DiscreteEventDynamicSystems,6, 275-306 (1996) O 1996KluwerAcademicPublishers,Boston. Manufacturedin The Netherlands.
Hierarchical Control of Timed Discrete-Event Systems K.C. WONG AND W.M. WONHAM
kcwong,
[email protected]
Dept. of Electrical and Computer Engineering, University of Toronto Received November 21, 1994; Revised September 5, 1995 Abstract. An abstract hierarchical control theory is developed lbr a class of timed discrete-event systems (TDES) within the discrete-event control architectural framework proposed earlier by the authors. For this development, a control theory for TDES is introduced in the spirit of a prior theory of Brandin. A notion of time control structures is introduced, and on its basis a general property of hierarchical consistency is achieved by establishing control consistency - namely preservation of time control structures through the aggregation mapping in a two-level hierarchy.
Keywords: timed discrete-event systems, time control structures, hierarchical consistency
1.
Introduction
A control theory (RW) for a general class of discrete-event systems (DES) - systems or processes that are discrete in time and state space, generative, and possibly nondeterministic - was initiated by Ramadge and Wonham (1982,1983) (for a review see Ramadge and Wonham (1989)). Within this framework, architectural concepts such as modular, decentralized, and hierarchical control have been investigated by a number of workers. In Ramadge and Wonham (1987), Wonham and Ramadge (1988), Lin and Wonham (1988), Cieslak et al. (1988), Willner and Heymann (1990), Inan (1992), and Rudie and Wonham (1992), the authors investigate horizontal decomposition in supervisory control through notions of modular and decentralized control. Dually, Zhong and Wonham (1990,1992) study vertical modularity in supervisory control. They consider a two-level hierarchy, in which the high-level system is an abstract, simplified model of a low-level process and is driven by it through an information channel, and introduce the concept of hierarchical consistency to ensure that tasks of the high-level supervisor are realized through implementation by the low-level agent. For greater modelling capacity and realism, the original RW framework has been extended to incorporate additional features such as time. A control theory of timed discrete-event systems (TDES) with its own control technology - the specific manner in which control is exercised - is developed in the spirit of the original RW theory in Brandin (1993) and Brandin and Wonham (1994). Within this theory, time is modelled with the special event tick, as the tick of a clock. As in RW, certain events can be disabled and hence prevented from occurring; in addition, a class of forcible events is available for the supervisor to preempt a tick
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event, modelling the situation in which an event is forced to occur before a given time bound. To achieve a more unified development of the control architectural ideas independent of any specific control technology, a framework was proposed in Wong and Wonham (1992), and further developed in Wong (1994). This framework is based on the concepts of control structures and observers. A control structure is an abstract generalization of families of controllable languages in the RW setting. With this, we obtain a general version of Zhong's hierarchical consistency by first establishing control consistency, namely the preservation of control structures through the information channel (mapping). Since the development is independent of any specific technology, it can be adapted to different extensions of the RW theory such as the control theory of TDES (Brandin and Wonham 1994). Within the framework of Wong (1994), a finer notion of hierarchical consistency in which the nonblocking property is preserved is investigated on the basis of observers, or congruences of suitably defined dynamics, along the lines of Wonham (1976). Hierarchical control of (untimed) DES based on the notions of control structures and observers is presented in Wong and Wonham (1996). In this paper we adapt the abstract hierarchical control of Wong and Wonham (1996) to (a variation of) the timed setting of Brandin and Wonham (1994). It is found that the class of TDES there with its timing semantics is not closed under control (at least the control proposed in Brandin and Wonham (1994)). Thus an alternative definition of TDES and a more refined timed control technology are introduced, for which the latter induces a control structure in our sense. Next, the abstract concept of time control structure, one with the additional requirement that time be not stopped by control, is introduced. Then as in the standard case, hierarchical consistency in the timed setting is achieved by establishing control consistency, the preservation of time control structures through the information channel (mapping). An earlier version of this work was reported in Wong and Wonham (1993); the ideas are more fully developed in Wong (1994). The rest of the paper is organized as follows: In section 2, we recall the concept of control structures and the general schema of Zhong's twolevel hierarchy (Zhong and Wonham 1990), and give a brief overview of the relevant ideas in Wong and Wonham (1996). In Section 3, an alternative definition of TDES is proposed, leading to a framework that refines the control technology of Brandin and Wonham (1994). Then the notion of time control structures is defined. On its basis hierarchical control of TDES is developed in Section 4; as in the untimed case we achieve hierarchical consistency by establishing control consistency. In Section 5 we present conclusions and topics for future research.
2.
Preliminaries
Let ~ be an alphabet of event labels and ~* be the set of all finite sequences of event labels, including the empty string e (~ ~). A subset of ~* is called a language over ~. Let s, u E ~*. Then s is a prefix of u, written s _ ]Pt (H) I = ttick*l = INI, i.e., IHI = c~. []
It follows from Lemma 1 that, for two nonempty time behaviours H and F with H C F, .A/~H ~ .A/~F.
We now consider the composition of time behaviours. Let TL denote the set of time behaviours that are sublanguages of a given language L. Suppose L1, L2 E Tz*, i.e., L1 and L2 are time behaviours. Then La and L2 are time synchronous if LIlt~L2 ~ 7-E* Here ]is is the synchronous product of languages (Ramadge and Wonham 1989), and L111sL2 represents the concurrent behaviour of two TDES governed by the same global clock. If L1 and L2 are not time synchronous, then the two TDES cannot cooperate under the same time frame as illustrated by the following example.
Example: Let G1 and G2 be given as in Figure 4. Clearly L(G1) and L(G2) are time behaviours. Here L(G1)IIsL(G2) = {~} ~ Tz*. In G1 c~ is required to occur before the first tick of the clock; whereas in G2 c~ must occur after the first tick. Thus G1 and G2 cannot cooperate under the same global clock. []
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K,C. WONG AND W.M. WONHAM
G2
•
tick
)
) tick
ir °~
tick
Figure 4.
Non-time synchronous TDES.
We notice that any two time behaviours with disjoint activity event labels are always time synchronous. Our definition of the composition of TDES differs from that of Brandin and Wonham (1994) in that here time is global, i.e., all the behaviours are set in a single time frame and are measured with respect to it; whereas in Brandin and Wonham (1994) time has a more local meaning as we see in Appendix A. We continue with the train and gate example.
Example: Let the time behaviours of Train and Gate be given as in Figure 3. In this case these time behaviours are time synchronous and their synchronous product, TrainGate, is displayed in Figure 5. [] Next we introduce the time control technology. Let L E 7:~z* - {13}, i.e., L is a nonempty time behaviour, representing the behaviour of a TDES. We fix the meaning of L for the remainder of this section. Let H be a prefix-closed language. Define
T+:H~7)(H):s,
~{stick n E H ] n E N
+}
Here N + is the set of positive natural numbers and tick n : : tick .~-tick. The set T+(s) n
represents the time future of s in H, i.e., the tick's that the system can undergo before it must execute an activity event. Following Brandin and Wonham (1994), let
EligH : 2* ~
79(E) " s~
, {~ E ~,I.sa e g }
for H c_ Z*. Here EligH(s) is the set of eligible events of s in H. Now we introduce the set of events at any point in L which can be prevented from occurring. Define ~m6 : L ~ 7~(2ac~) satisfying (Vs E L) T+(s) = ~ ~
Emb(S) C EligL(s)
(1)
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HIERARCHICAL C O N T R O L OF T I M E D D I S C R E T E - E V E N T SYSTEMS
tick tick
...
tick tick
tick tick tick
~ick
tick
0~2 OL2
tick tick tick
Figure 5. TrainGate, the concurrent time behaviour of T r a i n and Gate.
with strict inclusion. The set Ehib(S) denotes the set ofprohibitible events at s in L. If TL+ (s) = !?, then the events in E l i 9 L (s) are imminent, i.e., one of these events must occur before the next tick. Thus at least one of the eligible events must not be prohibitible. For all s E L, let E ~ ( s ) := Eact - E m v ( s ) be the set of uncontrollable events at s in L. The other means of control in TDES is forcing. Define E for : L ", P ( E ~ t ) with (2) The set Efor (s) denotes the set o f forcible events at s in L. We notice that in Brandin and Wonham (1994) events could be both prohibitible and forcible; but we can remodel such events so that our disjointness condition is satisfied, as illustrated by the following example. Example: Let G be given as in Figure 6, in which E for (e) = Emb (e) = {a} and E for (s) = Emb(S) = 0 for s E L ( G ) - {e}. Since an event cannot be disabled and forced at the same
time, which control option is to be exercised depends on the unmodelled "circumstances". To recover the disjointness condition, we model those "circumstances" explicitly as in G r in Figure 6 via/3 and 7- Here E for(/3) -- { a } and E r o s ( s ) = 0 for any s E L ( G ' ) - {/3}, and Emb(7) = {a} and Emb(S) = 0 for s E L ( G ' ) - {7}. [] We now introduce the definition of controllable languages in TDES based on the control technology defined in the previous paragraph. But first we restrict ourselves to the following subclass of time behaviours in L. Let
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K.C. WONG AND W,M. WONHAM
G
GI
~
t i c k / , ~ ¢ 0 ~ tick
tick tick
~0/
tick tick
Figure 6. Prohibitible and forcible events.
be the set ofpropertimebehavioursof L. Thus a proper time behaviour is a time behaviour with the property that at any point an uncontrollable event is eligible when tick is not; intuitively in a proper time behaviour control cannot stop the clock. We remark that ~ and L are in ~ L trivially. Following the spirit of Brandin and Wonham (1994), we introduce a notion of controllability. Let H E ~ L and K C_ H . Then K is controllablewithrespect to H if for all s C K ,
EligH(s) N (Eune(S)U{tick}) ifEligK(s) N Efo~(s) = (3) EligK(s) D EtigH(s)N Eunc(s) ifEligK(s) n ~~for(8) • 0 Thus K controllable means that, after any string in the closure of K , an event (r that is eligible in H must occur in the closure of K if: either a is uncontrollable, or a is tick and no forcible event is eligible in K ; thus tick can be excluded only if a forcible event is eligible in K (to preempt it).
Example: Let
G be given as in Figure 7, where
Ehib(Ce)=
{a},
Efor(a) =
{~}, and
G ~
tick
F
"1
!
!
a
L.
tick ..
"~~
! !
!l Controllable .t
tick
Figure 7. A controllable l a n g u a g e in TDES.
Ehib(S) = ~for(S) = 0 for s E L(G) -
{a}. The automaton enclosed in the box in Figure
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HIERARCHICAL CONTROL O F TIIVIED DISCRETE-EVENT SYSTEMS
7 is controllable as the second a can be disabled and the first tick event after the sequence a can be preempted by the forcible event/3. []
Example: We continue with the train and gate example introduced before. To ensure that safety can be enforced at the railroad crossing, we equip L(TrainGate) with the following control, where TrainGate is the automaton given in Figure 5: let Emb(S) = 0
tick
( ~
~ t i c k
q
tick
,,..
tick tick
