Control of vector discrete-event systems. I. The base model ...

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 8, AUGUST 1993

Control of Vector Discrete-Event Systems I-The Base Model Yong Li, Member, IEEE, and W. M . Wonham, Fellow, IEEE Abstract-A vector discrete-event system (VDES) is a discreteevent system model in which the system state is represented by a vector with integer components,and state transitions by integer vector addition. This paper investigates such structures from the viewpoint of control theory. The objective is to specialize to VDES the general automaton-basedcontrol theory of Ramadge and Wonham (RW). It is shown that within the VDES h m e work a useful class of control systems can be compactly modeled, and some of the associated RW control problems more efficiently solved.

I. INTRODUCTION HE objective of this paper and its sequel is to specialize the control theory of discrete-event systems (DES) developed in [27], [28], [341, [351 to a setting of vector addition systems. In the cited references, a DES (plant to be controlled) is modeled as an automaton that generates a formal language over a finite alphabet, say C, whose elements label the automaton’s transitions, or events. Events labeled by elements in a fixed subset C, of C are declared to be controZluble, namely can be disabled by an external controller or supemisor. Disablement is made to depend on the past history (string) of generated events, in such a way that a design specification of the controlled system behavior is satisfied. Under suitable conditions the control law can be optimized, in the sense of minimally restricting plant behavior. In this setting, which we call RW, various control-theoretic ideas have been explored, including controllability [27], observability [51, [6], [201, [261, stability [31, 1231, modular control [35], hierarchical control 1361, and timing issues [21, [15]. It is natural to enhance the abstract automaton model of RW by exploiting algebraic regularity of internal system structure when it exists. An obvious instance of such structure is arithmetic additivity over the integers. For instance, the state of a manufacturing workcell might be the current contents of various buffers and the numbers of machines in various modes of operation: thus, when a machine completes a work cycle, the status vector of

T

.

Manuscript received March 30, 1992; revised September 4, 1992 and November 24, 1992. This work was supported by the Natural Sciences and Engineering Research Council of Canada. Paper recommended by Associate Editor C. D. Cassandras. Y. Li is with the Department S906, Northern Telecom Canada Ltd., Brampton, Ontario, Canada L6V 2M6. W. M. Wonham is with the Department of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada MSS 1A4. IEEE Log Number 9210361.

machines and the vector of buffer contents would be suitably incremented. Similar examples are furnished by various kinds of traffic systems. Following [19] we introduce a class of vector DES (VDES) and specialize some of the general ideas of RW to the VDES setting. It will be shown here that VDES can offer advantages of compact modeling; in the sequel paper it will be shown how to utilize integer programming in computing optimal supervisory controls. Of course system modeling by vector addition systems is a long-standing technique [ll], [12], especially in the setting of Petri nets [lo], [21], [221, [291. For us, however, Petri nets will serve only as a graphical representation tool, and we make no use of net theory as such. For applications of Petri net theory to control, see for instance [71, [81, [lo], [13], [21], [24], [31]. Most relevant to our work are [SI, [13], [31], which translate to Petri nets some of the RW state-feedback logic of [28]. Our work differs from that cited in two ways. First, we extend the state-feedback logic to express the concepts of controllability, observability, dynamic control and closed form (i.e., VDES-type) controller. Second, our treatment is self contained, using only elementary tools of linear algebra and (in the sequel) integer programming. The base model is defined in Sections I1 and 111. A standard control-invariance problem of state feedback is developed in Section IV, along with the appropriate versions of controllability, observability, and modularity. A dynamic (memory-dependent) extension is provided in Section V. In Section VI we provide a tutorial example of VDES modeling and VDES control for a manufacturing workcell. 11. PREDICATES Following [28], we use logic terminology for some aspects of VDES behavior. A predicate on a set B is a function P: B + {O,l}. Let P ( d ) be the family of all predicates on d. The operators “ 1”(negation), “ A ” (conjunction), and “ V ” (disjunction) are defined as follows: (7

(PI P , ) ( q )

P ) ( q ) = 1* P ( q )

=

1

=

0

Pl(q) = 1 and P2(q) = 1

P1VP2 = T ( ( T P ~ ) A ( ~ P ~ ) ) .

men p(&), ~,A , V )

is a ~~~l~~~ algebra, which is isomorphic to the Boolean algebra (2“, -, n , U ), with

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LI AND WONHAM: VECTOR DISCRETE-EVENT SYSTEM

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the identification of P E P with the subset

integers. For each event a the state transition function 6( a,.) is a "'displacement" function in the state space. Q p := {qlq E Q and P ( q ) = 1). For simplicity we assume from now on that for each In this paper, we write Q p , the induced subset of Q by P, a E C, Fa = - E , and X, 2 0 unless otherwise stated. simply as P and employ the expressions "q E P7'and With this assumption the occurrence condition becomes: " P ( q ) = 1" interchangeably. X+E,rO. (1) Predicates 1', 2' E are said to be An important property of the model is that the final relative to P E P ( d ) , written P, = P,(rel P), if P, A P = state after the occurrence of an event string is determined P, h P. solely by the numbers of instances of events in the string, BY the foregoing a partial Order is induced On and is independent of how these events are interleaved. P ( d ) according to a2,***, am}, and let V, To make this precise, let C = {a1, P, d P2 e P , A P , = P I . be an m-dimensional integer vector defined for an event string w E C* as: Then ( P ( d ) ,6 )forms a complete lattice. V,(i> := lwlal Let R be a unary relation on Q. Then R induces a where Iw I, is the number of occurrences of the event a, predicate PR on Q: in w. V, is the occurrence vector of w. We can think of PR(q) = 1 e R ( q ) = TRUE. 5.)as a morphism from the monoid C* (under string concatenation) to the module Pt:=Zm. We call the ZIn this case we will simply write PR as R. Of interest in VDES theory are the predicates given by module Y the occurrence space. We then have the followlinear inequalities on n-dimensional integer linear space ing proposition which allows us to compute the state transition function of a VDES algebraically. Z": Proposition I: Let 6 be the extended state transition P = ( a , z , + a , z, + +a, z, Ib ) function on C* xZ, and w E C* be any event string. or Then for any state X 2 0 P = ( A Z Ib ) S ( w , X ) ! (Vt E G ) X + EF; 2 0 (2) where ai, b E Z and A := [a,, a 2 , * *a,]. ~ , Denote the clo- and 6(w,X ) = X + EV, (3) sure of all linear predicates on Z" under the operators 1,A and V by P , ( Z " ) .Then ( P , ( Z " ) , l ,A , V ) constiwhere E := [Ea1,E,z,*-.,E,,] and iij denotes the set of all tutes a proper subalgebra of ( P ( Z " ) ,1, A , V 1. prefixes of w. 0 Instead of directly working with C* and a state set Q as III.VECTORDES in the general RW framework, we will concentrate in The state of a VDES is described by state variables VDES control on the two more structured objects-modx,, x,;~~,x, ranging over integers Z . The state space of d e s 7 and 2 ' .The matrix E is a 2-linear transformation the VDES is defined as the set of n-dimensional integer from the module Pt to the module 2. If we consider EV, vectors: as the displacement vector for the event string w ,then (3) asserts that the effect of the occurrence of w can also be regarded as a displacement from a state, just as in the ease of ari individual event. A VDES with occurrence condition (1) can be represented as a Petri net (PN) [25] which has distinct transition labels and does not contain self-loops. The state As in general RW, we assume that the event set C is variables of the VDES are represented by the places of partitioned into two disjoint subsets, C, and C,, called the the PN, and the events by the transitions. The states of the controllable-event subset and uncontrollable-event subset, VDES correspond to the markings of the PN, and the respectively. The state transition is described by a partial initial state to the initial marking. (A VDES with general occurrence-condition vectors Fa may not be representable function 6: C x 2 ' 4 2 by a PN, unless a more general definition of PN is used.) 6 ( a , X )= X + E , Conversely, a PN with distinct transition labels (possibly where E, E Lz" is the displacement vector for a . N a , X )is with self-loops) can be modeled by a VDES with general defined, denoted by 6( a,X ) ! , if X 2 Fa, where F, E Lz" is occurrence-condition vectors Fa. As an illustration, consider the following PN graph' called the occurrence-condition vector for a. A VDES is (see Fig. 1) of a simple manufacturing system of three formally defined as a 4-tuple: machines feeding a buffer. 27 = (E,%, 6,X , ) where X, is the initial state. The state space Lz" is a Z-module under vector addition and multiplication by

'For convenience, we will simply draw undirected lines (instead of with its input places in a pN directed lines) a graph.

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IEEE TRANSACTIONSON AUTOMATIC CONTROL,VOL. 38, NO. 8, AUGUST 1993

Fig. 1. xl: # of idle machines x 2 : # of working machines xj: buffer contents al:a machine starts working a*:a machine finishes working

Assume that initially all machines are idle and the buffer is empty, and that a1 is controllable. The corresponding VDES is then specified as follows:

Fig. 2.

I

Ea,:=

2 ;

[-E], [ -i] [;I. E,,

:=

I

0;I

x,:=

I

2

I1

I

, I

I

Figure 2 shows some state transitions of the system. For this system -1

I I

t x2

1

Fig. 3.

Let ut

= cyI a1a2a2 and

i].

w 2 = a1a2 al az.The Vwl=

Remark 4: The VDES model was first introduced in

Vw,= These two strings will lead to the same end [16]. The occurrence vector and (the transpose of) the E-matrix defined in this section have been called the firing state X from X,: count vector and the incidence matrix respectively in the

[a] [-:-$][i]

x = x,+ EVW1= x,+ EVw* =

+

Petri net literature [22]. Equation (3) was called the state 0 equation in 1211. IV. STATICSTATE-FEEDBACK CONTROL OF VDES In general RW theory a controller is driven by events occurring in the plant. We call this configuration eventfeedback control (EFC). In VDES control, we will work with a slightly different configuration called state-feedback control (SFC). An SFC issues a new control pattem upon each state change. (See Fig. 4(a)-(b).) Formally a state-feedback controller for a VDES 5 is a function

=

This situation is shown in Fig. 3. Remark 2: With the assumptions that Fa = -E, and X, 2 0 all reachable states of a VDES are nonnegative (component-wise). 0 Remark 3: We do not exclude the possibility that E, = 0 for some event: In this case, we call a a free event; a forms a self-loop at each state in the state-transition diagram of the VDES. With the general occurrence-condition vectors Fa, the VDES model also includes the possibility of self-loops in its PN graph: a is a self-loop at where the place x i , in the PN graph if F,(i) > 0 and E,(i) = 0. 0

r is the set of control patterns defined as follows:

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LI AND WONHAM: VECTOR DISCRETE-EVENT SYSTEM

cate P E P ( 2 ) by: if S ( a , X ) ! and S ( a , X ) E P, or if not S ( a ,XI! otherwise

1

M,(P)(X)

:=

0 EFC

The transformation N, : P ( 2 ) + P ( 2 ) , the strongestpostcondition [28], is defined for P E P ( 2 ) by:

(a)

N,(P)(X)

SFC @)

Fig. 4.

An event (Y is enabled at a state X if a E f(X>; otherwise, a is disabled. Notice that for a E C, it is always

true that

(VX

E2

) a Ef= X + E , 6( a,X ' )

= X'

+ E, _____

LI AND WONHAM: VECTOR DISCRETE-EVENT SYSTEM

then

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Thus T ( S ( a ,X I ) = T ( X ) + T ( E , )

S ( ( u , X ’ )E D ( a , P > A N , ( P ) . 0 From observability of P , 6 ( a , X ‘ ) E P . = T ( X ‘ )+ T(E,) The following theorem asserts that the controllable and observable predicates characterize the sets of reachable = T ( 6 ( a ,X ’ ) ) states of a system controlled by controller in F,. as claimed. Theorem 13: For a predicate P on 2’with X , E P , The following technical definition of balanced controller there exists a state-feedback controller f E F, for i7 such will simplify our discussion. that R ( f / i 7 ) = P if an only if P is both controllable and Definition 10: A controller f is balanced if 0 observable with respect to S. ( V X , X’ E X , a E C) 6 ( a , X I ! and To prove this theorem we need two lemmas. For a controllable and observable predicate P we first S ( a ,X ) = X ’ and X , X’ E R ( f / F ) * f , ( X ) = 1. define a DES, g , ( P ) , which can be regarded as the 0 projection of g in 2,with respect to P : A balanced controller is the one which, among cong o ( p ) := (E,% U { y d ) , 6’7 x,,,) trollers synthesizing the same closed-loop predicate, en:= T ( X , ) ; ables most events at any reachable state. It can be checked where Yd is a dummy state adjoined to %; X0,, that any controller may be replaced by a balanced con- and for any a E C and Y E % troller without changing the reachable set ([19, A.21). By T ( S ( a ,X ) ) if ( 3 X E P ) T ( X ) = Y and concentrating on balanced controllers only we will gain 6 ( a ,X ) ! and S ( a , X ) E P some technical convenience (without sacrificing generalS ’ ( a , Y ) := Yd if ( 3 X E P ) T ( X ) = Y and ity) in stating and proving our results. 8 ( a , X ) ! and S ( a ,X ) E P Denote the class of balanced controllers which satisfy undejked otherwise the condition (6) by F,. Towards a characterization of reachable states in a VDES controlled by a controller in with 6’(a,Yd) undefined. It is not hard to see that the F,, we need the definition of observable predicate. above DES g , ( P ) is well-defined since P is observable. A Definition 11: A predicate P on 2 is observable with commutative diagram and a graph are given in Figs. 5 and respect to B if 6. The following result is now straightforward. (Va E C , ) P k D ( a , P ) h N , ( P ) Lemma 14: If P is controllable and observable, then where D ( a , P> := T-’(T(N,(P)A P)). 0 T ( P ) is controllable with respect to F,(P). The definition is rendered more intuitive by the following. Now, by Lemma 14 and Theorem 6, we construct a Proposition 12: A predicate P is observable with re- state-feedback controller f’ for g o ( P )such that spect to g if and only if for any CY E C, and x,x‘ E P R(f’/%(P)) = T ( P ) T ( X ) = T ( X ’ ) and S ( a , X ) ! and with S ( a , X ’ ) ! and S ( a , X ) E P =$ S ( a , X ’ ) E P . f: = M , ( T ( P ) ) (re1 T ( P ) )( C YE Xc). Proofi (IF) Let X * E D ( a , P I A N J P ) for some a We extend this higher level controller f’ to a controller f E C,. Then X * E N,(P) and there exists X ’ E P such for 5: that S( a,X ‘ ) = X * . Moreover, since f,W = f : ( T ( X N ( a E C,). X * E D(a,P ) = T - ’ ( T ( N , ( P ) A P ) ) For the controller f thus defined, we have Lemma 15: If P is controllable and observable with there exists X ” E P such that T ( X * ) = T ( X ” )N C YX, ) = X” for some X E P . But by (9,we know that T ( X ’ ) = X , E P,then R ( f / g ) = P. Proofi Similar to the first part of the proof of TheoT ( X ) . Then, by the hypothesis, applied to a,X‘, we have rem 6. 0 that X * = N C YX, ’ ) E P . This proves that We now prove Theorem 13. P b D ( a ,P I A N,(P) Proof of Theorem 13: (IF) Assume that P is both controllable and observable with respect to s.Then by i.e., P is observable. Lemma 15, we have R ( f / i 7 ) = P for the controller f in (ONLY IF) Assume that P is observable. Let a E C, Lemma 15. It is easy to check that f is balanced. Since f and X , X ’ E P with is the extension of f’ we have that f E F,. (ONLY IF) Assume that for a controller f E F,, we T ( X ) = T ( X ’ ) and S ( a , X ) ! and have R ( f / S ) = P . Then as in the second part of the N c Y , ~ ‘ )and ! S(a,X) E P. proof of Theorem 6, we can show that P is controllable. It Then S(a,X ’ ) E N J P ) , and S(a,X ‘ ) E D(a,P ) since then remains to show that P is observable. For X , X’ E P T ( S ( a ,X ‘ ) ) = T ( S ( a ,X I ) and S ( a , X ) E N , ( P ) A P . and CY E C, assume that T ( X ) = T ( X ’ ) ,S(a,X I ! , and

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 8, AUGUST 1993

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X

which satisfies

,x

6

R ( f i / F ) = sup C ( P , ) . A modular controller f:2+ r is then formed from f as

T

J

Theorem 17: Assume that all subcontrollers fi(i = 1,2,..., k ) are balanced. Then the modular controller A f= fi is also balanced and satisfies

(i*,

--

S(a,X’)! with S(a,X) E P . From

S ( a , X )E P

Afi/S

R

>

1

=supC(P).

[ P i ] := (Xl(tlw E E;) 6 ( w , X ) !* S ( w , X ) E P i ) .

We need to show that

=R(Sf,2)

we have that f J X ) = 1 (since f is balanced), and that f,(X’) = 1 from (6) (since f E F,). It follows that From the fact that S ( a , X‘)E R ( g f , 2 )= P . 0 From Proposition 12, P is observable. Remark 16: Definition 11 and Theorem 13 first appeared in [17].Our version of observability is static in the sense that it takes into account the current plant state only. Some definitions of dynamic observability, which incorporate the entire state trajectory up to the current plant state, have also been reported [41,[141, [261.

C. Modular State-Feedback Control

and the fact that R ( f i / F ) s Pi we have k

It follows that

Assume that the legal specification for a VDES g is given by a predicate P E P ( 2 ) of the form k

P

:=

A pi i= 1

where Pi E P(@ are called subspecifications. For each subspecification we synthesize a subcontroller fi:%+ r

since R( A f= fi/g)is controllable. It remains to prove the reverse inequality, or in light of Theorem 8, to prove

LI AND WONHAM: VECTOR DISCRETE-EVENTSYSTEM

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It is easy to check that

A

The synchronousproduct of two DES gi = (E,%, Si,X i , o ) (i = 1,2), denoted by ~lllsS’z, is formed as (C,Z1 x Z,, 6 , (Xl,o,X,,.)) where Z1X Z, is the Cartesian product of Z1 and 2,; S ( a , ( X l , X 2 ) ) !iff 8, ( a , X l ) ! and 6 , ( a , X , ) ! , and then 6 ( a ( X 1 ,X , ) ) := (81(a,X , ) , S,(a, X,)). Definition 19: Let K G C* be a closed language, and M = (C, y , 6 , Yo)be an automaton DES satisfying

k

[L1pi] =

[pi].

So we are done if it can be proved that

Let X E R ( g , A f=l[Pil>. Then there exist X j ( j 1,2,..., m) and aj(j = 0,1,2,-.-, m - 1) such that 1) S(a.,Xi)! and S(aj,X i ) = - 15;

L W ) 2L(B’).

=

( J = 0,1,2,..., m

Let P be a predicate on the state space of 9‘ := gllsA which satisfies

-

L(B”,P )

=K

nL ( 5 ) .

Then we say that K and P are compatible wrt F, written K 8r P, and that M is a memory for K. In other words, A is a memory for a closed language K ( i = 1,2,..-, k ; j = 0,1,2,-.., m - 1) if A does not constrain the behavior of 27 when they are Xj E R(fJ5) coupled together and if K can be recorded (or, memoimplying rized) in sIlsd by a state subset P c 2 X 9. ( i = 1,2,.*-,k ; j = 1,2,-.., m - 1) f i , u j < X j= ) 1 After a memory M is obtained, we can proceed to construct a state feedback controller f enforcing P in Y. since all f, are balanced. Thus We call the pair (f,M )a dynamic controller, (see Fig. 7 ) to highlight the analogy to the dynamic compensator in A f i = 1 ( j = 0,1,2,.-., m - 1) linear control theory [32]. Denote (f,A) by F , and write (ir, )aj L(F/B) in place of L ( f / g ’ ) ,the closed-loop language. implying A natural question now arises: What is the closed-loop language L ( F * / g ) when f* synthesize sup C(P) in g’? It turns out that L ( F * / g ) is precisely the supremal controllable sublanguage of K n L ( g ) wrt 9, denoted by In particular, X = X , E R ( A f=,/S’).Thus (8) is proved, SUP CL(K n L(B)). Theorem 20: Assume d is a memory for K with K as required. 0 Remark 18: A slightly different version of Theorem 17 P (g’ := SllsA).Let f * be a balanced state feedwas initially reported in [18]. back controller synthesizing sup C ( P ) in s’: In view of (7) we have

-

st

R(f*/Yj

V.DYNAMIC STATE-FEEDBACK CONTROL OF VDES The state-feedback controllers discussed in the previous section are static in the sense that their control actions are fully determined by the current state of the plant and are independent of how this state is reached. Such “memoryless” controllers are sufficient in forcing the plant to behave in accordance with a legal-behavior specification given in the form of a legal-state range, i.e., a predicate on 2.However, in many cases, legal-behavior specifications also refer to the history of the plant, thus cannot be formalized merely as predicates on 2’.Controllers accessing only the current plant state are not powerful enough to enforce such “dynamic” specifications-Auxiliary information about the plant history must be made available to controllers. Auxiliary devices, called memories, can be constructed to record the required history information. This leads to dynamic control structure. A. Memory and Dynamic Controller

Denote by L ( F , P ) the strings in L ( s ) which visit only states satisfying P, a predicate on the state space of 27:

LW,P )

:=

{ t E L ( g ) ) ( V wE t ) S ( w , X 0 ) E P}.

=

supC(P).

Then L ( F * / ~=) SUP CL(K n L ( g ) ) .

Proofi [30, theorem 21 states that L(Z’,supC(P))

=

SUPCLFI(L(B’,P))

-

where sup CL,(L( s’,P ) ) denotes the supremal controllable sublanguage of L(s’,P ) wrt F’. Since K we have L ( F ,P )

=K

n Us).

Since f* is balanced, it can be checked ([19, A.31) that L ( f * / S ’ ) = L W ’ , R(f*/F))= L ( S ‘ ,sup C(P)).

so

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Note that

yzx

Z = Z X y 1x

Zi = Z X

Denote by

zi.Let

'*'

xy

k

g..

Ti: Z +Zi the natural projection from

Z to

k

p

:=

A T-l(Pi). i=l

Proofs of the following two results can be found in [19, Section 9.21. is a memory for L with Claim 22: d 2 1 1 .-sIIs This result establishes the optimality of the dynamic L P. 0 controller F* in terms of the closed-loop language. Let fi be a controller defined for 5, and satisfying To end this subsection we give the following proposition, stating that there exists a memory for any closed R ( L / q ) = supC(P,). language. Thus dynamic state feedback can be utilized for We extend fi to a controller L! defined for 5': any legal specification in the form of a closed language. Corolfay 21: For any nonempty closed language K E,, ( Z ) :=f i , ,(Ti(Z)). there exist a memory A and a dynamic controller F* Claim 23: such that Fig. 7. Dynamic controller.

o

p

t

L ( F * / F ) = sup C L ( K n I!,(.!?)).

R

Root Let Jf be a recognizer for K with jY as (possibly infinite) state space. Attach a dump state Y# to where A f= JV so that the transition function is total. Denote the new recognizer by A. Then it is easy to see that P = Z X jY is compatible with K , and A is a memory for K. The second part of the corollary follows from Theorem 20. 0

)

A E/5' = supC(P) (i*,

(9)

is the conjunctive controller defined by

(i-6)

k

:= a

A f;,

( aE

C).

i=l

0

B. Modular Dynamic-ControllerSynthesis Let A f = I := ( A f l y AlIIs A z I I s I l s A k ) . By virtue of Claim 22 and Theorem 20, (9) can be rephrased Let the overall legal specification be given in the form by saying that the dynamic controller A f= Fi synthesizes of the meet of k closed languages: sup C L ( K ) in g. k Therefore, when combining controllers which operate L = nL1. optimally by accessing their local memories, we end up 2=1 with a controller which is optimal among controllers acEach subspecification L, needs a memory 4 to record cessing all memories. the relevant history of the plant, and can be expressed by a compatible predicate PI on the state space of 511s4. C. Linear Dynamic SpeciJicationand Linear Memory of The monolithic approach to controller synthesis is to KDES construct a controller enforcing all specifications and acIn this subsection, we restrict attention to the class of cessing the states of all memories. For the sake of flexi- linear dynamic specifications, which will be shown to lead bility and computational efficiency, however, it is more to a more structured dynamic control. attractive to adopt a modular approach. Construct a conWe first consider an extended VDES (EVDES), which troller for each subspecification which only accesses the will be used to model the synchronous product of a VDES state of the memory relevant to this particular subspecifi- plant and linear memories (see Fig. 8). The state space of cation; then combine these controllers to satisfy the over- the extended model is defined as the direct sum all specification. For i = 1,2;.., k , let & = (E, %, e,, Y;, Z=Z@y J be a memory for L, with with 2' and y being generated by state variables L, q$"qP I . xl, x z , - - - , xnl and y l , y 2 ; * + ,y,,, respectively. The event (Y E C can occur at Z = X e Y,namely 6 ( a ,Z ) ! , if Let2

g'=

(c7

2,8,

2 0 ) := F ? l l sA 1 1 1 s

AzIls

"'

11s Ak

X+El,,20

q = (c,2;,e , , ~ ~:=, as ~ ) 4.

for some El,a E 3.The extended state transition function is described as

2With a slight abuse of notation, here ''3 is"not intended to denote the set of integers, but the abstract state set of F'.

S(a,Z) =z +E,

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Lemma 24:

L ( 9 , Po) * 5’ P,. I

Proot

I

M Ii r ’ l

+ p -J -I

L ( Y , P,)

=

{ t E L ( 9 ” ) J ( V w E t ) S’(w,Z,) E P,}

=

{t

=

=

Fig. 8. Modular dynamic controller.

i i t

t

E

L(9“)I(VWE

t>5 ( W , Y o ) 2 01 m

EL

( ~ ” ) J ( VEWi) d -

C(i)V,(i) 2

o

i= 1 m E

L(F’)((VwE i)

C(i)V,(i> I d i= 1

I

= { t E L(B‘)l(VW E t ) CV, Id } with E, = El, EB E2,,for an Ez,,E y.It is easy to see = {t E L ( S ) l ( V w E i)CV, 5 d } that all reachable states will have nonnegative x-components if X o 2 0 (even though the y-components of these (L(B’) = L(B)) states may become negative). Occurrences of events in = L ( 9 , Po> EVDES will not depend on the state variables yl,y2,**-, ynZ; the occurrence condition of an event is = L ( 9 ,Po>n L(B) ( L W ,Po>G L W ) . solely determined by the x-components of the state Z . 0 When n2 = 0 an EVDES reduces to a VDES as originally Let f* be the optimal state feedback controller enforcdefined in Section I11 (with occurrence Condition 1). Then . the result below follows from Lemma A linear dynamic specification is given by a linear ing P, in 9’ 24, Theorem 20. L ( 9 , Po) = L(9’,Po). predicate Po on the occurrence space 7 : Theorem 25: Po = (c1 U 1 + cz U z + *.. +c, U, s d ) L ( F * / 9 ) = sup C L ( L ( B ’ Po>>. , or Po = ( C V s d ) with C := [c1,c2,.*-, cm]. Here all ci 0 and d are integers. Po induces a language L ( 9 , Po) G E*: We now consider the case when the legal specification is the conjunction of finitely many linear specificationson L ( 9 , Po) := {t E L ( 9 ) J ( V wE i)V, E Po}.

We now show that for any linear predicate on 7 there exist a one dimensional memory A and a linear predicate P, on the state space of 911sd which satisfy

-

L ( 9 , Po) rfP,

d = ( Y ,1

7 5 ,

&xj,Y)

=

Y-

5 is

ai),

and the initial state Yo = d . The composition of the plant

B and the memory A is the following EVDES:

c,

9’ = (2,a‘, Z , )

where

z:=zEBy S’(a,Z)!-X+ E, 2 0 s ’ ( a i , Z )= z + E,, @ [C(i>l

zo= x oEB Yo. Define a predicate P, on Z by P, O), namely (Y 2 0).

=

=

A ( C i V < di).

(11)

i=l

k

(10)

Yo)

k

P,

Thus Po characterizes a region in 7 bounded by k “hyperplanes.” We can easily check that

where 9‘ := .!Ylls A. For Po = (CV I d ) this memory A is given as where is the set of integers; the transition function total and specified as

T-:

([O, O,..., 0,112 2

L W , pol =

n L ( R ( c ys d i n

i= 1

The modular dynamic control discussed in Section V-B now comes into play. For each subspecification Pi = (CiV I d i ) we construct a memory Aiin the same way as we constructed A in (lo), and obtain an optimal control :f enforcing this subspecification. Then, by virtue of Claim 23, the conjunction of all controllers :f optimally enforces the overall legal specification (11). Finally, let us consider the legal specifications given by VDES. We will show that a VDES describes the same legal language as a conjunction of linear dynamic specifications. Therefore the modular dynamic control can be applied when the specification is given by VDES. Theorem 26: i) For any predicate Po on 7 of the form (11) there exists a VDES with the event set C such that L ( B ,P,)

=~

( 9 n L(&. )

(12)

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IEEE TRANSACTIONSON AUTOUATICCONTROL, VOL. 38, NO. 8, AUGUST 1993

ii) For any VDES AT with the event set C there exists a predicate Po on ’T of the form (11) such that (12) holds. Roo& 1) Let k

Po A ( C , V S d , ) . I=

Then w w

1

E L ( g , Po) if

EL

V. EWLE

( g ) and (VI

E

W)(Vi

E

&) C,V, I d,

(k := { 1 , 2 , 3 , * * k* ,} ) or

w

E

L ( g ) and (VI

E

@ ) ( V i E k) d , - C,V, 2 0

or w ~ L ( i 7 )and

(VI E WNVi E

&) d ,

+

m

( - C , ( j ) ) V , ( j >2 0 j=1

or w ~ L ( 5 )and m

(VIE i ~ ) (E~k)i YoGI +

~ ~ , ( i ) ~2, (o j ) I= 1

or w

EL

Remark 27: Intuitively, each state variable of a VDES can be regarded as a memory state variable, and the occurrence condition of a k-dimensional VDES imposes the condition that all k memory state variables should 0 always be nonnegative. Remark 28: All results in this section are documented in [19]. 0

( g ) and (VI

E

m

E)Yo +

Ha, V , ( j > 2 0 j=1

We consider the modeling and control of a production network, which is a variation of the network first studied in [l]. A Petri-net graph of the network is shown in Fig. 9. The system operates as follows: Machines in Groups 1 and 2 receive parts from a nondepleting inventory and deposit the finished parts in Buffers 1 and 2, respectively. Machines in Group 3 fetch parts from Buffers 1 and 2 for assembling. Upon completion, the finished part is deposited in Buffer 3 to be further processed by Group 4, and the part produced by Group 4 is sent to an inspection unit which can either output the part as a finished product or reject it for reworking by Group 4. We use a modular approach to modeling this production network. We first model modules of the network individually and then compose them to form the model of the complete system [19]. Modeling of the Machines The state of each machine group can be described by four state variables x i , x;, x ; , x ! , and the state space of a machine is:

for vectors Yo and Ha, defined as follows:

Yo(i):= d , , H a p := - C , ( j ) (i

=

1,2,..., k ; j

=

1,2;.-, m ) .

It follows from 2) that

w

E

L ( 9 , Po) 3 w

E

L(F) n L(

for A := (y, C, 5,Yo),which has displacement vectors Ha ( j = 1,2,..., m) and the initial state Yo. h) Let AT = (y, C, 6, Yo)be a k-dimensional VDES, with displacement vectors Ha. Then w E L ( g ) n L ( A ) if

w

EL

( 5 ) and (VI

E

The state transition function for Group i is with

4:Ci X

S i ( a ! , X i ) = X i +Ei+,,

(i

=

1,2,5,7; j

=

1,2,3,4,5)

where

E) ,$(/,Yo)!

or w

E

L ( g ) and (VI

E

m

E) Yo +

Ha, V , ( j > 2 0 j=

1

or w ~ L ( g )and m

(VI

E

W)(Vi E

k) - Y J i ) -

Ha{i)V,(j) I 0. j=

1

or w E L ( S ,Po)

for Po as defined in (11) and d , := Yo(i),Ct(J) := -Ha,(i) = 1,2,*..,k ; j = 1,2,..., m).

(i

+

g

Initially, all machines are in the “READY” state:

bl

LI AND WONKAM: VECrrOR DISCRETE-EVENTSYSTEM

Group 1

1225

Buffer 1

-._ .._.._____ __

have

L.$.-----.J

&roup 2

L.-,

Buffer 2

Fig. 9. a:: A machine in Group i starts working a:: A machine in Group i finishes processing a?:A machine in Group i discharges a processed part a:: A machine in Group i breaks down a:: A machine in Group i is repaired

Composition

Finally we compose the above components to obtain the VDES model of the production network =

( Z , Z 8 , XO)

=Il$-l

q

where The vector models of these machines are then

8

8

e = iU ci,cc= iU = 1 =l

q = (E,,%, si,X i , , ) where

ci= { a ; , a?, a:, a;, ai”) with Ci,c = {a:), i.e., we assume that only the event “Finish Processing” is controllable. Modeling of the Inspection Unit

with 8

Ea!

= @ k = l Ek,a!

where we assume that Ek,n,l= 0 if a/ is not in c k . The connections of the system modules are displayed in Fig. 10. Note that the connection of S1 and 5,is serial, that of g,and g4parallel, and that of g6 and g8feedback. Control of the Production Network We now discuss how to synthesize a modular controller to satisfy the performance specification of the system. The specification is that no buffer overflow and that at most one part be inspected at a given time. We assume that Buffer i has capacity ki and the buffer inside the inspection unit (xl) has capacity 1. This specification can be formalized as a predicate on the state space 2

The model for the inspector is

and

with

with

Modeling of Buffers

m e three buffers can be modeled as scalar systems. For Buffer 1, we have 5 2 =

(e,,%.,8 2 , X2,O)

with

For the purpose of this example we just list the optimal subcontrollers for the above linear specifications;the details of the procedure for obtaining such subcontrollers will be presented in a sequel paper devoted to computational methods. f i , a ( X )= 0

e, = { a ; , a:), 2,=z

e,, = {a:) X,,O

8,(a:, X , ) = x,

=

0

+1

a

=

a; and

xt 2 k ,

f , , , ( X > = 0 e a = a: and x: 2 k, f 3 , = ( X )= 0 e ( a = a: or a = ai) and x i + x i 2 k , f 4 , a ( X )= 0

e

a

=

a; and

X:

2 1.

L-JL-7

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IEEE TRANSACTIONSON AUTOMATIC CONTROL,VOL.38, NO. 8, AUGUST 1993

Fig. 10.

The conjunction of these subcontrollers is 5

f:=

M

AA.

Fig. 11.

i= 1

It is easy to check that all subcontrollers in f are balanced. Therefore, this modular controller is optimal in the sense that it synthesizes a largest reachable state set among all controllers which enforce the specification 4

P = Api i= 1

as asserted by Theorem 17. The above modular controller can lead to deadlock of the controlled system. To see this, consider the state at which xk = k,, X: = k,, X: = k,, x i = 1 and xi” = g , ( i = 1,3,5,7), with all other state variables being 0. At this state all controllable events are disabled and no uncontrollable event can occur. One way to remove the deadlock in the system is to add another subspecification which ensures that the deadlock state cannot be reached: P6=(

Xf+

i=2,4,6,8

x? i=1,3,5,7

r 1+

gi i=1,3,5,7

+

ki). i=1,2,3

To illustrate dynamic control, let us consider the following linear dynamic specification: la531 -/ai1 I k (13) which specifies that the total number of rejected parts by the inspection unit does not exceed an integer k. Here I a31 (i = 5,s) denotes the number of occurrences of a:. A one-dimensional memory can be easily constructed from this specification and is shown in Fig. 11. The dynamic specification is then equivalent to a static specification

y r k

on the state space z e y with y being the one-dimensional state space of d .Intuitively (and as will be verified in the sequel paper), the optimal controller f enforcing this static specification can be defined as

f=( We (13).

0 if a = a: and y 2 k 1 otherwise

the pair (&,f) a dynamic

enforcing

VII. CONCLUSION A vector discrete-euent system (VDES) is a structured DES model, which is economical in modeling systems with inherent additive structure-especially, flexible manufacturing systems and some computer and communication systems. In this paper we developed basic results for a statefeedback theory of VDES. The state-feedback logic in RW theory was extended to include such concepts as controllability, observability, and dynamic control. Controllability is an extension of control invariance in RW theory and embodies a notion of reachability. Together with controllability, observability characterizes the closed-loop predicates synthesized by controllers accessing only partial state information. A memory device was proposed to capture the history of a VDES plant when the control specification invokes legal event-strings; the memory along with a static controller accessing the current states of both the plant and the memory constitutes a dynamic controller. Many results in this paper can actually be applied to the general automaton DES model, since they are not based on the special VDES structure. In particular this is true of Sections IV, V-A, and V-B. In the sequel paper, however, we shall concentrate on exploiting VDES structure to obtain computationally efficient implementation of the general results presented here. REFERENCES 111 R. Y.Al-Jaar and A. A. Desrochers, “A modular approach for the performance analysis of automated manufacturing systems using generalized stochastic Petri nets,” REPORT RAL #116, Robotics Automation Lab., Rensselaer Polytechnic Institute, Troy, NY, 1988. [2] Y.Brave and M. Heymann, “Formulation and control of real-time discrete-event processes,” in Proc. 27th ZEEE Conf. Decision con@.,Dec. 1988, Austin, TX,1988, pp. 1131-1132. [31 -, 1990. “Stabilization of discrete-event urocesses.” Znt. J . Con@.,vol. 51, no. 5, pp. 1101-1117, 1990. [4] P. E.Caines, R. Greiner, and S. Wang, “Dynamical logic observers for finite automaton,” in Proc. 27th ZEEE Conf. Deciswn Contr., Austin, TX,Dec. 1988, pp. 226-233. [51 H.Cho and S . I. Marcus, “On the suoremal laneuaees of sublanguages that arise in supervisor synthesis probikms with partial observation,” Mathematics Con@., Signals Syst., vol. 2, no. 2, pp. 47-69, 1989.

LI AND WONHAM: VECTOR DISCRETE-EVENT SYSTEM

R. Cieslak, C. Desclaux, A. S . Fawaz, and P. Varaiya, “Supervisory control of discrete-event processes with partial observations,” ZEEE Tmns. Automat. Contr., vol. 33, no. 3, pp. 249-260, 1988. M. J. Denham, “A Petri-net approach to the control of discreteevent systems,” in M. J. Denham, k J. h u b , Eds., Advanced Computing Concepts and Techniques in Control Engineering, NATO AS1 Series, vol. F47. Berlin: Springer-Verlag,pp. 192-214, 1988. L. E. Holloway and B. H. Krogh, “Synthesis of feedback control logic for a class of controlled Petri nets,” ZEEE Tmns. Automat. Contr., vol. 35, no. 5, pp. 514-523, May 1990. N. Q. Huang, “SupeIvisory Control of Vector Discrete-Event Systems,” M.kSc. thesis, Dept. of Electrical Eng., Univ. of Toronto; also available as Tech. Rep. #9105, Systems Control Group, Dept. of Electrical Eng., Univ. of Toronto, Apr. 1991. A. Ichikawa and K Hiraishi, “Analysis and control of discrete-event systems represented by Petri nets,” in Discrete Event Systems: Models andApplications, IIASA Conf., Sopron, Hungary, Aug. 3-7, 1987, P. Varaiya and A. B. Kurzhanski, Eds., Lecture Notes in Control and Information Sciences, Vol. 103. New York SpringerVerlag, 1988. R. Karp and R. Miller, “Parallel program schemata,” J. Comput. Syst. Sci., vol. 3, no. 4, pp. 167-195, 1969. R. Keller, “Formal verification of parallel programs,” CommunicationsACM, vol. 19, no.7, pp. 371-384, 1976. B. H. Krogh, “Controlled Petri nets and maximally permissive feedback logic,” in Proc 25th Annual Allerton Conf., Univ. of Illinois at Urbana-Champaign, Oct. 1987, pp. 317-326. R. Kumar, V. K Garg, and S . I. Marcus, “Using predicate transformers for supervisory control,” in Proc 30th ZEEE Conf. Decision Contr., Brighton, UK, Dec. 1991, pp. 98-103. Y. Li and W. M. Wonham, “On the real-time supervisory control of discrete-event systems,” Info. Sci., vol. 46, no. 3, pp. 159-183, 1988. -, “A state-variable approach to the modeling and control of discrete-event systems, in “Proc.26th Annual Allerton Conf., Univ. of Illinois at Urbana-Champaign,Sept. 1988, pp. 1502-1507. -, “Controllability and observability in the state-feedback control of discrete-event systems,” in Proc. 27th ZEEE Conf. Decision Contr., Austin, TX,Dec. 1988, pp. 203-208. -, “Composition and modular state-feedback control of vector discrete-event systems, in Proc. 23rd Annual Conf. Info. Sci. Syst., Johns Hopkins Univ., Baltimore, MD, Mar. 1989, pp. 103-110. Y. Li, “Control of vector discrete-event systems,” PLD. thesis, Dept. of Electrical Eng., Univ. of Toronto; also available as Tech. Rep. #9106 (revised), Systems Control Group, Dept. of Electrical Eng., Univ. of Toronto, May 1991. F. Lin and W. M. Wonham, “On observability of discrete-event systems,” Info. Sci., vol. 44,no. 2, pp. 173-198, 1988. T. Murata, “State equation, controllability, and maximal matchmgs in Petri nets,” ZEEE Trans. Automat. Contr. vol. 22, no. 3, pp. 412-416, June 1977. -, “Petri nets: properties, analysis, applications,” Proc. ZEEE, vol. 77, no. 4, pp. 541-580, Apr. 1989. C. M. Ozveren and A. S. Willsky, “Output stabilizability of discrete-event dynamic systems,“ ZEEE Trans. Automat. Contr., vol. 36, no.8, pp. 925-935, Aug. 1991. K. M. Passino and P. J. Antsaklis, “Planning via heuristic search in a Petri net framework,” in Proc. 3rd ZEEE Symposium Intelligent Contr., Arlington, VA, Aug. 1988, pp. 626-631. J. Peterson, Petri Net Theory and the Modeling of Systems. Englewood Cliffs, NJ: Prentice-Hall, 1981. P. J. Ramadge, “Observability of discrete event systems,” in Proc. ZEEE Conf. Decision Contr., Athens, Greece, Dec. 1986, pp. 1108-1112. P. J. Ramadge and W. M. Wonham, “Supervisory control of a class of discrete event processes,” S U M J. Contr. Optitnu., vol. 25, no. 1, pp. 206-230, 1987. -, “Modular feedback logic for discrete event systems,” SLAM J. Contr. Optimiz., vol. 25, no. 5, pp. 1202-1218, 1987. J. Sifakk, “Structural properties of Petri nets,” in Mathematical Foundations of Computer Science, Lecture Notes in Computer Science. New York Springer-Verlag, pp. 474-483, 1978.

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T. Ushio, “On controllable predicates and languages in discrete event systems, in Proc. 28th ZEEE Conf. Decision Contr., Tampa, FL, Dec. 1989, pp. 123-124. T. Ushio and R. Matsumoto, “State feedback and modular control synthesis in controlled Petri nets,’’ in hoc. 27th IEEE Conf. Decision Contr., Austin, TX,Dec. 1988, pp. 1502-1507. W. M. Wonham, Linear Multivariable Control: A Geomehic Approach, 3rd ed. New York Springer-Verlag, 1985. -, “A control theory for discrete-event systems,” in Advanced Computing Concepts and Techniques in Control Engineering, M. J. Denham and A. J. h u b , Eds., NATO AS1 Series, vol. F47. Berlin: Springer-Verlag, pp. 129-169, 1988. W. M. Wonham and P. J. Ramadge, “On the supremal controllable sublanguage of a given languages,” S M J . Contr. Optimiz., vol. 8, no. 3, pp. 637-659, 1987. -, “Modular supervisory control of discrete event systems,” Maths. Contr., Signals Syst., vol. 1, no. 1, pp. 13-30, 1988. H. Zhong and W. M. Wonham, “On consistency of hierarchical supervision of discrete event systems,” IEEE Tmns. Automat. Contr., vol. 35, no. 10, pp. 1125-1134, 1990.

Yong Li was born in Sichuan, China, in 1962. He obtained the B.Eng degree in Computer Science and Automation from Chongqing University, Sichuan, in 1982; and the M.A.Sc. and Ph.D. degrees from the University of Toronto, Ontario, Canada, in 1986 and 1991, respectively, both in electrical engineering. In 1990, he joined the Digital Switching Division, Northem Telecom, Brampton, Ontario. His current research interests include discrete-event systems and telecommunication. Dr. Li has been a recipient of the Connaught Scholarship from the University of Toronto.

W. Murray Wonham (M’64-SM’76-F’77) received the B.S. degree in engineering physics from McGill University, Montreal, P.Q., Canada, in 1956, and the Ph.D. degree in control engineering from the University of Cambridge, Cambridge, England, in 1961. From 1961-1969, he was associated with the Control and Information Systems Laboratory at Purdue University, Lafayette, IN, the Research Institute for Advanced Studies (RIAS) of the Martin Marietta Co., the Division of Applied Mathematics at Brown University, Providence, RI, and (as a National Academy of Sciences Research Fellow) with the Office of Control Theory and Application of NASA’s Electronics Research Center. In 1970, he joined the Systems Control Group of the Department of Electrical Engineering at the University of Toronto, Ontario, Canada. He currently holds the J. Roy Cockburn Chair. In addition, he has held visiting academic appointments with the Department of Electrical Engineering at Massachusettes Institute of Technology, Cambridge, MA, the Department of Systems Science and Mathematics at Washington University, St. Louis, MO, the Department of Mathematics of the University of Bremen, the Mathematics Institute of the Academia Sinica, Beijing, the Indian Institute of Technology, Kanpur, and other institutions. His research interests have lain in the areas of stochastic control and filtering, the geometric theory of linear multivariable control, and more recently in discrete event systems from the viewpoint of formal logic and language. He has authored or coauthored about sixty research papers as well as the book Linear Multivariable Control: A Geometric Appmach. Dr. Wonham is a Fellow of the Royal Society of Canada. In 1987, he was the recipient of the IEEE Control Systems Science and Engineering Award, and in 1990, was Brouwer Medallist of the Netherlands Mathematical Society.