Designing Transitory Routing Controllers for Hybrid Plants

Claudia Fiedler, Wolfgang Meyer, Arne von DrathenHamburg University of Technology

claudia.fiedler@tu-harburg.de, w.meyer@tu-harburg.de,

von.drathen@tu-harburg.de

Abstract— We describe a generic routing controller for non-linear flow shops, and the respective design and verification toolbased on coloured timed Petri nets. The design methodologyintegrates logical and performance-related control for discreteevent systems by introducing a time invariant process V. Vstands for the cycle time of a repetitive production system andfor the stationary sequence of transport and move operations.Application site is a complex plant with 26 sequential/parallelworkstations and process plans of 28 operations includingmultiple loops. The transport system is the bottleneck resourceperforming as a non-linear GMEC (general mutual exclusionconstraint) on plant behaviour.

Index Terms— Hybrid systems, Petri nets, net invariantprocesses, scheduling

I. PROBLEM STATEMENT

A. The Manufacturing Problem

The application considered is a parallel/sequential plant,consisting of 26 specialized workstations or resources, someof them in parallel, which perform different operations,continuous as well as discrete ones. The flexibility of thisplant is provided by an advanced overhead rail transportsystem which directly connects all workstations. It offers analmost unlimited number of possible routings and, therewith,sequences of operations. Travel times depend on distanceswhich leads to a non-linear, hybrid problem. The processplan (Fig. 1) demands that there should be no slack timebetween two subsequent operations, i.e. buffers are notallowed between two workstations. The precedence and timespecifications posed upon the production process by theprocess plan are complemented by constraints of the plantitself, mainly capacity constraints of shared resources, i.e. re-sources required by a set of concurrently running processes.The processes include multiple loops and parallel operationswhich make planning and control a NP-hard problem.

B. The Control Problem

In the above scenario, the process plan is realized bythe transport system. By suitable transport control, themanufacturing problem is solved: to send parallel/sequentialprocesses over limited parallel/sequential resources in sucha way that the production goals are achieved. These are:Maximize productivity P, minimize flowtime f, minimizework-in-process WIP. WIP, f and P=WIP/f (the slope of theoutput curve) can be obtained from the plant throughputdiagram Fig. 2a [1]. The diagram displays a stationary orbalanced case, as input curve (of processes entering the

resources

time

process P1

process P2

resources

time

process P1

process P2

transport operations

Fig. 1. Process Gantt chart for two processes P1, P2

plant) and output curve (of products leaving the plant) are inparallel, indicating balanced plant operation. In a balancedplant with repetitive production mode, processes are releasedto the plant at a fixed time interval V. Solving the Cyclicstatic deterministic job shop problem means: find a schedulefor the transport operations which serves several concurrentprocesses and which minimizes V.

We aim at the Non-stationary non-linear job shop problemto be solved by transitory lot switching control. In termsof the non-stationary plant throughput diagram of Fig. 2bwe obtain a routing schedule for the transitory region V12which is different from the two stationary regions character-ized by their process invariants V1 and V2. The transitoryregion ranges somewhere between one and two flowtimesf. For small release times V12, productivity is high andthe transport system is heavily loaded which may lead todeadlock behaviour especially if buffer space is limited asin our application of strongly coupled processes. In fact,the two transporters themselves behave as multiple sharedresources and act as a major constraint on system flexibilityand throughput, causing deadlocks.

In our design method for DES (discrete event system)controllers, deadlock detection is used offline to rule outimpossible schedules and allocation decisions. Input for thecontroller design tool is an assorted list of process planslike the ones of Fig. 1. Output is the routing list for thetransporter, its destinations and associated transit times. Therouting list is verified and directly fed into a PLC program(programmable logical controller) which controls the trans-

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TB6.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 934

processes

P2P1P1P1P1

time

V1

V12V2

P2P2

transitory region

input

curve

output

curve

processes

P1 WIPP1P1P1P1P1

timef

V

P=WIP/f

processes

P2P1P1P1P1

time

V1

V12V2

P2P2

transitory region

input

curve

output

curve

processes

P1 WIPP1P1P1P1P1

timef

V

P=WIP/f

Fig. 2. Plant throughput diagram (a) stationary case; P productivity, fflowtime, WIP work in progress, V release time (b) non-stationary case: lotswitching transitory region; V1, V12, V2 release times

port system. Conceptually, the event driven simulation ofplant and transport system is converted into a time drivenrouting plan to be implemented as a cyclic control algorithm.

II. STATE OF THE ART

Though equipped with a variety of parallel machines inmultiple loops, production of our plant is characterized byhighly ordered, linear workflows with nearly fixed processingtimes only allowing for small deviations from the pre-specified process plan. With increasing productivity, i.e.concurrent production, mutual exclusion of shared limitedresources poses the main operational problem. For suchapplications, OR (operations research) has coined the termHoist Scheduling, providing several steady state algorithms[2] [3] [4]. In DES theory, the notion of sequential RAS(resource allocation system) has been established in the lastdecade as a branch of discrete systems control which dealswith the resolution of conflicts arising from allocation re-quests or claims from several processes at various processingstages. For recent reviews on RAS see [5] and [6]; theextension to unreliable resources can be found in [7] and[8] including important references.

However, almost all RAS methodologies deal with logicalcontrol or supervision only, and do not consider time. Thestructural control logic essentially defines the feasibilityspace for any performance-optimising control policy, notmore. Controller synthesis in our approach means to narrowthe feasibility space of the logical plant model by addingtime-constraints until the language generated by the timedmodel becomes a single string of (transport) events for agiven input, that is for a sequence of processes releasedto the plant. In this paper, we extend the notion of placeinvariant control [9] [10] [11] to time invariant control.The definition of a time invariant coupling process V(t) iscrucial for the integration of logical and performance-relatedcontrol. V(t) stands both for the cycle time of a repetitivesystem and for the stationary sequence of operations, heretransport operations respectively routing decisions. In thesubsequent sections, we illustrate the design procedure ofthe DES routing controller with a simple example in steadystate. Finally, we apply the method to the real plant and dealwith non-steady state behaviour.

III. PRINCIPLES AND METHODOLOGY

A. Scheduling DES (discrete event systems)

Parallel processes over shared resources create two typesof control problems: sequencing and allocation problems.Both together need scheduling: Scheduling = Sequencing +Allocation. Sequencing is the selection of a job sequenceout of a job queue in front of one machine. Allocation is theselection of one machine out of several parallel resources.Scheduling is the selection of machine and job sequence incase of a job queue in front of several parallel resources.

Plant and transport controller are modelled as Petri nets[12] [13]. The representational power of Petri nets for plan-ning and control tasks mainly derives from the fact that thebasic selection or choice decisions can easily be representedin a few simple net structures. Fig. 3a - 3c gives a selectionof control structures for regular (black) Petri nets whichwill be used in the forthcoming to constrain the reachablemarkings of the transport net. The notation is based on a wellacquainted text book about linear constraints [9]. Figs. 3d,eadd precedence relations to the list. Finally, time constraintsaccording to Fig. 3f turn out to be of great value to coordinateparallel production and transport processes.

B. Methodology of controller synthesis

Starting points for controller synthesis are process planand factory layout accompanied by a clear statement aboutbusiness goals like flexibility and diversification, or aboutproduction goals like productivity, flowtime, WIP. By turningthese specifications into a Petri net step by step, we arriveat a constrained model of the controlled (transport) system.Once the system shows desirable behaviour, the net canbe inverted in order to perform as the controller itself.Alternatively, after simulation and verification, the desiredbehaviour respectively the routing plan is fed into a PLC

c1

p1

p2

c1 p2

t5

c1 p3c1

p1

p2

c1 p2

t5

c1 p3

(b) (c)(a)

p1

p4

p2

p5

p3

p6

p2p4

p3 dp1

dp2

t1 t2

c2

c1

p1

p1

p4

p2

p5

p3

p6

p2p4

p3 dp1

dp2

t1 t2

c2

c1

p1

(d) (f)(e)

Fig. 3. Constraining Petri nets: (a) using sequential resources exclusively(b) limiting buffer overflow (c) prohibiting state-event combinations (d) pri-oritising events (e) sequencing operations (f) implementing time constraints.dp1 + dp2 = V; dp time duration

935

which guaranties stable operation. Our modelling approachbuilds upon the fundamental work of [14] and is extendedtowards repetitive production subject to time constraints.Specifically, we extend the notion of A-path towards timedA-path. The methodology follows these steps:I. Represent the multiple process Gantt chart as a Petri net

1) Arrange the operations (including transport operations)by the precedence relations as given in the process plan (Fig.1). Multiple occurrence of one type of operations is possibleand points to loops in the work flow.

2) For each operation of one process: create and label aplace to represent the status of the operation. Add a transition(start activity) and another transition (stop activity).

3) For each new process released, repeat step 3.4) Insert precedence constraints among operations of dif-

ferent processes which capture the logical plant structure(RAS structure). – Result is the logical process net forconcurrent processes and sequential/parallel resources.

5) Add operation times (durations d) and transport timesto the places according to the process plan. For a linearsystem: transport times are equal and fixed. For a nonlinearsystem: transport times depend on distance and therefore onthe sequence of routing decisions. – Result is the time-logicprocess net for concurrent processes and sequential/parallelresources.

6) For each operation in order: if such a place has notbeen created yet, create and label a place for each resourcewhich must be available to start the operation.

7) Specify the initial marking of the system.II. Define stationary coupling constraints among processes

8) Insert a fixed coupling constant or time invariant Vamong processes of the same type. V equals the order releasetime (see Fig. 2). – Result is the time-logic controller net forstationary operation of the plant.

9) If lot switching occurs, that is if two subsequent pro-cesses are not of the same kind, then V is not a fixed numberand must be calculated individually for each resource.III. Determine the time invariant process V(t) by simulation

10) Within the time window of V, we may find the cyclicrouting plan for stable plant operation by simulation with theaim to avoid deadlocks.

11) In case no acceptable values for V can be obtained,slight changes in the process plan may help. For this purpose,an extended type of simulation based on requests is presentedin Section 5 including a verification tool.

IV. ILLUSTRATIVE EXAMPLE

A. Control task

The simplified plant to be controlled resembles the onedescribed in [15]. It is an electrolytic production line whichprovides these resources (Fig. 4): 3 basins for electrolyticplating (BE1, BE3, BE5), 3 basins for rinsing (BR2, BR4,BR6), 1 transporter (T), several product carriers (PC). Ac-cordingly, these operations are performed on the PCs: plating(E1, E2, E3), rinsing (R2, R4, R6), transporting (loaded; T12,T23, . . . ), moving (unloaded; M12, M21, . . . ). The processesmust obey these constraints on plant behaviour: (i) Basins

TS1

R2

T12

E1

S0

availability of

transporter

T

availability

of basins

BE1

BR2

TS1

R2

T12

E1

S0

availability of

transporter

T

availability

of basins

BE1

BR2

availability

of basins

BE1

BR2

0

1

2

3

4

5

6

7

distance (m)

input

storage S0

electrolytic

basin BE1

rinsing

basin BR2

electrolytic

basin BE3

rinsing

basin BR4

electrolytic

basin BE5

rinsing

basin BR6

output

storage S7

transporter

(a) (b)

Fig. 4. Electrolytic line (a) plant layout (b) Petri net model

can only serve one process at a time. (ii) The transportercan carry only one PC at a time. (iii) The transporter is notallowed to wait wherever as long as it is loaded.

The transport controller takes care of these control prob-lems: (i) Synchronization among transporter and PC must beachieved. (ii) Competition among PCs for transporter must beresolved. (iii) Competition among PCs for basins must be re-solved. – For solving these problems, the transport controllerperforms these tasks: (i) collecting requests (transport jobs)for the transporter from different positions, within a specifiedtime window; (ii) sequencing transport jobs according topriority rules; (iii) triggering when and where to move ortransport.

In the Petri net representation of Fig. 4, multiple tokensstand for concurrent processes respectively product carriers.The availability nodes for transporter and basins monitorand control the sequence of operations both of transportand plant processes (the A-path). In this small example,we do not investigate any sophisticated sequence control.Instead, processes are released to the plant at equidistant timeintervals V beginning with the bottleneck operational time.Upon the first collision of transport requests, the programstops and a slightly increased release interval is selected untilthe processes run smoothly through all resources includingthe transport resources.

B. Integrated process Gantt charts

In the resource Gantt chart of Fig. 5, both the transportoperations (T12, . . . ) and the chemical plant operations (E11,. . . ) as dictated by the process plan, are integrated for 3identical processes. Obviously, basin BE3 is the bottleneckwith the longest processing time, determining the smallestpossible repetition period V of the processes in the line,or, according to the throughput diagram of Fig. 2, thesmallest possible release time V of orders for the line. Inthis hierarchical view, a Gantt chart is a decomposition of athroughput diagram.

Fig. 6 focusses at the transport operations between loca-tions 0 to 6 (input storage S0 to rinsing basin BR6; compare

936

BE1

BE5

BR4

BE3

BR2

BE1

BE5

BR4

BE3

BR2

resources

time

T

R41

E51

E11

R21

E31

R42

E52

E12

R22

E32

R43

E13

R23

E33

P1

time window V

dTS1 dE11 dT12 dBE1 transport operations plant operations

Fig. 5. Integrated process-transport Gantt chart; dE11 is the duration ofoperation E11. dBE1 is the time interval when resource BE1 is availablethat means basin BE1 is empty

Fig. 4). The operations are projected onto the only resourceof the transport system, transporter T. There is no collisionconflict shown in this example.

C. Petri net representation of Gantt charts

For large systems, Gantt charts have their limits as vi-sual planning and control tools. Therefore, we implementthem as Petri nets (PN) at the computer. Fig. 7 shows thelogical/structural (untimed) PN representation of the plantcapturing the precedence constraints of the process plan aswell as the mutual exclusion constraints of the basins. Forinstance, the token in Fig. 7, place BE1, is also shown inthe Gantt chart of Fig. 5 bottom left placed inside the timewindow. The token indicates the availability of basin BE1for another operation (E12).

Considering the time axis, the sum of basin BE1 occupied(dE11; d for duration) plus BE1 non-occupied (dBE1),added by two transport times, equals the time invariant V.This invariant equation: dE11+dBE1+2dT = V holds for allresources in the same way, e.g. for BR2: dR21+dBR2+2dT= V , etc. This set of equations coordinates the competitionof parallel processes for shared resources. The equationsare incorporated into the structural net as time constraintsaccording to Fig. 3f thus turning it into a timed PN. Uponcompletion, the time process net contains V as the only(unknown) variable which is determined by simulation insuch a way that no collisions occur for transport requests.- The above approach works for parallel resources as welland for loops up to a certain extent. Subsequent different

TS0

T45

T34

T23

T12

(resources)

operations

(T)

T56

P1 P2

stationary region

time

P4time window VP3

Fig. 6. Gantt chart for transport operations. Projection on top line(transporter resource) illustrates the mutual exclusive use of the transporter

sequential processes

per basin

sequential

operations

per process

E11

R41

E31

R21

dT12

dT23

dT34

BE1

BR2

BE3

BR4 BR4

BR2

BE1

BE3

P3 P1P2

S0

dTS1

E13

R43

E33

R23

dT12

dT23

dT34

E12

R42

E32

R22

dT12

dT23

dT34

BE1

BR2

BE3

BR4

dTS1 dTS1

S0S0

dBE1

dE11

Fig. 7. Logical Petri net of three processes P1, P2, P3 synchronizedby precedence relations according to the Gantt chart of Fig. 5. Eachtoken represents one product carrier. The thick-lined triangular structureimplements the time constraint dTS1+dE11+dT12+dBE1=V which can alsobe obtained from Fig. 5

processes P1, P2 can also be incorporated into the time-logicprocess matrice similar to Fig. 7. However, non-linearities oftransport times cannot be captured, and the move operationshave been omitted at all. Additionally, the model is tooinflexible to be used for process re-design.

V. NON-LINEAR ROUTING CONTROL

A severe consequence of the concurrent use of a transportsystem is that unloaded moves among different processesbecome a necessity. The feasibility space of (unloaded)movements among a small subset of locations (just 3 insteadof 26 as in the real plant) is shown in Fig. 8 as a Petri net.The hybrid properties of the move operations are captured bytimed transitions of the net. They are modelled as constantnumbers (linear case) or dynamic and space dependentfunctions (non-linear case).

By adding control places to all or some transitions ofthe Petri net (Fig. 8, right), we are able to select a specialrouting by defining a train of markings for the controlplaces M12, M23, etc. These markings are provided by anA-path net similar to Fig. 7. The process and controller

3

2

1

transition

times dt

locations M12 M21 M13 M31

movement commands M32 M23

1

3

2

M13 M31

M12 M21

M23 M32

3

2

1

transition

times dt

locations M12 M21 M13 M31

movement commands M32 M23

3

2

1

transition

times dt

locations M12 M21 M13 M31

movement commands M32 M23

1

3

2

M13 M31

M12 M21

M23 M32

1

3

2

1

3

2

M13 M31

M12 M21

M23 M32

(a)

Fig. 8. Petri net model of move operations: (a) uncontrolled (b) controlled

937

(a) (b) (c)

reso

urc

es

time

Fig. 9. Simulation Gantt chart for 20 processes of two different types P1, P2 showing the broad transitory region

nets are implemented as one coloured timed Petri net ofabout 1000 places. It enables the rigid implementation of thesend-receive principle as well: Operations call for resourceslike basins and transporters and are served on request – ordelayed. The scheduler allows for a maximum delay of onetransport time for some non-critical resources which is notmore than a few percent of the process time and acceptedby the process designers in our application. The outcomeof the simulation is graphically displayed as dynamic Ganttchart, Fig. 9, which is the solution to the non-stationary, non-linear job-shop problem as outlined in Fig. 2. The Gantt chartshows 20 processes with 28 plant operations each and abouttwice as many transport and move operations flowing throughthe factory within an 8 hours shift. There is a broad transitoryregion at the center of the chart framed by stationary regionsof P1 and P2 undisturbed. The routing list is contained in thetwo uppermost lines (Umsetzer 1 and 2 stand for transporters1 and 2). A closer look at the pattern reveals two differentbut periodic routings of the two transporters within theirrespective time windows V1 and V2, whereas a cutting fromthe transitory central part does not indicate any regularities(Fig. 10). Please note: The stationary windows display eachdifferent transport operation only once in a fixed sequencebut serve different processes per window.

Because of the complexity and the non-linearities captured

V1

V2

(b)

(c)

(a)

Fig. 10. Enlargement of the time windows from Fig. 9 detailing the cyclicschedules for P1 (a) and P2 (c) and the non-cyclic transitory schedule inbetween (b)

by this model, a lot of interesting dynamic effects have beenobserved in the Gantt charts. Sometimes some regions startbreathing suddenly, then they die out again. Most of thesephenomena have not opened up to us yet as the stabilityanalysis of discrete event systems is still in its infancy.

938

VI. SCHEDULE VERIFICATION

The principle underlying the verification tool is explainedon the basis of a regular place/transition net for two processesonly and a very short routing list. In Fig. 11, the A-pathnet for P1 and P2 has been stripped off from all timeinformation thus becoming an un-timed net. Via send/receiveplaces (SR) and test arcs, the V-process delivers the timeengine which drives the structural A-path net through time.V is the acronym both for verification and time invariantV which describes the routing sequence within the timewindow (V1 in Fig. 10, for instance), along a precise timeaxis. In terms of the Petri net representation of Fig. 11, Vis a cyclic process of time intervals dT, dt in a queue. Thetransport times dT are fixed numbers determined by the plantlayout (dT12 being the travel time between locations 1 and2, for instance, including loading and unloading operations),and dt are the slack times between two transport operationsincluding move operations as read from the respective timewindow. In this example, there are 28 different transport jobsper window dispatched to both transporters.

During the verification run, the V-process in alternatingrepetitive mode drives the (20 or more) processes P1, P2through the plant and creates the verification Gantt chartwhich, in case of a valid schedule, equals the simulationGantt chart of Fig. 9 with one but important distinction:Spurious modes of instability have been filtered out by thegenerate-and-test procedure. Reason: The verification Ganttchart is the outcome of time driven control, whereas thesimulation Gantt chart is the outcome of the event drivencontroller design.

dT12

dt1

dt2

dT45

E11

R21

T12

T23

P1 V

E32

R42

T34

T45

P2

SR

SR

SR

SR

C

dT12

dt1

dt2

dT45

E11

R21

T12

T23

P1 V

E32

R42

T34

T45

P2

SR

SR

SR

SR

C

Fig. 11. Principle of verification: The timeless A-paths for two processesP1, P2 are coupled to the cyclic process generator V by send-receive places(SR) and test arcs

VII. CONCLUSION

In principle, through our controller design procedure,we transform a non-linear time-variant model of the plantand transport system into a linear time-invariant controlledmodel. In other words, we pose time-driven behaviour upona discrete event system by control (which is a very generalformulation of the DES control problem as such). Theoutcome of the design procedure is a stationary cyclic

schedule if only one type of processes P1 is entering theplant, added by intermediate transitory schedules if differentconcurrent processes P1, P2 are considered. At the heart ofthe scheduler, a collision and deadlock avoidance mechanismis used based on the proper choice of a time invariant V.The Petri net implementation of the design tool offers thepossibility to relax some of the constraints, e.g. to enhancethe processing times of some non-critical operations. This, inturn, leads to spurious instabilities of the controlled net whichappear in a dynamic Gantt chart as small variations of thetime windows V with respect to interval length leaving thesequence unchanged in case of stable solutions. In the finaldesign step, we chose one of the time windows V(t) as thevalid schedule. After offline verification, V(t) is used onlinefor transport control. The proper choice of V also solves theproduction planning problem: by minimizing the time invari-ant, the plant productivity is maximized. Therefore, V(t) maybe the missing link between logical and performance-relatedcontrol as asked for in Section 2.

REFERENCES

[1] H.-P. Wiendahl, Load-Oriented Manufacturing Control. Berlin,Heidelberg, New York: Springer-Verlag, 1995.

[2] L. Phillips and P. Unger, “Mathematical programming solution ofa hoist scheduling program,” in AIIE Transactions: Industrial Engi-neering Research and Development, American Institute of IndustrialEngineers, New York, 1976, pp. 219–225.

[3] S. C. Dutilleul and J.-P. Denat, “P-timed Petri nets and the hoistscheduling problem,” in IEEE Int. Conf. on Systems, Man and Cy-bernetics, San Diego, California, USA, 1998, pp. 558–563.

[4] R. Rodosek and M. Wallace, “A generic model and hybrid algorithmfor hoist scheduling problems,” in 4th International Conference onPrinciples and Practice of Constraint Programming. London, UK:Springer-Verlag, 1998, pp. 385–399.

[5] S. A. Reveliotis, “Liveness enforcing supervision for sequential re-source allocation systems,” in Synthesis and Control of Discrete EventSystems. Boston, Dordrecht, London: Kluwer Academic Publishers,2002, pp. 203–212.

[6] M. P. Fanti, “Deadlock control methods in automated manufacturingsystems,” IEEE Trans. on Systems, Man and Cybernetics, vol. 34,no. 1, pp. 5–21, 2004.

[7] F.-S. Hsieh, “Fault-tolerant deadlock avoidance algorithm for assemblyprocesses,” IEEE Trans. on Systems, Man and Cybernetics, vol. 34,no. 1, pp. 65–79, 2004.

[8] Z. Li and M. Zhou, “Elementary siphons of Petri nets and theirapplication to deadlock prevention in flexible manufacturing systems,”IEEE Trans. on Systems, Man and Cybernetics, vol. 34, no. 1, pp. 38–51, 2004.

[9] J. O. Moody and P. J. Antsaklis, Supervisory Control of DiscreteEvent Systems using Petri Nets. Boston, Dordrecht, London: KluwerAcademic Publishers, 1998.

[10] F. Basile, P. Chiacchio, and A. Giua, “Optimal Petri net monitordesign,” in Synthesis and Control of Discrete Event Systems. Boston,Dordrecht, London: Kluwer Academic Publishers, 2002, pp. 141–153.

[11] A. Giua and C. Seatzu, “Deadlock characterization for Petri netscontrolled using GMEC’s and observers,” in IEEE American ControlConference, Denver, Colorado, 2003, pp. 320–325.

[12] C. G. Cassandras and S. Lafortune, Introduction to Discrete EventSystems. Boston, Dordrecht, London: Kluwer Academic Publishers,1999.

[13] R. David and H. Alla, Discrete, Continuous, and Hybrid Petri Nets.Berlin, Heidelberg: Springer-Verlag, 2005.

[14] M. Zhou and F. DiCesare, Petri Net Synthesis for Discrete EventControl of Manufacturing Systems. Boston, Dordrecht, London:Kluwer Academic Publishers, 1993.

[15] P. Bonhomme, P. Aygalinc, and S. Calvez, “Firing instant approachto control time critical systems in multi-product processing,” in 6thInt. Workshop on Discrete Event Systems, Zaragoza, Spain, 2002, pp.97–102.

939