arresting gears,p33is associated landing equipment,p34denotes sliding area,p10～p13are aircraft elevators,p22～p25are ejection firing areas,p14,p16,p18,p20are ejection firing devices,p15,p17,p19,p21are jet blast deflectors,and p26presents command system.In this model,51 unknown weights are concerned,the fuzzy belief of each transition ti（i=1,2,…,36）is set to be 1,and threshold of transitions ti（i=1,2,…,35,i≠13,16,36）is 0.

Using damage tree analysis method,different damage tree are required to established for different damage levels.However,only one damage model is constructed by adjusting thresholds of transitions,r1，r2， r3in this paper,to simulate varied damage levels with one FPN model.In this way,application of FPN greatly decreases workload of damage description for complex systems.

Fig.3 The proposed FPN for vulnerability assessment system of aircraft guarantee system

4.2 Solution steps of vulnerability assessment

After accomplishment of vulnerability evaluation model of aircraft guarantee system,this model can be applied to predict the system damage status under various operational environments with all the weights of equipments and subsystems.In this paper,the improved PSO algorithm is integrated in FPN model to train the model with sample data to give referenced weights,with which the evaluation model can predict the system vulnerability under varied conditions.The procedures of vulnerability assessment of aircraft guarantee system based on improve FPN are as follows:

(1)According to the function diagram of aircraft guarantee system,construct FPN model by fuzzy deductive rules,establish connections between places by definitions of transitions.

(2)The fuzzy belief of each transition is set to 1,viz.,f（t）=1.The threshold of transitions tiis set to 0 except ti（i=13,16,36）,which is set according to system damage levels.Initialize all the marked values of initial input places with the fuzzified damage probability of equipments and components,set zero to other places.

(3)Take all the parameters requiring learning as the search scope of particles,specify the search band and maximum velocity.

(4)The initial position and velocity for each particle should be generated randomly with consideration of constraint condition Eq.(5).

(5)Update velocity and position with Eqs.(6)-(11)of each particle.

(6)For each particle with r sets of training data,compute the marked value of concluding place of vulnerability evaluation model for aircraft guarantee system,and take the difference of computed value and actual value as objective function values.Determine whether or not to update Pbest and Gbest.

(7)If the current status reaches predetermined terminal rule,then stop the search procedure and go to step(8),otherwise go to step(5).

(8)Substitute the trained weights to FPN model,initiate the marked value of initial input places with fuzzified damage probabilities of equipments under some specified operational environments,then go to step

(9)Compute the marked value of concluding places through a forward pass to give the system damage probability.

(10)Output results,program end.

5 Experiments and analysis

According to the vulnerability assessment model illustrated in Fig.3,different damage models corresponding to varied damage level are modeled by changing thresholds,and improved FPN is utilized to compute the system damage status.The training instances contain 50 sets of data,where the fuzzified damage probabilities of equipments based on fuzzy membership function under varied attack cases are referenced from relevant internal report,and the system damage probability is reasoned by expert knowledge.In this paper,xmin=0,xmax=1,c1=c2=2,the number of particles is 20,the predetermined number of iteration is 150.A plot of the objective function value of the given model versus training cycles is presented in Fig.4.It can be evidently seen from this figure that the value of the objective function settles at the minimum after about 110 iterations.

Besides,the trained weights are of actual meaning.The weights of various equipments will approach ideal value by validity analysis.Furthermore,the learned weights are rather accurate for the threshold and fuzzy belief already specified in this paper[26].After the training of this presented model is over,we can use it for application in system vulnerability assessment.We experimented with aircraft guarantee system under different operational environment.Tab.2 contains damage information of main equipments in aircraft guarantee system.And the obtained results are listed in Tab.3,compared with results of traditional damage tree analysis method.

Fig.4 Convergence performance of presented training algorithm for the best solution

Tab.2 Training instances by damage probabilities of aircraft guarantee system

Tab.3 System vulnerability evaluation by the presented model in this paper and damage tree analysis method

In this paper,the different dedications of various equipments to upper system are taken into account through the construction of evaluation model and training procedure with sample data.However,the traditional damage tree analysis holds that the equipments on the same layer contribute equivalent effect to the upper system.Hence,as shown in Tab.2,different results are yielded under the same operational environment.Taking the redundant equipments,arresting gears,for example,damage tree analysis treats all the arresting gears at different location identically,but the No.3 arresting gear is rather more important to the aircraft guarantee system than No.1[28].However,this ignorance of damage tree analysis disagrees with practical situation,while the presented model based on improved FPN model in this paper is more comprehensive and in accordance with physical circumstances.So we can conclude that the results obtained by the evaluation model in this paper is more reliable.

6 Conclusion

Considering the fuzzy random damage characteristics of ship equipments and varied effects of different components to upper system as well,a vulnerability assessment system based on advanced FPN model for aircraft guarantee system is presented in this paper.On the basis of standard FPN,the improved PSO algorithm is directly constructed on FPN model to enable it parameter learning ability.The PSO algorithm is improved in order to make the trained parameters more agreeable with actual condition.The presented evaluation model is convenient to obtain large numbers of weights by learning ability,and successfully evaluate the system vulnerability of aircraft guarantee system under various operational environment.

The vulnerability assessment model for aircraft guarantee system based on improved FPN is proved to be convenient and reasonable,with sufficient consideration of damage characteristics of equipments,which is fit for vulnerability evaluation of complicated ship system,especially for the system with large numbers of unknown parameters,this presented model is of unique advantage.

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