Discrete-Event Systems in a Dioid Framework: Control Theory

Chapter 1

Discrete-Event Systems in a Dioid Framework:

Control Theory

Laurent Hardouin, Olivier Boutin, Bertrand Cottenceau, Thomas Brunsch and Jörg Raisch

Abstract In this chapter, we will recall some contributions to dioid theory dealing with control that were achieved during the last two decades. Just like in classical control engineering, control is to be understood as having an action on the inputs so as to adapt to given specifications. For instance, one could aim at finding an optimal control in order to track an a priori known output trajectory. Since inversion, which would be necessary for such a computation, is not available in a dioid framework, we will also present notions of residuation theory, which introduces pseudo-inverses that are suitable to our needs. Provided a model of a system and a desired output for it, it can be shown that there exists a greatest input that leads to an output which is lower than or equal to the desired one. Practically, this greatest solution implies that all the events occur as late as possible while ensuring that the output events occur before the ones given by the desired output. In a production management context, this comes down to delaying as much as possible the input of raw parts in the manu- facturing system, while ensuring a predefined performance; hence the internal stock is reduced as much as possible. The control strategy is then optimal according to the just-in-time criterion. This chapter will give the results allowing to synthesize this optimal control. But this kind of open-loop strategy does not take the real time Laurent Hardouin, Bertrand Cottenceau and Thomas Brunsch

Laboratoire d’Ingénierie des Systèmes Automatisés, Université d’Angers, 62 Avenue Notre-Dame du Lac, 49000 Angers, France.

e-mail:{laurent.hardouin,bertrand.cottenceau}@istia.univ-angers.fr Olivier Boutin

e-mail:olivier.research@gmail.com Thomas Brunsch and Jörg Raisch

Technische Universität Berlin, Fachgebiet Regelungssysteme, Einsteinufer 17, 10587 Berlin, Ger- many,

e-mail:{brunsch,raisch}@control.tu-berlin.de Jörg Raisch

Fachgruppe System- und Regelungstheorie, Max-Planck-Institut für Dynamik komplexer technis- cher Systeme, Sandtorstr. 1, 39106 Magdeburg, Germany

1

response of the system into account. So we will also extend control strategies to closed-loop ones, which allow to react to possible disturbances. For each control strategy an illustrative example dealing with a High-Throughput Screening system, which has served as a case study within the DISC project, will be given.

1.1 Motivation

This chapter deals with the control of discrete-event systems which admit linear models in dioids (or idempotent semirings), introduced in the previous chapter (see also the books [1, 17]). These systems are characterized by synchronization and delay phenomena. Their modeling is focused on the evaluation of the occurrence dates of events. The outputs correspond to the occurrence dates of events produced by the system and the inputs are the occurrence dates of exogenous events acting on it. The control of the system inputs will lead to a scheduling of these occurrence dates such that a specified objective for the system be achieved. For example, in a manufacturing management context, the controller will have to schedule the input dates of raw materials or the starting of jobs in order to obtain desired output dates for the produced parts. The nature of the considered systems is such that they cannot be accelerated, indeed the best performance of the system is fixed and known. The only degree of freedom is to delay the inputs (i.e. to decrease its performance).

For example, in a manufacturing system, the maximal throughput of a machine is obtained with its fastest behavior, hence the only thing which can be done is to delay the starting of jobs or decrease its production speed. It is the same in transportation networks. E.g. for a train, the maximal speed is known, the only available action is to decrease the speed or to delay the starting time. Hence, the control synthesis presented in this part will aim at controlling the input dates in the system in order to get the outputs at the desired date. More precisely, it will be shown that the proposed control strategies will aim at delaying the inputs as much as possible while ensuring that the outputs occur before the desired dates. This kind of strategy is rather popular in an industrial context, since it is optimal with regard to the just-in-time criterion, which means that the jobs will start as late as possible, while ensuring they will be completed before the desired date. The results presented in this chapter were mainly introduced in [16, 26, 5, 6, 24, 22], considering the just-in-time criterion as objective function for the controller synthesis. In the related literature some other points of view were considered. In [9, 29], the authors considered the so-called model predictive control in order to minimize a quadratic criterion by using a linear programming approach. The same authors have extended their control strategy to a stochastic context (see [32]) in order to take uncertain systems into account. In [20, 21] system uncertainties are taken into account by extending interval analysis to dioids (see [12, 23]), the controls proposed are then robust to variations of the systems. This chapter will be organized as follows. In a first step, some theoretical results will be given; they come in addition to the ones introduced in Chapter 20.

Then the control strategies will be presented, starting with open-loop strategies and

concluding with closed-loop ones. For both, an illustration based on the example introduced in Chapter 20 will be provided.

1.2 Theory and Concepts

The systems considered and the algebraic context were described in the previous chapter. Classically, the control of a system has to do with system inversion. Hence we recall in this section some algebraic results allowing us to invert linear equations in dioids,i.e.we focus on the resolution of a systemA⊗X =B, whereA,X and B are matrices with entries in a given dioid. The product law of semirings does not necessarily admit an inverse, nevertheless it is possible to consider residuation theory to get a pseudo inverse. This theory (see [2]) allows us to deal with inversion of mappings defined over ordered sets. In our algebraic framework, this will be useful to get the greatest solution of inequality A⊗X B. Indeed, a dioid is an ordered set, because the idempotency of the addition law (a⊕b=a⇔ab) defines a canonical order. Hence this theory is suitable to invert mappings defined over dioids. The main useful results are recalled in a first part. Another useful key point is the characterization of solutions of implicit equations such asx=a⊗x⊕b. It will be recalled in a second step that the latter admits a smallest solution, namelya^∗b, where∗is the Kleene star operator. Some useful properties involving this operator will then be recalled.

1.2.1 Mapping Inversion Over a Dioid

Definition 1 (Isotone mapping) An order-preserving mapping f :D→C, where DandC are ordered sets, is a mapping such that: xy⇒ f(x) f(y). It is said to beisotonein the sequel.

Definition 2 (Residuated mapping) LetDandC be two ordered sets. An isotone mappingΠ:D→C is said to be residuated, if the equationΠ(x)b has a greatest solution inDfor all b∈C.

The following theorems yield some necessary and sufficient conditions to charac- terize theses mappings.

Theorem 3 ([3, 1]) LetDandC be two ordered sets. LetΠ:D→C be an isotone mapping. The following statements are equivalent.

(i) Π is residuated.

(ii) There exists an isotone mapping denoted Π^] :C →D such that Π◦Π^] Id_C and Π^]◦Π Id_D.

Theorem 4 LetD and C be two ordered sets. LetΠ :D →C be a residuated mapping; then the following equalities hold.

Π◦Π^]◦Π=Π Π^]◦Π◦Π^]=Π^]

(1.1) Example 5 In [1], the following mappings are considered:

L_a:D→D

x7→a⊗x (left product by a), R_a:D→D

x7→x⊗a (right product by a),

(1.2)

whereDis a dioid. It can be shown that these mappings are residuated. The corre- sponding residual mappings are denoted:

L_a^](x) =a\x^◦ (left division by a),

R^]_a(x) =x/a^◦ (right division by a). (1.3) Therefore, equationa⊗xbhas a greatest solution denotedL^]a(b) =a\b. In the^◦ same way,x⊗abadmitsR^]a(x) =b/a^◦ as its greatest solution. LetA,D∈D^m×n, B∈D^m×p,C∈D^n×pbe some matrices. The greatest solution of inequalityA⊗X Bis given byC=A\B^◦ and the greatest solution ofX⊗CBis given byD=B/C.^◦ The entries of matricesCandDare obtained as follows:

C_{i j} =

m V k=1

(A_ki\B^◦ _{k j}) D_{i j}=

p V k=1

(B_ik/C^◦ _jk),

(1.4)

where∧is the greatest lower bound. It must be noted that∀a∈D,ε⊗xaadmits ε\a^◦ =>as its greatest solution. Indeed,εis absorbing for the product law. Further- more,> ⊗xaadmits>\a^◦ =εas a solution, except ifa=>. Indeed, in the latter case>\>^◦ =>, that is to say> ⊗x >admits>as a greatest solution.

Example 6 Let us consider the following relation AXB with A^t= 2ε 0

5 3 7

and B^t = 6 4 9

being matrices with entries in Zmax. By recalling that in this dioid the product corresponds to the classical sum, the residuation corresponds to the classical minus. That is to say, 5⊗x8 admits the solutions set X ={x|x 5\8,^◦ with5\8^◦ =8−5=3}, with3being the greatest solution of this set¹. Hence, by applying the computation rules of Equation(1.4), the following result is obtained:

A\B^◦ =

2\6^◦ ∧ε\4^◦ ∧0\9^◦ 5\6^◦ ∧3\4^◦ ∧7\9^◦

= 4

1

.

1Residuation achieves equality in the scalar case.

This matrix is the greatest one which ensures that A⊗XB. See the verification below.

A⊗(A\B) =^◦



 2 5 ε3 0 7



 4

1

=



 6 4 8







 6 4 9



=B.

According to Theorem 4, the reader is invited to check that:

A\^◦ A⊗(A\B)^◦

=A\B,^◦ A⊗ A\(A^◦ ⊗B)

=A⊗B. (1.5)

Practically, this illustrates that this kind of systems admits an inverse when matrix Bis in the image of matrixA, which is equivalent to say that there exists a matrix L∈D^n×psuch thatB=A⊗L. As in the classical algebraic context, in the absence of ambiguity, the operator⊗will be omitted in the sequel.

Below, some properties about this “division” operator are recalled. Actually, only the ones that are useful in control synthesis are recalled. The reader is invited to consult [1, pp. 182-185] and [8] to get a more comprehensive presentation.

Property 1.

Left product Right product

a(a\x)^◦ x (x/a)a^◦ x (1.6)

a\(ax)^◦ x (xa)/a^◦ x (1.7) a a\(ax)^◦

=ax (xa)/a^◦

a=xa (1.8)

(ab)\x^◦ =b\(a^◦ \x)^◦ x/(ba) = (x^◦ /a)^◦ /b^◦ (1.9)

(a\x)b^◦ a\(xb)^◦ b(x/a)^◦ (bx)/a^◦ (1.10) Proof. Proofs are given for the left product, the same hold for the right one.

Equations (1.6) and (1.7): Theorem 3 leads toL_a◦L^]_aId_D (i.e.,a(a\x)^◦ x) and L_a^]◦L_aId_D(i.e.,a\(ax)^◦ x).

Equation (1.8): Theorem 4 leads toL_a◦L^]_a◦L_a=L_a(i.e., a a\(ax)^◦

=ax).

Equation (1.9): It is a direct consequence of the associativity of the product law.

Equation (1.10): According to associativity of the product law, a(xb) = (ax)b then a (a\x)b^◦

= a(a\x)^◦

b and a\^◦

a (a\x)b^◦

=a\^◦

a(a\x)^◦ b

. Accord- ing to Equation (1.7), (a\x)b^◦ a\^◦

a (a\x)b^◦

and according to Equation (1.6) a\^◦

a(a\x)^◦ b

a\(xb)^◦ . Then, inequality (1.10) holds.

1.2.2 Implicit Equations Over Dioids

Definition 7 LetDbe a dioid and a∈D. The Kleene star operator is defined as follows:

a^∗,e⊕a⊕a²⊕...=^M

i∈N0

aⁱ, with a⁰=e and aⁿ=a⊗a⊗ · · · ⊗a

| {z }

n times

.

Theorem 8 The implicit equation x=ax⊕b admits a^∗b as smallest solution.

Proof. By repeatedly inserting equation into itself, one can write:

x=ax⊕b=a(ax⊕b)⊕b=a²x⊕ab⊕b=aⁿx⊕aⁿ⁻¹b⊕ · · · ⊕a²b⊕ab⊕b.

According to Definition 7, this leads tox=a^∞x⊕a^∗b. Hencexa^∗b. Furthermore, it can be shown thata^∗bis a solution. Indeed:

a(a^∗b)⊕b= (aa^∗⊕e)b=a^∗b, which implies thata^∗bis the smallest solution.

Now we give some useful properties of the Kleene star operator.

Property 9 LetDbe a complete dioid.∀a,b∈D,

a^∗a^∗=a^∗ (1.11)

(a^∗)^∗=a^∗ (1.12)

a(ba)^∗= (ab)^∗a (1.13)

(a⊕b)^∗= (a^∗b)^∗a^∗=b^∗(ab^∗)^∗= (a⊕b)^∗a^∗=b^∗(a⊕b)^∗ (1.14) Proof. Equation (1.11):a^∗⊗a^∗= (e⊕a⊕a²⊕. . .)⊗(e⊕a⊕a²⊕. . .) =e⊕a⊕ a²⊕a³⊕a⁴⊕. . .=a^∗.

Equation (1.12):(a^∗)^∗=e⊕a^∗⊕a^∗a^∗⊕. . .=e⊕a^∗⊕a^∗.

Equation (1.13):a(ba)^∗=a(e⊕ba⊕baba⊕. . .) =a⊕aba⊕ababa⊕. . .= (e⊕ ab⊕abab⊕. . .)⊗a= (ab)^∗a.

Equation (1.14): x= (a⊕b)^∗ is the smallest solution of x= (a⊕b)⊗x⊕e= ax⊕bx⊕e. On the other hand, this equation admitsx= (a^∗b)x⊕a^∗= (a^∗b)^∗a^∗ as its smallest solution. Hence(a⊕b)^∗= (a^∗b)^∗a^∗. The other equalities can be ob- tained by commutingaandb.

Theorem 10 ([30], [6]) LetD be a complete dioid and a,b∈D. Elements(a\a)^◦ and(b/b)^◦ are such that

a\a^◦ = (a\a)^{◦ ∗},

b/b^◦ = (b/b)^{◦ ∗}. (1.15)

Proof. According to Equation (1.7),a\(ae) =^◦ a\a^◦ e. Furthermore, according to Theorem 4,a\(a^◦ ⊗(a\a)) =^◦ a\a. Moreover, according to Inequality (1.10),^◦ a\(a^◦ ⊗ (a\a))^◦ (a\a)^◦ ⊗(a\a). Hence,^◦ e(a\a)^◦ ⊗(a\a) = (a^◦ \a)^{◦ 2}(a\a). By recalling^◦ that the product law is isotone, these inequalities can be extended toe(a\a)^{◦ n} (a\a). And then by considering definition of the Kleene star operator,^◦ e(a\a)^{◦ ∗}= (a\a). The same proof can be developed for^◦ b/b.^◦

Theorem 11 LetDbe a complete dioid and x,a∈D. The greatest solution of

x^∗a^∗ (1.16)

is x=a^∗.

Proof. First, according to the Kleene star definition (see Definition 7), the following equivalence holds:

x^∗=e⊕x⊕x²⊕. . .a^∗⇔





















 ea^∗ xa^∗ x²a^∗ . . . xⁿa^∗ . . .

. (1.17)

Let us focus on the right hand side of this equivalence. From Definition 7, the first inequality holds. The second admitsa^∗as its greatest solution, which is also solution to the other ones. Indeed, according to Equation (1.11),a^∗a^∗=a^∗and(a^∗)ⁿ=a^∗. Hencex=a^∗satisfies all these inequalities and is then the greatest solution.

1.2.3 Dioid M

J γ , δ K

In the previous chapter, the semiring of non decreasing series in two variables with exponents inZand with Boolean coefficients, denotedM_in^axJγ,δK, was considered to model timed event graphs and to obtain transfer relations between transitions. In this section, the results introduced in the previous chapter are extended. Formally, the considered series are defined as follows:

s=^M

i∈N0

s(i)γⁿⁱδ^ti, (1.18)

with s(i)∈ {e,ε} and n_i,t_i ∈Z. The support of s is defined bySupp(s) ={i∈ Z|s(i)6=ε}. The valuation inγofsis defined as:val(s) =min{n_i|i∈Supp(s)}. A seriess∈M_in^axJγ,δKis said to be a polynomial ifSupp(s)is finite. Furthermore, a polynomial is said to be a monomial if there is only one element.

Definition 12 (Causality) A series s∈M_in^axJγ,δK is causal if s=ε or if both val(s)≥0and sγ^val(s). The set of causal elements ofM_in^axJγ,δKhas a complete semiring structure denotedM_in^ax+Jγ,δK.

Definition 13 (Periodicity) A series s∈M_in^axJγ,δKis said to be periodic if it can be written as s=p⊕q⊗r^∗, with p=^Ln

i=1

γⁿⁱδ^ti, q=^Lm

j=1

γ^Njδ^Tj are polynomials and r=γ^νδ^τ, withν,τ∈N, is a monomial depicting the periodicity and allowing to define the asymptotic slope of the series as σ∞(s) =ν/τ . A matrix is said to be periodic if all its entries are periodic.

Definition 14 (Realizability) A series s∈M_in^axJγ,δK is said to be realizable if there exists three matrices A, B and C with entries in N∪ {−∞,+∞} such that s=CA^∗B. A matrix is said to be realizable if all its entries are realizable.

In other words, a seriessis realizable if it corresponds to the transfer relation of a timed event graph².

Theorem 15 ([4]) The following statements are equivalent:

A series s is realizable.

A series s is periodic and causal.

Definition 16 (canonical form of polynomials) A polynomial p=^Ln

i=1

γⁿⁱδ^ti is in canonical form if n1<n2<· · ·<n_nand t1<t2<· · ·<t_n.

Definition 17 (canonical form of series) A series s=p⊕q⊗r^∗, with p=

n L i=1

γⁿⁱδ^ti, q=^Lm

j=1

γ^Njδ^Tj and r=γ^νδ^τis inproper form, if

• p and q are in canonical form

• n_n<N₀and t_n<T₀

• N_m−N₀<νand T_m−T₀<τ

furthermore, a series is incanonical formif it is in proper form and if

• (n_n,t_n)is minimal

• (ν,τ)is minimal

Sum, product, Kleene star, and residuation of periodic series are well defined (see [11], [18]), and algorithms and software toolboxes are available in order to handle periodic series and to compute transfer relations (see [7]). Below, useful computa- tional rules are recalled

2Timed event graphs constitute a sub class of Petri nets as introduced in the previous chapters.

δ^t1γⁿ⊕δ^t2γⁿ=δ^max(t1,t2)γⁿ, (1.19) δ^tγⁿ¹⊕δ^tγⁿ² =δ^tγ^min(n1,n2), (1.20) δ^t1γⁿ¹⊗δ^t2γⁿ² =δ^t1+t2γⁿ¹⁺ⁿ², (1.21) (δ^t1γⁿ¹)\(δ^{◦ t2}γⁿ²) =δ^t2−t1γⁿ²⁻ⁿ¹, (1.22) δ^t1γⁿ¹∧δ^t2γⁿ²=δ^min(t1,t2)γ^max(n1,n2). (1.23) Furthermore, we recall that the order relation is such thatδ^t1γⁿ¹ δ^t2γⁿ² ⇔n₁≤ n₂andt₁≥t₂. Letpandp⁰be two polynomials composed ofmandm⁰monomials respectively, then the following rules hold:

p⊕p⁰=

m M

i=1

γⁿⁱδ^ti⊕

m⁰ M

j=1

γⁿ

0 jδ^t

0

j, (1.24)

p⊗p⁰=

m M

i=1 m⁰ M

j=1

γⁿⁱ⁺ⁿ

0 jδ^ti+t

0

j, (1.25)

p∧p⁰=

m M

i=1

(γⁿⁱδ^ti∧p⁰) =

m M

i=1 m⁰ M

j=1

γ^max(ni,n

0 j)

δ^min(ti,t

0 j)

, (1.26)

p^0◦\p= (

m⁰ M

j=1

γⁿ

0 jδ^t

0

j)\p^◦ =

m⁰

^

j=1

(γ^−n0jδ^−t

0

j⊗p) =

m⁰

^

j=1 m M

i=1

γⁿⁱ⁻ⁿ

0 jδ^ti−t

0

j. (1.27) Letsands⁰be two series, the asymptotic slopes satisfy the following rules

σ_∞(s⊕s⁰) = min(σ_∞(s),σ_∞(s⁰)), (1.28) σ∞(s⊗s⁰) = min(σ_∞(s),σ∞(s⁰)), (1.29) σ_∞(s∧s⁰) = max(σ_∞(s),σ_∞(s⁰)), (1.30) ifσ∞(s)≤σ∞(s⁰) then σ∞(s^0◦\s) =σ∞(s), (1.31)

else s^0◦\s=ε.

Theorem 18 ([6]) The canonical injectionId_|Max in

+

Jγ,δ_K:M_in^ax+Jγ,δK→M_in^axJγ,δK is residuated and its residual is denotedPr+:M_in^axJγ,δK→M_in^ax+Jγ,δK.

Pr₊(s)is the greatest causal series less than or equal to seriess∈M_in^axJγ,δK. From a practical point of view, for all seriess∈M_in^axJγ,δK, the computation ofPr₊(s)is obtained by:

Pr_+M

i∈I

s(n_i,ti)γⁿⁱδ^ti

=^M

i∈I

s+(n_i,t_i)γⁿⁱδ^ti where

s+(n_i,ti) =

(e if(n_i,t_i)≥(0,0)

ε otherwise .

1.3 Control

1.3.1 Optimal Open-Loop Control

As in classical linear system theory, the control of systems aims at obtaining control inputs of these systems in order to achieve fixed objectives. The first result appeared in [4]. The optimal control proposed therein tracks a trajectorya prioriknown in order to minimize the just-in-time criterion. It is an open-loop control strategy, well detailed in [1], and some refinements are proposed later in [27],[28]. The solved problem is the following.

The model of a linear system in a dioid is assumed to be known, which implies that entries of matricesA∈D^n×n,B∈D^n×p,C∈D^q×nare given and the evolution of the state vectorx∈Dⁿand the output vectory∈D^qcan be predicted thanks to the model given by:

x=Ax⊕Bu

y=Cx (1.32)

whereu∈D^pis the input vector. Furthermore, a desired outputz∈D^qis assumed to be knowna priori. This corresponds to a trajectory to track. It can be shown that there exists a unique greatest input denotedu_optsuch that the output corresponding to this control, denotedyopt, is lower than or equal to the desired outputz. Practically, the greatest input means that the inputs in the system will occur as late as possible while ensuring that the occurrence dates of the output will be lower than the one given by the desired outputz. Controlu_opt is then optimal according to the just- in-time criterion, i.e., the inputs will occur as late as possible while ensuring that the corresponding outputs satisfy the constraints given byz. This kind of problem is very popular in the framework of manufacturing systems: the desired output z corresponds to the customer demand, and the optimal controlu_opt corresponds to the input of raw parts in the manufacturing system. The latter is delayed as much as possible, while ensuring that optimal outputy_optof the processed parts occurs before the customer demand. Hence the internal stocks are reduced as much as possible.

According to Theorem 8, the smallest solution of System (1.32) is given by:

x=A^∗Bu

y=CA^∗Bu. (1.33)

This smallest solution represents the fastest evolution of the system. Formally, the optimal problem consists in computing the greatestusuch thatyz. By recalling that the left product is residuated (see example 5), the following equivalence holds:

y=CA^∗Buz⇔u(CA^∗B)\z.^◦ In other words the greatest control achieving the objective is:

u_opt= (CA^∗B)\z.^◦ (1.34)

1.3.1.1 Illustration

High-Throughput Screening (HTS) systems allow researchers to quickly conduct millions of chemical, genetic or pharmacological tests. In the previous chapter, an elementary one was considered as an illustrative example. Below, its model in the semiringM_in^axJγ,δKis recalled.

x(γ,δ) =

































ε ε γ ε ε ε ε ε ε ε δ⁷ε ε ε e ε ε ε ε ε ε δ ε ε ε ε ε ε ε ε ε ε ε ε ε ε γ² ε ε ε ε eε δ ε ε ε ε ε ε ε ε ε ε δ ε ε ε eε ε ε ε ε ε δ¹² ε ε ε ε ε ε ε ε ε ε ε ε ε γ ε ε ε ε ε ε ε δ⁵ε ε ε ε ε ε ε ε ε ε δ ε

































x(γ,δ)⊕































 eε ε ε ε ε ε ε ε ε eε ε ε ε ε ε ε ε ε ε ε ε e ε ε ε ε ε ε































 u(γ,δ)

y(γ,δ) = ε ε ε ε ε ε eε ε ε x(γ,δ)

(1.35)

This model leads to the following transfer matrices, where the entries are periodic series inM_in^axJγ,δK:

x(γ,δ) =



































(e) γ δ⁸∗

γ δ² γ δ⁸∗

γ³δ¹⁹ γ δ⁸∗

δ⁷ γ δ⁸∗

(δ) γ δ⁸∗

γ²δ¹⁸ γ δ⁸∗

δ⁸ γ δ⁸∗

δ² γ δ⁸∗

γ²δ¹⁹ γ δ⁸∗

γ²δ²⁰ γ δ⁸∗

e⊕ γ²δ¹⁴ γ δ⁸∗

γ²δ¹⁷⊕ γ³δ²³ γ δ⁸∗

δ⁷ γ δ⁸∗

(δ) γ δ⁸∗

γ²δ¹⁸ γ δ⁸∗

δ⁸ γ δ⁸∗

δ² γ δ⁸∗

δ⁵⊕ γ δ¹¹ γ δ⁸∗

δ²⁰ γ δ⁸∗

δ¹⁴ γ δ⁸∗

δ¹⁷⊕ γ δ²³ γ δ⁸∗

ε ε (e) γ δ⁶∗

ε ε δ⁵

γ δ^6∗

ε ε δ⁶

γ δ⁶∗

































 u(γ,δ)

y(γ,δ) = δ²⁰

γ δ⁸∗

δ¹⁴ γ δ⁸∗

δ¹⁷⊕ γ δ²³

γ δ⁸∗ u(γ,δ).

(1.36) The following reference trajectory is assumed to be known:

z(γ,δ) =δ³⁷⊕γ δ⁴²⊕γ³δ⁵⁵⊕γ⁴δ^+∞. (1.37) It models that one part is desired before or at time 37, 2 parts before (or at) time 42 and 4 parts before (or at) time 55. It must be noted that the numbering of parts starts at 0³. According to Equation (1.34) and the computation rules given in the previous section, the optimal firing trajectories of the three inputs are as follows:

3Following the convention defined between remarks 5.22 and 5.23 in [1, Section 5.4].

u_opt(γ,δ) =





δ⁶⊕γ δ¹⁴⊕γ²δ²²⊕γ³δ³⁵⊕γ⁴δ^+∞

δ¹²⊕γ δ²⁰⊕γ²δ²⁸⊕γ³δ⁴¹⊕γ⁴δ^+∞

δ¹¹⊕γ δ¹⁹⊕γ²δ²⁵⊕γ³δ³⁸⊕γ⁴δ^+∞



, (1.38)

and the resulting optimal output is:

y_opt(γ,δ) =CA^∗Bu_opt(γ,δ) =δ²⁸⊕γ δ³⁶⊕γ²δ⁴²⊕γ³δ⁵⁵⊕γ⁴δ^+∞, (1.39) which means that the first part will exit the system at time 28, the second one at time 36, the third one at time 42 and the fourth one at time 55. This output trajectory is the greatest in the image of matrixCA^∗B, that is the greatest reachable output, such that the events occur before the desired dates. This example⁴was computed with library Minmaxgd (a Scilab toolbox, see [7]).

1.3.2 Optimal Input Filtering

In the previous section, the proposed control law leads to the optimal tracking of a trajectory, which is assumed to bea prioriknown. In this section, the aim is to track a reference model in an optimal manner. This means that the desired output is a priori unknown and the goal is to control the system so that it behaves as the ref- erence model which was built from a specification. Hence, this problem is a model matching problem. The reference model describing the desired behavior is assumed to be a linear model in the considered dioid. The objective is to synthesize a filter controlling the inputs in order to get an output as close as possible to the one you would obtain by applying this input to the reference model.

Formally, the desired behavior is assumed to be described by a transfer matrix denotedG∈D^q×mleading to the outputsz=Gv∈D^qwherev∈D^mis the model input. The system is assumed to be described by the transfer relationCA^∗B∈D^q×p and its outputyis given byy=CA^∗Bu∈D^q. The control is assumed to be given by a filterP∈D^p×m,i.e.,u=Pv∈D^p. Therefore the aim is to synthesize this filter such that, for allv:

CA^∗BPvGv, (1.40)

which is equivalent to

CA^∗BPG. (1.41)

Thanks to residuation theory, the following equivalence holds.

CA^∗BPG⇔P(CA^∗B)\G^◦ (1.42)

4 The sources are available at the following URL: www.istia.univ-angers.fr/

~hardouin/DISCBookChapterControlInDioid.html.

Hence the optimal filter is given by P_opt = (CA^∗B)\G^◦ and it leads to the control u_opt=P_optv. Furthermore, ifGis in the image of the transfer matrixCA^∗B(i.e.,∃L such thatG=CA^∗BL), then the following equality holds:

G=CA^∗BPopt. (1.43)

From a practical point of view, an interesting particular case consists in choosing the reference model such thatG=CA^∗B. Obviously, this reference model is reach- able since it is sufficient to take a filter equal to the identity matrix to satisfy the equality. The optimal filter is given byP_opt= (CA^∗B)\(CA^{◦ ∗}B)and Equality (1.43) is satisfied. Formally G=CA^∗B=CA^∗BP_opt=CA^∗B (CA^∗B)\(CA^{◦ ∗}B)

. This means that output ywill be unchanged by the optimal filter, i.e. y=CA^∗Bv=CA^∗BPv hence the outputs occurrence of the controlled system will be as fast as the one of the system without control. Nevertheless the controlu_opt=P_optvwill be the greatest leading to this unchanged output. In the framework of manufacturing systems, this means that the job will start as late as possible, while preserving the output. Hence the work-in-progress will be reduced as much as possible. This kind of controller is called neutral due to its neutral action on the output behavior. The benefit lies on a reduction of the internal stocks.

1.3.2.1 Illustration

By applying these results to the previous example, the optimal input filter is given by:

P_opt(γ,δ) = (CA^∗B)\(CA^{◦ ∗}B) =





γ δ⁸∗

δ⁻⁶ γ δ⁸∗

δ⁻⁵ γ δ⁸∗

δ⁶ γ δ⁸∗

γ δ⁸∗

δ¹ γ δ⁸∗

δ³ γ δ⁸∗

δ⁻³ γ δ⁸∗

e⊕ γ δ⁶ γ δ⁸∗



 (1.44) It must be noted that some powers are negative (see the computational rule 1.27).

This means that the filter is not causal, and therefore not realizable (see Theorem 15). Practically, it is necessary to project it in the semiring of causal series, denoted M_in^ax+Jγ,δK. This is done thanks to the rule introduced in Theorem 18.

By applying this projector to the previous filter, the optimal causal filter is given by:

P_opt⁺(γ,δ) =Pr₊(P_opt) =





γ δ⁸∗

γ δ² γ δ⁸∗

γ δ³ γ δ⁸∗

δ⁶ γ δ⁸∗

γ δ⁸∗

δ¹ γ δ⁸∗

δ³ γ δ⁸∗

γ δ⁵ γ δ⁸∗

e⊕ γ δ⁶ γ δ⁸∗



. (1.45) This filter describes a 3-input 3-output system of which a realization can be ob- tained either in temporal or in event domain. As explained in the previous chapter, in the temporal domain, the variables, and their associated trajectory, will represent the maximal number of events occurred at a timet. Dually, in the event domain,

they will represent the earliest occurrence dates of thek^thevent. The adopted point of view will depend on the technological context leading to the implementation.

Indeed, the control law can be implemented in a control command system or in a PLC, which can be either event driven or synchronized with a clock. In both cases, the following method may be used to obtain a realization of the control law. First, we recall the expression of the control lawu_i=

p L j=1

P_{opti j}v_j withi∈[1..p], where each entryP_{opti j} is a periodic series which can be written as follows:P_{opti j} =p_{i j}⊕q_{i j}r^∗_{i j}. This leads to the following control law:

u_i=

p M

j=1

P_{opti j}v_j (1.46)

with

P_{opti j}v_j=p_{i j}v_j⊕q_{i j}r_{i j}^∗v_j.

The last line can be written in an explicit form, by introducing an internal variable ζi j(we recall thatx=a^∗bis the least solution ofx=ax⊕b, see Theorem 8):

ζi j =r_{i j}ζi j⊕q_{i j}v_j

P_{opti j}v_j =p_{i j}v_j⊕ζ_{i j} . (1.47) This explicit formulation may be used to obtain the control law, either in the time domain or in the event domain. Below, the control law (Equation (1.45)) is given in the time domain:

ζ11(t) =min(1+ζ11(t−8),v1(t)) ζ12(t) =min(1+ζ12(t−8),1+v2(t−2)) ζ13(t) =min(1+ζ13(t−8),1+v3(t−3)) ζ21(t) =min(1+ζ21(t−8),v₁(t−6)) ζ22(t) =min(1+ζ22(t−8),v₂(t)) ζ23(t) =min(1+ζ23(t−8),v₃(t−1)) ζ31(t) =min(1+ζ31(t−8),v₁(t−3)) ζ32(t) =min(1+ζ32(t−8),1+v₂(t−5)) ζ33(t) =min(1+ζ33(t−8),1+v₃(t−6)) u₁(t) =min(ζ₁₁(t),ζ12(t),ζ13(t)) u₂(t) =min(ζ₂₁(t),ζ₂₂(t),ζ₂₃(t)) u₃(t) =min(ζ₂₁(t),ζ₂₂(t),ζ₂₃(t),v₃(t)).

(1.48)

1.3.3 Closed-Loop Control

The two previous sections have proposed open-loop control strategies. In this sec- tion, the measurement of the system outputs are taken into account in order to com- pute the control inputs. The closed-loop control strategy considered is given in Fig- ure 1.1. It aims at modifying the dynamics of systemCA^∗B∈D^q×p by using a

controllerF∈D^p×qlocated between the outputy∈D^qand the inputu∈D^pof the system and a filterP∈D^p×mlocated upstream of the system, as the one considered in the previous section. The controllers are chosen in order to obtain a controlled system as close as possible to a given reference model G∈D^q×m. The latter is assumed to be a linear model in the considered dioid.

Fig. 1.1 Control architecture, the system is controlled by output feedbackFand filterP.

Formally, the control input is equal tou=Fy⊕Pv∈D^pand leads to the follow- ing closed-loop behavior:

x=Ax⊕B(Fy⊕Pv)

y=CxGv ∀v. (1.49)

By replacingyin the first equation, one can write

x= (A⊕BFC)x⊕BPv= (A⊕BFC)^∗BPv

y=Cx=C(A⊕BFC)^∗BPvGv ∀v. (1.50)

Hence the problem is to find controllersFandPsuch that:

C(A⊕BFC)^∗BPG. (1.51)

By recalling that(a⊕b)^∗=a^∗(ba^∗)^∗= (a^∗b)^∗a^∗(see Equation (1.14)), this equation can be written as:

CA^∗(BFCA^∗)^∗BP=CA^∗B(FCA^∗B)^∗PG. (1.52) According to the Kleene star definition(FCA^∗B)^∗=E⊕FCA^∗B⊕(FCA^∗B)²⊕. . ., so Equation (1.52) implies the following constraint on the filterP:

CA^∗BPG, (1.53)

which is equivalent to:

P(CA^∗B)\G^◦ =P_opt, (1.54)

whereP_optis the same as the optimal filter introduced in Section 1.3.2. By consid- ering this optimal solution in Equation (1.52), the following equivalences hold:

CA^∗B(FCA^∗B)^∗P_optG⇔CA^∗B(FCA^∗B)^∗G/P^◦ _opt

⇔(FCA^∗B)^∗(CA^∗B)\G^◦ /P^◦ _opt=P_opt/P^◦ _opt. (1.55) By recalling that(a/a) = (a^◦ /a)^{◦ ∗}(see Theorem 10) and thatx^∗a^∗admitsx=a^∗ as its greatest solution (see Theorem 11), this equation can be written:

(FCA^∗B)P_opt/P^◦ _opt. (1.56) By using the right division, the optimal feedback controller is obtained as follows:

F(P_opt/P^◦ _opt)/(CA^{◦ ∗}B) =P_opt/((CA^{◦ ∗}B)P_opt) =F_opt. (1.57) And the causal feedback is given by:F_opt⁺ =Pr₊(F_opt). It can be noted that, as for the optimal filter, the reference model describes the desired behavior. From a practical point of view, an interesting choice is G=CA^∗B. This means that the objective is to obtain the greatest control inputs, while preserving the output of the system without control,i.e.to delay the input dates as much as possible while preserving the output dates.

1.3.3.1 Illustration

The example of section 1.3.2.1 is continued,i.e., the reference model isG=CA^∗B.

Then, the optimal filter is given by Equation (1.45) and the feedback controller is obtained thanks to Equation (1.57). The practical computation yields:

F_opt⁺ =



 γ³δ²

γ δ⁸∗

γ² γ δ⁸∗

γ³δ⁵ γ δ⁸∗



. (1.58)

Then the control law,u=P_opt⁺ v⊕F_opt⁺yis obtained by adding the feedback control to law (1.48). By using the same methodology as in section 1.3.2.1, the control law in the time domain can be obtained. An intermediate variableβis considered:

β11(t) =min(1+β11(t−8),3+y(t−2)) β₂₁(t) =min(1+β₂₁(t−8),2+y(t)) β₃₁(t) =min(1+β₃₁(t−8),3+y(t−5)),

(1.59)

then the following control law is obtained:

u₁(t) =min(ζ₁₁(t),ζ₁₂(t),ζ13(t),β11(t)) u₂(t) =min(ζ₂₁(t),ζ₂₂(t),ζ₂₃(t),β₁₁(t)) u₃(t) =min(ζ₂₁(t),ζ₂₂(t),ζ₂₃(t),v₃(t),β₁₁(t)),

(1.60)

where variableζis given in Equation (1.48) andβin Equation (1.59). This equation can easily be implemented in a control system. It can also be realized as a timed event graph, such as the one given in Figure 1.2.

Fig. 1.2 Realization of the optimal filterP_opt⁺ and of the optimal feedbackF_opt⁺.

1.4 Conclusion

In this chapter, the control of systems in a dioid framework has been presented. The aim is to modify the behavior of the system in order to achieve some objectives fixed by a specification. It was shown that both open-loop and closed-loop control can be considered for this purpose. The synthesis is based on residuation theory, which allows us to approximately invert order preserving mappings defined over ordered sets. The common characteristic of all the synthesized controllers is that they are the greatest ones, according to the order of the considered dioid, that is to say opti- mal with regard to the just-in-time criterion. This kind of control leads to the greatest reduction of internal stocks in the context of manufacturing systems, as well as wait- ing times in a transportation network, and also to avoid useless buffer saturation in a computer network. One should notice that some other classical standard problems can be considered from this point of view. Indeed, it is possible to synthesize a linear observer in Luenberger’s sense, which allows us to estimate the unmeasured state

of the system by considering its output measurement (see [15, 14]). Another related problem consists in the disturbance decoupling problem (see [13, 19]), which aims at taking into account disturbances acting on the system to compute the control law.

About uncertainty, the reader is invited to consider dioids of intervals allowing the synthesis of robust controllers in a bounded error context (see [12, 20]). The identi- fication of model parameters was also considered in some previous work (see e.g., [10, 25, 31]). To summarize, automatic control of discrete-event systems in a dioid framework is still an improving topic with many problems yet to solve.

References

1. F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat. Synchronization and Linearity, An Algebra for Discrete Event Systems. John Wiley and Sons, New York, New York, USA, 1992.

Available atwww-rocq.inria.fr/metalau/cohen/SED/SED1-book.html.

2. T. S. Blyth.Lattices and Ordered Algebraic Structures. Springer Verlag, 2005.

3. T. S. Blyth and M. F. Janowitz. Residuation Theory. Pergamon press, Oxford, United King- dom, 1972.

4. G. Cohen, P. Moller, J.-P. Quadrat, and M. Viot. Algebraic Tools for the Performance Evalua- tion of Discrete Event Systems.Proceedings of the IEEE, 77(1):39–58, January 1989. Special issue on Discrete Event Systems.

5. B. Cottenceau, L. Hardouin, J.-L. Boimond, and J.-L. Ferrier. Synthesis of Greatest Lin- ear Feedback for Timed Event Graphs in Dioid. IEEE Transactions on Automatic Control, 44(6):1258–1262, 1999. Available atdoi:10.1109/9.769386.

6. B. Cottenceau, L. Hardouin, J.-L. Boimond, and J.-L. Ferrier. Model Reference Control for Timed Event Graphs in Dioids.Automatica, 37(9):1451–1458, September 2001. Available at doi:10.1016/S0005-1098(01)00073-5.

7. B. Cottenceau, L. Hardouin, M. Lhommeau, and J.-L. Boimond. Data processing tool for calculation in dioid. InProceedings of the 5^th International Workshop on Discrete Event Systems, WODES 2000, Ghent, Belgium, 2000. cf.www.istia.univ-angers.fr/

~hardouin/outils.html.

8. R. A. Cuninghame-Green.Minimax Algebra, volume 166 ofLecture Notes in Economics and Mathematical Systems. Springer-Verlag, 1979.

9. B. De Schutter and T. J. J. van den Boom. Model Predictive Control for max-plus Linear Discrete Event Systems.Automatica, 37(7), July 2001.

10. F. Gallot, J.-L. Boimond, and L. Hardouin. Identification of simple elements in max-algebra:

application to SISO discrete event systems modelisation. InProceedings of the 4^thEuropean Control Conference, Brussels, July 1997. CD-ROM.

11. S. Gaubert. Théorie des Systèmes Linéaires dans les Dioïdes. Thèse, École des Mines de Paris, July 1992.

12. L. Hardouin, B. Cottenceau, M. Lhommeau, and E. Le Corronc. Interval systems over idempo- tent semiring.Linear Algebra and its Applications, 431(5-7):855–862, August 2009. Available atdoi:10.1016/j.laa.2009.03.039.

13. L. Hardouin, M. Lhommeau, and Y. Shang. Towards Geometric Control of Max-Plus Linear Systems with Applications to Manufacturing Systems. InProceedings of the 50^thIEEE Con- ference on Decision and Control and European Control Conference, pages 171–176, Orlando, Florida, USA, December 2011. Available atistia.univ-angers.fr/~hardouin/

YShangDDPCDC2011.pdf.

14. L. Hardouin, C. A. Maia, B. Cottenceau, and M. Lhommeau. Max-plus Linear Observer:

Application to manufacturing Systems. InProceedings of the 10^thInternational Workshop on Discrete Event Systems, WODES’10, pages 171–176, Berlin, September 2010. Available at istia.univ-angers.fr/~hardouin/Observer.html.

15. L. Hardouin, C. A. Maia, B. Cottenceau, and M. Lhommeau. Observer Design for (max,plus) Linear Systems. IEEE Transactions on Automatic Control, 55(2), February 2010. doi:

10.1109/TAC.2009.2037477.

16. L. Hardouin, E. Menguy, J.-L. Boimond, and J.-L. Ferrier. SISO Discrete Event Systems Con- trol in Dioids Algebra.Journal Européen des Systèmes Automatisés, 31(3):433–452, 1997.

17. B. Heidergott, G. J. Olsder, and J. van der Woude.Max Plus at Work – Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications. Princeton University Press, 2006.

18. D. Krob. Complete systems ofB-rational identities.Theoretical Computer Science, 89:207 – 343, 1991.

19. M. Lhommeau, L. Hardouin, and B. Cottenceau. Disturbance Decoupling of Timed Event Graphs by Output Feedback Controller. InProceedings of the 6^thInternational Workshop on Discrete Event Systems, WODES’02, Zaragoza, Spain, October 2002. Available atistia.

univ-angers.fr/~hardouin/Wodes02.pdf.

20. M. Lhommeau, L. Hardouin, B. Cottenceau, and L. Jaulin. Interval Analysis and Dioid: Appli- cation to Robust Controller Design for Timed Event Graphs.Automatica, 40(11):1923–1930, November 2004. Available atdoi:10.1016/j.automatica.2004.05.013.

21. M. Lhommeau, L. Hardouin, J.-L. Ferrier, and I. Ouerghi. Interval Analysis in Dioid : Applica- tion to Robust Open Loop Control for Timed Event Graphs. InProceedings of the 44^thIEEE Conference on Decision and Control and European Control Conference, pages 7744–7749, Seville, Spain, December 2005. Available atdoi:10.1109/CDC.2005.1583413.

22. M. Lhommeau, L. Hardouin, R. Santos Mendes, and B. Cottenceau. On the Model Refer- ence Control for Max-Plus Linear Systems. InProceedings of the 44^thIEEE Conference on Decision and Control and European Control Conference, pages 7799–7803, Seville, Spain, December 2005. Available atdoi:10.1109/CDC.2005.1583422.

23. G. L. Litvinov and A. N. Sobolevski˘ı. Idempotent Interval Analysis and Optimization Prob- lems. Reliable Computing, 7(5):353 – 377, 2001. Available atarxiv.org/abs/math/

0101080.

24. C. A. Maia, L. Hardouin, R. Santos Mendes, and B. Cottenceau. Optimal Closed-Loop Control for Timed Event Graphs in Dioid. IEEE Transactions on Automatic Control, 48(12):2284–

2287, 2003. Available atdoi:10.1109/TAC.2003.820666.

25. C. A. Maia, R. Santos Mendes, and L. Hardouin. Some Results on Identification of Timed Event Graphs in Dioid. InProceedings of the 11^thIEEE Mediterranean Conference on Control and Automation, MED’03, Rhodes, Greece, June 2003.

26. E. Menguy, J.-L. Boimond, and L. Hardouin. A feedback control in max-algebra. InProceed- ings of the 4^thEuropean Control Conference, Brussels, July 1997.

27. E. Menguy, J.-L. Boimond, and L. Hardouin. Optimal control of discrete event systems in case of updated reference input. InIFAC Conference on System Structure and Control, pages 601–607, Nantes, France, July 1998.

28. E. Menguy, J.-L. Boimond, L. Hardouin, and J.-L.Ferrier. Just in time control of timed event graphs: update of reference input, presence of uncontrollable input. IEEE Transactions on Automatic Control, 45(11):2155–2159, 2000. Available atdoi:10.1109/9.887652.

29. I. Necoara, B. De Schutter, T. J. J. van den Boom, and H. Hellendoorn. Stable model predictive control for constrained max-plus-linear systems. Discrete Event Dynamic Systems: Theory and Applications, 17(3):329 – 354, September 2007.

30. M. Plus. Second Order Theory of Min-Linear Systems and its Application to Discrete Event Systems. InProceedings of the 30^thIEEE Conference on Decision and Control, volume 2, pages 1511 – 1516, Brighton, United Kingdom, December 1991.

31. G. Schullerus, V. Krebs, B. De Schutter, and T. J. J. van den Boom. Input signal design for identification of max-plus-linear systems.Automatica, 42(6):937–943, June 2006.

32. T. J. J. van den Boom and B. De Schutter. Model predictive control for perturbed max-plus- linear systems: A stochastic approach.International Journal of Control, 77(3):302–309, 2004.