Holding Time Maximization Preserving Output Performance for Timed Event Graphs

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Holding Time Maximization Preserving Output Performance for Timed Event Graphs Xavier David-Henriet, Laurent Hardouin, J¨org Raisch, and

Bertrand Cottenceau

Abstract—The trade-off between energy consumption and execution time (i.e., for a given task, the faster it is achieved, the higher its energy consumption is) is investigated for systems modelled by timed event graphs. In this paper, we aim to increase execution times (and, consequently, lower energy consumption) while preserving input-output and perturbation-output behaviors. Under this condition, the optimal solution is independent of the considered cost functions and is obtained using residuation theory.

Index Terms—Dioid, discrete event systems, manufacturing process, Petri nets.

I. INTRODUCTION

For some systems (e.g., in the field of manufacturing processes or transport networks), the occurrence of an event never disables the occurrence of other events. Such systems are called decision-free: the interesting question is not what the next event is, but when the next events happen. A possible model for such systems is an event graph, a particular Petri net, where each place has exactly one upstream transition and one downstream transition and all arcs have weight 1 [1]. To capture time in event graphs, timed event graphs (TEG) are built by equipping each place with a holding time (i.e., duration a token must spend in a place before enabling the firing of the next transition). It is a well known fact that, over some dioids (or idempotent semirings), the time/event behavior of TEG, under the earliest functioning rule (i.e., transitions with input places fire as soon as they are enabled), is expressed by linear state-space representations [2]. This last property leads to the development, by analogy with classical control theory, of control methods for TEG (e.g., optimal control [3], linear feedback [4], model predictive control [5]).

In the following, the problem of holding time maximization while maintaining acceptable performance is addressed (e.g., for manufacturing process, it could mean reducing the rate of some machines without decreasing the overall production rate). This problem is relevant to reduce energy consumption. Indeed, for a large class of systems, there is a trade-off between energy consumption and execution time: the energy consumption associated with a task (modelled by a place in the TEG) decreases, when its execution time (i.e., the holding time of the place) increases.

Some work has already been done on this problem. In [6], assuming that input dates and deadlines are known, holding times are adjusted at each cycle to minimize a convex energy criterion. The algorithm proposed by the authors is very efficient: the complexity is linear with the number of tasks. In [7], a structural approach is considered. The optimization is done for all inputs, this leads to a modification of the holding times once for all. The performance criterion is on the system throughput (periodicity of the transfer function matrix), which must remain greater than a certain value (specification). After some transformations, the author manages to X. David-Henriet and J. Raisch are with Fachgebiet Regelungssysteme, Technische Universit¨at Berlin, Einsteinufer 17, 10587 Berlin, Germany, and with Control Systems Group, Max Planck Institute for Dynamics of Complex Technical Systems (e-mail: [email protected], [email protected])

L. Hardouin and B. Cottenceau are with Laboratoire d’Ingénierie des Systèmes Automatisés, ISTIA, Université d’Angers, 62, avenue Notre Dame du Lac, 49000, Angers, France (e-mail: [email protected], [email protected]).

obtain an integer linear programming problem leading to the mini- mization of the number of resources and/or the maximization of the holding times. It yields a new system with a throughput above the specification.

In this paper, a structural approach is also considered, and the specification is to preserve the input-output and perturbation-output behavior of the system, not only to maintain the system throughput above a certain value as in [7]. Therefore, in our approach, the transient of the input-output behavior is not modified, which is relevant for systems prone to repetitive start-up procedures. Besides, as the perturbation-output behavior is preserved, the system after holding time maximization and the initial system have the same behavior in case of additive perturbations on the state.

The paper is organized as follows. Necessary algebraic tools are given in section II. In section III, TEG modeling over the dioid M^ax_in^vγ, δ^wis recalled. A first contribution, namely theγ-projection onto a series, is introduced in section IV. In section V, a residuation- based approach to solve the problem is presented.

II. ALGEBRAICPRELIMINARIES

The following is a short summary of basic results from dioid theory and residuation theory. The reader is invited to consult [2], [8], [9]

for more details.

A. Dioid Theory

A dioidDis a set endowed with two internal operations denoted

`(addition) and^b(multiplication, often denoted by juxtaposition), both associative and both having a neutral element denoted ε and e respectively. Moreover, ^`is commutative and idempotent (a ^P D, a^`aa),^bis distributive with respect to^`, andεis absorbing for^b(a^PD, ε^baa^bεε).

The operation ^` induces an order relation ^¨ on D, defined by

a, b^PD, a^©ba^`ba. According to this order relation,a^`b is the least upper bound of^ta, b^u. A dioidDis said to be complete if it is closed for infinite sums and if multiplication distributes over infinite sums.

By analogy with linear algebra,^`and^bare defined for matrices with entries in a dioid. Consider matrices A, B ^P Dⁿ^m and C ^P D^m^p,^pA^`B^q_ijAij^`Bij and ^pA^bC^q_ij^À^m_k

1AikCkj. Endowed with these operations, the set of square matrices with entries in a complete dioid is also a complete dioid.

Definition 1. A closure (resp. dual closure) mappingfis an isotone (i.e., order-preserving) projection (i.e.,ff f) from a dioidD into itself, greater than or equal to (resp. less than or equal to) the identity mappingId, i.e.,x^PD,f^px^q^©x(resp.f^px^q^¨x).

The following theorem gives the least solution to some implicit equations in complete dioids. It plays a fundamental role for the study of TEG behavior under the earliest functioning rule.

Theorem 1 (Kleene star theorem). The implicit equationxax^`b defined over a complete dioid admitsxabas the least solution witha

À

i¥0aⁱ (Kleene star), wherea⁰eandaⁱa^baⁱ¹ fori^¥1.

Remark 1 ([4]). x^ÞÑx, defined over a complete dioid, is a closure.

Proposition 1 ([4]). In a complete dioidD, the greatest solution to inequalityx^¨a isxa.

Proposition 2 ([10]). Leta, bbe two elements in a complete dioid D.a^©bis equivalent toabbaa.

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B. Residuation Theory

In ordered sets, like dioids, equationsf^px^qbmay have either no solution, one solution, or multiple solutions. In order to give always a unique answer to the problem of mapping inversion, residuation theory [11], [12] provides, under some assumptions, the greatest solution (in accordance with the considered order) to the inequality f^px^q^¨b.

Definition 2 (Residuation). Letf:E ^ÑF, with^pE,^¨qand^pF,^¨q ordered sets. An isotone mappingf is said to be residuated if for all y^PF, the least upper bound of the subset^tx^PE^|f^px^q^¨y^uexists and lies in this subset. It is denotedf⁷^py^q, and mappingf⁷is called the residual off.

If the considered ordered sets are complete dioids,La:x^ÞÑa^bx (left-product bya), respectivelyRa:x^ÞÑx^ba(right-product bya), is residuated. Its residual is denoted by L⁷a^px^qa^zx(left-division bya), resp.R⁷a^px^qx^{a(right-division bya). Therefore,a^zx(resp.

x^{a) denotes the greatest solutionyof the inequalitya^by^¨x(resp.

y^ba^¨x). As left- and right-products are extended to matrices with entries in a complete dioid, left- and right-divisions are also extended to matrices with entries in a complete dioid.

The following theorem gives a fundamental link between Kleene star and left-division (or right-division).

Theorem 2 ([4]). LetDbe a complete dioid andA^PD^pⁿ. Then, A^zA ^P Dⁿⁿ and A^{A ^P D^p^p. Moreover, A^{A ^pA^{A^q and A^zA^pA^zA^q.

III. TEG DESCRIPTION

The behavior of a TEG may be represented by transfer relations in some particular dioids. Hereafter, such a dioid is briefly presented, namelyM^axin^vγ, δ^w(see [2], [9] for more details), and TEG description in this dioid is recalled.

A. DioidM^ax_in^vγ, δ^w

DioidM^ax_in^vγ, δ^wis formally the quotient dioid ofBvγ, δ^w(the set of formal power series in two commutative variablesγ and δ, with Boolean coefficients and with exponents in Z), by the equivalence relation xRy γ^pδ¹^qx γ^pδ¹^qy. Dioid M^axin^vγ, δ^w is complete.

AsM^axin^vγ, δ^wis a quotient dioid, an element ofM^axin^vγ, δ^wmay admit several representatives inBvγ, δ^w. The representative which is minimal with respect to the number of terms is called the minimum representative.

A simple geometrical interpretation of the previous equivalence relation is available in the^pγ, δ^q-plane. Consider a monomialγ^kδ^t^P Bvγ, δ^w, its south-east cone is defined as k¹, t¹

|k¹^¥k and t¹^¤t

(

. The south-east cone of a series in Bvγ, δ^w is defined as the union of the south-east cones associated with the monomials composing the considered series. For two elements s¹ and s² in Bvγ, δ^w,s¹Rs²(i.e.,s¹ands²are equal inM^ax_in^vγ, δ^w) is equivalent to the equality of their south-east cones. Direct consequences of the previous geometrical interpretation are:

simplification rules inM^ax_in^vγ, δ^w

γ^k^`γ^lγ^min^p^k,l^qand δ^k^`δ^lδ^max^p^k,l^q (1)

a simple formulation of the order relation for monomials γⁿδ^t^¨γⁿ

1

δ^t

1

n^¥n¹ andt^¤t¹

The dater canonically associated with the seriessinM^axin^vγ, δ^wis the unique non-decreasing functionds :ZÑZYt8, ^8usuch

thats

À

k^PZγ^kδ^d^s^p^k^q. A simple interpretation of the variablesγ andδfor daters is available:

multiplying a seriessbyγis equivalent to shifting the argument of the associated dater function by1

multiplying a seriess byδ is equivalent to shifting the values of the associated dater function by1

Example 1. Consider the seriessγδ^`γ³δ⁴^`γ⁴δ³ represented by dots in Fig. 1. The south-east cone ofsis colored in grey in Fig. 1.

The minimum representative ofs inM^axin^vγ, δ^wisγδ^`γ³δ⁴. This result could be obtained using the simplification rules of (1).

0 1 2 3 4

1 2 3 4 5 γ

δ

Fig. 1. sand its south-east cone (grey)

Besides, s

à

k^¤0

γ^kδ⁸^`

à

k1,2

γ^kδ^`

à

k^¥3

γ^kδ⁴ Therefore, the daterds associated withs is given by

ds^pk^q

$

&

%

8ifk^¤0 1ifk1,2 4ifk^¥3

B. Linear state-space representation of TEG inM^axin^vγ, δ^w From now on, we only consider TEG with at most one place from a transition to another transition. This assumption is not restrictive, as it is always possible to transform any TEG in an equivalent TEG with at most one place from a transition to another transition.

The dynamics of a TEG may be captured by associating each transition with a seriess ^P M^axin^vγ, δ^w, where ds^pk^q is defined as the time of firingkof the transition. Therefore, for TEG,γis a shift operator in the event domain, where an event is interpreted as the firing of the transition, andδis a shift operator in the time domain.

The transitions of a TEG are divided into three categories:

state transitions (x¹, . . . ,xn): transitions with at least one input place and one output place

input transitions (u¹, . . . ,up): transitions with at least one output place, but no input places

output transitions (y¹, . . . , ym): transitions with at least one input place, but no output places

Under the earliest functioning rule (i.e., state and output transitions fire as soon as they are enabled), with respect to a place with initially mtokens and holding timet, the influence of its upstream transition on its downstream transition is a positive shift in the time domain of ttime units and a negative shift in the event domain of mevents.

The complete shift operator is coded by the monomial γ^mδ^t in M^ax_in^vγ, δ^w. Therefore, consider the place upstream from transition ti and downstream from transition tj, the influence of transition tj

on transitiontiis coded by the monomialfijinMâx_in^vγ, δ^wdefined byfijγ^mîjδ^τîj wheremijis the initial number of tokens in the place andτij is the holding time of the place.

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Consequently, a TEG admits a linear state-space representation in M^ax_in^vγ, δ^w

"

xAx^`Bu^`q

yCx (2)

where x ^P Mâxin^vγ, δ^wⁿ is the state, u ^P Mâxin^vγ, δ^w^p the input, y ^P Mâxin^vγ, δ^w^m the output, and q ^P Mâxin^vγ, δ^wⁿ the additive perturbation on the state. The perturbation q models, for example, unexpected failure, delays or uncertain parameters such as task duration (see [13]). A ^PMâxin^vγ, δ^wⁿⁿ, B ^P Mâxin^vγ, δ^wⁿ^p, and C ^PMâx_in^vγ, δ^w^mⁿ are matrices with monomial entries describing the influence of transitions on each other.

According to Th. 1, under the earliest functioning rule, the input- output (resp. perturbation-output) transfer function matrix H (resp.

G) of the system is equal toCAB (resp.CA).

yCABu^`CAqHu^`Gq (3) Therefore, the condition for holding time maximization (preserving the input-output and perturbation-output behaviors) is rephrased in terms of transfer function matrices. The condition is now to preserve the input-output and perturbation-output transfer function matrices.

When an element s of M^ax_in^vγ, δ^w is used to code information concerning a transition of a TEG, then a monomialγ^kδ^twithk, t^¥0 may be interpreted as at most k events occur strictly before timet (i.e.,ds^pk^q^¥t). An elementsofM^axin^vγ, δ^w, used to code a transfer relation between two transitions of a TEG (e.g., an entry ofH), is causal (i.e., no anticipation in the time/event domain: all exponents are non-negative) and periodic (i.e.,sp^`qr with polynomials p, q and a monomialre). For a periodic seriesswithrγ^νδ^τ, its asymptotic slope σ^ps^qis defined as ^ν_τ.

Example 2. A manufacturing system,composed of three machines M¹ (with transitions x² and x³),M² (with transitions x⁴ andx⁵) and M³ (with transitions x⁶ and x⁷), is considered. The system is modelled by the TEG represented in Fig. 2.

u

1

x2 3

2

x4 1

1 8 2

x7 y

0 2

x6

1 x3 1

x5 x1

0

Fig. 2. Manufacturing system

The matrices of the state-space representation are

A

ε ε ε ε ε ε ε

δ² ε γδ ε ε ε ε

ε δ³ ε ε ε ε ε

δ² ε ε ε γ²δ ε ε

ε ε ε δ ε ε ε

ε ε δ ε ε ε γδ²

ε ε e ε γδ γδ⁸ ε

Æ

B

e ε ε ε ε ε ε

Æ

C ε ε ε ε ε ε e

The input-output transfer function matrix, which is computed with the C++ library described in [14], is equal to

HCABδ⁵^` γδ¹⁴^`γ²δ¹⁸

γ²δ¹⁰

The asymptotic slope ofH is0.2, i.e., the average production rate of the system is at most1piece every5time units.

IV. γ-PROJECTIONONTO ASERIES

In this section, a new mapping from M^ax_in^vγ, δ^w to M^ax_in^vγ, δ^w is introduced to combine the event behavior of a series with the time behavior of another series. Its aim is to formalize the following condition: only holding times should differ between the initial system and the system after holding time maximization. First, a preliminary notion is defined:γ-discontinuity.

Definition 3. Given a series s^PM^axin^vγ, δ^wandds its canonically associated dater. A γ-discontinuity of s is an event k such that ds^pk^q ds^pk1q. The set of γ-discontinuities associated with s (i.e.,^tk^PZ|ds^pk^qds^pk1qu) is denotedKs.

As ds is a non-decreasing function, ds^pk^q ^¡ ds^pk1q for k ^PKs. Thus, in TEG, a γ-discontinuity is an event which occurs strictly later than the previous event. It represents a piece of information not given by the previous values of the dater (the south-east cone ofγ^kδ^d^s^p^k^q is not included in the south-east cone generated by the previous events). Therefore, considering theγ-discontinuities is necessary and sufficient to completely define the series. This assertion is formally shown in the following proposition.

Proposition 3. Given a seriess^PM^ax_in^vγ, δ^wandds its associated dater,

À

k^PK_sγ^kδ^d^s^p^k^q is the minimum representative ofs.

Proof: This result is a direct consequence of Th. 5.20 in [2,

§5.4.2.4].

Second, the new mapping itself is defined: theγ-projection onto a series.

Definition 4. Given a seriess⁰^PM^ax_in^vγ, δ^w. Theγ-projection onto seriess⁰, denotedPr^γ_s

0, of a seriess^PM^axin^vγ, δ^wis defined as Pr^γ_s

0^ps^q

à

k^PK_s0

γ^kδ^d^s^p^k^q

with Ks0 the set of γ-discontinuities of seriess⁰ and ds the dater canonically associated with seriess.

Theγ-projection of series sonto seriess⁰ is a series combining the event behavior ofs⁰ represented by its set of γ-discontinuities with the time behavior ofsrepresented by its associated daterds.

A direct extension of the γ-projection to the matrix case is possible: consider S⁰ ^PM^ax_in^vγ, δ^wⁿ^m,Pr^γ

S0 is defined as, for all S^PM^ax_in^vγ, δ^wⁿ^m, Pr^γ

S0^pS^q

ijPr^γ

pS0^qij

pSij^q. Proposition 4. Considersands⁰ inM^ax_in^vγ, δ^w

k^PZ, d_Pr^γ_s0ps^q^pk^qds

max

lPK_s0

tl^|l^¤k^u

Proof: Let us denote d˜the non-decreasing mapping, defined by d˜^pk^q ds max_l

PK_s0^tl^|l^¤k^u

, and ki, ki 1 two successive elements ofKs0 withkiki 1. Then, asd˜^pk^qds^pki^qforki^¤

kki 1,

à

kPZ

γ^kδ^d^˜^p^k^qPr^γ_s

0^ps^q

As the dater canonically associated with a series s is the unique non-decreasing function such that s ^À_k

PZγ^kδ^d^s^p^k^q, the dater canonically associated withPr^γ

s0^ps^qisd.˜

In the following proposition, the behavior of theγ-projection with respect to causality and periodicity is investigated.

Proposition 5. Consider s and s⁰ two causal and periodic series with

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sp^`qrandrγ^νδ^τ

s⁰p⁰^`q⁰r0andr⁰ γ^ν⁰δ^τ⁰ Pr^γ_s

0^ps^qis causal and periodic and, ifs⁰ is not a polynomial, then σ Pr^γ_s

0^ps^q

σ^ps^q.

Proof: For causality, the previous result is obvious. Therefore, only periodicity is considered in the following. If s or s⁰ are polynomials (i.e., ν 0, τ 0, ν⁰ 0, or τ⁰ 0), Pr^γ_s

0^ps^q is also a polynomial and the proposition holds. Thus, only the non- degenerated case is considered: ν, τ, ν⁰, and τ⁰ are greater than 0.

On the one hand, the periodicity ofs⁰ determines the structure of Ks0:

Ks0Kp0^Y

¤

j^¥0

Kq0,j withKq0,j^ta jν⁰^|a^PKq0^u

Kp0 (resp.Kq0) denotes the set ofγ-discontinuities ofp⁰(resp.q⁰).

On the other hand, the periodicity ofsimplies the periodicity of ds:

DK^¥0,k^¥K, ds^pk ν^qτ ds^pk^q

Next, consider j¹ ^P N such that K¹ minKq0,j¹ ^¥ K and k^¥K¹, according to Prop. 4,

d_Pr^γ_s0ps^q^pk νν⁰^qds k¹

withk¹ max

lPK_s0

tl^|l^¤k νν⁰^u (4) Furthermore,k^¥K¹^ñk νν⁰^¥K¹ νν⁰and sinceK¹^PKs0, K¹ νν⁰^PKs0 too, i.e.,

d_Pr^γ_s0ps^q K¹ νν⁰

ds K¹ νν⁰

(5) k^¥K¹ combined with (4) and (5) leads to

d_Pr^γ_s0psq

pk νν⁰^qds k¹

¥dPr^γ^psq K¹ νν⁰

¥ds K¹ νν⁰

It impliesk¹^¥K¹ νν⁰minK_q₀_,j¹ _ν. Then k¹ max

l^P_j

¥j¹ νK_q0,j

tl^|l^¤k νν⁰^u

max

l^P

j^¥j¹K_q0,j

tl νν⁰^|l νν⁰^¤k νν⁰^u

νν⁰ ˜kwith˜k max

l^P

j^¥j¹K_q0,j

tl^|l^¤k^u

Hence˜k^¥K¹^¥Kandk˜^PKs0 and by using periodicity ofs:

d_Pr^γ_s0ps^q^pk νν⁰^qds k¹

ds

νν⁰ k˜ ds

˜k τ ν⁰

According to Prop. 4, since˜k max_l

PK_s0^tl^|l^¤k^u, ds

k˜ d_Pr^γ_s0ps^q^pk^q, then,

d_Pr^γ_s0ps^q^pk νν⁰^qd_Pr^γ_s0ps^q^pk^q τ ν⁰ Consequently, Pr^γ

s0^ps^qis periodic and σ Pr^γ_s

0^ps^q

νν⁰ τ ν⁰ σ^ps^q

Example 3. Let s⁰ γδ¹⁰^`γ³δ¹³ γ³δ³

and sδ γ²δ⁵

. Then, Ks0 is equal to ^t1,3kwithk^¥0u and Pr^γ

s0^ps^q is equal to γδ ^` γ³δ⁶^`γ⁶δ¹⁶

γ⁶δ¹⁵

. In Fig. 3, the series s, s⁰ and Pr^γ_s

0^ps^q are represented in the^pγ, δ^q-plane. As expected (see Prop. 5), the throughputs of s and Pr^γ

s0^ps^q are both equal to 0.4, however the periodicities are different: γ²δ⁵ for s but γ⁶δ¹⁵ for Pr^γ_s

0^ps^q.

The aim of the following propositions is to characterize the series Pr^γ

s0^ps^q as the optimal solution of a problem conserving the set of

0 30

25

20

15

10

5

15 10

5 γ

δ

Fig. 3. Representation ofs₀ (dashed line),s(continuous line), Pr^γs0^psq

(dots), andK_s₀ (dotted line)

γ-discontinuities ofs⁰. First, some important properties ofPr^γ

s0 are proven.

Proposition 6. Given a series s⁰ in M^ax_in^vγ, δ^w, Pr^γ_s

0 is a dual closure.

Proof: Obviously,Pr^γ_s

0 is isotone (i.e.,s¹^©s²^ñPr^γ_s

0^ps¹^q^© Pr^γ

s0^ps²^q). Besides, due to the idempotency of^`,

s^PM^ax_in^vγ, δ^w, s^`Pr^γ_s

0^ps^q

à

kPZ

γ^kδ^d^s^p^k^q^`

à

kPK_s0

γ^kδ^d^s^p^k^q

s Therefore,s^PM^axin^vγ, δ^w,Pr^γ_s

0^ps^q^¨sor equivalentlyPr^γ_s

0 ^¨Id. It remains to show thatPr^γ

s0 is a projection (i.e.,Pr^γ

s0Pr^γ

s0 Pr^γ

s0).

According to Prop. 4, fork ^PKs0, we haved_Pr^γ_s0ps^q^pk^q ds^pk^q. Thus,

Pr^γ_s

0 Pr^γ_s

0^ps^q

à

k^PK_s0

γ^kδ^d^Pr^γ^s0^p^s^q

pk^q

à

k^PK_s0

γ^kδ^d^s^p^k^q

Pr^γ_s

0^ps^q

Second, an interpretation, in terms of γ-discontinuities, of the image of theγ-projection ontos⁰,Im Pr^γ_s

0

, is presented.

Proposition 7. Given a series s⁰ inM^ax_in^vγ, δ^w, Im Pr^γ_s

0

is the set of all seriess^PM^axin^vγ, δ^whaving a set ofγ-discontinuitiesKs

included inKs0.

Proof: On the one hand, consider s such that Ks Ks0. Then, as Ks Ks0 Z, s ^À_k

PK_s0γ^kδ^d^s^p^k^q (see Prop. 3).

Consequently,sPr^γ_s

0^ps^qands^PIm Pr^γ_s

0

. On the other hand, considers^PIm Pr^γ_s

0

, then there existss¹such thatsPr^γ_s

0 s¹

. Thus,s ^À_k

PK_s0γ^kδ^d^s^1p^k^q and, according to Prop. 3,KsKs0.

The following proposition presents an interesting result for dual closures.

Proposition 8. Consider a dual closure φ on a dioid D and an elements^PD.φ^ps^q is the greatest solution of

"

x^¨s x^PIm

pφ^q (6)

Proof: Asφ^¨Id,φ^ps^qis a solution of (6). Consider a solution xof (6), asx ^P Impφ^q, xφ^px^q ^¨s. Besides, as φ is isotone, xφ^px^q^¨φ^ps^q.

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Finally, combining the previous propositions (Prop. 6,7,8) leads to the interpretation of Pr^γ

s0^ps^q as the greatest solution of a problem preserving the set of γ-discontinuities ofs⁰.

Proposition 9. Consider two seriessands⁰inM^axin^vγ, δ^w,Pr^γ_s

0^ps^q is the greatest series less than or equal tosand having a set of γ- discontinuities included in the set of γ-discontinuities of s⁰, Ks0, i.e.,Pr^γ

s0^ps^qis the greatest solution of

"

x^¨s KxKs0

Proof: Consider a seriesx^PM^ax_in^vγ, δ^w. According to Prop. 7, x^PIm Pr^γ_s

0

is equivalent toKxKs0. Therefore,

"

x^¨s KxKs0

"

x^¨s x^PIm Pr^γ

s0

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As Pr^γ_s

0 is a dual closure (see Prop. 6), Pr^γ_s

0^ps^q is, according to Prop. 8, the greatest solution of (7).

The previous proposition is extended to matrices.

Proposition 10. ConsiderS andS⁰ inM^axin^vγ, δ^w^mⁿ,Pr^γ

S0

pS^qis the greatest element inM^axin^vγ, δ^w^mⁿless than or equal toS such that the set of γ-discontinuities of an entry ofPr^γ

S0^pS^q is included in the set of γ-discontinuities of the corresponding entry of S⁰.

Proof: As Pr^γ

S0

pS^q

ij Pr^γ

pS0^qij

pSij^q, the result is an obvious consequence of Prop. 9.

V. HOLDINGTIMEMAXIMIZATION

In this section, a method to obtain the greatest holding time while preserving the input-output and perturbation-output transfer function matrices is introduced. First, we look for the greatest state-matrix AM greater than or equal toAand preserving the input-output and perturbation-output transfer function matrices. Formally, this means finding the greatest state-matrixAM such thatAM ^©A,CA_MB CAB, andCA_M CA.

Proposition 11. Consider a TEG represented by matrices^pC, A, B^q, the greatest state-matrixAMsuch thatAM ^©A,CA_MBCAB, andCA_M CA is

AM CA^zCA

Proof: Preserving the perturbation-output transfer function ma- trix implies preserving the input-output transfer function matrix:

CA_M CA ^ñ CA_MB CAB. Therefore, the problem is to find the greatest state-matrixAM ^©A such thatCAM CA. As AM ^© A impliesCA_M ^© CA, it is equivalent to find the greatest state-matrixAM such thatAM ^©Aand CAM ^¨CA. If AM ^© A, then A_M ^© A. Thus, A_M AA_M (see Prop. 2) and CAM ^¨ CA CAAM ^¨ CA. Using residuation theory, CAA_M ^¨ CA A_M ^¨ CA^zCA. Finally, according to Th. 2 CA^zCA is a star, then, according to Prop. 1, A_M ^¨CA^zCA AM ^¨CA^zCA. Clearly,CA^zCA is a solution. Therefore, AM CA^zCA.

Remark 2. This problem can be rephrased as an optimal feedback control problem: find the greatest feedback F from state to state preserving the perturbation-output transfer function matrix. Formally, this is equivalent to finding the greatest feedback F such that C^pA^`F^q^¨CA (see [4]).

Prop. 11 is combined with theγ-projection onto matrices to obtain a TEG differing from the original TEG only for the holding times.

For the state-matrix A, a coefficientAij represents the influence of transitionxjon transitionxi. According to section III, consider the

place upstream from transitionxi and downstream from transition xj,Aij is the monomialγ^mîjδ^τîj wheremij is the initial number of tokens in the place andτijis the holding time of the place, hence, KA_ij ^tmijû. Furthermore, conserving the places and their initial markings (i.e., constraint of the considered problem) is equivalent to maintaining the set of γ-discontinuities in comparison to the one of the initial system. Consequently, the problem of holding time maximization is to find the greatest solution A¹ ^P Mâx_in^vγ, δ^wⁿⁿ of "

A¹^¨AM

i, j K_A¹

ij

KA_ij (8) Next, we show that, withAM CA^zCA, the previous problem comes down to finding the greatest solutionA¹^PM^ax_in^vγ, δ^wⁿⁿof

"

A¹^¨AM

i, j K_A¹

ij

KA_ij (9) According to Prop. 9 and Prop. 10, (9) admitsAoptPr^γ

A^pAM^q

as the greatest solution. We check that Aopt is a solution of (8).

Indeed, as the entries ofAare monomials, KA_ij is either empty or a singleton. IfKA_ij ^H, thenKA_opt,ij KA_ij. Otherwise,KA_ij

A^pA^q A. Therefore, Aij ε impliesAopt,ijε. In terms ofγ-discontinuities,KA_ij ^Himplies KA_opt,ij^H. Then,

HKA_opt,ij KA_ij

Furthermore, asKA_ij is a singleton,KA_opt,ij KA_ij. Consequently, as a solution of (8) is a solution of (9),Aoptis the greatest solution of (8).

Therefore, the greatest state-matrix modifying only the holding times and preserving the perturbation-output transfer function matrix is Aopt. Besides, the initial system and the system after holding time maximization have the same input-output behavior and the same response in case of perturbations.

Remark 3. The complexity of the calculation ofAoptis inO n³

elementary operations on periodic series withnthe number of state transitions in the considered TEG. In [7], holding time optimization is solved with an integer linear programming problem with n² constraints. However, the problems are not comparable since the maintained characteristics of the system are different between the two approaches. Therefore, the most suitable method might depend on the considered application.

Example 4. Consider the system presented in Ex. 2, the calculation is done with the C++ library described in [14], and the source code is available in [15]. Theγ-projection ontoAis applied toCA^zCA. For example, CA^zCA

45δ¹ γ²δ¹⁰

andA⁴⁵γ²δleads to

Pr^γ

A CA^zCA

45Pr^γ

A45 CA^zCA

45

γ²δ⁹ The complete solution is

AoptPr^γ

A CA^zCA

ε ε ε ε ε ε ε

δ² ε γδ ε ε ε ε

ε δ³ ε ε ε ε ε

δ¹² ε ε ε γ²δ⁹ ε ε

ε ε ε δ ε ε ε

ε ε δ ε ε ε γδ²

ε ε e ε γδ γδ⁸ ε

Æ

In this very simple example, the maximal admissible holding time modifications are