Performance and tolerance evaluation with respect to forbidden state

IFAC [:0[>

Copyright Cl IFAC Fault Detection. Supervision and Safety of Technical Processes. Washington. D.C .• USA. 2003

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PERFORMANCE AND TOLERANCE EVALUATION WITH RESPECT TO FORBIDDEN STATE Collart Dutilleal SimOB and Craye EtienDe

UIL, Ecole Centrale de Lille Cite Scientifique, BN8, France 59651 Villeneuve d'Ascq Simon.CoIlart [email protected]. [email protected]

Abstract. The main proposition of the current paper is to provide a performance evaluation of a secure control in a repetitive functioning. The tolerance of the control towards time disturbances at a given functioning point is also analyzed. The methodology is based upon the collaboration of two different kinds of Petri Net models. Temporal Petri Nets are used for the contro] synthesis. Then, P-time PN provide a performance evaluation integrating the system and its control. Some analytical properties may be used in order to prove some functioning margins. Copyright © 2003 IFA C

Keywords: Petri Net, DEDS, Manufacturing, Control

The ability of Temporal Petri nets to face some forbidden state control problem have been already pointed out [1]. Moreover, a monitoring methodology has been proposed in order to face dynamic temporal uncertainties [2]. This approach is based upon "minimal or maximal remaining time in a location". It just so happened that Time PN had been introduced to provide fonnal specification about staying time constraints. The structural properties of this modeling tool provide strong analytical results. For example, the validity margins of an Event Graph in periodic functioning can be computed [3). This particularity may be used to solve some practical problem of Flexible Manufacturing System (FMS) [4], (5). The main proposition of the current paper is to provide a performance evaluation of a secure control in a repetitive functioning. The tolerance of the control towards time disturbances at a given functioning point will be analyzed too. The methodology is based upon the collaboration of two different kind of Petri Net models

Strongly Connected Event Graph (SCEG). Some structural properties are presented in order to provide an analytical characterization of the functioning validity ranges.

1. FORBIDDEN STATE A VOI DANCE AT mE CONTROL LEVEL. The discrete event control theory is based upon automata theory developed by Ramadge and Wonham [7]: In this paper a transition firing depends on the occurrence of events which cm be controllable and observable. The behavior of the uncontrolled plant is described by the automata and is called the open loop system. Then a control loop is added and the bebavior of the plant is called the closed-loop bebavior. One of the conveniences of this approach is the clear separation between the process and its control, permitting the introduction of theory about controllability and observability and the use of language analysis. Controlled PN have been first introduced by Krogh [8]. Some results dealing with forbidden state problem have been presented (9). The Wonham's formalism have been integrated afterwards in PN control theory [10]. Then the temporal PN (TtPN) have been introduced to face with temporal aspect of control theory ofPN (1).

The first part presents how the temporal Petri Net tool is used in order to provide some temporal specification. This step focuses on the control point of view. Then, the temporal specifications are described with the p-time PN (6). The closed loop system in periodic functioning is modeled by a

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An Arc transition in the PN marking graph corresponds to an activated clock in a given location. An activated Clock described a validated transition in the PN model. Let us denote Yu this Clock corresponding to the transition t" : a,. ~Yu~ b,. . Let us denote QUj a clock vector associated to each arc of the automata, where , and ~ are respectively the source and the destination locations. Obviously the dimension of this vector is equal to the number of clocks.

1.1 T-temporal Petri Net (TtPN) A t-1Jme-RdP is a couple where R is a marked Petri Net [11),[12]. IS: T-+(Q+uO)x(Q+u oo) ,Q+isthesetof positive rational numbers.

t; -+IS; =

[a; ,b; ].withO~a; ~bi

/S, is a static interval (to make a difference with

~ilj = ( [alilj ,h.ilj Dt.

the dynamic interval which will be defined in the following). A transition t must be validated during a minimum time ~ before it can be fired and can not be fired after a maximum duration ~. The firing of a transition has no duration.

The index h corresponds to the Clock. O
Computation o/the valid Clock intervals. The behavior of a three stated automaton will be considered.

A T-temporal Petri Net is a Rime PN that has been already defmed where T=Tc Vfuc [1]. Tc is the set of directly controllable transitions (we are able to inhibit these transitions) and Tuc the set ofuncontroIlable transitions . Let us consider G=(N, U, ~) . N describes the open-loop system and G the closedloop system as it has been explained in the state of the art. U is the external control based on predicates on external events and internal observable events, and Uo is the initial value of the control. A transition must be validated by the state of the system and by the control to be fired There is a control validation when the associated predicate is true.

Figure 1. A three stated automaton Let us denote "jlk Ilia the minimal staying time in the location ~ before the evolution towards the location Ik. tljlk miD

=max (0, 8!jll< -h.ilj)

Then, "jlk lUll is the maximal value of the staying time in the location ~ before the evolution towards the location It.

1.2 Analysis with continuous automata [2]

ltjlk mu = (hljll< Jllilj)

When there is a choice structure on the location the formula of the maximwn staying time is :

A continuous time automata is a 4uplet where: A =, where S is a finite set of location, X is a finite set of associated real-valued variable named Clock. E is a finite set of discrete transitions (s, "', ,(,5') : s,s'eS where s is the source transition and s' the destination transition '" is a constraint associated to the state transition. It is expressed by boolean combinations of linear constraint L(X'). X'cX '( is a function that assign a real value to a given clock. The automaton state is described by a location and a clock value. A discrete transition produce a change of the current location, but a continuous transition does not.

~

,

The clock vector deal with the two following case:

- 'Vh e H, (h fi R(Jt1j)/\ h e Y/i ("\ ~j) ,

where R(~ ~) is the set of clock variable which appears in the constraints of the evolution from li to ~ :

[aljll< ,bljlk]+- [alilj +tljlk ~iD

~jlk= Falj+[ tljlk .....

- 'Vhe H,(he R(/;l j

,

)/\

,bliti +tlj"'" it ~

he Ylj),

in [aljlk ,bljlk]+- [alilj.+ tljlk .. , blilj + tl/'1

E.jlk= (Elilj+[ tljt mill

The behavioral analysis of a TtPN model, is based upon a conversion into continuous automata [13]. A Clock variable in automaton is assign to each transition in TtPN.

•

ltj~8jjlk Ajlk]

The behavior construction considers all the possible evolution and computes the valid staying time ranges.

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1.4 Backfrom automata to temporal PN

Forbidden state avoidance The problem is now to avoid the forbidden state. It is decomposed into two different control rules: -Let us assume that the location ~ is a structural choice. The first scenario corresponds to an evolution between ~ and \c which is controllable. In this case, qjlk is restricted such a way that the forbidden evolution never occurs. When Ir is the forbidden location and b 1jlk new is the new upper bound value Ofqjlk, The conditions are :

A first proposition is to translate back the automata model into a temporal PN one which has a restricted behavior. This methodology does not need to build a new PN structure because the initial one may be used. A problem proceeds from the staying time specification imposed by the forbidden state avoidance. Actually, the forward and the backward control rules provide some temporal restriction. Unluckily, there is no practical way to express these constraints. It points out the modeling tool limitations. 1.5 Exampie

When it is not possible to fulfill the above specification, the second control rule has to be applied. -In the second case, the first rule, named forward control rule does not provide any result. There is no other controllable evolution on ~. or the analytic conditions of this rule did not match. Then, the constraints of the arcs leading to ~ are considered. The constraints associated to the evolution ~~ are modified in order to avoid the evolution to the forbidden state: such a rule is named a backward control rule.

Figure 2. the workshop Let us consider a synchronization constraint. For example, let us focus on a robot which has to deal with a product. The ware is on a conveyor which goes ahead slowly, but never stops. When the product arrives at the end of the conveyor, it falls and crashes down on the floor. Therefore, this state is forbidden and the robot has to catch the ware (figure 2).

When this two rules does not achieve the location Ir avoidance, the location lj is considered as a forbidden one. Consequently, the algorithm will try the first and second control rules with ~ and so on.

1.3 Building a cyclic scheduling Building the control for a repetitive production is a particular case of the control Jl'oblem which has been described above. In fact, the same methodology is applied but there is an additional constraint which requires the beginning state to be the same as the ending one. The location has to be the same, and the difference between the corresponding clock values has to be equal to the functioning period. Nevertheless, the control seeking step does not take into account the perfonnance evaluation. Moreover, there is no proof of a valid solution existence provide by the PN model itself. This leads to look for some more efficient tools for the specifications validation. The problem is to express the coupling between several Clock values and the staying times constraints which ensue from the forbidden state avoidance.

Robot ready to pick up a

ware.

Tc = { tl , t3 , ts , ~ } Tuc = { t2 , t4 } Figure 3. the t-Time PN model

On the figure 4, the arc which arrives to ll; is controllable, but the evolution towards the forbidden location Ir is not. The evolutions towards

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These last remarks lead us to introduce the definition of a live~oken sequenc e. A Jttime PN has a live token sequenc e if there exists a sequenc e where for all places : 'v'p.3q. ! ai S;qiS; b ,

are controllable, because the behavio r of the robot is under control. Obviou sly, the robot has to be disposa ble when a product arrives to the end of the convey or. When the condition (a)>b.t) is true, there is no solution for a valid product ion. Even if this static conditio n is not fulfilled, there exist some validity conditions ensuing from the dynami c intervals. ~I

where Cl is the staying time of a mark in the place Pi·

2.1 Modeling staying time constraints ware picked up

Conseq uently an input transitions OfoPi of places Pi immedi ately before a synchro nization transition must be fired in compat ible time with their interval s of staying time as it is shown from figure 5.

Drop of the ware

S"pJ (n) + al S; SpO(n) SpO(n) S; S"pJ(n) + bl S"p2(n) + 82 S; SpO(n) SpO(n) S; SOp2(n) + b 2

Figure 4. the automat on model From

~I

the forward control rules gives:

From ~2 the same rule provide the forbidd en state avoidance if (a) < a2-+14. b l ) is true. Then the appropr iate constraints evolutio n is :

If the last assertio n is false, the backwa rd control rule will be applied. The solution is non unique, but we propose :

Figure 5 : A synchronization in P-time PN These two last relation s give:

([a),b)] ~ [a3, a3]hll.J2 ([a),bd ~ [al . a2-+-14. b 3]) li2ljl

Max{S"pJ(n) + al; S"p2(n) + az}S; SpO(n) SpO(n) S; min{S"pI(n) + b l ; SOp2(n) + b 2} the left band part of this relation is the earliest nt. validati on date of transitions pOi, the right hand part is its latest one.

A valid control as been founded, but this control is not unique and it there is no way to affiIlD that it is the best onc from a production rate point of view.

Obviou sly, this conditio n is not satisfied , when the earliest firing instant is bigger than the latest one. Moreov er, an earliest functioning mode can produce a violation of the staying time constraint in a place before a synchro nization transition.

2. P-time PN Khansa W. has propose d P-time Petri nets [6]. A P-time Petri net is a couple where: R : is a marked Petri net I : P-i(Q+ U)«Q+ \..)Do)

pi-+li~bi] with ~Sbi

2.2 Properties of a strongly Connected Event

'v'i,~m,m=Card(p).

Graphs (SCEG)

Ii defines the interval of staying time of a mark in place Pi (Q+ is the set of positive rational numbers). A mark in the place It is taken into ac<:OUDl in transition validations when it has stayed in It at

With the assumption sufficie ntly unusual changes of production, the system arrives in a steady repetitiv e functioning. In this case, the process and its comma nd can be modele d by a Strongl y Connec ted Event Graph (SCEG) [14]. This lead us to present some useful properti es of P~ SCEG.

least a duration 8j and no longer than hi (after the hi duration the marked is said to be death).

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tackle with. Actually, the P~ime PN model is not well fitted to handle such a problem. The design of the P~ime PN model of the plant is a SCEG (Figure 6). Consequendy, the validity range of the production rate is directly computed with a polynomial algorithm (Theoreml). Until this step, it is possible to choose a valid production and the firing instants are computed.

Theorem 1: the existence sufficient and necessary conditions of live repetitive firing sequence are sufficient and necessary condition for monoperiodic functioning too. The demonstration can be consulted in [15]. A valid functioning can be computed with a polynomial time [4].

As a consequence computation of staying time restrictions and functioning frequency are the same. Theorem 2 (4): Let G be a P-time-SCEG functioning with a cyclic functioning of period C , [CmiD(G) , ~G>l is the interval of the time cycle of G. A necessary and sufficient condition of a periodic functioning existence is : C E [CmiD(G), c.....(G)]

2.3 Specifying the firing instants

ware.

onveyor

The fuing instants can be calculated with a polynomial algorithm [4]. A graph G' is associated with the Strongly Connected P-time Event Graph G in l-periodic functioning mode of C period: Figure 6. the P-time PN model • •

the nodes ofG' are the transitions ofG, the arcs ofG' are obtained from the places ofG: two arcs are associated with each place p. the first one from 0p to po is valued by: vp = lip - C.op

For a given functioning Cycle C, the developed graph associated to the SCEG of the figure 6 is described on the figure 7. Earliest and a lowest fuing instants will be compute on the considering the oriented path on the last graph. Moreover the functioning margins can be computed in order to integrate some uncertainties on operation durations [16].

- the second one from po to 0p is valued by : v'p

=-bp + C.op

A periodic control of the firing instants is obtained with the following algorithm - choose a transition t. associate St.(I) = 0 with ts associate with each transition tu E T

Stu(l) =

IIfiX LV p ••

where

~u

~I ..

is an elementary directed path from s to u.

This last algoritlun is in O(n\ The control is not unique and for a given time cycle an earliest and a lowest firing instant may be compute.

Figure 7 Associated valued Graph

2.4 Modeling a Forbidden temporal state

Let us assume that St~~ :

Then, the model P-time PN model corresponding to the figure 3 do not include any choice structure, because the drop of the ware is not represented. The easy computation of the cycle duration is the advantage of this matter of fact. By the contrary, when the temporal constraints are not fulfilled, there is no marking modeling the failure to be

St]max=max( aI, a2+a4b3) => ([a •.bd +- [ai, a2+at. b1]) lil'] ActuaIly, the same result as in the section 1.5 is obtained. Nevertheless, this last problem

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[3] Khansa, W. P. Aygalinc, J.P. Denat. ,1996, Structura1 analysis of p Time Petri Nets, CESA. '96,

corresponds to the computation of a longest path in a graph. This is a well known polynomial problem.

Lille [4] Col1art-Dutilleu1 S., Deruit J.P., "P-time Petri

To sum up, the analysi; based upon Time PN is made in a polynomial time. It allows to find some valid instant of transition firing taking into account both production rate and functioning margins. By the contrary the t-Tune PN based synthesis use some iterative procedures which complexity is

Nets and The Hoist Scbeduling Problem", Proc. of the IEEE Coni on Systems, Man, and Cybernetics (SMC'98), San Diego, CA, USA, October 1998, pp. 558-563. [5] Collart-Dutilleul S., Chetouaoe F., "Human integration and Participation in Time Constraint Workshop with Limiting Transportation Resources", Proc. of the 3rd IEEFJIFIP Int. Conf. , Information Technology for Balanced Automation System in Manufacturing (BASYS '98), Prague, Czech Republic, August 1998, pp.579-586. [6] Khansa W., Denat J.P.,Collart Dutilleul S., "Ptime Petri Nets for Manufacturing Systems", Proc. of the lEE International Workshop on Discrete Event Systems (WODES'96), Edinburgh, Scotland, August 96, pp. 94-102. [7] Ramadge P J.G and Wonbam W.M. "The Control of Discrete Event Systems", 1989 Proc. of IEEE, Vol 77, N° l,p 81-98. [8] Krogh B.H. "Controlled Petri nets and maximal1y permissive feedback logic", Allerton Conf.,October 1987. [9] Krogh B.H., Magott J. et Holloway L.E. " On the Complexity of Forbidden State problems for Controled Marked Graphs", IEEE TrallS. on Automatic Control, December 1991. [10] Ushio T. and Matsumoto R. "State feedback and modular control synthesis in controlled Petri Net", 27th conference IEEE on Decision and Cootrol. [11] P. Merlin, "A Study of the Recoverability of Communication Protocols", Ph.D. Thesis, Computer Science Dep., University of California, lrvine,1974. [12] B. Berthomieu and M. Diu, "Modeling and Verification of Time Dependent Systems Using Time Petri Nets-, in IEEE Trans. on Software Eng., Vol. 17, No. 3, 1991, pp259-273. [13] Culita J. "Contributii la sinteza sistemJor de rnonitorizare pcntru fabricatia integrata cu calculatorul" PhD Thesis, U.P. Bucarest Romania [14] Laftit, S. ,1992. Optimisation of invariant Criteria for Event Graph. IEEE Transaction on Automatic Control, Vol 37,N)S, Mayl992, pp547-

unknown.

Nevertheless, the choice structure on the figure 3 is not a functional choice, as it corresponds to a constraint. The P-time PN are Specification tool to model the desired bebavior. Temporal PN is rather a control tool. In the state of the art t-Tune PN may be used as a scenarios generator [17]. Moreover, there are strong results concerning control theory based upon automata. The temporal Petri Net use this powerfulness in order to face strong control problems including some time constraints.

3 CONCLUSIONS The ability of temporal Petri Nets to face some foxbidden state control have been shortly described in this paper. In the particular case of staying time constraint, a short example has shown the efficiency of the P-time PN as a specification tool. Nevertheless, this last analysis does not provide a result concerning the maximally pennissive control. It is just a formal specification tool. The firing. instants validity ranges are computed without taking ioto account whether tramitions are controllable or not. For these reasons we propose to use a mixed approach based upon model transformation. At a upper level, P-time PN may be used to solve some cyclic Scheduling problems. Then the global control may be faced with temporal PN. Nevertheless, local temporal tolerance and performance evaluation will be made with a more accurate way with P-time PN, as it is shown on the short example of the presented paper.

4. REFERENCES

[1] Caramihai S.I. and Alia H." On the synthesis of a controllable supervisor for discrete processes modeled by temporal Petri Nets" In Proceedings of the IEEE conference on Robotic and Automation, 1997, vol 1, pp 257,262. [2] Janet1a Culita, Simona lCaramibai, A.M.Stanescu "Monitoring the IJesynchronized /Jysfondions in Flexible Manufacturing Systems ", IFAC LSS200 I Symposium, Bucarest, Romania, July 2001

555. [IS] Khansa, W. ,1997, Reseaux. de petri Ptemporels contribution a I' etude des systemes a evenements discrets. PhD Thesis, March 1997. [16) Collart Dutilleul S., Denat J.P., Integration of the chemist expertise in electroplating line automation, ECC'99, Karlsruhe, Germany. [17] AK.A Toguyeni, E Craye and E. Castelain Robotics and Flexible Manufacturing Systems, 13th IMACS World Congress, Dublin (Irlande), Proc. of the 13th !MACS World Congress, Juillet 1991.

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