Available via license: CC0

Content may be subject to copyright.

Supervisory Control of Multi-Agent

Discrete-Event Systems

with Partial Observation

Yingying Liu, Jan Komenda, and Zhiwu Li ∗†

March 22, 2021

Abstract

In this paper we investigate multi-agent discrete-event systems with

partial observation. The agents can be divided into several groups in each

of which the agents have identical (isomorphic) state transition structures,

and thus can be relabeled into the same template. Based on the template

ascalable supervisor whose state size and computational cost are indepen-

dent of the number of agents is designed for the case of partial observation.

The scalable supervisor under partial observation does not need to be re-

computed regardless of how many agents are added to or removed from

the system. We generalize our earlier results to partial observation by

proposing suﬃcient conditions for safety and maximal permissiveness of

the scalable least restrictive supervisor on the template level. An example

is provided to illustrate the proposed scalable supervisory synthesis.

1 Introduction

In manufacturing, logistical or similar technological systems one often encoun-

ters the need of not specifying the number of subsystems a priori, since their

number can vary even without an upper bound. Very often, such components

are just instantiations of a ﬁnite number of template subsystems, where each

group of components is isomorphic to a template. Such systems, often called

multi-agent discrete-event systems (DES), are used in situations where several

entities (e.g. robots, machines) perform the same type of jobs, while their num-

ber can vary in time. In multi-agent DES the agents (modeled as subsystems)

can be divided into several groups, and within each group the agents have similar

or identical state transition structures.

∗The work of Jan Komenda is supported by RVO 67985840 and GAˇ

CR grant 19-06175J.

†Yingying Liu is with Shanghai Jiao Tong University, Shanghai, 200240, China, Zhiwu Li is

with School of Electro-Mechanical Engineering, Xidian University, Xi’an, 710071, China, and

Jan Komenda is with Institute of Mathematics, Academy of Sciences of the Czech Republic,

Prague, Czech Republic.

arXiv:2103.10877v1 [eess.SY] 19 Mar 2021

The ﬁrst work about multi-agent DES is found in [11], where multi-agent

DES with a single group of isomorphic agents (i.e. single template) are consid-

ered. A more general multi-agent DES framework has been presented in [12],

where a broadcasting-based parallel composition rule is used to describe both

cooperative and competitive interactions of agents. In [5] the authors have stud-

ied modular discrete-event systems that are formed as synchronous products of

components that allow several isomorphic agents. The authors exploit the sym-

metry using state tree structures for achieving a larger computational beneﬁt.

In [13] all local requirements are also instantiated from a given requirement

template. A control protocol synthesis is investigated, which assumes that each

private alphabet is only observable to the corresponding agent, but the global

alphabet is accessible by all agents.

In [10] the case of several templates is studied under the assumption that

there are no shared events inside the groups (among isomorphic agents), but also

no shared events among diﬀerent groups (represented by templates). Moreover,

the method in [10] only deals with the complete observations case. Our goal is

to synthesize supervisors for templates that guarantee a speciﬁcation regardless

of the number of agents corresponding to templates (i.e. number of agents in

diﬀerent groups).

This work extends the results of [10] in several direction. Firstly, we gener-

alize the computation of scalable supervisors to the case of partial observation,

where observability of the speciﬁcation language is needed to synthesize the su-

pervisors. Secondly, we compare permissiveness of a monolithic supervisor with

a scalable, template based, supervisor under partial observation. We investigate

under which conditions a scalable least restrictive supervisor based on supremal

relatively observable sublanguage on the template level is not more restrictive

than the monolithic supervisor. Finally, we relax the assumption about ab-

sence of shared events and allow shared events among diﬀerent groups, i.e. the

templates are allowed to have shared events.

The paper is organized as follows. The next section contains preliminaries

about supervisory control and recalls basic notions and results used throughout

this paper. Section 3 is devoted to investigation of suﬃcient conditions for

existence of scalable safety supervisor based on the supervisor for the relabeled

system. In Section 4 we propose suﬃcient conditions for the scalable least

restrictive supervisor to be as permissive as the monolithic supervisor. Section

5 reaches conclusions.

2 Preliminaries

Let the DES plant to be controlled be modeled by a generator G= (Z, Σ, δ, z0),

where Σ is a ﬁnite event set, Zis the ﬁnite state set, z0∈Zthe initial state,

and δ:Z×Σ→Zthe (partial) transition function. Extend δin the usual

way such that δ:Z×Σ∗→Z. The closed behavior of Gis the language

L(G) := {s∈Σ∗|δ(z0, s)!}, where the notation δ(z0, s)! means that δ(z0, s) is

deﬁned. A string s1∈Σ∗is a preﬁx of another string s∈Σ∗, written s1≤s,

if there exists s2∈Σ∗such that s1s2=s. The length of string sis denoted by

|s|. We say that L(G) is preﬁx-closed if every for string s∈L(G) every preﬁx

s1≤sis also in L(G).

For partial observation, let the event set Σ be partitioned into Σo, the ob-

servable event subset, and Σuo, the unobservable subset (i.e. Σ = Σo˙

∪Σuo). A

(natural) projection P: Σ∗→Σ∗

ois deﬁned according to

P(ε) = ε, ε is the empty string;

P(σ) = (ε, if σ /∈Σo,

σ, if σ∈Σo;

P(sσ) = P(s)P(σ), s ∈Σ∗, σ ∈Σ.

In the usual way, Pis extended to P:P wr(Σ∗)→P wr (Σ∗

o), where P wr(·)

denotes powerset. The inverse image of P, denoted by P−1:P wr (Σ∗

o)→

P wr(Σ∗), is deﬁned as P−1(s) = {w∈Σ∗|P(w) = s}. The deﬁnitions

can naturally be extended to languages. The projection of a generator Gis a

generator P(G) whose behavior satisﬁes L(P(G)) = P(L(G)) and L(P(G)) =

P(L(G)). More details about partially observed DES can be found in [4].

Fixing a reference sublanguage C⊆L(G), we introduce relative observabil-

ity of language K. Let K⊆C⊆L.Kis C-observable with respect to Land

Σoif

(∀s, s0∈Σ∗)(∀σ∈Σ)(sσ ∈K&s0∈C&s0σ∈L(G)

&P(s) = P(s0)) ⇒s0σ∈K.

The following Lemma, transitivity of relative observability, is needed in the

proof of Theorem 2).

Lemma 1 Suppose K⊆N⊆L=L(G)⊆Σ∗and reference languages C⊆

C0. If Kis relatively observable with respect to Cand Nand Nis relatively

observable with respect to C0and L, then Kis relatively observable with respect

to Cand L.

Proof Let sb ∈K,s0∈C,P(s) = P(s0),b∈Σ, and s0b∈L. We need to show

that s0b∈K. Since K⊆N, i.e. K⊆Nas well, we have sb ∈N. Similarly,

C⊆C0implies C⊆C0. Thus, we have s0∈C0. Since P(s) = P(s0)and

s0b∈Lit follows from Nbeing relatively observable with respect to C0and L

that s0b∈N. Finally, we obtain from relative observability of Kwith respect to

Cand Nthat s0b∈K, which shows that is relatively observable with respect to

Cand L.

It is proved in [2] that relative observability is closed under arbitrary set

unions. Given a generator Gwith Σ = Σo˙

∪Σuo. Let E⊆Σ∗be a speciﬁcation

language for G. The family of all sublanguages of Ethat are E-observable with

Σ∗

T∗Σ∗

o

T∗

o

R

PR

P

R

Figure 1: Schematic of natural projection Pand relabeling map R

respect to Land Σois O(E, L) := {K⊆E|Kis E-observable wrt Land Σo}.

Then O(E, L) has a unique supremal element [2], i.e.

sup O(E, L) = ∪{K|K∈ O(E, L)}.

To describe the structure of a multi-agent plant G, we brieﬂy review a

concept, called relabeling map. We refer the reader to [10] and reference therein

for a more complete treatment for the relabeling map. Let Tbe a set of new

events, i.e. Σ ∩T=∅, and R: Σ →T. Deﬁne a relabeling map R: Σ →

Tsuch that is surjective but need not be injective. We require that events

σ1, σ2∈Σ with the same R-image, i.e. R(σ1) = R(σ2) = τ∈T, have similar

physical meaning and are just two instantiations of τin diﬀerent but isomorphic

subsystems. We extend Rby morphism to R: Σ∗→T∗according to

(i) R(ε) = ε,εis the empty string;

(ii) R(sσ) = R(s)R(σ), σ∈Σ, s∈Σ∗.

Note that R(s)6=εfor all s∈Σ∗\ {ε}. Further extend Rfor languages, i.e.

R:P wr(Σ∗)→P w r(T∗), and deﬁne R(L) = {R(s)∈T∗|s∈L}, L ⊆Σ∗.

The inverse-image function R−1of Ris given by R−1:P wr(T∗)→P w r(Σ∗):

R−1(H) = {s∈Σ∗|R(s)∈H},H⊆T∗. Note that RR−1(H) = H,H⊆

T∗; while R−1R(L)⊇L,L⊆Σ∗. We say that L⊆Σ∗is (G, R)-normal if

R−1R(L)∩L(G)⊆L.

We now discuss computation of Rand R−1by generators. Let

G= (Q, Σ, δ, q0) be a generator. First, relabel each transition of Gto obtain

GT= (Q, T, δT, q0), where δT:Q×T→Qis deﬁned by δT(q1, τ ) = q2iﬀ (∃σ∈

Σ)R(σ) = τ&δ(q1, σ) = q2.Hence L(GT) = R(L(G)). However, GTas

given above may be nondeterministic [14]. Thus apply subset construction [14]

to convert GTinto a deterministic generator H= (Z, T , ζ, z0), with L(H) =

L(GT).

We require that Rpreserves observability status of events in Σ. Thus, To:=

{R(σ)|σ∈Σo},Tuo := {R(σ)|σ∈Σuo}, and T=To˙

∪Tuo. Let P: Σ∗→Σ∗

o

and PR:T∗→T∗

obe natural projections, their relationships with Rare shown

in Fig. 1.

Large complex DES are built out of the small ones using concurrent compo-

sition known as synchronous product. Given languages Li⊆Σ∗

i, i = 1, . . . , n,

their synchronous product (parallel composition) is deﬁned by L1k. . . kLn=

Tn

i=1 P−1

i(Li)⊆Σ∗, where Pi: Σ∗→Σ∗

iare projections to local alphabets.

There are corresponding deﬁnitions in terms of generators, i.e. L(G1kG2) =

L(G1)kL(G2)[4].

Σ∗

T∗Σ∗

i

T∗

i

R

Pi|R

Pi

R

Figure 2: Schematic of local projection Pi(i= 1, . . . , l), Pi|R, and relabeling

map R

3 Scalable Supervisor under Partial Observation

Let R: Σ∗→T∗be a relabeling map, and the plant Gcan be divided into l(>

1) groups of component agents. In each group Gi(i∈ {1, . . . , l}) being a similar

set of generators under the given relabeling map R, i.e. Gi={Gi1,...,Gi ni}

(ni≥1) and there is a template generator Hisuch that

(∀j∈ {1, . . . , ni})R(Gij) = Hi.(1)

Let Gijbe deﬁned on Σij = Σij,o ˙

∪Σij,uo, templates Hion Ti=Ti,o ˙

∪Ti,uo,

and group languages Gion Σi=Sj=1,...,niΣij . Then R(Σij ) = Ti,R(Σij,o ) =

R(Σi,o) = Ti,o , and R(Σij,uo) = R(Σi,uo) = Ti,uo for all j∈ {1, ..., ni}. We

emphasize that the number niof generators (agents) in group iis not ﬁxed but

may vary in time. We denote the alphabet of Giby Σi, i.e. Σi=∪ni

j=1Σij . We

require that Rpreserves local status of events in Σ; namely R(σ) is an event in

Tiif and only if σ∈Σi. Thus Ti:= {R(σ)|σ∈Σi}and T=Si=l

i=1 Ti. Similarly,

Pi|R:T∗→T∗

i, for i= 1,2, are projections to local relabeled event sets. The

relationships of R, projections Pi, and Pi|Rare shown in Fig. 2.

Now we make the following assumptions.

(A1) The speciﬁcation language Eis preﬁx closed and only generated languages

are considered.

(A2) The speciﬁcation language Eis (G, R)-normal, i.e. R−1R(E)∩L(G)⊆E.

Due to (A1) all component agents are then automatically nonblocking. Note

that blocking issue can be handled in a similar way as in hierarchical control

and has been solved for relabeling in [6] by adapting an observer property (OP)

in so called relabeled OP (ROP), but in this paper we focus on another ex-

tension of the framework in [10], namely to partial observations and maximal

permissiveness.

Given plant Gand speciﬁcation E, the monolithic supervisor under partial

observation is computed based on supremal relatively observable sublanguages.

The whole plant Gis computed as synchronous product of all component agents:

G=||i=1,...,lGi,where Gi=||j=1,...,niGij.(2)

Recall that G=||i=1,...,lGi=Ti=1,...,l P−1

i(Gi) for Pi: Σ∗→Σ∗

i. Then

supremal relatively observable sublanguage is computed:

L(SUPo) = sup O(E∩L(G), L(G)).(3)

The supervisor SUPomust be recomputed or reconﬁgured in order to adapt

to the change of the number of agents (increases when more agents are added

into the system to enhance productivity or decreases when some agents mal-

function and are removed from the system). Therefore, in this paper we aim

to synthesize scalable supervisors under partial observation whose state size is

independent of the number of agents.

3.1 Scalable supervisory control with partial observation

In this subsection we design a scalable supervisor SSUPothat is independent

of the number niof agents for all i∈ {1, . . . , l}and satisﬁes {} ⊂ L(SSUPo)∩

L(G)⊆L(SUPo), while the opposite inclusion (maximal permissiveness) is

studied next.

Consider the plant Gas described in (2). Let Σ = Σo˙

∪Σuo) be the event

set of G, and R: Σ →Ta relabeling map. The procedure of designing a

scalable supervisor under partial observation is in steps (P1)-(P4), which ﬁrst

synthesizes a supervisor for ‘relabeled system’ under Rand then inverse-relabel

the supervisor.

(P1) First compute the relabeled plant. Let ki∈ {1,2, . . . , ni}be the number of

agents in group iallowed to work in parallel, and compute the template Mi:=

R(||j=1,...,kiGij). Then compute the relabeled plant Mas the synchronous

product of the template generators Mi, i.e.

M:= ||i=1,...,lMi.(4)

The event set of Mis T=To˙

∪Tuo, where To=R(Σo), and Tuo =R(Σuo ).

Note that kishould be much smaller than ni(the number of agents in group i) for

computational eﬃciency, it is trade-oﬀ between expressiveness and complexity

. When ki= 1, we have Mi=Higiven by (1). Note that once kiare ﬁxed, the

state sizes of Miand Mare ﬁxed as well, and thus independent of the number ni

of agents in group i. Recall that ki>1 is desirable for allowing agents to work in

parallel as R(Gi1kGi2) can be strictly larger than R(Gi1)kR(Gi2) = HikHi=

Hi, which distinguishes relabeling (surjective mask) from natural projection.

(P2) Compute relabeled speciﬁcation F:= R(E), where E⊆Σ∗is the speciﬁca-

tion imposed on G.

(P3) Synthesize a relabeled supervisor under partial observation RSUPo(a non-

blocking generator) such that

L(RSUPo) = sup O(F, L(M)) ⊆T∗.

(P4) Inverse-relabel RSUPoto derive scalable supervisor SSUPo, i.e.

SSUPo:= R−1(RSUPo) (5)

Notice that the computations involved in the above procedure are indepen-

dent of the number ni(i∈ {1, ..., l}) of agents. In (P1), once kiare ﬁxed, the

state sizes of Miand Mare ﬁxed and independent of the number niof agents in

group i(although dependent on ki). In (P3), the number of states of RSUPois

independent of the number of agents as the state size of Mis so. Finally in (P4),

inverse-relabeling does not change the number of states. Therefore SSUPohas

the same number of states as RSUPo. It then follows that the state size of

SSUPois independent of the number of agents in plant G.

Arguably, with large numbers of agents kiin the templates Miand for a

large number lof modules (groups) it may be computationally challenging to

compute the global template M. In such a situation we propose to combine

the results of this paper with coordination control approach, see e.g. [8], which

consists in conditionally decomposing the template speciﬁcation F, construct

the corresponding coordinator and compute local supervisors for local tem-

plates Micombined with the coordinator. More formally, instead of computing

sup CO(F, L(M)) we can ﬁrst ﬁnd a coordinator alphabet Σkcontaining at least

shared events Σs=∪i,i0=1,...,l, i6=i0Σi∩Σi0such that F=ki=1,...,lPi+k(F) is con-

ditionally decomposable with respect to “augmented” local alphabets Σi∪Σk,

where Pi+k: Σ∗→(Σi∪Σk)∗are the corresponding natural projection. Then

the underlying coordinator is Mk=ki=1,...,lPk(Mi), where Pk: Σ∗→Σ∗

kis

natural projection. Instead of computing supervisor for the whole template, i.e.

sup CO(F, L(M)), we can compute supervisors sup O(Pi+k(F), L(MikMk)). In

general we only have ki=1,...,l sup CO(Pi+k(F), L(MikMk)) ⊆sup CO(F, L(M)),

but under some conditions (e.g. mutual observability between coordinated tem-

plates or conditions used in hierarchical control with partial observation) the

equality holds.

Now we compare the designed scalable supervisor for multi-agent systems

with the monolithic one. We emphasize that unlike modular or hierarchical

control, where modular or abstracted safety supervisor is always included in

the monolithic one, the situation here is more complicated, because scalable

supervisor is computed with respect to the template and not with respect to

the relabeling of the plant R(L(G)), which would correspond to hierarchical

control with relabeling being an abstraction. This is because the relabeling

map does not distribute with the synchronous product. In multi-agent systems

we naturally use templates for control synthesis. Note that the inclusion

L(M) = kl

i=1L(Mi) = kl

i=1R(||j=1,...,kiGij)⊆R(L(G))

holds under the assumption that there are no shared events inside the group, and

allow to share events among templates. The above inclusion is typically strict

as the relabeling behaves diﬀerently than natural projection with respect to

the synchronous product. Hence, due to anti-monotonicity of supremal control

operators in the plant argument, the scalable supervisor computed with respect

to Mas a plant can be larger than the supervisor (computed using R(G).

Therefore the inclusion studied in the result below that establishes safety is also

non trivial.

Theorem 2 Suppose that (A1) and (A2) hold. If L(M)is relatively observable

with respect to R(E∩L(G)) and R(L(G)), then L(SSUPo)∩L(G)⊆L(SUPo).

Proof Since L(SUPo) = sup RO(E∩L(G), L(G)), it suﬃces to prove that

(i) L(SSUPo)∩L(G)⊆E∩L(G)and (ii) L(SSUPo)∩L(G)is relatively

observable with respect to E∩L(G)and L(G). For (i) we have

L(SSUPo)∩L(G) = R−1(L(RSUPo)) ∩L(G)

(by (P4))=R−1sup RO(R(E), L(M)) ∩L(G)⊆R−1R(E)∩L(G) = E∩L(G)

by assumption (A2).

For (ii), let sb ∈L(SSUPo)∩L(G),s0∈E∩L(G)⊆E∩L(G), and

s0b∈L(G). We need to show that s0b∈L(SSUPo)∩L(G). We have

sb ∈L(SSUPo)∩L(G) = R−1sup RO(R(E), L(M)) ∩L(G)thus R(sb)∈

sup RO(R(E), L(M)) ∩L(G). We also have R(s0)∈R(E), and R(s0b)∈

R(L(G)). Since R(E∩L(G)) ⊆R(E)and sup RO(R(E), L(M)) is relatively

observable with respect to R(E)and R(L(G)), we get that sup RO(R(E), L(M))

is relatively observable with respect to R(E∩L(G)) and L(M)[2]. By Lemma 1,

combining it with relative observability of L(M)with respect to R(E∩L(G))

and R(L(G)), it implies that sup RO(R(E), L(M)) is relatively observable with

respect to R(E∩L(G)) and R(L(G)). We have R(s0b)∈sup RO(R(E), L(M)).

Therefore, s0b∈R−1R(s0b)⊆R−1sup RO(R(E), L(M)) ∩L(G) = L(SSUPo)∩

L(G) = L(SSUPo)∩L(G), which proves (ii).

Note that in this paper we only discuss the observability problem, since the

controllability of the scalable supervisor has been well studied in [9]. Theorem 2

provides a suﬃcient condition under which the scalable supervisor is always

included in the monolithic one. This condition is L(M) is relatively observable

with respect to R(E∩L(G)) and R(L(G)). As shown above, this condition

is essential in proving relative observability of L(SSUPo)∩L(G) with respect

to E∩L(G) and L(G). However, the computation of Gis required in this

condition, which needs to be avoided. We will make an additional assumption

and give the following result.

(SEF) We assume that the event sets of systems inside each group are pair-

wise disjoint, i.e. for all i∈ {1, . . . , l}and j, j0∈ {1, . . . , kl}, Σij ∩Σij 0=∅.

Note that unlike [10] we allow shared events between templates.

Proposition 3 Let (SEF) hold. If for each group i∈ {1, . . . , l}and Gi1,Gi2∈

Gi,L(Hi)is relatively observable with respect to R(Pi(E)∩L(Gi1kGi2)) and

R(L(Gi1kGi2)), then L(M)is relatively observable with respect to R(E∩L(G)

and R(L(G)).

The proof of Proposition 3 is referred to Section 5. Proposition 3 indicates that

relative observability of L(M) with respect to R(E∩L(G)) and R(L(G)) can be

veriﬁed by checking relative observability of L(Hi) for each group with respect to

only two (arbitrarily chosen) component agents. Therefore, the computational

eﬀort of checking the suﬃcient condition in Theorem 2 is low. We recall that

algorithms for checking relative observability and computing supremal relatively

observable sublanguages are proposed in [3] and [1]. An illustrative example is

given below.

Example 4 Consider a small factory consisting of n1input machines

G11,...,G1n1and n2output machines G21,...,G2n2, linked by a buﬀer in the

middle. The generators of the agents are shown in Fig. 3. Based on their

diﬀerent roles, the machines are divided into 2 groups: G1={G11,...,G1n1}

and G2={G21,...,G2n2}. Let the relabeling map Rbe given by

R(1i0) = 10, R(1i1) = 11, R(1i2) = 12, R(1i3) = 13,

R(2j0) = 20, R(2j1) = 21, R(2j2) = 22, R(2j3) = 23

with template generators H1and H2respectively, where

Σ = Σo˙

∪Σuo ={1i0,1i1,1i3,2j0,2j1,2j3}˙

∪{1i2,2j2},i∈[1, n1], and j∈

[1, n2].

Since the event sets of agents in the same group are pairwise disjoint (Σ1∩

Σ2={1i0,1i1,1i2,1i3}∩{2i0,2i1,2i2,2i3}=∅), Assumptions (A1) and (SEF)

hold for this example. The speciﬁcation is to avoid underﬂow and overﬂow

of buﬀer with two capacities, which is enforced by a generator Eand shown

in Fig. 4 (lower part). It can be veriﬁed that E=R−1R(E), we thus have

Assumption (A2) holds. Now we design templates for each group. Let k1=

2, k2= 1. The templates are L(M1) := L(R(G11kG12)) (for the input group)

and L(M2) := L(R(G21)) (for the output group). Note that in this example we

have L(M1) = R(L(G11kG12))) is diﬀerent from L(H1) = R(L(G11)), while

L(M2) = L(H2).

Now we need to check the suﬃcient condition of Theorem 2; namely, L(M)

is relatively observable with respect to R(E∩L(G)) and R(L(G)). If this con-

dition holds, then we can employ procedure (P1)-(P4) proposed in Section 3.1

to compute the scalable supervisor. By Proposition 3, we need to check if for

i= 1,2,L(Hi)is relatively observable with respect to R(Pi(E)∩L(Gi1kGi2))

and R(L(Gi1kGi2)). We compute supCO(R(Pi(E)∩L(Gi1kGi2)),L(Hi)and

get that L(Hi)is relatively observable with respect to R(Pi(E)∩L(Gi1kGi2))

and R(L(Gi1kGi2)). We also get that R(Pi(E))∩R(L(Gi1kGi2)) = R(Pi(E)) ∩

R(L(Gi1kGi2)). Hence by Proposition 3, we have that L(M)is relatively ob-

servable with respect to R(E∩L(G)) and R(L(G)). Therefore the suﬃcient

condition of Theorem 2 is satisﬁed.

In procedure (P1)-(P4), we design a scalable supervisor SSUPo, displayed in

Fig. 5. It can be veriﬁed that the scalable supervisor is included in the supremal

monolithic one, i.e. L(SSUPo)∩L(G)⊆L(SUPo).

4 Maximal permissiveness of the scalable super-

visor

In the last section, we provid the conditions that the obtained relatively ob-

servable sublanguage is a sublanguage of the supremal relatively observable

sublanguage, i.e. L(SSUPo)∩L(G)⊆L(SUPo). In this suction, we will pro-

pose conditions for the opposite inclusion, i.e. the maximal permissiveness of

the scalable supervisor. To give our main result, the following conditions are

needed.

ij1

ij0

Hi(i= 1,2)

ij2

ij3

i1

i0

i2

i3

G11

BUFFER

111

110

G1n1

1n11

1n10

G1j(j= [1, n1])

G2j(j= [1, n2])

ij1

ij0

ij2

ij3

G21

G2n2

211

2n21

210

2n20

Figure 3: Small factory: system conﬁguration and component agents. Events

1j1 (j∈[1, . . . , n1]) and 1j2 (j∈[1, . . . , n2]) mean that machine Gijstarts to

work by taking in a workpiece; events 1j0 and 2j0 mean that Gijﬁnishes work

and outputs a workpiece; events 1j2 and 2j2 mean that Gijis broken down;

events 1j3 and 2j3 mean that Gijis repaired. Convention: the initial state of a

generator is labeled by a circle with an entering arrow. The same notation will

be used in subsequent ﬁgures.

110,...,1n10

211,...,2n21

110,...,1n10

211,...,2n21

E(= R−1R(E))

*

∗= 1i1,1i2, , 1i3,2j0,2j1,2j3i∈[1, n1]; j∈[1, n2]

*

*

Figure 4: Small factory: templates and speciﬁcation generator

0

1

2

5

8

13

18

9

28

24

17

15

22

27

21

3

23

19

26

10

14

16

25

29

7

12

20

4

6

11

a

b

e

h

d

e

h

c

b

d

b

e

h

d

f

a

c

h

e

d

b

h

e

e

h

h

e

a

d

b

g

h

e

b

c

h

e

f

h

f

e

c

g

a

d

a

b

d

h

e

d

g

b

d

g

b

a

g

c

b

c

d

b

g

c

a

d

b

c

g

a

b

d

a

c

g

g

a

c

g

c

a

a= 110,...,1n10

b= 111,...,1n11

c= 112,...,1n12

d= 113,...,1n13

e= 210,...,2n20

f= 211,...,2n21

g= 212,...,2n22

h= 213,...,2n23

Figure 5: Small factory: SSUPo

∀s∈LP(s) = P(s0)∃s0∈L

Rs PR(Rs) = PR(t0)∀t0∈RL

P P

R

PRPR

Figure 6: Illustration of ROC condition

Deﬁnition 5 ( Relabeling observational consistency) A relabeling map R

is said to be Relabeling observation consistent (ROC) with respect to plant lan-

guage L=L(G)⊆Σ∗and natural projection Pif for all strings s∈Land

t0∈R(L)such that PR(R(s)) = PR(t0), there exists a string s0∈Lsuch that

R(s0) = t0and P(s) = P(s0).

As we can see from ﬁgure 6, ROC condition of a language requires that for every

string sin the language and every string t0in the relabeling of this language

that looks the same as the relabeling of s, there exists another string s0in the

language that looks the same as sand its relabeling equals t0. We will see in

section 5 that ROC always holds if there are no shared events between diﬀerent

agents in the same group.

Deﬁnition 6 (Local relabeling observer consistency) A relabeling map R

is said to be Locally relabeling observation consistent (LROC) with respect to

plant language L=L(G)⊆Σ∗and natural projection Pif for all strings

s, s0∈Lwith P(s) = P(s0)and unobservable events b, b0∈Σuo such that

R(b) = R(b0)we have

(sb ∈L∧s0b0∈L)⇒s0b∈L.

The following result formulates suﬃcient conditions for maximal permissive-

ness of the scalable supervisor.

Theorem 7 Let L=L(G)be ROC and LROC with respect to relabeling map

Rand natural projection Pand let speciﬁcation E⊆Σ∗be preﬁx-closed and

(G, R)-normal and let L(M)⊆RL(G). Then L(SUPo)⊆L(SSUPo)∩L(G)

Proof We need to show that

sup RO(E∩L(G), L(G)) ⊆R−1sup RO(R(E), L(M)) ∩L(G).

Since by deﬁnition sup RO(E∩L(G), L(G)) ⊆E∩L(G), it suﬃces to show

that R(sup RO(E∩L(G), L(G))) ⊆sup RO(R(E), L(M)). Note that due to

the assumption L(M)⊆RL(G)every language N⊆T∗that is relatively

observable wrt R(E)and RL(G)is also relatively observable wrt R(E)and

L(M)). Therefore, sup RO(R(E), RL(G)) ⊆sup RO(R(E), L(M)), a standard

antimonotonicity property of supremal sublanguages of supervisory control in

the plant argument. Thus, we only need to show R(sup RO(E∩L(G), L(G))) ⊆

sup RO(R(E), RL(G)). The latter inclusion then amounts to show that

(i) R(sup RO(E∩L(G), L(G))) ⊆R(E), which is obvious, and

(ii) R(sup RO(E∩L(G), L(G))) is relatively observable wrt R(E)and RL(G).

To show (ii), let t, t0∈T∗be such that PR(t) = PR(t0), and let a∈Tbe such

that ta ∈Rsup RO(E∩L(G), L(G)),t0∈R(E), and t0a∈RL(G). We have to

show that t0a∈R(sup RO(E∩L(G), L(G))).

From ta ∈R(sup RO(E∩L(G), L(G))) we have that there exists

sb ∈sup RO(E∩L(G), L(G)) such that R(sb) = ta. Since t0a∈RL(G)and

PR(R(sb)) = PR(t0a), ROC (applied to sb playing the role of sand t0aplaying

the role of t0) implies that there is a v0∈L(G)such that R(v0) = t0aand P(v0) =

P(sb). Then v0=s0b0for some s0∈L(G)and b0∈Σsuch that R(b0) = aand

P(b0) = P(b). This means that if b0∈Σothen b0=b. Note that R(s0) = t0∈

R(E), i.e. by normality assumption (A2) we have s0∈R−1R(E)∩L(G) = E.

Hence, in case b=b0∈Σow have that sb ∈sup RO(E∩L(G), L(G)),s0∈

E∩L(G),s0b∈L(G)), and P(s0) = P(s). Thus, from relative observability of

sb ∈sup RO(E∩L(G), L(G)) with respect to E∩L(G)and L(G)we obtain s0b∈

sup RO(E∩L(G), L(G)), whence t0a=R(s0b)∈R(sup RO(E∩L(G), L(G))).

Now, if both b, b0∈Σuo are unobservable, then we have s, s0∈L(G)∧P(s0) =

P(s)∧sb ∈L(G)∧s0b0∈L(G)∧R(b) = R(b0). From LROC we get s0b∈L(G)

as well. Thus, we can continue in the same way as with b, b0both observable.

Note that the condition L(M)⊆RL(G) in Theorem 7 always holds under

the assumption (SEF), which is shown by the following lemmas.

Lemma 8 Let (SEF) hold. For any string s∈Σ∗

i, natural projections Pi:

Σ∗→Σ∗

i,PR,i :T∗→T∗

i, and the relabeling map R: Σ∗→T∗, we have

RP −1

i(s) = P−1

R,i R(s).

Proof By induction, intuitively RP −1

i() = P−1

R,i R() = . Suppose that RP −1

i(s) =

P−1

R,i R(s)holds. Then for any σ∈Σi, we need to show RP −1

i(sσ) = P−1

R,i R(sσ).

Since RP −1

i(sσ) = R(P−1

i(s)P−1

i(σ)) = RP −1

i(s)RP −1

i(σ)and P−1

R,i R(sσ) =

P−1

R,i (R(s)R(σ)) = P−1

R,i R(s)P−1

R,i R(σ). By recalling RP −1

i(s) = P−1

R,i R(s), we

only need to show RP −1

i(σ) = P−1

R,i R(σ). Since Pi: Σ∗→Σ∗

iand σ∈Σi, we

have that RP −1

i(σ) = R((Σ \Σi)∗σ(Σ \Σi)∗) = (T\Ti)∗R(σ)(T\Ti)∗. Sim-

ilarly, PR,i :T∗→T∗

i, so we get P−1

R,i R(σ) = (T\Ti)∗R(σ)(T\Ti)∗. Thus,

RP −1

i(s) = P−1

R,i R(s).

Lemma 9 L(M)⊆RL(G)always holds under the assumption (SEF).

Proof For any string t∈L(M)⊆T∗we need to show t∈RL(G). By

t∈L(M) = kl

i=1L(Mi) = Tl

i=1 P−1

R,i (L(Mi)), we have PR,i (t)∈L(Mi) =

R(kki

j=1L(Gij )). Then there exist strings si∈Σ∗

isuch that si∈ kki

j=1L(Gij )and

PR,i(t) = R(si). Then we get that kl

i=1si∈ kl

i=1kki

j=1L(Gij ). Since kl

i=1si=

Tl

i=1 P−1

i(si), we obtain Tl

i=1 P−1

i(si)∈ kl

i=1kki

j=1L(Gij ). We apply Ron

both sides, so we get that R(Tl

i=1 P−1

i(si)) ∈R(kl

i=1kki

j=1L(Gij )). It follows

by lemma 8 that

R(

l

\

i=1

P−1

i(si)) =

l

\

i=1

R(P−1

i(si)) =

l

\

i=1

P−1

R,i R(si)∈R(kl

i=1kki

j=1L(Gij )).

We know that Tl

i=1 P−1

R,i R(si) = kl

i=1R(si) = kl

i=1PR,i (t)⊇tby R(si) = PR,i(t).

We recall that R(kl

i=1kki

j=1L(Gij )) ⊆R(kl

i=1kni

j=1L(Gij )) = R(L(G)) always

holds under assumption (SEF). We thus get that s∈R(L(G)). Therefore,

L(M)⊆RL(G)is proved under the assumption (SEF).

4.1 observability between the template level and the orig-

inal system level

For scalability reasons the controller synthesis should be done only at the tem-

plate level, in this subsection we will study the preservation of the property that

is essential for computing supervisors under partial observation, i.e. observabil-

ity between the template level and the original system level. Before giving

our result, we need the following Lemma known as transitivity of observability,

which will be used to proof our main result (Theorem 11).

Lemma 10 Suppose K⊆N⊆L=L(G)⊆Σ∗. If Kis observable with respect

to Nand Nis observable with respect to L, then Kis observable with respect

to L.

Proof Let s, s0∈K,P(s) = P(s0),b∈Σ,sb ∈K, and s0b∈L. We need to

show that s0b∈K. Since K⊆N, i.e. K⊆Nas well, we have s, s0∈N. It

now follows form the observability of Nwith respect to Lthat s0b∈N. Finally,

it is obtained from the observability of Kwith respect to Nthat s0b∈K, which

shows that Kis observable with respect to L.

Now the result about preserving observability is ready to be stated. We assume

that the speciﬁcation Eand the plant L(G) are non conﬂicting, i.e. E∩L(G) =

E∩L(G).

Theorem 11 Let Lbe a generator language over an event set Σwith relabeling

R(L)over an event set Tand let E⊆Σ∗be a (G, R)-normal speciﬁcation and

L(M)is observable with respect to R(L(G)) and PR. If R(E)is observable with

respect to L(M)and PR, then E∩L(G)is observable with respect to L(G)and

P.

Proof Let R(E)be observable with respect to L(M)and PR. It will be shown

that E∩L(G)is observable with respect to L(G)and P. Assume that s, s0∈

E∩L(G), for some b∈Σ, such that sb ∈E∩L(G),s0b∈L(G), and P(s) =

P(s0). We have to show that s0b∈E∩L(G). We have then R(s), R(s0)∈

R(E∩L(G)) = R(E∩L(G)) ⊆R(E),R(P(s)) = R(P(s0)). Let us denote

R(b) = a∈T, then R(s)a∈R(E∩L(G)) ⊆R(E)and R(s0)a∈R(L(G)).

Since Rpreserves observability status of events in Σ, we have PR(R(s)) =

PR(R(s0)).

From Lemma 10 it follows that R(E)is observable with respect to R(L(G))

and PR. Using this observability we conclude that R(s0)a∈R(E). It implies

that s0b∈R−1R(s0)a⊆R−1R(E) = R−1R(E). We thus have s0b∈R−1R(E)∩

L(G) = R−1R(E)∩L(G) = E∩L(G)by (G, R)-normality of Eassumption.

Now the observability of Ewith respect to L(G)is proved.

5 Eﬃcient Veriﬁcation of Suﬃcient Conditions

in Theorem 2 and Theorem 7

. In this section we use assumption(SEF), i.e. shared events between diﬀerent

agents in the same group are excluded. We ﬁrst address the veriﬁcation of the

suﬃcient condition used in Theorem 2. As shown in section 3.1, this condition

requires computation of Gwhich is dependent on the number of agents. Thus

Proposition 3 is proposed to avoid computing G. To prove Proposition 3 we

need the following lemmas and propositions.

Lemma 12 For each group i∈ {1, . . . , l}and arbitrary Gi1,Gi2,∈ Gi, if L(Hi)

is relatively observable with respect to R(Pi(E)∩L(Gi1kGi2)) and

R(L(Gi1kGi2)), then L(Hi)is relatively observable with respect to R(Pi(E)∩

L(Gi1kGi2kGi3)) and R(L(Gi1kGi2)).

Proof It follows directly from deﬁnition of relative observability.

In some proof below, we employ the same state transition structure of iso-

morphic generators to ﬁnd a new string t0that has a similar property with t,

but with one less agent. For all agents in the same group we have the following

”go down” property, an illustration is below the remark.

Remark 13 Consider arbitrary `agents Gi1,Gi2,...,Gi`∈ Gi. If we have a

string t∈R(L(k`

j=1Gij)) and t /∈R(L(k`−1

j=1Gij)) for `∈ {1, . . . , ni}, then there

exists a string s∈Σ∗such that s∈L(k`

j=1Gij)and for all ˜swith R(˜s) = t

including sitself we have ˜s /∈L(k`−1

j=1Gij). Let s0=Pi`(s)for Pi` : Σ∗

i→

(Σi\Σi`)∗. We thus have s0∈L(kj=`−1

j=1 Gij)and s0/∈L(kj=`−2

j=1 Gij). The latter

is obtained from that if s0∈L(k`−2

j=1Gij), then we have s∈L(k`−1

j=1Gij)which

is conﬂict with t /∈R(L(k`−1

j=1Gij)). Then we denote a string t0∈T∗

isuch that

R(s0) = t0. It implies that t0∈R(L(k`−1

j=1Gij)) and t0/∈R(L(k`−2

j=1Gij)). If

there exist strings s00 with R(s00) = t0such that s00 ∈L(k`−2

j=1Gij), which implies

that t0∈R(L(kj=`−2

j=1 Gij)). Then from s0/∈L(k`−2

j=1Gij)we see that s00 6=s0=

Pi`(s). Therefore, for s0=Pi`(s)we have R(s0) = t0∈R(L(k`−1

j=1Gij)) and

t0/∈R(L(k`−2

j=1Gij)).

For the small factory example in Fig. 3. If we take t= 11.10.11.10.11.10 =

R(111.110.121.120.131.130), then we have t∈R(L(Gi1kGi2kGi3)),

t /∈R(L(Gi1kGi2)), and string s= 111.110.121.120.131.130 with R(s) = t. Let

s0=Pi3(s) = 111.110.121.120. Then we have t0=R(s0) = R(111.110.121.120) =

11.10.11.10 for Pi3: Σ∗

i→(Σi\Σi3)∗. It can be veriﬁed that t0∈R(L(Gi1kGi2))

and t0/∈R(L(Gi1)). Now we proceed in a similar way with increasing the plant

component.

Lemma 14 For each group i∈ {1, . . . , l}and Gi1,Gi2,Gi3∈ Gi, if L(Hi)

is relatively observable with respect to R(Pi(E)∩L(Gi)) and R(L(Gi1kGi2))

for Pi: Σ∗→Σ∗

i, then R(L(Gi1kGi2)) is relatively observable with respect to

R(Pi(E)∩L(Gi)) and R(L(Gi1kGi2kGi3)).

The following claim is needed.

Proposition 15 For each group i∈ {1, . . . , l}and Gi1,Gi2∈ Gi, if L(Hi)is

relatively observable with respect to R(Pi(E)∩L(Gi)) and R(L(Gi1kGi2)) for

Pi: Σ∗→Σ∗

i, then L(Mi)is relatively observable with respect to R(Pi(E)∩

L(Gi)) and L(Gi).

Proof Extending Lemma 14 inductively, it is derived that if L(Hi)(i∈ {1, . . . , l})

is relatively observable with respect to R(Pi(E)∩L(Gi)) and R(L(Gi1kGi2)),

then L(Mi)is relatively observable with respect to R(Pi(E)∩L(Gi)) and

R(L(||j∈{1,...,ki+1}Gij)). Again by applying the transitivity lemma (Lemma 1)

inductively, we have L(Mi)is relatively observable with respect to R(Pi(E)∩

L(Gi)) and L(Gi).

Now we are ready to prove Proposition 3.

Proof Let t, t0∈T∗,a∈T,ta ∈L(M),PR(t) = PR(t0),t0∈R(Pi(E)∩L(G)),

and t0a∈R(L(G)). We will show that t0a∈L(M). From ta ∈L(M)we derive

ta ∈L(M) = L(||i=1,...,lMi) = \

i=1,...,l

P−1

i|RL(Mi),

where Pi|R:T∗→T∗

i.We thus get that Pi|R(ta)∈L(Mi). Since t0∈R(E∩

L(G)), there exists a string s0∈Σ∗such that s0∈E∩L(G)and R(s0) = t0. By

L(G) = L(||i=1,...,lGi) = Ti=1,...,l P−1

i(L(Gi)) with Pi: Σ∗→Σ∗

i, we have

s0∈E∩\

i=1,...,l

P−1

i(L(Gi)) ⊆E∩P−1

i(L(Gi)).

Then we get that Pi(s0)∈Pi(E)∩L(Gi)which implies that R(Pi(s0)) = Pi|R(R(s0)) =

Pi|R(t0)∈R(Pi(E)∩L(Gi)). Recall that t0a∈R(L(G)), then there exists an

event b0∈Σsuch that R(b0) = aand

s0b0∈ kl

i=1L(Gi) =

l

\

i=1

P−1

i(L(Gi)).

We have s0b0∈P−1

i(L(Gi)), i.e. Pi(s0b0)∈L(Gi). Then R(Pi(s0b0)) =

Pi|R(R(s0b0)) = Pi|R(t0a)∈R(L(Gi)). It follows form PR(t) = PR(t0)that

PR(Pi|R(t)) = PR(Pi|R(t0)). By Proposition 15 it directly follows that if L(Hi)

(i∈ {1, . . . , l}) is relatively observable with respect to R(Pi(E)∩L(Gi1kGi2))

and R(L(Gi1kGi2)), then L(Mi)is relatively observable with respect to

R(Pi(E)∩L(Gi)) and R(L(Gi)). Therefore, Pi|R(t0a)∈L(Mi), i.e. t0a∈

P−1

i|RL((Mi)). It is derived that t0a∈ ||l

i=1L(Mi) = L(M).

Next we will show that the conditions in Theorem 7 can be checked with

low computation eﬀort.

Under shared event free (SEF) assumption between diﬀerent agents in the

same group relabeled observation consistency (ROC) condition in Theorem 7

always holds true. We need the following lemma that is used in the proof of

Proposition . It is technical and states that it is possible to replace a string s0

by a similar string w0with the same projection as a given string s.

Lemma 16 Consider Ggiven in (2). Consider strings s= ˜sσb ∈L(G)and

s0= ˜s0σ0b0∈L(G)with σ, σ0∈Σ∗

uo such that P(˜s) = P(˜s0)and R(b) = R(b0)

Then there exists a string w0∈L(G)such that P(w0) = P(s)and R(w0) =

R(s0).

Proof First of all, from R(b) = R(b0)it follows that b, b0are either both ob-

servable or both unobservable. Since P(s) = P(˜sσb) = P(˜s)P(b)and P(s0) =

P(˜s0σ0b0) = P(˜s0)P(b0), we have that b, b0∈Σuo implies P(s) = P(s0). Then we

can take w0=s0. Similarly, if b, b0∈Σoand b=b0we also get P(s) = P(s0).

Assume that b6=b0and b, b0∈Σo. Then P(s)6=P(s0).

Let ˜s0=α1β1α2β2, . . . , αnβnfor α1, . . . , αn∈Σuo and β1, . . . , βn∈Σo. Now

we consider a set

W={w∈R−1R(α1)β1R−1R(α2)β2, . . . , R−1R(αn)βn}.

By similar structure of agents in groups we have from s0= ˜s0σ0b0∈L(G)that

W R−1R(σ0)b∩L(G)6=∅. Thus, there exists a string w0∈WR −1R(σ0)b∩L(G),

i.e. w0= ˜w0˜σ0b∈L(G)with ˜w0∈W. We then have: P( ˜w0) = P(˜s0) =

β1β2, . . . , βnand R( ˜w0) = R(˜s0) = R(α1)R(β1)R(α2)R(β2), . . . , R(αn)R(βn).

Note that ˜σ0∈Σ∗

uo, because ˜σ0∈R−1R(σ0). Hence, we have P(w0) = P(s)and

due to R(b) = R(b0)we also have R(w0) = R(s0).

Now we are ready to state the result concerning ROC.

Proposition 17 Consider Ggiven in (2). Under (SEF) assumption ROC con-

dition always holds.

Proof The proof goes by structural induction with respect to the observable

string PR(Rs) = PR(t0)∈T∗

o, where s∈L(G) = Land t0∈R(L). The base

step is proven below. If PR(Rs) = PR(t0) = εthen we need to show that there

exists a string s0∈Lsuch that R(s0) = t0and P(s) = P(s0). Since R(Σo) = To

we have from t0=R(s0)that s0∈Σ∗

uo. Similarly, from PR(Rs) = ε, i.e.

Rs ∈T∗

uo we have that s∈Σ∗

uo, i.e. we always have that P(s) = P(s0). Thus, it

suﬃces to take any string s0∈Lsuch that t0=R(s0)that exists from t0∈R(L).

The induction hypothesis consists in assuming that the ROC condition holds

for all ˜s∈Land ˜

t0∈R(L)such that PR(R(˜s)) = PR(t0) = w∈T∗

o. In

the induction step we will show that ROC condition also holds for s∈Land

t0∈R(L)with PR(Rs) = PR(t0) = wa ∈T∗

o, namely that there exists a string

s0∈Lsuch that R(s0) = t0and P(s) = P(s0).

Note that every t0∈R(L)with PR(t0) = wa is of the form t0=˜

t0τ0afor some

˜

t0∈R(L)with PR(˜

t0) = wand τ0∈T∗

uo. Similarly, by denoting t=Rs, we

have t∈R(L)with PR(t) = wa, i.e. tis of the form t=˜

tτa for some ˜

t∈R(L)

with PR(˜

t) = wand τ∈T∗

uo. Therefore, scan be decomposed as s= ˜sσb, where

R(˜s) = ˜

t,R(σ) = τ, and R(b) = a. We recall that PR(˜

t0) = w=PR(R(˜s)).

Therefore, from the induction hypothesis we know that there exists a string ˜s0∈

Lsuch that R(˜s0) = ˜

t0, and P(˜s) = P(˜s0).

We will ﬁrst show existence of a string s0∈Lsuch that R(s0) = t0and

P(s) = P(s0). We will search s0of the form s0∈˜s0R−1(τ0)R−1(a)so that

R(s0) = t0holds. At the end of the proof we will replace string s0by w0based on

Lemma 16.

We will show that there exists an event b0∈Σoand a string σ0∈Σ∗

uo such

that for s0= ˜s0σ0b0we have R(s0) = t0and P(s) = P(s0).

From R(˜s) = ˜

tand R(˜s0) = ˜

t0with PR(˜

t) = PR(˜

t0)we obtain that R(PR(˜s)) =

PR(R(˜s)) = PR(R(˜s0)) = R(PR(˜s0)).

Let us consider a candidate string s0= ˜s0σ0b0∈Lwith R(s0) = t0=˜

t0τ0a.

We will show that if P(s0)6=P(s)then there exists a w0∈Σ∗with R(w0) =

R(s0)and P(s) = P(w0). Since R(s) = tand R(s0) = t0, we must have that

R(σb) = τ a and R(σ0b0) = τ0a. Since P(˜s) = P(˜s0), we must have P(b0) =

P(σ0b0)6=P(σb) = P(b).Therefore, P(b)6=P(b0). By Lemma 16 from R(b) =

R(b0)we obtain that by replacing s0with w0, i.e. by taking w0= ˜s0˜σ0bwith

P( ˜w0) = P(˜s0)and R( ˜w0˜σ0) = R(˜s0σ0) = t0we have w0∈Lwith P(w0) = P(s)

and R(w0) = R(s0) = t0. We then choose this w0as the new s0that satisﬁes the

conditions.

Finally, we will show how to eﬃciently check LROC condition in Theorem 7.

We notice that due to R(b0) = R(b) we necessarily have that band b0belong

to the same group of isomorphic agents, say b, b0∈Σi=Sj=ni

j=1 Σij . If band b0

belong to the same agent then b=b0, in which case LROC is trivially satisﬁed.

Let us assume b∈Σij and b0∈Σij0for some j6=j0. Let s, s0∈Lwith P(s) =

P(s0) and b, b0∈Σuo such that R(b) = R(b0) then sb ∈Lmeans Pij(s)b∈Lij

and s0b0∈Lmeans Pij0(s0)b0∈Lij0Veriﬁcation of LROC condition then consists

in checking s0b∈L, i.e. Pij(s0)b∈Lij . This suggests that LROC condition is

similar to observability. Indeed, observability of P−1

ij Lij and P−1

ij0Lij0means

that for all s, s0∈P−1

ij (Lij ) such that P(s) = P(s0), sa ∈P−1

ij (Lij ) and s0a∈

P−1

ij0(Lij0) observability means that s0a∈P−1

ij (Lij ).

Note that s, s0∈Limply that in particular s, s0∈P−1

ij (Lij ). Pij (s)b∈Lij

mean that sb ∈P−1

ij (Lij ), because Pij (sb) = Pij (s)b∈Lij . Similarly, s0b0∈

L, i.e. Pij 0(s0)b0∈Lij0implies that s0b0∈P−1

ij0(Lij0), because Pij 0(s0b0) =

Pij0(s0)b0∈Lij0. Finally, checking LROC consists in checking if Pij (s0)b∈Lij ,

which is equivalent to s0b∈P−1

ij (Lij ) using the same argument (namely Pij(b) =

b∈Σij ).

It follows from the above analysis that LROC can be checked in the same way

as observability of P−1

ij Lij with respect to P−1

ij0Lij0, where string sis extended

by b, but we allow instead of the same bthe event s0to be extended by a

diﬀerent event b0with the same relabeling. Moreover, LROC can be viewed

as relabeling counterpart of a similar condition from hierarchical supervisory

control with partial observations, called local observational consistency (LOC),

that was shown checkable in [7].

6 Conclusion

We have studied multi-agent DES with partial observation, where the agents

can be divided into several groups, and within each group the agents have simi-

lar state transition structures and can be relabeled into the same template. We

have designed a scalable supervisor under partial observation whose state size

and computational cost are independent of the number of agents. We have com-

pared permissiveness of the scalable supervisor with the monolithic supervisor,

and have proposed suﬃcient conditions, which guarantee that our scalable least

restrictive supervisor is not more restrictive than the monolithic one. More-

over, we have proved that all suﬃcient conditions proposed in this paper can

be veriﬁed with low computational eﬀort. In a future work we will integrate

these partial observation results with already existing results on complete ob-

servations and on nonblockingness. Note that this paper is based on relabeling

based abstraction of modular (multi-agent) DES and relabeling is a special case

of mask type abstraction as well as natural projection is another special case

of abstraction. By integrating the results we can obtain results for hierarchical

control under general, mask based, abstraction that can both rename and delete

events.

References

[1] M. Alves, L. Carvalho, and J. Basilio. New algorithms for veriﬁcation of rela-

tive observability and computation of supremal relatively observable sublan-

guage. IEEE Transactions on Automatic Control, 62(11):5902–5908, 2017.

[2] K. Cai, R. Zhang, and W. M. Wonham. Relative observability of discrete-

event systems and its supremal sublanguages. IEEE Transactions on Auto-

matic Control, 60(3):659–670, 2015.

[3] K. Cai, R. Zhang, and W. M. Wonham. Characterizations and eﬀective

computation of supremal relatively observable sublanguages. Discrete Event

Dynamic Systems, 43:269–287, 2018.

[4] C. Cassandras and S. Lafortune. Introduction to discrete event systems.

Springer, 2008.

[5] T. Jiao, Y. Gan, G. Xiao, and W. M. Wonham. Exploiting symmetry of

state tree structures for discrete-event systems with parallel components.

International Journal of Control, 90(8):1639–1651, 2017.

[6] T. Jiao, Y. Gan, G. Xiao, and W. M. Wonham. Exploiting symmetry of

discrete-event systems by relabeling and reconﬁguration. IEEE Transactions

on Systems, Man, and Cybernetics: Systems, 50(6):2056–2067, 2020.

[7] J. Komenda and T. Masopust. Conditions for hierarchical supervisory con-

trol under partial observation. In 15th International Workshop on Discrete

Event Systems (WODES), pages 75–82, 2020.

[8] J. Komenda, T. Masopust, and J. H. van Schuppen. Coordination control of

discrete-event systems revisited. Discrete Event Dynamic Systems, 25:65–94,

2015.

[9] Y. Liu, K. Cai, and Z. Li. On scalable supervisory control of multi-agent

discrete-event systems. In Workshop on Discrete-Event System, pages 25–30,

Italy, 2018.

[10] Yingying Liu, Kai Cai, and Zhiwu Li. On scalable supervisory control of

multi-agent discrete-event systems. Automatica, 108, 2019.

[11] K. Rohloﬀ and S. Lafortune. The veriﬁcation and control of interacting

similar discrete-event systems. SIAM Journal on Control and Optimization,

45(2):634–667, 2006.

[12] R. Su. Discrete-event modeling of multi-agent systems with broadcasting-

based parallel composition. Automatica, 49(11):3502–3506, 2013.

[13] R. Su and B. Lennartsson. Control protocol synthesis for multi-agent sys-

tems with similar actions instantiated from agent and requirement tem-

plates. Automatica, 79:244–255, 2017.

[14] W. M. Wonham and K. Cai. Supervisory control of discrete-event systems.

Springer, 2019.