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Modeling of Weighted Timed Event Graphs in Dioids

Seminaire LARIS

Bertrand Cottenceau

Juin 2015

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 1 / 29

Timed Event Graphs (TEG)

TEG : subclass of Timed Discrete Event Systems

3 TUA S D

I events are associated to transitions

I A TEG gives a graphical notation to describe constraints on events

Graphical model for manufacturing systems, transportation systems,computer networks,...

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 2 / 29

Example (animation1)

I Each place (circle) has exactly 1 input and 1 output transition

I A sojourn time is associated to places

u1

u2

x1

x2

x33

2

1

1

I Internal transitions (x1, x2, x3) are fired As Soon As PossibleA transition consumes 1 token in each input place and produces 1token in each output place.

I Input (exogenous) events (u1, u2)

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 3 / 29

Example (animation2)

Production rate : 2 output events / 4 time units

u1

x1 x2 3 yx31 2

1

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 4 / 29

Timed Event Graphs as Linear Systems

Context

I Discrete Event Systems on max-plus algebra.

TEGs can be modeled by linear equations on some dioids (Baccelli etal (1992), Heidergott et al (2006))1.

I Similarities between the classical Control Theory and the theory ofLinear Systems on (max,+) algebras

keywords max-plus linear systems, linear systems on idempotentsemirings, linear systems on dioids

1Baccelli, Cohen, Olsder, Quadrat (1992) Synchronisation and Linearity, WileyHeidergott, Olsder,van der Woude, (2006) Max Plus at Work: Modeling and Analysis ofSynchronized Systems

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 5 / 29

Input-output models for TEGs (transfer function)

3 TUA S D

I Every TEG can be turned into a formal series in an algebraic structureMax

in [[γ, δ]]

δτ : time-shift operator γν : event-shift operator

Kleene star : x∗ = e ⊕ x ⊕ x2 ⊕ x3 ⊕ ...{D = δ3SS = γ2D ⊕ A

⇒ D = δ3(γ2δ3)∗A= (δ3 ⊕ γ2δ6 ⊕ γ4δ9 ⊕ ...)A

I The transfer series H = δ3(γ2δ3)∗ completely describes the TEG.

I Linearity ⇐⇒ Hδ1 = δ1H and Hγ1 = γ1H

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 6 / 29

The transfer series of TEGs are ultimately periodic2 (Gaubert (1992))

s =⊕

i γni δti

= p ⊕ q(γνδτ )∗

p= transient behaviorq = periodic pattern

s = γ0δ0 ⊕ γ1δ1 ⊕ γ2δ2 ⊕ γ4δ3...⊕ γ7δ6(γ2δ2)∗

Software Library MinMaxGD C++/Scilab (LARIS L. Hardouin) :computes the main operations on transfer series

2S.Gaubert (1992) Theorie des systemes lineaires sur les dioıdes. PhDBertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 7 / 29

Main GoalModeling of Weighted TEGs (integer weights) with similar tools.

63

2

3

4u y

The weights allow us to model newphenomena

I Batching/Lot making : synchronization onconsecutive occurrences of the same event

I Unbatching : one input event producesseveral output events

⇒ towards an input-output representation for Weighted TEGs

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 8 / 29

Example (animation 3)

Weighted TEG : the weights represent how many tokens areconsumed/produced when a transition is fired

yx2 x3x1 11 2

2 3 4

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 9 / 29

Outline

Timed Event Graphs as Linear Systems

Modeling toolsCounter functionOperatorsWeighted TEGs and Operators

Weight-Balanced TEGs

WBTEGs and formal series

Conclusion

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 10 / 29

Modeling tools : event list

Event occurrences

1

2

3

1

2

3

A

S

1

2

3D

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

time shift

time

time

time

2 occurences occurence of A at t=1

3 TUA S D

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 11 / 29

Modeling tools : Counter function

Counter : cumulated number of event occurrences

1

2

3

A

1 2 3 4 5 6 7 8 9 10

t

CA

1

2

3

1 2 3 4 5 6 7 8 9 10

4

5

6events A

t

Counter of events A

Counter associated to events A:

CA(t) : Z→ Z, t 7→ number of events A up to time t

In this context, a counter function plays the role of signal.

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 12 / 29

Modeling tools : operators

Σ: set of counters Operator : mapping on Σ.

2

2 4 126 148 1610

4

6

t

8

10

input counter x1

2

2 4 126 148 1610

4

6

t

time and event shift

8

10

18

output counter x2

operator

Example : operator h(x1) = x2 is a time-shift of 2 and an event-shift of 1.Additive operator : an operator h is said additive if ∀x1, x2 ∈ Σ,

h(x1 ⊕ x2) = h(x1)⊕ h(x2)

.

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 13 / 29

Modeling tools : operators

Operators γn and δt

γν : x 7→ γνx , {(γνx)(t) = x(t) + ν}δτ : x 7→ δτx , {(δτx)(t) = x(t − τ)}

2

2 4 126 148 1610

4

6

t

counter x1

counter x2

8

18 20

event shift

time shift

x2 = γ1(δ2x1) = δ2(γ1x1) = (γ1 ◦ δ2)x1 = (δ2 ◦ γ1)x1

γν and δτ can commuteBertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 14 / 29

Input-output models for TEGs

3 TUA S D

δ3S ⇐⇒ events S time-shifted by 3

γ2D ⇐⇒ events D event-shifted by 2{D = δ3SS = γ2D ⊕ A

⇒ D = δ3(γ2δ3)∗A= (δ3 ⊕ γ2δ6 ⊕ γ4δ9 ⊕ ...)A

I TEGs are described by formal series in a dioid 3 denoted Maxin [[γ, δ]],

where γ and δ are shift operators.

3Baccelli, Cohen, Olsder, Quadrat (1992) Synchronisation and Linearity, WileyBertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 15 / 29

Weighted TEGs and operators1 event a → 2 events b3 events c → 1 event d

unbatcha b

c d

2

3

a b

c dbatch

Equations with counters

Cb(t) = 2× Ca(t) (unbatch)Cd(t) = bCc(t)/3c (batch)

I unbatch ⇐⇒ multiply a counter by an integer

I batch ⇐⇒ integer division of a counter

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 16 / 29

Weighted TEGs and operatorsNew operators to model weights

[unbatch] µm : x 7→ µmx , {(µmx)(t) = x(t)×m}[batch] βb : x 7→ βbx , {(βbx)(t) = bx(t)/bc)}

x2 = µ2x1 x3 = β2x1∀t, x2(t) = x1(t)× 2 ∀t, x3(t) = bx1(t)/2c

2

2 4 126 148 10

4

6

8

10

input x1

output x2

2

2 4 126 148 10

4

6

8

10

input x1

output x3

time time

input

output

input

output

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 17 / 29

Transfer of a Weighted TEG= rational expressions on {γn, δt , µm, βb}.

µm cannot commute with γn or βbβb cannot commute with γn or µmδt can commute with γn, βb and µm.

63

2

3

4u y

v

w

w = β3γ1u

v = µ2u ⊕ γ1δ3vy = γ1v ⊕ δ4µ6w

⇒ y =(γ1(γ1δ3)∗µ2 ⊕ δ4µ6β3γ1

)u

= Hu

WTEGs are not Linear ⇐⇒ Hγ1 6= γ1HWTEGs are time invariant ⇐⇒ Hδ1 = δ1H

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 18 / 29

A subclass of Weighted TEGs

Weighted TEGs ⇒ liveness/boundedness issues.

Let us define a subclass with a structural constraint.

Gain of a path Γ: the product of multipliers divided by the product ofdivisors

y

2

34

3

u

x1

x2

Γ(u → x1 → y) = Γ(γ1β2γ1) = 1/2

Γ(u → x2 → y) = Γ(µ3β4γ1) = 3/4

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 19 / 29

Weight-Balanced TEG

A WTEG is said Weight-Balanced if for all pair of transitions ti , tj , thenall paths ti → tj have the same gain.

y2

34 63

3u

2

3

4u y

Not Weight-Balanced Weight-Balanced

x1

x2

x1

x2

Gain 1 for a loop

12 6=

34 2 = 6

3

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 20 / 29

2D Graphical representation of balanced E-operatorsE-operators : finite composition of operators in {γν , µm, βb}Counter to Counter (C/C) function Fw : ki 7→ ko

Fw1⊕w2 = min(Fw1 ,Fw2)Fw1w2 = Fw1 ◦ Fw2

w a balanced E-operator ⇐⇒ Fw is periodic.

β2γ1µ3 grey dots γ4µ3β2 black dots

Fβ2γ1µ3(ki ) = b(ki × 3 + 1)/2cFγ4µ3β2(ki ) = bki/2c × 3 + 4

3 2

32

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 21 / 29

3D Graphical representation

The input-output transfer of a WBTEG is a formal series with variables δwhere the coefficients are periodic E-operators

H =⊕i

wiδti s.t.

wi ∈ E [coefficient] periodic E-operatorti ∈ Z [exponent]

3D drawing

β2γ1µ3δ

2 ⊕ γ4µ3β2δ5 ⊕γ7µ6β4γ

1δ7

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 22 / 29

3D Graphical representation

Example The WBTEG below has the next transfer seriesβ2γ

1µ3δ2 ⊕ γ4µ3β2δ5 ⊕ γ7µ6β4γ1δ7

All the behavior of the WBTEG is contained in this 3D drawing.

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 23 / 29

Ultimately periodic series

Proposition

The transfer series of WBTEGs are (ultimately) periodic4.

Example H = p ⊕ q(γνδτ )∗ = µ3β2δ3 ⊕ µ3β2γ1δ4(γ1δ2)∗

H = µ3β2δ3

⊕ µ3β2γ1δ4

⊕ µ3β2γ1δ4(γ1δ2)

⊕ µ3β2γ1δ4(γ2δ4)

⊕ ...

4Cottenceau, Hardouin, Boimond (2014) Modeling and Control of Weight-BalancedTimed Event Graphs in Dioids, IEEE TAC

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 24 / 29

Ultimately periodic series and WBTEGs

The class of Weight-Balanced TEGs is an extension of the class of TEGsfor which periodic phenomena are still prevailing

TEGs : 2D formal series

Operators :γ δTransfer : p ⊕ q(γνδτ )∗

WBTEGs : 3D formal series

Operators :γ δ µ βTransfer : p ⊕ q(γνδτ )∗

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 25 / 29

Animation 4

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 26 / 29

Conclusion

I TEGs : formal series s =⊕γnδt (2D representation)

I Weight-Balanced TEGs : formal series s =⊕

wδt (3Drepresentation) where w is a periodic Event Operator

I Control problems are solved for these Discrete Event Systems.

I Perspectives : software tools, extension to WTEG with newtime-operators

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 27 / 29

Merci de votre attention!

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 28 / 29

Formal equalities on operators / Weighted TEGs

δ3δ1 = δ3+1

γ3γ2 = γ3+2

δ1 ⊕ δ3 = δmax(1,3)

γ1 ⊕ γ3 = γmin(1,3)

µ2γ1 = γ2µ2

β3γ4 = β3γ

3γ1

= γ1β3γ1

1 3 4

1

3

3

2 2

3 3

Bertrand Cottenceau Modeling of Weighted Timed Event Graphs in Dioids Juin 2015 29 / 29