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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
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10
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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
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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