Decentralized Supervisory Control of Discrete Event Systems

Moving Decisions Closer to Actions

Ahmed Khoumsi and Hicham Chakib

Department of Electrical & Computer Engineering, University of Sherbrooke, Sherbrooke, Canada

Keywords:

Decentralized Supervisory Control, Discrete Event Systems, Local Supervisors, Fusion Modules, Sensors,

Actuators, Moving Decision Closer to Actions, Mixed Architecture, Mixed-observability.

Abstract:

In decentralized control of discrete event systems, two main agents contribute to the computation of decisions:

local supervisors and fusion modules. The local supervisors process the information detected on the plant

and its environment, and transmit their results to the fusion modules. The latter process what is received

from the local supervisors in order to decide actions to be applied to the plant. In the existing decentralized

control architectures, the local supervisors execute complex operations, while the fusion modules execute

simple operations. In the present article, we propose to move the decision computation complexity from

local supervisors to fusion modules, that is what we term: moving decisions closer to actions. We justify

this movement of decision and develop a simple architecture based on it. With the proposed architecture, the

local supervisors are simple local observers, while all decisions are computed by the fusion modules. We

characterize the class of languages achievable with the new architecture and compare it with the classes of

languages achievable with the existing decentralized architectures and the centralized architecture.

1 INTRODUCTION

In supervisory control, the behavior of a plant, mod-

eled as a discrete event systems (DES), is restricted

by a supervision system (or supervisor) so that it re-

spects a global speciﬁcation (Ramadge and Wonham,

1987). In decentralized supervisory control (more

brieﬂy: decentralized control) of DES, observation

and control tasks are shared among several agents

that cooperate to take adequate decisions that restrict

the behavior of the plant as desired. In this article,

we consider decentralized control as it has been initi-

ated in (Cieslak et al., 1988; Lin and Wonham, 1988;

Lin and Wonham, 1990; Rudie and Wonham, 1992)

and then continued in many studies, such as (Rudie

and Willems, 1995; Prosser et al., 1997; Ricker and

Rudie, 2000; Overkamp and van Schuppen, 2001;

Jiang and Kumar, 2000; Yoo and Lafortune, 2002;

Yoo and Lafortune, 2004; Ricker and Rudie, 2007;

Kumar and Takai, 2007; Chakib and Khoumsi, 2011).

In decentralized control, two categories of agents

contribute to the computation of decisions, which we

call local supervisors and fusion modules. The local

supervisors process detected information on the plant

and its environment, and transmit their results to the

fusion modules. The latter process what is received

from the local supervisors in order to decide enable-

ment/disablement of events of the plant.

Figure 1 outlines the conceptual structure of de-

centralized control, by an example with three local

supervisors (Sup

)

i=1···3

and two fusion modules FM

and FM

. Each Sup

observes the behavior of the

plant through a mask (or projection) P

, and processes

the observed information before providing the result

to one or both fusion modules. Each fusion module

combines what it receives from two local supervisors,

in order to take enablement/disablement decisions.

Sup

Plant

Figure 1: Decentralized control.

Deﬁnition 1. A local supervisor or fusion module is

said powerful if it has a high processing capacity and

models of the dynamics of the plant and the speciﬁca-

tion. On the other hand, it is said powerless if it has

a minimal processing capacity and no model of the

dynamics of the plant and the speciﬁcation.

280

Khoumsi A. and Chakib H..

Decentralized Supervisory Control of Discrete Event Systems - Moving Decisions Closer to Actions.

DOI: 10.5220/0005010702800287

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 280-287

ISBN: 978-989-758-040-6

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

A typical minimal processing capacity is to

be able to execute simple operations, for example

boolean operations. A typical high processing ca-

pacity is to be able to manipulate complex automata-

based models of DES, by executing various automata-

based operators.

The following assumption has been made by the

existing architectures of decentralized control:

Assumption 1. The local supervisors are powerful

and the fusion modules are powerless.

Due to assumption 1, the local supervisors use

complex operations (e.g. automata-based) to compute

local decisions which are made accessible to fusion

modules. Due to assumption 1, the fusion modules

merge by simple (e.g. boolean) operations the local

decisions of the local supervisors, in order to com-

pute global decisions of enabling/disabling control-

lable events which are effectively applied to the plant.

In a word, the most complex part of decision com-

putation is done by powerful local supervisors, while

the simplest part is done by powerless fusion mod-

ules. A possible justiﬁcation of this approach is that

it is consistent with hierarchical organizations, con-

sidering that fusion modules are at a higher level than

local supervisors. Besides, this approach gives a good

insight of the limit of control that can be reached with

powerless fusion modules. In the sequel, decentral-

ized control based on assumption 1 (which includes

the previously mentioned references) will be referred

to as conventional decentralized control.

Since most of the important results in decentral-

ized control have been obtained in conventional de-

centralized control, assumption 1 has been consid-

ered almost as a standard. But we had a long reﬂec-

tion which led us to think that assumption 1 is not

always indispensable and may sometimes have dis-

advantages. This is conﬁrmed by our examination

of several examples used in the literature of conven-

tional decentralized control, in which we see no prac-

tical reason that obliges to make assumption 1. This

assumption is even sometimes artiﬁcial, for example

when the separation between local supervisors and

fusion modules is logical but not physical. Indeed,

we found examples where the same physical module

hosts a local supervisor and a fusion module. We will

return on this point in Section 2.5.

The above observation has motivated this article,

where our objective is to propose a new decentralized

architecture without the limitative assumption 1. Our

ﬁrst idea has then been to use powerful local supervi-

sors as well as powerful fusion modules. But as we

will explain in Section 2.5 and prove in Section 5.2,

if the fusion modules are powerful, then it is not re-

strictive to have powerless local supervisors That is,

the following assumption is not restrictive:

Assumption 2. The local supervisors are powerless

and the fusion modules are powerful.

We use the following ﬁgurative deﬁnition:

Deﬁnition 2. To move decisions closer to actions

means to replace assumption 1 by assumption 2, be-

cause such a replacement implies moving the decision

computation complexity from the local supervisors to

the fusion modules.

The remainder of the paper is organized as fol-

lows: In Section 2, we present a reﬂection which led

us to deduce that moving decisions closer to actions

is realistic and has advantages. A decentralized ar-

chitecture based on this displacement of decisions is

proposed in Section 3. The proposed architecture is

called mixed architecture, because it has similarities

with both decentralized and centralized architectures.

In Section 4, we study the existence of solutions and

deﬁne the notion of mixed-observability which, with

controllability, characterizes the class of languages

achievable under the mixed architecture. Section 5

compares this class with the classes of languages

achievable under conventional decentralized control

and centralized control. Finally, Section 6 contains a

conclusion and propositions of future work.

2 MOVE DECISIONS CLOSER TO

ACTIONS: REALISTIC AND

ADVANTAGEOUS

This section presents a reﬂection on decentralized

control, which leads us to deduce that assumption 1

is not always indispensable. More precisely, our de-

duction is that moving decisions closer to actions (i.e.

replacing assumption 1 by assumption 2) is realistic

and has advantages. The most important advantage is

that such a movement of decision will permit to obtain

a simpler and more general control architecture.

2.1 Structure based on Sensors and

Actuators

In practice, a control system interacts with the plant

under control through two categories of modules: sen-

sors and actuators.

• Sensors are parts of the plant through which the

control system detects information on the behav-

ior and the environment of the plant. Examples of

sensors: an obstacle detector that warns when an

object is too close to the plant, and an accelerom-

eter that informs of the acceleration of the plant.

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• Actuators are parts of the plant through which the

control system can inﬂuence the behavior of the

plant. Examples of actuators: an electric motor

that gives motion to a car, and a valve actuator

that opens and closes a valve of a water system.

In decentralized control, several sensors and actu-

ators are distributed at several parts of the plant, ac-

cording to given requirements which may be: physi-

cal, i.e. to satisfy mandatory objectives due to physi-

cal considerations; or logical, i.e. to satisfy desirable

objectives, e.g. to decrease cost or increase efﬁciency.

Let us present concrete sensor and actuator neighbor-

hoods and their usual theoretical models in conven-

tional decentralized control.

2.1.1 Sensors Neighborhood

A sensor plays the role of an interface through which

the control system obtains information on the be-

havior and the environment of the plant. Accord-

ing to given requirements, sensors are regrouped by

sites identiﬁed by numbers i = 1 ···n. Each site i

has a given memory and processing capability (more

brieﬂy: processing capacity) to process information

obtained from sensors. The processed information is

then made accessible to controllers of actuators. Let

us make the link with the usual theoretical models:

• The information obtained in site i from sensors is

modeled as the alphabet of events observable at

site i , usually denoted Σ

o,i

• The sensors of site i are modeled as a mask (or

projection) P

that shows (all and only) the events

of Σ

o,i

, among the events executed by the plant.

• The processing capacity of site i (that processes

information obtained from sensors of site i) is

modeled as a local supervisor Sup

Hence, the neighborhood of the sensors of a site

consists of the local supervisor of that site.

2.1.2 Actuator Neighborhood

An actuator is an interface through which the con-

trol system can inﬂuence the behavior of the plant.

Each actuator is directly controlled by the means of

a given processing capacity which, from information

provided by local supervisors, decides which enable-

ment/disablement actions to apply to the actuator. Let

us make the link with the usual theoretical models.

• An actuator is associated to a set of control-

lable events and is modeled as a module that re-

ceives orders of enabling/disabling these control-

lable events. For example, a motor that receives

the commands “start moving” and “stop moving”.

• The processing capacity used to control an actu-

ator is modeled as a set of fusion modules, typ-

ically one fusion module FM

for each control-

lable event σ associated to the actuator.

• Each fusion module FM

merges (i.e. collects and

processes) information from sites (more precisely,

from local supervisors), in order to compute (en-

ablement/disablement) decisions on σ that are ap-

plied to the actuator.

• The set of controllable events whose fusion mod-

ules are connected to (i.e. receive information

from) a given site i, is speciﬁed by the The con-

nectivity between local supervisors and fusion

modules can also be speciﬁed by the index set of

a controllable σ: I

= {i |σ ∈ Σ

c,i

}. That is, I

speciﬁes to which local supervisors is connected

the fusion module FM

Hence, the neighborhood of an actuator consists

of the set of fusion modules acting on it.

2.2 Knowledge of the Plant and

Processing Capacity

We have seen that in decentralized control, two types

of processing systems are used: local supervisors and

fusion modules. Hence, to evaluate the capacity of

control that can be achieved, it is necessary to spec-

ify, for each local supervisor and each fusion module:

what knowledge it has of the plant and the speciﬁca-

tion, and what is its processing capacity.

2.2.1 Knowledge of the Plant and the

Speciﬁcation

It is necessary to specify what knowledge of the plant

and the speciﬁcation, each local supervisor and fusion

module has. When it is not null, such a knowledge is

usually in the form of automata-based models of the

dynamics of the plant and the speciﬁcation.

In conventional decentralized control, due to as-

sumption 1, only the local supervisors have a knowl-

edge of the plant and the speciﬁcation.

2.2.2 Processing Capacity

It is necessary to specify what is the processing ca-

pacity of each local supervisor and fusion module. By

processing capacity, we mean memory and processing

capability. The fusion modules associated to the same

actuator must be assumed having the same processing

capacity, because they are physically in the control

module of the same actuator. A typical minimal pro-

cessing capacity is to be able to execute simple op-

erations, for example boolean operations. A typical

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high processing capacity is to be able to manipulate

complex automata-based models of DES by execut-

ing various automata-based operators.

In conventional decentralized control, due to as-

sumption 1, only the local supervisors have a high

processing capacity, while the fusion modules have

a minimal processing capacity.

2.3 Functioning

Let Σ

∪

i=1···n

c,i

denote the set of all controllable

events. Each Sup

(modeling the processing system

in site i) uses its processing capacity and its knowl-

edge of the plant and the speciﬁcation, to process in-

formation taken from sensors of site i (modeled by the

events of Σ

o,i

). The processing result of Sup

is then

an information accessible by some means to every fu-

sion module FM

connected to Sup

(i.e. such that

σ ∈ Σ

c,i

, or equivalently i ∈ I

Each FM

(which models the processing system

that decides on enablement/disablement of σ ∈ Σ

)

uses its processing capacity and its knowledge of the

plant, to process information from supervisors Sup

connected to it. The processing result of FM

is a

command that is applied to an actuator of the plant to

disable/enable σ.

In conventional decentralized control, due to as-

sumption 1, powerful local supervisors compute lo-

cal decisions from their knowledge of the plant and

the speciﬁcation, and powerless fusion modules com-

pute effective decisions by combining the local deci-

sions using operations independent of the plant and

the speciﬁcation.

2.4 Illustrative Example

Figure 2 outlines by an example our [sensor,actuator]-

based vision of decentralized control. The plant

has three sensors and two actuators. The three

sensors detect information on the plant and trans-

mit it in the form of events a

,{a

},c

, respec-

tively. The two actuators are controlled by enable-

ment/disablement of the events {σ,µ} and ρ, respec-

tively. Hence, Σ

= {σ,µ,ρ}. Dec(x) denotes the en-

ablement/disablement decision taken on an event x.

The control system consists of three supervisors

and three fusion modules. The three supervisors

(Sup

)

i=1···3

process the events obtained from the

three sensors, respectively. These events are mod-

eled by the local observable alphabets Σ

o,1

= {a

o,2

= {a

} and Σ

o,3

= {c

}, An arrow connecting

Sup

to FM

means that the processing result of Sup

accessible to FM

Hence, FM

and FM

have access

to processing results from Sup

and Sup

, and FM

has access to processing results from Sup

and Sup

This connectivity is modeled by the controllable al-

phabets Σ

c,1

= {σ,µ}, Σ

c,2

= {σ,µ,ρ} and Σ

c,3

= {ρ}.

This connectivity is also modeled by the index sets:

= I

= {1,2} and I

= {2,3}.

Actuator

Dec( )

Sup

Actuator

Dec( )

Plant

Sensor

Site 1

Site 3

Site 2

Figure 2: Example of our [sensor,actuator]-based vision of

decentralized control.

2.5 Discussion and Objective

Intuitively, σ ∈ Σ

c,i

(or equivalently: i ∈ I

) means

that Sup

is involved in some way in the decision

of enabling/disabling σ, by providing information to

which takes the effective decision.

Deﬁnition 3. Sup

is said actively involved in a de-

cision (of enablement/disablement) on σ ∈ Σ

, if it

computes a local decision on σ which is used by FM

to compute the effective decision on σ. On the other

hand, Sup

is said passively involved in a decision on

σ, if it just provides an observed information which is

used by FM

to compute the effective decision on σ.

Active involvement of a local supervisor is ap-

propriate if it is powerful (Def. 1), because it can

use its high processing capacity and its knowledge of

the plant and the speciﬁcation, to compute local de-

cisions. Passive involvement of a local supervisor is

appropriate if it is powerless (Def. 1).

In conventional decentralized control, due to as-

sumption 1, the (powerful) local supervisors are ac-

tively involved by computing local decisions (using

complex automata-based operations), and the (pow-

erless) fusion modules combine the local decisions

(using simple operations) in order to compute global

decisions of enabling/disabling controllable events.

Since most of the important results in decentralized

control have been obtained in conventional decentral-

ized control, assumption 1 has been practically con-

sidered as a standard in decentralized control. But our

[sensor-actuator]-based vision shows no concrete rea-

son that obliges to make assumption 1. Concretely,

we see no reason why sensor environments must be

powerful, while actuator environments must be pow-

erless. Our impression is conﬁrmed by the fact that

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the literature of conventional decentralized control

presents no concrete justiﬁcation of the indispensabil-

ity of assumption 1. In some examples, the separation

between local supervisors and fusion modules is log-

ical but not physical. For example, an illustrative ex-

ample of a decentralized trafﬁc control is presented in

(Yoo and Lafortune, 2004), where the same physical

module hosts a local supervisor and a fusion module.

In addition to the fact that assumption 1 is not in-

dispensable, we think that it is sometimes restrictive,

from the following point of view: In conventional

decentralized control, the local supervisors convert a

precise information (their local observations) into a

coarse information (their local decisions) before send-

ing it to the fusion modules.

Our ﬁrst objective has therefore been to remove

assumption 1, by using powerful local supervisors as

well as powerful fusion modules. But if the fusion

modules are powerful, it is not indispensable to have

powerful local supervisors. The explanation is that

the maximal information which is available to decide

of enabling/disabling a controllable event σ, is the set

of events observed by the local supervisors connected

to FM

. Such information can be provided by power-

less local supervisors which just play the role of ob-

servers that forward their observations to the fusion

modules. Hence our objective in Sections 3 and 4 is:

Objective 1. To develop a decentralized control ar-

chitecture which uses assumption 2 (instead of as-

sumption 1).

3 CONTROL ARCHITECTURE

BASED ON ASSUMPTION 2

3.1 Principle of the Proposed

Architecture

We will make the following assumption which has

also been made in all the control architectures:

Assumption 3. The plant is slower than the control

system, in the sense that between the executions of

any pair of consecutive events by the plant, the con-

trol system has the time to compute and apply its en-

ablement/disablement decisions.

For every controllable event σ, we denote by Σ

the set of events observable by the local supervisors

connected to FM

. Formally:

Deﬁnition 4. For every controllable event σ, the ob-

servable alphabet of FM

is Σ

∪

i∈I

o,i

In the proposed architecture, each fusion module

knows the observable alphabets of the local su-

pervisors connected to it, that is, it knows every Σ

o,i

such that i ∈ I

(or equivalently: σ ∈ Σ

c,i

). Also, FM

is informed of the occurrence of every event in Σ

the local supervisors connected to it. For these rea-

sons and due to assumption 3, FM

has conceptually

a partial observation of the plant through the alpha-

bet Σ

, that is, FM

can determine the order of the

occurrences of the events of Σ

. From the latter infor-

mation, FM

determines its decisions on σ. In other

words, each FM

is conceptually a centralized super-

visor under partial observation of the plant through

the alphabet Σ

. Hence, the proposed architecture is

equivalent to a collection of centralized supervisors,

each one controlling a single controllable event σ. We

call it mixed architecture, because it has similarities

with both decentralized and centralized controls.

Note that with the mixed architecture, the local

supervisors are (powerless) observers and the fusion

modules are (powerful) event controllers. But to fa-

cilitate a comparison with conventional decentralized

control, we will keep the usual designations local su-

pervisor and fusion module.

Figure 3 outlines the conceptual representation of

the mixed architecture for the example of Figure 1,

where P

and P

denote the projections in the alpha-

bets Σ

and Σ

, respectively.

Plant

Figure 3: Conceptual representation of the mixed architec-

ture for the example of Fig. 1.

3.2 Mixed Supervisor

As already explained, each FM

is a centralized su-

pervisor under partial observation through the alpha-

bet Σ

. As usual, consider an automaton G modeling

the plant and the regular language K modeling the

speciﬁcation, both deﬁned over the same alphabet Σ.

Here are a few necessary deﬁnitions and notations:

• Σ

∗

is the set of all ﬁnite sequences of events of Σ,

including the empty sequence.

• For two sequences λ,µ ∈ Σ

∗

, µ is said a preﬁx of

λ, if there exists ρ ∈ Σ

∗

such that λ = µρ.

• L(A) is the preﬁx-closed language of an automa-

ton A, that is, the set of sequences starting in the

initial state of A and terminating in any state of A.

• L

(A) is the marked language of A, that is, the set

of sequences starting in the initial state of A and

terminating in a marked state of A.

• L consists of the sequences of L and their preﬁxes.

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• Σ

∪

i=1···n

c,i

and Σ

= Σ \ Σ

are the sets of

controllable and uncontrollable events, resp.

• Σ

∪

i=1···n

o,i

is the set of observable events.

• P (resp. P

, P

) is the natural projection that hides,

from any λ ∈ Σ

∗

, the events which are not in Σ

(resp. Σ

o,i

, Σ

• E

= {λ ∈ K |λσ ∈ K }.

• D

= {λ ∈ K |λσ ∈ L (G) \K }.

• En and Dis denote the decisions of enabling and

disabling an event, respectively.

Intuitively, E

(resp. D

) contains the sequences

executed by the plant after which σ must be enabled

(resp. disabled) by FM

to guarantee that (all and

only) the sequences in the speciﬁcation are accepted.

A supervisor of a control architecture can be de-

ﬁned as a function SUP that associates one of the

decisions En or Dis to every pair (λ,σ), where λ is

a sequence executed by the plant and σ is an event.

SUP(λ,σ) is the decision taken by SUP for σ when

the plant has executed λ. As usual, SUP(λ, σ) = En

when σ ∈ Σ

A supervisor of a mixed architecture, which we

call mixed supervisor, is the set of the fusion modules

used in the mixed architecture. We consider that after

the execution of a sequence λ by the plant, each fusion

module FM

takes its decision SUP(λ,σ) which de-

pends on two things: 1) the observed execution P

(λ),

and 2) its knowledge of the plant and the speciﬁcation,

which is formalized by P

) and P

). Hence,

a mixed supervisor is any supervisor that takes de-

cisions SUP(λ,σ) depending on the above points 1

and 2. Here is an example of mixed supervisor:

∀σ ∈ Σ

, SUP(λ,σ) =











En, if P

(λ) ∈ P

)\P

)

Dis, if P

(λ) ∈ P

)\P

)

En or Dis, if P

(λ) ̸∈ (P

)∪P

))

don’t know, if P

(λ) ∈ P

)∩P

)











(1)

Intuitively, the above mixed supervisor enables

(resp. disables) σ when it is certain that enabling

(resp. disabling) σ will not violate the speciﬁcation.

When FM

is certain that σ is impossible (i.e. σ is

not accepted by the plant), it can decide any of En

and Dis, because σ will not occur whatever the deci-

sion taken by FM

. When FM

is uncertain of the

adequate decision, it takes a “don’t know” decision

which may concretely correspond to an alarm. We

have considered this “don’t know” situation explic-

itly, because it is undesirable and we will present in

Section 4 the condition that avoids it.

3.3 Example

Consider the preﬁx-closed plant of Figure 4 where

all the states are marked. The speciﬁcation is

obtained by removing state 9. We have Σ

o,1

},Σ

o,2

= {a

},Σ

o,3

= {a

}, Σ

= Σ

o,1

∪ Σ

o,2

∪

o,3

= {a

}, Σ

c,1

= Σ

c,2

= Σ

= {σ}, Σ

c,3

0. Hence, I

= {1,2}. Therefore, the observable

alphabet of FM

is Σ

= {a

}. We compute

= {a

}, D

= {a

}, P

) = {a

}

and P

) = {a

}. If we apply Eq. (1) to this ex-

ample, we obtain the mixed supervisor of Eq (2).

∀σ ∈ Σ

, SUP(λ,σ) =







En, if P

(λ) = a

Dis, if P

(λ) = a

En or Dis, if P

(λ) ∈ {a

}







(2)

Figure 4: Example of plant and speciﬁcation.

4 EXISTENCE OF SOLUTIONS

4.1 Mixed-Observability

Consider an automaton G modeling the plant over an

alphabet Σ. We deﬁne a notion of observability, here

called mixed-observability, which (with controllabil-

ity) characterizes the class of languages achievable

under the mixed architecture.

Deﬁnition 5. Given a mixed architecture speci-

ﬁed by the alphabets (Σ

)

σ∈Σ

, a regular language

K ⊆ L

(G) is said mixed-observable if: ∀σ ∈ Σ

) ∩ P

) =

Intuitively, K is mixed-observable if every FM

has at any time enough information to decide on the

enablement/disablement of σ without violating the

speciﬁcation. This means that for every σ ∈ Σ

, the

decision “don’t know” OF Eq. 1 is never taken.

When Σ

= Σ

for every σ ∈ Σ

, mixed-

observability is equivalent to observability of the cen-

tralized architecture. This is not surprising, because

this case occurs when each FM

observes all the

events that are observable by a centralized supervisor.

4.2 Existence Result

A language K ⊆ L

(G) is said L

(G)-closed if K =

K ∩ L

(G), and it is said (L(G),Σ

)-controllable if

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K Σ

∩L(G) ⊆ K . Let L(SUP/G) denote the preﬁx-

closed language generated by the plant under the

control of a supervisor SUP, and let L

(SUP/G) =

L(SUP/G) ∩ L

(G) be the corresponding marked

language. SUP is said non-blocking if L

(SUP/G) =

L(SUP/G). L(SUP/G) is formally deﬁned as fol-

lows, where ε denotes the empty event sequence:

• ε ∈ L(SUP/G)

• [(λ ∈ L(SUP/G)) ∧(λσ ∈ L(G))∧(SUP(λ, σ) =

En)] ⇔ λσ ∈ L(SUP/G).

The following theorem states a necessary and suf-

ﬁcient condition for the existence of a mixed super-

visor SUP that controls a plant G so that it respects a

speciﬁcation K .

Theorem 1. Consider a nonempty K ⊆ L

(G).

There exists a nonblocking mixed supervisor SUP

satisfying L(SUP/G) = K and L

(SUP/G) = K

if and only if K is mixed-observable, (L(G),Σ

controllable and L

(G)-closed.

4.3 Example

We return to the example of Sect. 3.3 represented

in Fig. 4. The speciﬁcation is mixed-observable be-

cause P

)∩P

) = {a

}∩{a

} =

0. Since

the speciﬁcation is also (L(G),Σ

)-controllable and

(G)-closed, then from Theorem 1, there exists a

nonblocking mixed supervisor that controls the plant

so that it respects the speciﬁcation. An example of

such a mixed supervisor is given by Eq. (2).

5 COMPARISON WITH OTHER

ARCHITECTURES

As in Theorem 1 for the mixed architecture, the

class of languages achievable by any control archi-

tecture is characterized by three notions: (L (G), Σ

controllability, L

(G)-closure and a notion of observ-

ability. Since (L(G),Σ

)-controllability and L

(G)-

closure are independent of the considered architec-

ture, we can compare control architectures by com-

paring their respective observabilities.

5.1 Comparison with the Centralized

Control

The observability of a speciﬁcation K in a centralized

architecture, which we denote cent-observability, is

deﬁned as follows, where P is the projection on the

observable alphabet Σ

: ∀σ ∈ Σ

: P(E

)∩P(D

) =

Since Σ

⊆ Σ

for every σ ∈ Σ

, mixed-

observability implies cent-observability. Therefore,

the class of languages achievable with the centralized

architecture subsumes the class of languages achiev-

able with the mixed architecture.

We return to the example of Sect. 3.3 represented

in Fig. 4, where we make the following modiﬁca-

tions: Σ

c,1

= Σ

c,3

= Σ

= {σ} and Σ

c,2

0, and hence

= {1,3}. Since Σ

= {a

}, we have P(E

) =

= {a

} and P(D

) = D

= {a

}. Hence,

P(E

) ∩ P(D

) =

0, that is, the speciﬁcation is cent-

observable. The observable alphabet of FM

now Σ

= {a

}, P

) = {a

} and P

) =

}. Hence, P

) ∩ P

) = {a

} ̸=

0, that

is, the speciﬁcation is not mixed-observable.

Intuitively, since it observes

{

}

, a

centralized supervisor can distinguish the sequences

and a

, where distinct decisions must be

taken on σ. On the other hand, a mixed supervi-

sor cannot distinguish these two sequences because

it does not observe a

5.2 Comparison with Conventional

Decentralized Control

Let us consider any conventional decentralized archi-

tecture and denote by coobservability the correspond-

ing notion of observability. A sufﬁcient condition that

a speciﬁcation K is not coobservable is the existence

of a pair of sequences necessitating distinct decisions

and which cannot be distinguished by the local obser-

vations. More formally:

K coobservable ⇒ (∀σ∈Σ

)

(∀λ∈ E

,∀µ ∈D

)(∃i∈ I

) : P

(λ)̸= P

(µ)

(3)

On the other hand, from the fact that ∀σ ∈ Σ

,∀i ∈

: Σ

o,i

⊆ Σ

, we deduce that:

(∀σ∈ Σ

)(∀i∈ I

)(∀λ,µ ∈Σ

∗

) :

(λ) ̸= P

(µ)) ⇒ (P

(λ)̸= P

(µ))

(4)

Equations (3,4) imply the following equation:

K coobservable ⇒

(∀σ∈ Σ

)(∀λ∈ E

,∀µ ∈D

) : P

(λ)̸= P

(µ)

(5)

From Def. 5 and Eq. (5), we have: coobservabil-

ity implies mixed-observability, from which we de-

duce that the class of languages achievable with the

mixed architecture subsumes the class of languages

achieved by any conventional decentralized architec-

ture. Hence, assumption 2 is not restrictive.

Let us return to the example of Section 3.3 repre-

sented in Figure 4. We have shown in Section 4.3

ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics

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that the speciﬁcation is mixed-observable. On the

other hand, for λ = a

∈ E

,µ = a

∈ D

, we

have P

(λ) = P

(µ) and P

(λ) = P

(µ). From Eq. (3),

we deduce that the speciﬁcation is not coobservable

in conventional decentralized control. Therefore, no

conventional decentralized architecture can achieve

the speciﬁcation of this example.

6 CONCLUSION

We have proposed a decentralized controlled archi-

tecture, called mixed architecture, where the local su-

pervisors are simple local observers, while all com-

putations of decisions are made by the fusion mod-

ules which are conceptually equivalent to centralized

supervisors. We prove that this approach is realistic

and that it is simpler and more general than conven-

tional decentralized control. Actually, this idea has

already been studied in (Khoumsi and Chakib, 2008),

but the corresponding developed architecture was in-

decidable, due to the fact that the plant was not as-

sumed slower than the control systems.

In a near future, we intend to study the applica-

bility of the mixed architecture in complex real life

system.

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