Piece Linear-Aggregate Approach for Modelling and Analysis of Fuzzy Systems

—In this paper, we use a PLA model for performance and behaviour analysis of fuzzy system. This model makes connection between fuzzy logic and formalism. A case study contains an illustration how the proposed model can be fruitfully exploited to model traffic control systems based on fuzzy logic. Piece-Linear Aggregate model for traffic signal control system has been transformed into timed automaton for verification of safety, liveness, bounded-liveness and deadlock-freeness properties based on model checking. The system performance analysis was performed using Arena software package. A comparative analysis of traffic light controllers with fixed time and fuzzy logic algorithms is given.


I. INTRODUCTION
Formal methods are widely used for modelling and verification of complex systems.
By the increasing interest in a fuzzy logic a lot of scientists were interested to use formal methods to model fuzzy systems.Various methods are being used for modelling fuzzy systems, such as: DEVS [1], [2]- [8] Petrinets [3], Process algebra [4] and others.We think the researchers that use formal methods to model fuzzy systems pay not enough attention for the verification of these models.For formal modelling of fuzzy system we used piece linear-aggregate (PLA) formalism [9], which allows on the base of single formal description of system to create models for performance and behaviour analysis.The PLA formalism has been already applied to modelling computer network protocols, traffic systems [5], [10], medical applications [6], variable structure system, hybrid system [6], etc.
In this paper, we propose a general PLA model for modelling fuzzy systems.This model was used for performance and behaviour analysis of fuzzy traffic signal control system.For behaviour analysis the PLA model have been transformed to timed automata in order to verify the safety and liveness properties of a system using the UPPAAL model-checking tools.PLA formalism can be defined like a universal and general methodology that provides tools to simulate and verify systems which behaviour is based on discrete event [6].In the application of the aggregate approach for system modelling, the system is viewed as a whole of interacting aggregates.Each aggregate is represented as an object defined by a set of input signals X, output signals Y, events E and states Z.
A fuzzy logic systems (FLS) consists of four main parts: fuzzifier, rules, inference engine, and defuzzifier.These components and the general architecture of a FLS are shown in Fig. 1.In the first stage, the input parameters are fed to the fuzzification part to determine the degree of membership of crisp inputs in appropriate fuzzy sets.Following, the fuzzified input data are entered into the inference where the most appropriate rules are selected from the fuzzy rule base.Finally, the resulting fuzzy output is mapped to a crisp output in the defuzzification step.Using the scheme defined in Fig. 1 we created PLA model for modelling fuzzy systems.In PLA approach the fuzzy logic system aggregate is composed of several subaggregates (Distributor, Fuzzifiers, Connector, SMF and Deffuzifier) interacting with each other in order to implement a fuzzy system.Fig. 2 shows the aggregation model of fuzzy logic system.In aggregates Distributor, Fuzzifier and SMF there are no time management when input is received to the aggregate, it immediately generates an output.The model is completely generic, but depending on the application, the aggregates could be redefined.
A Fuzzifier transforms crisp values into membership grades of fuzzy sets.Formal specification of this aggregate in PLA formalism is presented below: 1.The set of input event = , ∈ ℝ; 2. The set of output event = , [0,1]; Piece Linear-Aggregate Approach for Modelling and Analysis of Fuzzy Systems

III. PLA MODEL OF THE TRAFFIC SIGNAL CONTROL SYSTEM
Case study demonstrates the application of PLA model, presented above, for modelling traffic signal control system.
We created a traffic signal control model for three-sided intersection, the exhaustive description could be found in [10].The traffic flows at three-sided intersection is regulated by three-phase traffic signals, indicating separate signal phases for westbound, south-west, and east-south traffic.The lights are operated by a controller which is able to control traffic flows adaptively based on traffic conditions.The control device controlling algorithm is based on fuzzy logic, which decides, whether to stay in current green phase or move to the next.
The PLA model of fuzzy traffic signal control system based on fuzzy logic is presented in Fig. 3.The system consists of three aggregates: Intersection, TLC (traffic light controller) and FLC (fuzzy logic controller).An input signal arrived from the aggregate TLC to aggregate Intersection causes an external transition and generates the output to the aggregate FLC.The output signal consists of the following attributes: 1) Average number of vehicles in the lanes of current green; 2) Arrival rate of current green phase; 3) Average number of vehicles in the lanes of next green.
Other output signal of this aggregate is the number of vehicles that passed the intersection during the green light.
The FLC receives signal from aggregate Intersection and produce the information for TLC to which state to transit after internal transition.When an internal transition occurs the TLC changes its state and produces an output signal to an aggregate Intersection.Fig. 4  :; = 0 ∧ (ℎ = 6 .

IV. VERIFICATION OF FUZZY TRAFFIC SIGNAL CONTROL SYSTEM
The traffic signal light controller has been verified with UPPAAL model checker [7], which is widely used for modelling, validation and verification of real-time systems.It is an integrated tool environment consisting of a graphical user interface, which allows system descriptions to be defined graphically as networks of timed automata, and a model checking engine that verify properties.It can check for invariant, reachability, and liveness properties of the system expressed in the UPPAAL Requirement Specification Language.This language is a subset of Timed Computational Tree Logic (TCTL).Table I specifies the queries available in the verifier.Because in this verifier the systems are modelled using timed automata, to this purpose the formal traffic light control model specification written in PLA was transformed to timed automata.The analyzed system in UPPAL environment consists of the following timed automata: queues (Q1, Q2 and Q3), traffic lights (TL1, TL2 and TL3) and traffic light controller (TLC).The state graph illustrated in figure 4 for an aggregate TLC is the same and for TLC timed automata.
A safety, liveness and bounded-liveness properties of TLC were checked during verification.The verified safety property states that whenever the traffic is allowed in one direction (the light is green), the light is red on the other directions.To prove that a system satisfies a safety property, we used an invariant.In order to define system invariant three symbolic states were identified, using the predicates ( , ( ? , ( G .In order to define this predicates the following auxiliary predicates were used: 1) H : HI = J Klight for the first approach is green; 2) H ? : HI ?= J K-light for the second approach is green; 3) H G : HI G = J Klight for the third approach is green; 4) : ./" , 0 ≠ ∞ -phase1 is active; 5) ?: ./?" , 0 ≠ ∞ -phase2 is active; 6) G : ./G " , 0 ≠ ∞ -phase3 is active.Using the defined auxiliary predicates we can formulate predicates ( , ( ? , ( G for symbolic states: 1) The expression ( means that the traffic lights in the first phase must be as follows: the light for the first approach is green, for second and third approaches the lights are red.( ? and ( G are interpreted similarly.Thus, the system invariant is: O = ( ⋁( ?⋁( G One liveness and three bounded liveness properties were checked.These properties written in timed computational tree logic are presented in Table II: 1) Phases liveness.Whenever traffic light controller activates Phase1, it will eventually activate Phase2 and Phase3.We check similar properties for the rest phases.
2) Minimum duration of the phases.The duration of the phase must be more than or equal to 10 seconds.
3) Maximum duration of the phases.Depending on traffic flows, the duration of each phase can be extended till 40 seconds.4) Traffic lights cycle time.The duration interval of the phase varies from 10 to 40 seconds, the length of the traffic lights cycle varies too, and its interval ranges between 30 to 120 seconds.Finally, the model checker confirmed that the model has no deadlocks in any points during a runtime.

SYSTEM
In this part a performance analysis of the fuzzy logic traffic lights control system was accomplished.A virtual environment for the junction based on the proposed modelling methodology was built using simulation software "Arena".Arena is a discrete event simulation software, which uses the SIMAN processor and simulation language.A comparative analysis between fuzzy logic control method and the fixed time controller has been made in traffic flows control under the same conditions without pedestrian crossing.The geometry of the intersection is illustrated in Fig. 5. Fig. 6 presents the phase order of the intersection model.The cycle sequence is 1-2-3-1-....During the simulation, arrival rate of vehicles in the lane varied between 0.08 and 0.19 vehicles per second (288-680 veh/hr/lane).The criterion used for the evaluation is the average stopped delay, i.e. the delay which occurs when a vehicle is fully immobilized.The best control strategy is the one that provides the lowest delays.The results of the system performance are shown in Fig. 7.During low and medium traffic flows the TLC based on fuzzy logic produced lower average stopped delay than fixed time technique.The fixed time controller produces similar time delay compared to FLC control strategy at heavy traffic flows, since the fixed time controller was optimized with Webster's equation to control traffic flows at intersection under heavy traffic conditions.The overall average stopped delay value was 11.6% lower using fuzzy logic controller as compared to fixed time controller.

VI. CONCLUSIONS
In this paper, we presented a PLA model for modelling fuzzy systems.The PLA model of fuzzy systems is completely generic.Depending on the application, the model has to be redefined.The created PLA model of fuzzy system could be used for analyzing its behaviour, safety, liveness and bounded liveness properties.In this paper for system behaviour analysis we used UPPAAL modelchecker.The PLA model of fuzzy system can be also used for creating simulators for system performance analysis.

Fig. 3 .
Fig. 3.The PLA model of traffic signal control system.
. PLA BASED MODEL OF FUZZY SYSTEMS.
Manuscript received March 19, 2012; accepted May 22, 2012.II shows a state graph for the aggregate TLC.Formal specification of aggregate TLC is ′ : /Input signal is received from aggregate FLC/:

Table I .
UPPAAL Property Specification.

TABLE II .
TCTL FORMULAS FOR ANALYSIS OF TLC SYSTEM.