Decentralized Supervisory Control of Discrete Event Systems: Using

Multi-Decision Control as an Alternative to Inference-Based Control

Ahmed Khoumsi and Hicham Chakib

University of Sherbrooke, Dep. Elec. & Comp. Eng., Sherbrooke, Canada

Keywords:

Discrete Event Systems, Multi-Decision Control, C&P-control, D&A-control, Inference-Based Control.

Abstract:

Some years ago, a decentralized architecture, qualiﬁed as inference-based, has been developed for supervisory

control of discrete event systems. One of its essential principles is to associate ambiguity levels to local deci-

sions. More recently, a decentralized control architecture, qualiﬁed as multi-decision, has been developed. Its

main principle is to use several decentralized control architectures in parallel. So far, multi-decision control

has been mostly studied as a solution to generalize inference-based control, by using several inference-based

architectures in parallel. In the present study, we regard multi-decision control with a different perspective.

Instead of using multi-decision control to generalize inference-based control, we will rather use it as an alter-

native to inference-based control. More precisely, our objective is to avoid using inference-based architectures,

by using instead several simpler architectures running in parallel.

1 INTRODUCTION

This paper is about decentralized control, where sev-

eral local supervisors cooperate in order to restrict the

behavior of a plant so that it respects a given speciﬁca-

tion. The authors of (Kumar and Takai, 2007) propose

an interesting decentralized control approach, called

inference-based control, where an ambiguity level is

associated to the decision of each local supervisor.

The principle is that among the decisions of the local

supervisors, the effective decision which is selected is

the one with the lowest ambiguity. In (Kumar and

Takai, 2007), it is shown that inference-based con-

trol generalizes several previous control architectures,

in particular C&P and D&A architectures (Rudie and

Wonham, 1992; Yoo and Lafortune, 2002), which are

the simplest interesting decentralized architectures.

The authors of (Yoo and Lafortune, 2002) also com-

bine C&P and D&A to obtain a so-called C&P∨D&A

architecture. C&P, D&A and C&P∨D&A are in

fact speciﬁc cases of inference-based control, where

uniquely the null ambiguity level is associated to the

local decisions.

Recently, a decentralized architecture for con-

trol, qualiﬁed as multi-decision, has been devel-

oped (Chakib and Khoumsi, 2011). Its principle is

to use several decentralized architectures in parallel

whose decisions are combined disjunctively or con-

junctively. Each of the architectures in parallel can

be any of the known decentralized architectures, for

example C&P∨D&A or inference-based. The ob-

tained multi-decision architecture generalizes all the

architectures in parallel, in the sense that it permits to

achieve, not only all the languages achievable by each

of the architectures in parallel, but also languages

achievable by none of the architectures in parallel. In

(Chakib and Khoumsi, 2011), multi-decision control

is studied as a solution to generalize inference-based

control, by using inference-based architectures run-

ning in parallel.

Compared to the multi-decision control, the

inference-based control has the advantage that its ar-

chitecture does not depend on the structures of the au-

tomata modeling the plant and the speciﬁcation. But

on the other side, the inference-based architecture is

relatively complex. So a question that arises is: Is

it possible to avoid using inference-based control, by

using instead multi-decision with several simple con-

trol architectures running in parallel, without losing

in generality ? Our study is an attempt to answer that

question. The simple architectures in parallel we will

consider are qualiﬁed as C&P∨D&A, which are the

simplest relevant decentralized architectures known

in the literature.

The rest of the paper is organized a follows. In

Section 2, we introduce decentralized control with

an emphasis on pertinent architectures for our study,

namely C&P∨D&A and inference-based controls.

Khoumsi, A. and Chakib, H.

Decentralized Supervisory Control of Discrete Event Systems: Using Multi-Decision Control as an Alternative to Inference-Based Control.

DOI: 10.5220/0005990102130220

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 213-220

ISBN: 978-989-758-198-4

213

Section 3 presents multi-decision control with an em-

phasis on pertinent results obtained in previous re-

search. In Section 4, we develop a procedure that

constructs a multi-decision architecture consisting

uniquely of C&P∨D&A architectures in parallel, in-

stead of inference-based architectures. Section 5 con-

cludes our study.

2 DECENTRALIZED CONTROL

Let alphabet denote a ﬁnite set of events, trace denote

a ﬁnite sequence of events, and language denote a set

of traces. A trace λ is said a preﬁx of a trace µ if there

exists a trace α such that µ = λα. L denotes the set of

preﬁxes of a language L. L \ K denotes the language

L without the traces of the language K. Let P

(X)

and P

−1

(X) denote the usual projection and inverse

projection of a language or an automaton X.

2.1 Supervisory Control

Supervisory control (or more succinctly: control)

consists in forcing a discrete event system (DES),

called plant, to achieve the language of a given spec-

iﬁcation. More precisely, the objective is to re-

strict the behavior of the plant so that it executes

uniquely traces accepted by the speciﬁcation. The

plant is modeled by a ﬁnite state automaton (FSA)

G = (Q, Σ, δ, q

, Q

), where Q is a ﬁnite set of states,

Σ is an alphabet, the transitions are speciﬁed by a

partial function δ : Q × Σ → Q, q

∈ Q is the initial

state, and Q

⊆ Q is the set of marked states. We

denote by L and L

the generated and marked lan-

guages of G , respectively. We have,

⊆ L. Intu-

itively, L (resp. L

) contains the traces of G start-

ing in q

and reaching Q (resp. Q

). In the same

way, we consider a speciﬁcation modeled by a trim

FSA K = (R, Σ, ξ, r

, R

), and we denote by K the

marked language of K . Since K is trim, its gener-

ated language is

K. We have the following notion of

-closure which is fundamental in control:

Deﬁnition 2.1. K is said L

-closed if K =

K ∩ L

The alphabet Σ is partitioned into Σ

and Σ

, the

set of controllable and uncontrollable events, respec-

tively. We deﬁne the following languages E

and D

Deﬁnition 2.2. For every event σ ∈ Σ, E

= {λ ∈

K|λσ ∈ K} and D

= {λ ∈ K |λσ ∈ L \ K}. Intu-

itively, E

is the set of traces of the speciﬁcation after

which σ is accepted by both the plant and the speci-

ﬁcation, while D

is the set of traces of the speciﬁca-

tion after which σ is accepted only by the plant.

The control is realized by the means of a supervi-

sor SUP that observes continuously the evolution of

the plant in order to decide to enable (i.e. permit) or

disable (i.e. forbid) events. We denote by Sup(λ, σ) ∈

{1, 0} the decision taken by SUP on an event σ when

the plant has executed a trace λ ∈ L. Sup(λ, σ) = 1

(resp. 0) means that σ is enabled (resp. disabled) after

the execution of λ. A fundamental property of control

is that: ∀λ ∈ L : σ ∈ Σ

⇒ Sup(λ, σ) = 1

Since E

and D

(Def. 2.2) are fundamental in the

formulation of multi-decision control, we will formu-

late several usual notions of control as function of E

and D

. First of all, the objective of SUP to force the

plant to respect the speciﬁcation can be formulated by

Eq. (1,2) for every σ ∈ Σ

. To be precise, we must say

that the objective of SUP is to satisfy Eq. (1,2) w.r.t

, D

), for every σ ∈ Σ

. Note that these equations

do not specify a decision when λ 6∈ (E

∪ D

). This

is because σ is accepted by the plant only in E

∪D

and hence enablement/disablement of σ has no effect

on the plant when λ 6∈ (E

∪ D

λ ∈ E

⇒ Sup(λ, σ) = 1 (1)

λ ∈ D

⇒ Sup(λ, σ) = 0 (2)

We have also the fundamental notion of control-

lable speciﬁcation which can be formulated by:

Deﬁnition 2.3. K is said L-controllable if ∀σ ∈ Σ

The following Def. 2.4 is convenient for the for-

mulation of multi-decision control that is presented in

Section 3.

Deﬁnition 2.4. A pair of languages (E, D) is called

the enabling-pair of a control architecture S for σ ∈

to mean that they are the greatest sublanguages of

L for which S satisﬁes Eq. (1,2) w.r.t (E, D).

2.2 Decentralized Supervisory Control

Decentralized control consists in using n local su-

pervisors (SUP

)

1≤i≤n

, where each SUP

has its own

set of observable events Σ

o,i

and own set of control-

lable events Σ

c,i

. We deﬁne Σ

= Σ

o,1

∪ ··· ∪ Σ

o,n

and

= Σ

c,1

∪ ··· ∪ Σ

c,n

. We denote by I = {1, ··· , n}

the indexing set of all local supervisors, and by I

{i ∈ I |σ ∈ Σ

c,i

} the indexing set of local supervisors

controlling σ ∈ Σ

. Let n

denote the cardinality of

Among the most known decentralized control ar-

chitectures developed in the literature, in the present

study we consider C&P∨D&A control (Yoo and

Lafortune, 2002) and inference-based control (Kumar

and Takai, 2007). We consider C&P∨D&A control

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

214

because it regroups the C&P and D&A control archi-

tectures, which are the simplest relevant decentralized

control architectures. We consider inference-based

control because, to our best knowledge, it is the most

general decentralized control before the development

of multi-decision control. Interestingly, the authors of

(Kumar and Takai, 2007) prove that C&P∨D&A con-

trol is a particular case of inference-based control (see

Sect. 2.4).

To every control architecture is associated a prop-

erty of observability, which is usually termed coob-

servability in decentralized control. Coobservability

is fundamental because it is a necessary condition for

the existence of a supervisor that satisﬁes Eq. (1,2).

More precisely, it is shown in the literature related to

every developed decentralized control, that the exis-

tence of supervisor satisfying Eq. (1,2) has as neces-

sary and sufﬁcient condition the conjunction of three

properties: L

-closure (Def. 2.1), L-controllability

(Def. 2.3), and coobservability. L-controllability and

-closure are classical notions that are independent

of the control architecture, and hence are not relevant

to compare control architectures. Therefore, without

loss of generality, we will consider uniquely coob-

servability as condition of existence of supervisor.

The following subsections 2.3 and 2.4 present suc-

cinctly C&P∨D&A control and inference-based con-

trol. In fact, we present only the notions that are indis-

pensable to make our paper self-contained. See (Yoo

and Lafortune, 2002; Kumar and Takai, 2007) for de-

tailed studies of these two architectures.

2.3 C&P∨D&A Control

Since C&P∨D&A control is the combination of C&P

control and D&A control, let us ﬁrst present each of

these two control architectures.

In C&P control (C for Conjunctive and P for Per-

missive):

1. Each local supervisor SUP

is permissive, in the

sense that it locally disables σ ∈ Σ

c,i

if and only if

it is certain that σ 6∈ E

. Formally, for any σ ∈ Σ

c,i

after the execution of λ ∈ L, SUP

observes P

(λ)

and computes its local decision Sup

(λ), σ) by:

Sup

(λ), σ) =



1, if P

(λ) ∈ P

)

0, if P

(λ) 6∈ P

)



(3)

2. The local decisions (Sup

(λ), σ))

i∈I

are fused

conjunctively, in order to generate the actual deci-

sion Sup(λ, σ). Formally:

Sup(λ, σ) =

i∈I

Sup

(λ), σ) (4)

The property of coobservability associated to

C&P control for some σ ∈ Σ

is given by D

[1] =

where D

[1] is deﬁned by Eq. (5). For any σ ∈ Σ

, we

say that (E

, D

) is C&P-COOBS if D

[1] =

0. We

have that “(E

, D

) is C&P-COOBS” is a necessary

condition so that Eqs. (3,4) guarantee the satisfaction

of Eqs. (1,2) w.r.t (E

, D

[1] =

i∈I

−1

) ∩ D

(5)

In D&A control (D for Disjunctive and A for Anti-

permissive):

1. Each local supervisor SUP

is anti-permissive, in

the sense that it locally enables σ ∈ Σ

c,i

if and

only if it is certain that σ 6∈ D

. Formally, for

any σ ∈ Σ

c,i

, after the execution of λ ∈ L, SUP

observes P

(λ) and computes its local decision

Sup

(λ), σ) by:

Sup

(λ), σ) =



0, if P

(λ) ∈ P

)

1, if P

(λ) 6∈ P

)



(6)

2. The local decisions (Sup

(λ), σ))

i∈I

are fused

disjunctively, in order to generate the actual deci-

sion Sup(λ, σ). Formally:

Sup(λ, σ) =

i∈I

Sup

(λ), σ) (7)

The property of coobservability associated to

D&A control for some σ ∈ Σ

is given by E

[1] =

where E

[1] is deﬁned by Eq. (8). For any σ ∈ Σ

, we

say that (E

, D

) is D&A-COOBS if E

[1] =

0. We

have that “(E

, D

) is D&A-COOBS” is a necessary

condition so that Eqs. (6,7) guarantee the satisfaction

of Eqs. (1,2) w.r.t (E

, D

[1] =

i∈I

−1

) ∩ E

(8)

We are in the presence of a C&P∨D&A control

if Σ

is partitioned in two alphabets Σ

and Σ

such that C&P control is applied to every σ ∈ Σ

and D&A control is applied to every σ ∈ Σ

. There-

fore, the property of coobservability associated to

C&P∨D&A control for some σ ∈ Σ

is that (E

, D

)

is (C&P∨D&A)-COOBS, which means (E

, D

) is

C&P-COOBS or D&A-COOBS.

2.4 Inference-Based Control

In Inference-based control, every local supervisor

SUP

generates a local decision associated to an am-

biguity level which is computed in a quite complex

(but systematic) way. The taken global decision is

the local decision with the lowest ambiguity level.

Inference-based control is based on the following it-

erative computations:

Decentralized Supervisory Control of Discrete Event Systems: Using Multi-Decision Control as an Alternative to Inference-Based Control

215

• Basis: E

[0] = E

and D

[0] = D

• Inductive step: for k ≥ 0

[k+ 1] = [

i=1···n

−1

[k])] ∩ E

[k]

[k+ 1] = [

i=1···n

−1

[k])] ∩ D

[k]

Inference-based control is denoted Inf

-control if

N is the maximum used ambiguity level. The property

of coobservabilityassociated to Inf

-controlfor some

σ ∈ Σ

is that E

[N+1] =

0 or D

[N+1] =

0. In such

a case, we say that (E

, D

) is Inf

-COOBS. It is

worth noting that (C&P∨D&A)-COOBS is equivalent

to Inf

-COOBS.

3 MULTI-DECISION CONTROL

As we have recalled it, for every control architecture,

the existence of supervisor is conditioned formally by

the satisfaction of three properties: L-controllability,

-closure, and coobservability. The ﬁrst two prop-

erties are independent of the architecture, they de-

pend uniquely on the plant and speciﬁcation models.

Therefore, they are not relevant to compare control

architectures. On the contrary, coobservability differs

for each control architecture, and hence is a good cri-

terion to compare control architectures. A control ar-

chitecture A is said more general than an architecture

B if coobservability associated to A is weaker than

coobservability associated to B, in the sense that the

latter implies the former. Intuitively, the generality of

A over B means that every language achievable by B

is also achievable by A.

For example inference-based control is more

general than C&P∨D&A control, because Inf

COOBS is weaker than (C&P∨D&A)-COOBS. In-

deed E

[1] =

0 or D

[1] =

0 implies E

[N + 1] =

and D

[N + 1] =

0 for every N ≥ 0.

In fact, we think that one of the main motivations

to research in decentralized control has been the de-

sire to discover more and more general architectures.

This is indeed the motivation of the development of

the multi-decision control. The intuition of the au-

thors of (Chakib and Khoumsi, 2011) was that a more

general architecture can be obtained by using several

decentralized architectures in parallel whose respec-

tive decisions are all combined to generate the effec-

tive decisions. Two combination operators have been

used: disjunction and conjunction, which we present

in the following subsections.

3.1 Disjunctive Multi-Decision Control

Consider several (say p) decentralized control archi-

tectures (S

)

j=1···p

such that, for every σ ∈ Σ

, the

global decisions of all S

are combined disjunctively

to issue the effective decision on σ. Formally, we have

Eq. (9), where Sup

(λ, σ) is the global decision taken

by S

and Sup(λ, σ) is the effective decision synthe-

sized from all (Sup

(λ, σ))

j=1···p

. The obtained archi-

tecture is named ∨-(S

, ·· · , S

Sup(λ, σ) =

j=1···p

Sup

(λ, σ) (9)

Let (E

, D

) be the enabling-pair of S

for any

σ ∈ Σ

, i.e. S

enables σ in E

and disables it in

. From Eq. (9), it is easy to deduce that the archi-

tecture resulting from the disjunctive combination of

)

j=1···p

has its enabling-pair (E

, D

) for σ ∈ Σ

speciﬁed by: E

j=1···p

and D

j=1···p

If we take D

= D

for every j = 1··· p, we obtain

that each S

has an enabling-pair (E

, D

) such that

(

j=1···p

, D

) is the enabling-pair of the resulting

architecture.

Consider now the opposite situation where we

have to ﬁnd a set of p architectures (S

)

j=1···p

such that ∨-(S

, ·· · , S

) has a given enabling pair

, D

) for every σ ∈ Σ

. This amounts to ﬁnd, for

every σ ∈ Σ

, a decomposition (E

)

j=1···p

of E

such

that every (E

, D

) is the enabling-pair of S

Finding a decomposition of an inﬁnite E

is in

general a difﬁcult problem if not undecidable. The au-

thors of (Chakib and Khoumsi, 2011) have proposed

that instead of decomposing E

, we decompose the

set of marked states of an FSA A

accepting E

Hence, the undecidable problem of decomposing an

inﬁnite set E

is transformed into a decidable prob-

lem of decomposing the ﬁnite set of marked states

of some FSA accepting E

. With this approach, the

possible decompositions are said authorized by A

and have the characteristic that each E

corresponds

to one or several marked states of A

. Note that

if some marked states of A

are split into several

equivalent states, more decompositionsare authorized

by the new obtained FSA than by A

3.2 Conjunctive Multi-Decision Control

There exist strong similarities between disjunctive

and conjunctive multi-decision controls. Indeed, the

fundamental difference when passing from the ﬁrst

one to the second one is that Eq. (9) is replaced by

Eq. (10), and decomposing E

w.r.t A

is replaced

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

216

by decomposing D

w.r.t an FSA A

accepting D

Hence, we obtain that each S

has an enabling-pair

, D

) such that (E

j=1···p

) is the enabling-

pair of the resulting architecture. The obtained archi-

tecture is named ∧-(S

, ·· · , S

Sup(λ, σ) =

j=1···p

Sup

(λ, σ) (10)

4 OUR PROPOSITION

We consider a plant modeled by an FSA G whose

generated and marked languages are L and L

, and

a speciﬁcation modeled by a trim FSA K whose

marked language is K (the generated language of K

K). We are also given the number n of sites and

their respective controllable and observable alphabets

(Σ

c,i

)

1≤i≤n

and (Σ

o,i

)

1≤i≤n

. As explained in Sec-

tions 2.2 and 3, we consider uniquely the property of

coobservability as condition of existence of supervi-

sor.

As mentioned in the introduction, our motivation

is to avoid using complex (namely, inference-based

Inf

) architectures by using instead several simple

(namely, C&P∨D&A) architectures running in par-

allel. Recall that C&P∨D&A is equivalent to Inf

that is, it corresponds the simplest case of inference-

based control, where ambiguity levels are restricted to

0. The price to be paid for this simpliﬁcation is an in-

crease of the number of architectures in parallel. For

example, it is possible that a given control objective

can be realized by any of the following three architec-

tures:

• one Inf

architecture;

• two architectures, a Inf

and a Inf

, running in

parallel;

• three Inf

(i.e. C&P∨D&A) architectures in par-

allel.

Our objective is therefore to satisfy Eqs (1,2) by

using uniquely Inf

(i.e. C&P∨D&A) control archi-

tectures in parallel. For the sake of clarity, we identify

them as Inf

, Inf

, ·· · . Hence, our study consists in

determining whether there exists a ∨-(Inf

, ·· · , Inf

)

or ∧-(Inf

, ·· · , Inf

) architecture that respects the ob-

jective formulated by Eqs. (1,2). But we present

here uniquely the case of ∨-(Inf

, ·· · , Inf

), because

of the strong similarities between the two architec-

tures as explained in Section 3.2. The architecture ∨-

(Inf

, ·· · , Inf

) is also denoted ∨-Inf

where p spec-

iﬁes the number of architectures in parallel. We may

also use the notation ∨-Inf

≥1

when p is unspeciﬁed.

To simplify the presentation of our procedure, we

consider a single controllable event σ. In the presence

of several controllable events, we must apply the same

procedure to each of them.

4.1 Running Example

We consider the example of Figure 1 that will be used

to illustrate each step of our proposition. The com-

plete automaton represents the plant G , and the spec-

iﬁcation K is obtained by removing the two dashed

self-loops of σ. All states are marked, hence G and

K are preﬁx-closed. We have two sites (i.e. n = 2),

and the local observable and controllable alphabets

are: Σ

o,1

= {a

, b

, c

}, Σ

o,2

= {a

, b

, c

, d

}, and

c,1

= Σ

c,2

= {σ}.

Figure 1: Plant G and Speciﬁcation K .

4.2 Step 1: Computing A

and A

Recall that A

denotes an automaton accepting a lan-

guage X. Since our control objective is based on E

and D

(through Eqs (1,2)), we compute A

and

. A

is computed from K by marking uniquely

the states where σ is accepted and then removing the

states from which no marked state is reachable. A

is computed in 3 steps: 1) we compute the synchro-

nized product of G and K , which is an automaton

whose states are deﬁned by (q, r), where q and r are

states of G and K , respectively; 2) we mark every

state (q, r) where we have σ enabled in the state q of

G and disabled in the state r of K ; and 3) we remove

the states from which no marked state is reachable.

Let X and Y denote the sets of states of A

and A

respectively. Let X

and Y

denote the sets of marked

states of A

and A

, respectively. The states of X

are identiﬁed as x

, x

, ·· · , and the states of Y

are

identiﬁed as y

, y

, ·· · .

Consider our example: A

is obtained from Fig-

ure 1 by removing states 3, 7 and 9, and marking

states 6 and 8. A

is obtained from Figure 1 by

removing states 4, 6 and 8, and by marking states

7 and 9. A

and A

are represented in Figures 2

and 3. The marked states of A

are x

and x

, and

the marked states of A

are y

and y

Decentralized Supervisory Control of Discrete Event Systems: Using Multi-Decision Control as an Alternative to Inference-Based Control

217

Figure 2: Automaton A

Figure 3: Automaton A

4.3 Step 2: Computing A

[1]

and A

[1]

Inf

-COOBS is based on D

[1] and E

[1] deﬁned by

Eqs. (5,8). Hence, FSA A

[1]

and A

[1]

accepting

[1] and D

[1] are computed from A

and A

by Eqs. (11,12), where

and × denote the synchro-

nized product of automata.

[1]

i∈I

−1

) × A

(11)

[1]

i∈I

−1

) × A

(12)

1. Every state of A

[1]

is deﬁned by u =

, ·· · , u

, u

), where u

∈ X and u

⊆ Y

for every i ∈ I

u is said marked if u

∈ X

and u

∩Y

0 for

every i ∈ I

u is said x

-marked if it is marked and u

= x

2. Every state of A

[1]

is deﬁned by v =

, ·· · , v

, v

), where v

∈ Y and v

⊆ X

for every i ∈ I

v is said marked if v

∈ Y

and v

∩ X

0 for

every i ∈ I

v is said x

-marked if it is marked and x

∈ v

for

every i ∈ I

Then, we remove from A

[1]

and A

[1]

the states

from which no marked state is reachable.

Consider our example: A

[1]

and A

[1]

are rep-

resented in Figures 4 and 5, where marked states are

dark and x

-marked states are indicated by x

Figure 4: Automaton A

[1]

Figure 5: Automaton A

[1]

4.4 Step 3: Checking if (E

, D

) is

∨-Inf

≥1

-COOBS w.r.t A

First of all, it is worth checking whether Inf

architecture (without multi-decision) can be used to

satisfy Eqs. (1,2) w.r.t (E

, D

). As explained in

Section 2.3, this amounts to determine if (E

, D

)

is Inf

-COOBS. The veriﬁcation is done as follows:

, D

) is Inf

-COOBS if and only if A

[1]

has no marked state.

If we compute that (E

, D

) is Inf

-COOBS,

Inf

-architecture can be used since it satisﬁes

Eqs. (1,2) w.r.t (E

, D

); we go to Step 5 (Sect. 4.6)

to compute the local and global decisions taken by the

architecture.

If we compute that (E

, D

) is not Inf

-COOBS

(i.e. both A

[1]

or A

[1]

have marked states), we

continue in this step 3.

Since (E

, D

) is not Inf

-COOBS, we have now

to verify if multi-decision can help. More precisely,

we have to determine whether there exists a ∨-Inf

≥1

architecture that satisﬁes Eqs. (1,2) w.r.t (E

, D

). As

explained in Section 3.1, this amounts to determine

whether there exists a decomposition (E

)

j=1···

of E

such that (E

, D

)

j=1···

are the respective enabling-

pairs of the Inf

architectures in parallel. From Sec-

tion 2.3, this amounts to determine whether there ex-

ists a decomposition (E

)

j=1···

of E

such that every

, D

) is Inf

-COOBS, i.e. E

[1] =

0 or D

[1] =

for every j = 1· ·· p, where D

[1] and E

[1] are de-

ﬁned as D

[1] and E

[1] in Eqs. (5,8), but by using

instead of E

We have explained in the last paragraph of Sec-

tion 3.1 why we consider uniquely decompositions

)

j=1···

authorized by A

, i.e. where each E

corresponds to one or several marked states of A

Therefore, by using Def. 4.1 below, our objective be-

comes to determine if (E

, D

) is ∨-Inf

≥1

-COOBS

w.r.t A

Deﬁnition 4.1. (E

, D

) is said ∨-Inf

≥1

-COOBS

w.r.t A

if there exists a decomposition (E

)

j=1···

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218

of E

which is authorized by A

, such that every

, D

) is Inf

-COOBS.

Let us consider the decomposition (in fact a par-

tition) of E

where each E

corresponds to a sin-

gle marked state x

∈ X

of A

. Such a partition

is called trivial partition of E

w.r.t A

. Note that

the number p of languages E

of such a trivial parti-

tion is the cardinality of the set X

of marked states of

. Clearly, if for such a trivial partition we have not

that every (E

, D

) is Inf

-COOBS, then (E

, D

) is

not ∨-Inf

≥1

-COOBS w.r.t A

, i.e. there exists no

other decomposition authorized by A

for which we

have that every (E

, D

) is Inf

-COOBS. Therefore,

, D

) is ∨-Inf

≥1

-COOBS w.r.t A

if and only if

for every E

corresponding to a state x

∈ X

of A

, D

) is Inf

-COOBS, i.e. E

[1] =

0 or D

[1] =

This can be veriﬁed as follows: Emptiness of E

[1]

(resp. D

[1]) is equivalent to that A

[1]

(resp. A

[1]

)

has no x

-marked state.

If we compute that (E

, D

) is ∨-Inf

≥1

-COOBS

w.r.t A

, the ∨-Inf

≥1

-architecture can be used since

it satisﬁes Eqs. (1,2) w.r.t (E

, D

). We go to Step

5 (Sect. 4.6) to compute the decisions taken by the

architecture.

If we compute that (E

, D

) is not ∨-Inf

≥1

COOBS w.r.t A

, we go to step 4 (Sect. 4.5).

Consider our example: Both A

[1]

and A

[1]

have marked states (dark in Figs. 4 and 5), hence

, D

) is not Inf

-COOBS. We consider the trivial

partition E

and E

corresponding to states x

and x

of A

of Fig. 2. The languages of A

correspond-

ing to x

and x

are, respectively, E

= {c

∗

} and

= {a

∗

, a

∗

, b

∗

}. x

is not problem-

atic because A

[1]

of Fig. 5 has no x

-marked state,

which means that D

[1] =

0. Therefore, (E

, D

) is

Inf

-COOBS. Hence, E

= {c

∗

} will be kept as

it is. x

is problematic because both A

[1]

and A

[1]

have a x

-marked state, which means that E

[1] 6=

and D

[1] 6=

0. Therefore, (E

, D

) is not Inf

COOBS. This problem is solved in the following step

(Sect. 4.5).

4.5 Step 4: When (E

, D

) is not

∨-Inf

≥1

-COOBS w.r.t A

If (E

, D

) is not ∨-Inf

≥1

-COOBS w.r.t A

, we

have to determine if we can obtain that (E

, D

) is

∨-Inf

≥1

-COOBS w.r.t another automaton which au-

thorizes more decompositions than A

. In fact, we

need to consider only the problematic states, i.e. the

states x

of A

that correspond to the E

for which

[1] 6=

0 and D

[1] 6=

0. The solution is to split

each problematic state x

into severalequivalent states

, ·· · , x

, which permits to decompose the corre-

sponding problematic E

language into several lan-

guages E

, ·· · , E

. Hence, we need to use an opera-

tor that splits states of an automaton without changing

its accepted language. Let us consider the example of

operator O given by Eq. (13).

O(A

) = P

−1

) × ···P

−1

) × A

(13)

Once O(A

) computed, we repeat Steps 2-3

(Sections 4.3 and 4.4), but by using O(A

) in-

stead of A

and by considering as marked states of

O(A

) uniquely the states equivalent to the problem-

atic states of A

. And so on, we may have to do

the iteration steps 2-3-4-2-...-3, and hence compute

, A

, ·· · . The iteration is stopped in Step 3 if

we obtain that (E

, D

) is ∨-Inf

≥1

-COOBS w.r.t the

current A

, or after a given number of iterations.

Consider our example: We have seen that for

the E

corresponding to state x

of A

(Fig. 2),

, D

) is not Inf

-COOBS. If we apply the oper-

ator of Eq. (13) to A

of Fig. 2, we obtain O(A

)

of Fig. 6. By this operation, the problematic state x

of A

has been split into the two states indicated

by x

and x

in Fig. 6. This amounts to decom-

pose the language E

= {a

∗

, a

∗

, b

∗

}

into the two languages E

= {a

∗

, a

∗

} and

= {b

∗

Since the language E

corresponding to x

needs

not be changed, we consider as marked uniquely the

states x

and x

. New A

[1]

and A

[1]

are computed

by applying Eqs. (11,12) to (O(A

), A

). We ﬁnd

that the new A

[1]

and A

[1]

have no x

-marked or

-marked automaton. This means that (E

, D

) and

, D

) are Inf

-COOBS, To recapitulate, we have

found a partition (E

)

j=1,2,3

of E

such that every

, D

) is Inf

-COOBS, i.e. for each j = 1, 2, 3

we have E

[1] =

0 or D

[1] =

0. Indeed, we obtain

[1] =

0 for j = 1, 2, 3, and E

[1] =

0 and E

[1] =

Therefore, (E

, D

) is ∨-In f

-COOBS w.r.t O(A

Figure 6: Automaton O(A

Decentralized Supervisory Control of Discrete Event Systems: Using Multi-Decision Control as an Alternative to Inference-Based Control

219

4.6 Step 5: After Steps 2-3-4-2-...2-3

If after Steps 2-3-4-2-...2-3, we obtain that (E

, D

)

is not ∨-Inf

≥1

-COOBS w.r.t the current A

, we

conclude that our method is not applicable with the

proposed operator O that computes A

, A

, ·· · .

Otherwise (i.e. (E

, D

) is ∨-Inf

≥1

-COOBS w.r.t

the current A

), we consider the decomposition

)

j=1···p

that has been constructed. We can use

the ∨-(Inf

, ·· · , Inf

) architecture such that each

, D

) is the enabling-pair of the Inf

-architecture.

The effective decision Sup(λ, σ) is computed by

Eq. (9) where each Sup

(λ, σ) is the decision taken

by Inf

. Each Sup

(λ, σ) is computed as follows:

• If D

[1] =

0: Therefore, Inf

is a C&P-

architecture and hence its global decision

Sup

(λ, σ) is computed by Eqs. (3,4), but by

adding a superscript j to E

, Sup

(λ), σ) and

Sup(λ, σ).

• If E

[1] =

0: Therefore, Inf

is a D&A-

architecture and hence its global decision

Sup

(λ, σ) is computed by Eqs. (6,7), but by

adding a superscript j to Sup

(λ), σ) and

Sup(λ, σ).

Note that the case where (E

, D

) is Inf

-COOBS

can be treated as above by taking p = 1.

Consider our example: we have found a parti-

tion (E

)

j=1,2,3

of E

such that every (E

, D

) is

C&P-COOBS. Recall that E

= {c

∗

}, E

∗

, a

∗

} and E

= {b

∗

}. Therefore,

we can use three C&P-architectures in parallel. Ta-

ble 1 represents the generated decisions for the traces

λ ∈ E

∪ D

, which are the only traces where a con-

trol decision on σ is relevant. Columns P

and P

contain the projections P

(λ) and P

(λ). For each

j = 1, 2, 3, we have three columns S

, S

that con-

tain the decisions of the j

architecture: S

and S

are the local decisions computed by Eq. (3) for each

of the two sites; S

is the global decision computed by

Eq. (4), i.e. S

is the conjunction of S

and S

. The

last column contains the effective decision computed

Eq. (9), i.e. S is the disjunction of S

, S

. We see

that the decision 1 is taken for the traces of E

(lines

1-4), and the decision 0 is taken for the traces of D

(lines 5-7).

5 CONCLUSION

Multi-decision control has been so far studied as a so-

lution to generalize inference-based control, by using

Table 1: The decisions taken by the ∨-Inf

architecture of

our example.

λ P

∗

1 1 1 0 0 0 0 0 0 1

∗

0 0 0 1 1 1 0 0 0 1

∗

0 0 0 0 0 0 1 1 1 1

∗

0 0 0 0 0 0 1 1 1 1

1 0 0 0 0 0 0 0 1 0

0 1 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0 0

several inference-based architectures in parallel. In

the present study, we have used multi-decision con-

trol with a different perspective. Instead of using it

to generalize inference-based control, we have rather

used it as an alternative to inference-based control.

More precisely, our objective has been to avoid using

inference-based architectures, by using instead sev-

eral simple architectures running in parallel. The sim-

ple used architectures are called C&P∨D&A, which

are the simplest relevant decentralized architectures

known in the literature.

A possible future work is to develop a more gen-

eral procedure that tries architectures with the small-

est nonnull ambiguity levels, if the control objective

cannot be reached by uniquely the ambiguity level 0.

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