Decentralized Supervisory Control: A New Architecture with a Dynamic Decision Fusion Rule Tae-Sic Yoo and St´ephane Lafortune Department of Electrical Engineering and Computer Science The University of Michigan, 1301 Beal Avenue, Ann Arbor, MI 48109–2122, U.S.A. ftyoo, [email protected]; www.eecs.umich.edu/umdes Abstract We consider a decentralized control architecture for discrete-event systems where local supervisors are allowed to change dynamically the manner in which their local decisions are combined globally. This is done by changing the default decisions regarding the enablement of controllable events. In previous work, this default was fixed a priori and remained constant throughout the operation of the system. We show that under dynamic decision fusion rules, a larger class of languages can be achieved as compared with architectures with static fusion rules. A dynamic version of the notion of co-observability appears in the necessary and sufficient conditions for the existence of supervisors in the new architecture. Dynamic co-observability relaxes (static) co-observability. The existence of a set of local dynamic default decision rules that ensures dynamic co-observability can be decided in polynomial time.

1 Introduction We consider control problems for discrete-event systems where a set of “local” supervisors, each with its own sensing and actuation capabilities, cooperate in order to achieve a given desired controlled behavior. Such decentralized control architectures are of considerable interest as they arise in a large variety of networked systems. Mobile ad hoc communication networks, integrated sensor networks, networked control systems, and automated vehicular systems are all examples of networked systems. Networked systems are informationally-decentralized and event-driven dynamic systems where groups of individual “agents” (i.e., local supervisors) interact in order to accomplish a common set of objectives. We consider control problems for networked systems that are posed in the framework of the theory of supervisory

control of discrete-event systems (cf. [9] and Chapter 3 in [2]). In the conventional conjunctive decentralized control architecture studied in supervisory control [3, 14], the control actions of local supervisors are fused using intersection of locally enabled events. Most of the results on decentralized supervisory control are based on the conjunctive architecture [1, 3–7, 10, 13–16]. In recent work, decentralized supervision with different fusion rules has been considered [8, 11, 18]. In these works, new event fusion rules are introduced and synthesis techniques guaranteeing the safety of the supervised language under various event fusion rules are developed. In particular, a generalized form of the conjunctive architecture is considered in [18]. In this architecture, the control actions of a set of supervisors are fused using both union and intersection of enabled events. Under this general architecture, local supervisors decide a priori that some controllable events are processed by “fusion by union” (of enabled events) and other controllable events are handled by “fusion by intersection” (of enabled events). We will refer to this architecture as the general architecture. The general architecture is more powerful than the purely conjunctive one in the sense that a relaxed version of co-observability appears in the necessary and sufficient conditions for the existence of a set of supervisors that achieves a given desired language [18]. In this paper, we go beyond the approach of [18] and consider a decentralized control architecture where supervisors are allowed to update dynamically the partitioning of controllable event between fusion by union and fusion by intersection upon the occurrence of observable events. We will refer to this architecture as the general architecture with dynamic decision fusion rules, or simply the dynamic general architecture. We propose to implement dynamic partitioning using a dynamic default decision rule for controllable events. Namely, the control action of the supervisors will consist of three parts: (i) events that must be enabled; (ii) events that must be disabled; and (iii) events whose “global”

default should be set to “enable”. By global default, we mean the default that should be used by the system when no supervisor explicitly enforces enablement or disablement of an event (parts (i) and (ii) of the control actions). We have found that the reformulation of the control actions of the supervisors in terms of the three parts mentioned above is the best manner in which to approach and investigate the general architecture with dynamic decision fusion rules. Prior work on the conjunctive, disjunctive, and general architectures for decentralized supervisory control can be reformulated in the above framework, although we do not explicitly do so in this paper; such a reformulation will follow as a special case of our results. The contributions of this paper are as follows. 1. The mechanism of (dynamic) default decision rules and of the fusion of the local control actions under these rules is introduced in Section 3. 2. Given a set of default decision rules, necessary and sufficient conditions for the existence of a set of local supervisors that achieves a given desired language are given in Section 4. These conditions characterize the class of languages achievable under the dynamic general architecture and introduce the notion of dynamic co-observability. 3. In Section 5, we show that the class of languages achievable under the dynamic general architecture strictly includes that of the (static) general architecture. 4. In Section 6, we introduce the notion of EDFpartitionability (Enable by DeFault) that is a necessary and sufficient condition for the existence of a set of local default decision rules satisfying dynamic coobservability. 5. A polynomial time algorithm to verify EDFpartitionability is given in Section 7. General knowledge of supervisory control and its most common notations are assumed. For introductory material, the reader is directed to Chapter 3 of [2]. Due to space limitations, proofs have been omitted from this paper; they are available in [17].

2 Preliminaries Let us consider the decentralized control architecture depicted in Fig. 1 where a set of supervisors jointly controls G by each observing subsets of
o (denoted by
o
i ) and controlling subsets of
c (denoted by
c
i ) in order to achieve a desired behavior K
L(G)


. We denote by
uo =
n
o and
uc =
n
c , the unobservable and uncontrollable event sets, respectively.

Decision Fusion System

Local Decisions

S

S

P1

P1

S

P2

G Pn

P2

Pn

Figure 1. Decentralized control architecture C&P co-observability is a key component of the necessary and sufficient condition for the existence of a decentralized control system that exactly achieves the desired behavior when the decision fusion block is the conjunction (of locally enabled events) [3, 10, 14]. We follow the definition that is presented in [1, 2]. In that definition, P i is the projection operation from

to

o
i and Pi;1 (s) := fs0 2

: 0 Pi (s ) = sg. Definition 2.1 A language K
M = M is said to be C&P co-observable w.r.t. M ,
o
1 ,
c
1 ,: : :,
o
n ,
c
n , if 8s 2 K and 8 2
c = ni=1
c
i s.t. s 2 M n K ,

(9 2 f1 g) i ( ) \ = ] ^  2
c
i ]] ;1 where i ( ) := i i( ) \ . i

E

:::n

s

P

E

P

s

s 

K





K

C&P co-observability is for the conjunctive architecture and hence the “C” in C&P co-observability refers to the conjunctive fusion rule for controllable events. Similarly, the “P” refers to the permissive decision strategy taken by local supervisors if there is insufficient local knowledge to determine the correct control action. The permissive local decision rule implies that the default control action for a supervisor under insufficient information is to “enable” an event. In [18], we considered a disjunctive architecture where the conjunctive fusion block is replaced by a disjunctive fusion block, meaning that an event is enabled whenever at least one of local supervisors enables it. The analogue of C&P co-observability for the disjunctive architecture is called D&A co-observability and it is defined as follows [18]. Definition 2.2 A language K
M = M is said to be D&A co-observable w.r.t. M ,
o
1 ,
c
1 ,: : :,
o
n ,
c
n , if 8s 2 K and 8 2
c = ni=1
c
i s.t. s 2 K ,

(9 2 f1 i

g) i ( ) \

:::n

E

s 

M


] ^  2
c
i ]] K



:

The “D” in D&A co-observability stands for disjunctive because D&A co-observability is formulated for the disjunctive architecture. Furthermore, the “A” in D&A coobservability stands for antipermissive because individual

events are always disabled by a local supervisor whenever that supervisor is unsure if the events should be enabled. The intuitive meaning of the antipermissive rule is to permit the occurrence of a controllable event after a trace s has occurred only if local supervisor has sufficient information to determine with certainty based on an “estimate” of the behavior, Ei (s) = Pi;1 Pi (s) \ K , that enabling the controllable event will be legal. In [18], a more general decentralized control architecture where the control actions of the individual supervisors are combined in a more flexible manner than pure conjunction or disjunction is considered. Namely, the supervisors agree a priori about choosing “fusion by union” (of enabled events) for certain controllable events (
c
d ) and “fusion by intersection” for the other controllable events (
c
e ), as shown in Fig. 2. This control architecture is more power-


c
e
2 , ,
o
n,
c
d
n,
c
e
n , if 1 is C&P co-observable w.r.t.
o
n
c
e
n 2 is D&A co-observable w.r.t.
o
n
c
d
n :::

: K

1 2

Local Decisions Sp1

Sp2

Spn

P1

P2

Pn

System

ful than the purely conjunctive and disjunctive ones in the sense that a relaxed version of co-observability appears in the necessary and sufficient conditions for the existence of a set of supervisors that achieves a given desired language [18]. In order to present this relaxed version of coobservability, let us define the following sets of events: For i 2 f1 : : : ng,


c
e
i :=
c
i \
c
e and
c
d
i :=
c
i \
c
d



where
c
d _
c
e =
c .
c
e
i is the set of locally controllable events whose default setting is enablement while
c
d
i is the set of locally controllable events whose default setting is disablement. We generalize C&P and D&A coobservability to embrace the partition of
c ; we call this generalized notion “co-observability” for the sake of simplicity. Definition 2.3 A language K co-observable w.r.t. M ,
o
1 ,


= is said to be
c
d
1,
c
e
1 ,
o
2 ,
c
d
2, M

M


o
1
c
d
1 

:::

:::


o
n
c
d
n
c
e
n = \ Lm ( ) 

K



:

G :

For the Sgs supervisor, gs stands for “General decentralized control law with Static partition”. The proof of Theorem 2.1 is given in [18]. It is constructive and the formula for the actions of the local supervisors composing S gs is given in the following equation.

gs P (Pi (s)) = f 2
c
d
i : Ei (s) \ L(G)
K g  f 2
c
e
i : Ei (s) \ K 6= g 
uc 
c
e n
c
i

S

Figure 2. The general decentralized control architecture.

M



is controllable w.r.t. L(G)
uc : is co-observable w.r.t. L(G)
o
1 
c
d
1 
c
e
1 

:::

: K

G

:


o
1
c
e
1

Theorem 2.1 Consider a language K
Lm (G) where K 6=  and consider a fixed partition of
c such that
c =
c
d _
c
e . There exists a nonblocking supervisor Sgs such that Lm (Sgs =G) = K and L(Sgs =G) = K iff the three following conditions hold:

3

Or Σ c,d



M

With this generalized notion of co-observability, the existence result of the general architecture can be presented [18]. The joint action of local supervisors S P1  ::: SPn in Fig. 2 is denoted by Sgs .

: K

And Σ c,e



: K

: K

Combined Fusion



i

(1)

The local supervisors using gs enable permissive events that might lead to legal behavior and disable antipermissive events that never lead to illegal behavior.

3 Dynamic General Architecture In the (static) general architecture, the default of the controllable events,
c
e and
c
d , is known a priori to all supervisors and remained constant throughout the operation of the system. The role of local supervisor is to decide which locally controllable events should be enabled and disabled. In this section, we consider a decentralized control architecture where
c
e and
c
d are allowed to change dynamically. We will refer to this as the dynamic general architecture. In order to retain a decentralized structure, we propose to implement dynamic changes of
c
e and
c
d via default decision rules issued by local supervisors. Specifically, local supervisors decide which locally controllable events should be enabled by default whenever local observations happen. Consequently, the components of the decisions of local supervisors are “enable by default”, enablement, and disablement decisions. That is,

P

S i

: i (

) ! 2  2  2 P

c
i

c
i

c
i



where SPi (Pi (s)) = (edfi (Pi (s)) ei (Pi (s)) di (Pi (s))), for i 2 f1 : : :  ng. Hence, edfi (Pi (s)), ei (Pi (s)), and di (Pi (s)) represent the “enable by default”, enablement, and disablement decisions of the ith local supervisor, respectively. The fusion of local decisions is performed as follows:

gd (P (s)) := F (Sn P1 (P1 (s)) : : :  SnP (Pn (s))) n := ( i=1 edfi (Pi (s)) i=1 ei (Pi (s)) i=1 di (Pi (s))) S

S

S

S

n

where P is the usual projection mapping that P :

!

o . For the Sgd supervisor, gd stands for “General decentralized control law with a Dynamic decision fusion rule”. We say that Sgd is control-nonconflicting w.r.t. L(G), K , and
c , if for all s 2 K and  2
c such that s 2 L(G),

2

 =


n

i=1

e


n

i (Pi (s))] \ 

i=1

i (Pi (s))]:

d

We update the partition
c
d and
c
e in the following manner: For s 2

,

2 Sni

 2
c
e ( ( ))] ,  =1
c
d( ( )) =
c n
c
e ( ( )) 

P s

P s



P s

edf

i (Pi (s))] and

:


c
d
i( ( )) :=
c
i \
c
d( ( )) and (3)
c
e
i ( ( )) :=
c
i \
c
e ( ( )) From (2) and (3), we obtain that for 2 f1 g and 2

,
c
e
i ( ( )) _
c
d
i ( ( )) =
c
i Note that
c
d
i ( ( )) and
c
e
i ( ( )) are functions of the P s

P s

P s

:

i

:::n

s

P s

P s

P s

1 2 L( gd 2 2 L( gd  2 L( gd f( 2
c
d( f( 2
c
e ( : 

:

P s

global projection P and the exact status of local partitions is not known to local supervisors in general. This is not a problem since the decision of local supervisor, S Pi (Pi (s)), will not directly depend on
c
d
i (P (s)) and
c
e
i (P (s)) as we shall see in the next section. The reason for defining
c
d
i(Pi (s)) and
c
e
i (Pi (s)) is to relate to the property of co-observability previously defined. In the (static) general architecture,
c
e represents the fusion by intersection of enabled events. Therefore,  2
c
e is globally disabled if there is a local supervisor that does not enable  . This can be equivalently described by the fusion by union with disablement decisions. Namely, if any local supervisor disables  2
c
e , then  is disabled. Otherwise,  is enabled by default. On the other hand,
c
d

S

: s s

)

=G

S

S

), )] ^  2 L(Sn)] ^ f 2
uc g_ ( ))) ^ ( 2 Si=1 i ( i ( ))g)g_ ( ))) ^ ( 2 ni=1 i ( i ( )))g]

=G

=G

s

G





P s



e

P

s



P s

 =

d

P

s

(4)

:

The implication of above definition is that a feasible transition  after trace s can occur, if any of the following three cases is satisfied:

  

(2)

Equation (2) means that local supervisors decide which locally controllable events should be enabled by default and the decisions are accepted globally. With the global partition functions,
c
e (P ( )) and
c
d (P ( )), we define local partition functions over the same domain as follows: For
i 2 f1 : : :  ng and s 2
, P s

represents the fusion by union of enabled events. Therefore, if any local supervisor enables  2
c
d , then  is enabled. Otherwise,  is disabled by default. With this in mind, we define the prefix-closed language L(S gd =G) generated in the context of the dynamic general architecture as follows:



is uncontrollable.

S

is disabled by default ( 2
c
d (P (s))) but it is enn abled by a local supervisor ( 2 i=1 ei (Pi (s))). 

 is enabled by default ( 2
c
e (P (s))) and it is not disabled by any local supervisor ( 2 = n d (P (s))). i=1 i i

S

The marked language is defined as usual: Sgd =G) \ Lm (G).

L(

Lm (

S

gd =G)

=

4 Supervisor Existence Result In this section, we assume that a set of local default decision rules is given. Therefore, the global partition and local partition functions are given by (2) and (3), respectively. Under this assumption, we investigate the issue of the existence of a nonblocking and control-nonconflicting supervisor Sgd that achieves the desired language exactly. We start by modifying the definitions of co-observability to reflect the dynamic partition of
c . First, let us define the control index function below: For  2
c ,

c ( ) := fi :  2
c
i g:

I

With this, dynamic C&P co-observability is defined as follows: Definition 4.1 A language K
M = M is said to be dynamically C&P co-observable w.r.t. M ,
o
1 ,
c
e
1 (P ( )), : : :,
o
n ,
c
e
n (P ( )), if 8s 2 K and 8 2
c
e (P (s)) s.t. s 2 M n K , there exists i 2 Ic ( ) s.t.

i (s) \ K = :

E

The dynamic D&A co-observability can be defined similarly.

Definition 4.2 A language K
M = M is said to be dynamically D&A co-observable w.r.t. M ,
o
1 ,
c
d
1 (P ( )), : : :,
o
n ,
c
d
n (P ( )), if 8s 2 K and 8 2
c
d (P (s)) s.t. s 2 K , there exists i 2 Ic ( ) s.t.

i (s) \ M

E




K:

The dynamic version of Definition 2.3 is: Definition 4.3 A language K
M = M is said to be dynamically co-observable w.r.t. M ,
o
1 ,
c
d
1 (P ( )),
c
e
1(P ( )),: : :,
o
n,
c
d
n(P ( )),
c
e
n (P ( )), if

1 2

: K

is dynamically C&P co-observable w.r.t.

M : K


o
1
c
e
1 ( ( )) 

P

:::


o
n
c
e
n( ( )) 

P



is dynamically D&A co-observable w.r.t.

M


o
1
c
d
1( ( )) 

P

:::


o
n
c
d
n( ( )) 

P

:

With the notion of dynamic co-observability, the existence result of the dynamic general architecture can be presented. Theorem 4.1 Consider a language K
Lm (G) where K 6=  and consider a set of default decision rules edfi (Pi ( )), for i 2 f1 : : :  ng. There exists a controlnonconflicting supervisor Sgd such that Lm (Sgd =G) = K and L(Sgd =G) = K iff the following conditions hold:

1 2

: K

: K

is controllable w.r.t. L(G) and
uc : is dynamically co-observable w.r.t.

L( )
o
1
c
d
1( ( ))
c
e
1( ( ))
o
n
c
d
n( ( ))
c
e
n( ( )) 3 is Lm ( )-closed G 





: K

P

P

G





P

P

:::

:

:

(Recall that the functions
c
d
i (P ( )) and
c
e
i (P ( )) in the condition 2 of the theorem are built from the given edfi (Pi ( )) function according to equations (2) and (3).) For a language satisfying the conditions in Theorem 4.1, the following local decision rules can be applied: For s 2 K , i 2 f1 : : :  ng,

In [18], the default of controllable events (
c
d and
c
d ) is assumed to be known to local supervisors and local decision rule (1) depends on this knowledge. For the dynamic general architecture, we do not assume that the status of the partition of
c is known to local supervisors. From (2), it is clear that the exact status of the partition of
c is not known to local supervisors and (5) shows that local enablement and disablement decision rules do not depend on partition information. In order words, while the necessary and sufficient condition of dynamic co-observability is inherently global, when this condition holds, it is possible to find local decision rules (namely, those in (5)) that exactly achieve the desired language K . Note that the general architecture in [18] can be reformulated as a special case of the dynamic general architecture. With this reformulation, local enablement and disablement decision rules do not have to depend on partition information.

5 Properties of the Architectures M

Let us define the following classes of languages where is assumed to be prefix-closed:

Lcen ( ) = f
: LDA ( ) = f
:

is observable w.r.t. M ,
o ,
c g is D&A co-observable w.r.t. M
o
1 ,
c
1 , : : :,
o
n ,
c
n g LCP (K ) = fL
K : L is C&P co-observable w.r.t. M
o
1 ,
c
1 , : : :,
o
n ,
c
n g Lgs (K ) = fL
K : 9
c
d and
c
e s.t.
c
d _
c
e =
c and L is co-observable w.r.t. K

L

K

L

K

L

K

L

M



:::

K



P

P



P



e

Proposition 5.1 In general,

Lcen ( ).

:::

P

We relate K above as follows.

As (4) shows, local decisions are always accepted globally. Therefore, local supervisors should enable/disable controllable events only if they are certain about it. This is the implication of local decision rule (5). If all local supervisors are uncertain about enabling or disabling a controllable event, the default of the controllable event at that point determines the outcome.




o
1
c
d
1( ( ))
c
e
1 ( ( ))
o
n
c
d
n( ( ))
c
e
n ( ( ))g Lgs ( ) and the classes of languages 

(5)



L K edfi (Pi ( )), for i = 1 : : :  n, such that L is dynamically co-observable w.r.t. M

where


o
n
c
d
n
c
e
n g

Lgd ( ) = f
: 9

i

i (Pi (s)) := f 2
c
i : Ei (s) \ L(G) 6=  Ei (s) \ L(G)
K g and di (Pi (s)) := f 2
c
i : Ei (s) \ L(G) 6=  Ei (s) \ K = g:



In [18], relations between the classes of languages defined above were investigated. Let us define the class of dynamically co-observable sublanguages of K :

gd P (Pi (s)) = (edfi (Pi (s)) ei (Pi (s)) di (Pi (s)))

S


o
1
c
d
1
c
e
1

defined

Lgs ( )
Lgd ( )
K

K

K

The relations between the classes of languages defined at the beginning of this section are summarized in Fig. 3.

6 Existence of Default Decision Rules: EDF-Partitionability (Enable by DeFault) In Section 4, we assumed that a set of local default decision rules is given. Under this assumption, we characterized

Lcen (K )

First, let us define the following “Violation of C&P coobservability” function:

Lgd (K ) Lgs (K )

vc

LCP (K )

LDA (K )

i (Pi (s)) := f 2
c
i : 9si  2 Ei (s) \ (L(G) n K ) 8j 2 Ic () Ej (si ) \ K 6= g:

For a language satisfying EDF-partitionability, the following local default decision rules can be applied: For i 2 f1 : : :  ng,

i (Pi (s)) =
c
i n vci (Pi (s)):

edf

Figure 3. Relation between the classes of languages

the necessary and sufficient conditions for the existence of a nonblocking and control-nonconflicting supervisor S gd that achieves the desired language exactly. Then, the following question arises: When can we find a set of local default decision rules satisfying dynamic co-observability? In order to address this question, we define the notion of EDFpartitionability below. Definition 6.1 We say that K is EDF-partitionable w.r.t. L(G),
o
1 ,
c
1 , : : :,
o
n ,
c
n, if 8s 2 K and 8 2
c s.t. s 2 K ,

(9 2 c ( )) i

I



E DF

]

(8 i 2 i ( ) \ (L( ) n )) 9 2 c ( ) j ( i ) \ = ] s 

j

E

I



s 

s:t: E

s

G

K



K

:

From (2), we know that a controllable event is disabled by default if no local supervisor decides to enable it by default. Since the local default decision is always accepted globally, local supervisors should be conservative when the default decisions are made. Therefore, “enable it by default” decisions are made only if no violation of C&P coobservability is found. This intuition is formalized in (7).

7 Verification of EDF-Partitionability In this section, we investigate the computational complexity of testing EDF-partitionability. The desired language is assumed to be generated by the trim finite-state deterministic automaton H . That is, K = Lm (H ) and K = L(H ) = Lm (H ). Following the approach of verifying C&P and D&A co-observabilities, we can build a nondeterministic automaton Medf (
c ) that marks the violation of EDF-partitionability. The results are stated for two local supervisors. However, the extension to any finite number of supervisors is straightforward. Then, we have the following result. Theorem 7.1 Lm (H ) is EDF-partitionable w.r.t.
o
1 ,
c
1 ,
o
2 ,
c
2 , iff Lm (Medf (
c )) = .

where E DF denotes the following condition: (6)

The E DF condition implies that for each illegal controllable continuation  that the ith supervisor estimates, there is a supervisor that can ensure that continuation with  is illegal. That is, the ith supervisor can infer that there is a supervisor (labeled j ) that can disable  with certainty. Therefore, “Enable by DeFault” is used for  . Hence, EDF-partitionability roughly implies the following: if  is a legal controllable continuation, there exists a local supervisor that can infer that  can be enabled by default. With this, the following result is stated. Theorem 6.1 Consider a language K
L(G). There exists a set of local default decision rules fedfi (Pi ( )) : i = 1 : : :  ng satisfying dynamic co-observability of K w.r.t. L(G),
o
1 ,
c
d
1 (P ( )),
c
e
1 (P ( )), : : :,
o
n ,
c
d
n(P ( )),
c
e
n (P ( )), iff K is EDF-partitionable w.r.t. L(G),
o
1 ,
c
1 , : : :,
o
n ,
c
n .

(7)

L( ), G

Given that the number of local supervisors is fixed, we can build Medf (
c ) in polynomial time w.r.t. jQH j and jQGj.1 Thus, Theorem 7.1 provides a polynomial-time test for EDF-partitionability. With the facts that controllability and Lm (G)-closure can be verified in polynomial time w.r.t. jQH j and jQG j, we have that the solvability of the dynamic general architecture can be verified in polynomial time. We can now state the main result of this paper, which addresses the solvability of the dynamic general architecture. Theorem 7.2 Let Lm (H ) be controllable and Lm (G)closed. There exists a control-nonconflicting supervisor Sgd such that L(Sgd =G) = L(H ) = Lm (H ) and Lm (Sgd =G) = Lm (H ) iff Lm (Medf (
c )) = . 1 It should be noted that the computational complexity of the construction of edf (c ) is exponential in the number of local supervisors, . In [12], it is shown that deciding C&P co-observability is NP-hard w.r.t. . It is clear that the verification of EDF-partitionability is clearly harder than that of C&P co-observability. Consequently, we have that deciding EDF-partitionability is NP-hard w.r.t. as well.

M

n

n

n

8 Conclusion In this paper, we considered a dynamic general architecture that generalizes further the general architecture of [18]. This architecture allows local supervisors to change the default decision regarding the enablement of controllable events. We showed that a proper choice of default decision rules results in a larger class of achievable languages than that of the general architecture of [18]. A polynomial time algorithm for verifying the existence of a set of local default decision rules that ensures the achievability of the desired behavior was presented. We developed a constructive methodology for updating dynamically the default decisions for the enablement of controllable events. For the issue of the realization of supervisors and proofs of results presented in this paper, we direct the reader to [17].

Acknowledgment This research is supported in part by NSF grant CCR0082784 and by the DDR&E MURI on Low Energy Electronics Design for Mobile Platforms and managed by ARO under grant ARO DAAH04-96-1-0377.

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