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Hierarchical control for structural decentralized DES
ELSEVIER
Copyright © IFAC Discrete Event Systems. Reims, France, 2004
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
HIERARCHICAL CO!'lTROL FOR STRVCTL'RAL DECE~TRALIZED DES Klaus Schmidt, Johann Reger, Thomas :\1oor
LelzrslUlzl fur Regelungstechllik. Unil'ersitiit £rlaflgefl-N unzberg CauersrrafJe 7. D-91 058 £rlallgell, Germany [email protected]
Abstract: In this contribution, we consider structural decentralized DES and supplement the existing control architecture with a two-level hierarchy. For the proposed overall system. we prove hierarchical consistency and that the closed- loop behavior is nonblocking. A comprehensive example demonstrates the computational benefit of our method. Copyright © 2004 fFAC Keyword s: supervisory control. structural decentralized control. hierarchical control.
1. I)JTRODUCTION
In hierarchical architecture s (Zhong and Wonham, 1990: da Cunha et a1.. 2002; Hubbard and Caines, 2002). controller synthesis is based on a plant abstraction (high-level model) , which is supposed to be less complex than the original plant model (lowlevel model ). Technicall y, abstractions can be defined as la nguage projection s. While from worst case scenarios projections are known to be of exponential computational complexity. application relevant cases with polynomial complexity are identified in Wong (1997 ). An important question is how to derive the plant abstraction. such that a high-level controller can be implemented by available low-level control actions (hierarchical consistency). A characterization of this property is given in Zhong and Wonham (1990).
In the past decade a great variety of ideas have been studied to reduce the complexity of synthesis algorithms fo r the supervisory control of discrete event systems (Rudie and Wonham. 1992: Jiang et aI., 2001: Yoo and Lafortune. 2000; Hubbard and Caines, 2002; da Cunha et a1.. 2002: Zhong and Wonham. 1990: Lee and Wong. 2002: Wo ng and Wonham. 1996). A key ingredient of promising approaches is to assume or to impose a particular contro l architecture. such that computationall y expensive product compositions of individual subsystems can be either avoided altogether or at least postponed to a more favorable stage in the design process. Our contribution builds on two recent results from this category. namely structural decentralized control and hie rarchical control.
In this paper, we consider the decentralized setting of Lee and Wong (2002), where the overall system is modelled by the synchronous product of the individual subsys tems. As an abstraction, we propose the natural language projection based on the shared events. We then show that this abstraction does comply with hierarchical consistency as defined in Zhong and Wonham (1990). Furthermore. th e high-level model can be computed by the synchronous product of the proJections of the individual subsystems (rather than the projection of the product of the individual subsystems ). Whenever the projections of the subsystems behave computati onally nicel y, this change of order
The decentralized control architecture proposed by Lee and Wong (2002) addresses plant models that are composed of a number of s ubsystems , which are coupled via shared events. Specifications are given for each subsystem individuall y. and the task is to synthesize indi vidual supervisors. As the subsystems are coupled, synthesis wi II in general need to refer to the sy nchronous product of all subsystems. Conditions under which such product can be avoided are given in Lee and Wong (2002).
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promises a substantial computational benefit. This is demonstrated by an example. The outline of the paper is as follows. Basic notations and definitions of supervisory control theory are recalled in Section 2. Section 3 and Section 4 introduce structural decentralized and hierarchical control of discrete e vent systems, respectivel y. In Section 5, both methods are combined so as to form a decentralized and hierarchical control architecture. A comprehensive example in Section 6 illustrates our contribution.
We recall basic facts from supervisory control theory. (Wonham, 2001: Cassandras and Lafol1une. 1999). For a finite alphabet I. the set of all finite strings over I is denoted I'. We write SI S2 E I ' for the concatenation of two strings SI, S: E L/ . We write SI :::; S when Si is a prefix of s. i.e. if there exists a string S2 E I' with S = 51S2. The empty string is denoted I' E I'. i.e. sE ES S for all S E I'. A lallguage over I is a subset H ~ I-. The prefix closure of H is defined by H:= {SI E I - .:::s E H S. t. SI:::; s}. A language H is prefix closed if H = H.
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A language H is said to be controllable w.r.t. L(G) if there exists a supervisor S such that H L(S/ G). The set of all languages that are controllable w.r.t. L(G) is denoted C(L(G )) and can be characterized by C(L(G)) {H ;; L(G)! 3S st. H L(S/ G)}. Furthermore, the set C(L(G)) is closed under arbitrary union . Hence . for every specification language E there uniquely exists a supremal conrrollable sublan· guage of E w.r.t. L (G), which is formally defined as KL (C ) (E ) := L:{K E C(L(G)) : K;; E}. A supervisor S that leads to a closed-loop behavior K L( c ) ( E ) is said to be ma;rimal permissive. A maximal permissive supervisor can be realized on the basis of a generator of KL(C) (E ). The latter can be computed from G and a generator of E. The computational complexity is of order O(N 2 M2 ), where N and M are the number of states in G and the generator of E , respectivel y.
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A language E is Lm-closed if En Lm E and the set of Lm(G) -closed languages is denoted FLo/(c) . For specifications E E FLo,( C ) ' the plant L(G) is non blocking under maximal permissive supervision.
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The nalural projection p , : I- -1 If' i L 2, for the I I L Iz is defined (not necessari Iy disjoint) union I iterati vely: (1) let Pil E) := E: (2) for s E I ' , G E I, let Pi(SG) := pi(S) G if G E I i , or Pi(SG ) := plI) otherwise. The set-valued inverse of Pi is denoted p,- I : Ii' -122:' , pil(l ) := {s E I' lp ,(s ) = T}. The sVllchronous product HI IHl ~ I ' of two languages Hi ~ I f is Hl' H2 = PI I (H J) n p2,1 (H2 ) ~ I ' .
3. STRUCTlJRAL DECEr\TRALlZED DES Structural decentralized DES as proposed in Lee and Wong (2002) are composed of subsystems. realized by finite state automata G i , i 1, 2 , ... ,11 with respecti ve alphabets I i . The synchronization of each two subsystems G, and G j is organized via shared events I , n I j '
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A filliTe aUTomaton is a tuple G (X,I , 8,xo ,Xm)' where X is the finite set of stales: I is the finite alphabet of evellls: 8 : X x I -1 X is the pal1ial transition jimCfion: Xo E X is the iniTial STate: and Xm ~ X is the set of marked STates. We write 8(x , G) ' if 8 is defined at (x, G). In order to extend 8 to a partial function on X x I ' , recursively let 8(x, E) := x and 8 (x , SG ) := 8(8(x,s ).G ), w henever both x' = 8(x,s ) and 8(x 1 O' )'. L(G) := {s El- : 8(xo , s)!} and LIn( G ) := {s E L(G) : 8(xo.s ) E Xm} are the closed and marked language generated by the finite automaton G. respectively. For a formal definition of the sy nchronous composition of two automata G , and Gl we refer to e.g. Cassandras and Lafortune (1999) and note that L (G I ll G2 )
Definition 3.1. A decenrralized col1trol system (DCS) consists of subsystems, modelled by finite state automata G" i 1, ... ,11 over the respective alphabets I i . The overall system is defined as G:= ! i~IG , over the alphabet I := U;'=I I i· The controllable and uncontrol lable events are Ii.c := I i r I e and L i.u := I i n I ll, respectively. where I e L.; I u I and I c r I u 0. For bre vity and convenience. let L:= L(G ), L o , := Lm(G;. Li := L(G,) . and L,," := Lm(G i ) .
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For our applications Vie assume local specifications 1, .... 17 for each subsystem G , . E, E FL I.", ~ I j, i Relative to the overall alphabet I, the Ei become (Pi) -I ( Ei ) . where Pi : I' -1 I i' denotes the natural projection. Taking into account the language L of G , the global specification is E r:'=l (p,) -l (E,) rL.
L(G I JI' L(G2 ).
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When L( Gl represents the plant behavior in a supervisory control context, we write I I , U I iJ • I , r Iu 0, to distinguish controllable (I ,) and ul7COlllrollable (L u) events. A control paTtern is a set '(. I u ;; Y ~ I, and the set of all control patterns is denoted r ~ 22:. A sllpervisor is a map S: L(G) -1 r , where S (s ) represents the set of enabled events after the occurrence of string s: i.e. a super;isor can disable controllable events only. The language L (S / G ) generated by G under supervision S is iteratively defined by ( J) I' E
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2. PRELlM]]',ARlES
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L(S/ G) and (2) SG E L(S/ G) iff sE L(S/ G),O' E S(s) and SG E L(G). Thus. L (S/G) represents the behavior of the closed· loop system. To take into account the marking of G. let Lm (S/ G) := L(S/G) n Lm( G). The closed-loop system is nonblocking if L In( S/ G) L(S/G) , i.e. if each string in L(S/ G) is the prefix ofa marked sning in Lm (S/ G).
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There are two approaches for generating a supervisor imp le menting the given set of specifications: the synthesis of a monolithic supervisor S for the overall specification E leading to the closed lang uage L(S/ G) = KJ.( E ) of the supervised system. and the synthesis of
280
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local supervisors Si for Gi and Ei, i 1, ... , 11 resulting in 1!;~lL(SiIGi ) n~ l (Pi)-I (KdE;)) n L: for the second approach see Figure I.
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G
.--
I
S
GIl l ,
L"I
~.
.
J...-"'T , G2
L2 L'
S2 ,
~~~ ~
; Sn
r-4
two levels are interconnected via Com hilo and In/ohi. The former allows Shi to impose high-level control on Slo , the latter drives the abstract plant G hi in accordance to the detailed model. From the perspective of the high-level supervisor. the forward path sequence Con/'ilo, Con lo is usuall y designated "command and control", while the feedback path sequence In!loh,. In!hi is identified with "report and advise". Formally. the high-level abstraction is defined as follows.
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(2)
Definition 4.1. (Hierarchical Abstraction). Let G (X ,L ,O,XO , Xm) be a finite automaton and L hi s; La set of high-level events. A reporter map 2 is a map 8: I' -+ (Ihi)' such that (1) 8 (£) £ and (2) either 8(s(j) = 8(s ) or 8(s(j) = 8(s)(jIIl, where (j E I. (jhi E Lhi. The high-level language is defined by Lhi := 8(L(G )) . The high-level marking is chosen S.t. L~ s; Lhi, where L~ is required to be regular. The canonical recognizer of L~~ is denoted G hi , and hence, L(Ghi ) = Lhi . Lm (Ghi ) L~~. Finally, high-level con-
Theorem 3. J. (Structural Decentralized DES Lee and
trollable and uncontrollable events are denoted L;" and I~i, respectively. where Lhi I~i U L~i, L;" n I~i 0.
Fig. I. Structural Decentralized Control Structure
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In Lee and Wong (2002), conditions under which both approaches lead to the same overall result are 1, ... , n the following holds: developed. Thus, for i
=
11
n (Pi)-1 (KLi(Ei))
= KdE) ,
(1)
i=)
pi( KdE )) is nonblocking w.r.t. Li.m.
=
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Wong (2002)). Using notation as in Definition 3.1, denote the natural projection (Ii L I j -t Li· Suppose that Ei E FLi.", and that for i, j 1, ... , n, i :p j
p;; :
(i)
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For our control architecture, not only the high-level events and the reporter map, but also the choice of contro!lable and uncontrollable events L~i and L~i are essential. For the synchronous composition of subsystems. we will show in Section 5 that this choice can be based on shared events.
Li.m marks I ,"'" L j , i.e. Lj (L,n L j)r Li.m
(ii)
r
s;
Li,m(Li;! L J )
(3)
r-:::l
Li and L j are mutually controllable, i.e. L;(Lj.llr Li)r p; j((p;)-I( Lj))
s;
Li
=
In!hi
~~ HCoi?ii~
(4)
COl11 hilo
hi G
In! I 0 h'I
t
r-~-
Then (I) and (2) hold. )
Con lo
In order to discuss the computational complexity, let N and M denote bounds for the number of states in G i and generators of E,. respectively. Then Nil and M n are the respective bounds for the number of states in the overa!l system. The computational complexity of the monolithic synthesis procedure is O(N2nM2n). whereas it is O(nN 2M2 ) for the local synthesis approach. It is clear that this benefit does not come for free. The approach in Lee and Wong (2002) requires the computation of the language projection (with exponential worst case complexity) in condition (4). However, G 1:':1 Gi need not be computed.
/ nf'°
~~
(b)
(a)
Fig. 2. Hierarchical Control Schemes For our fw·ther discussion, we need to define the interconnection of low- and high-level supervisors with the plant.
DefilliTion 4.2. (Hierarchical Control System). Referring to the notation in Definition 4.1 , a hierarchical conflvl SYSTem (HCS) consists of G, G hi, Shi and 51". where the high-level supervisor Shi and the lowlevel supervisor Slo fulfill the following conditions: Shi: L hi -t rh, with the high-level control patterns rhi := {YII~i s; y s; I.hi } ; and 510 : L(G) -t r with
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4. HIERARCHICAL CONTROL
s; L(Shi I Ghi ).
We consider the event-based hierarchical control scheme in Zhong and Wonham (1990) (Figure 2 (a)).
8(L(Slo I G))
The detailed plant model G and the supervisor 5 10 form a low-level closed-loop system, indicated by Con lo (co ntrol action) and Inf o (feedback information) . Similarly. the high-level closed loop consists of an abstract plant model G lli and the supervisor Sh'. The
Given a high-level specification Ehi EFL;;;' we can
J
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KLhi(Ehi) with synthesize Shi such that L(SIIII G hi ) a nonblocking high-level closed-loop. At this stage. the remaining task is to implement high-level control actions for the low-level plant by means of 510. 2 In the secuel. we focus our allention on the reponer map where is the natural language projection into (E'i)".
I'
'iOtc also that in :his case E EFL,,'
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p",
Definition 4.3. (Hie rarchical Control Problem). Given G:Ghi:S hi . find a low-level supervisor 5 10 as in Definition 4.2 such that the low-level controlled language of the HCS, L(Slo / G). is nonblocking.
U7=ILi. Relevant languages are L:= L(G). Lm := Lm (G). L, := L(Gi ), and LI.In := Lm(G,), controllable and uncontrollable events are L i.f Li n Le and Li,u = Li n Lu as in Definition 3.1. • Local low-level controllers are denoted 5 i : Li ~
On the one hand, there may not exist a low-level supervisor s in such that 8(L(SI0 /G )) L(Shi/ GllI) , and we end up with a strict subset relation. This is interpreted as an overoptimistic high-level supervisor expecting low-level behavior that is not possible. On the other hand. if the above equation turns out true for some s in, it provides a powerful tool to show that the the low-level is nonblocking. This is one motivation for the following notion of hierarchical consistency.
r i, where r, are the respective control patterns. Low-level closed-loop languages are denoted
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= L(SIIl/ G hi )
Lt:= L(Si/ Gi). Lj.m: = Lj' rLi,ln ' U:= T~ILj . = LCr: Lmf. Also let GCbe a genL(G ), L;~ Lm(Gf ) . erator s uch that U • For the abstraction the reporter map: 8 := plti is used. where phi : L' ~ (Llti) , denotes the natural projection and Lltl := L i,j,ii J( Li n L j ). The highlevel language is LIII := 8(U ) and 3 L~ := ! 1;~,L;:m
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t 0), with canonical recognizer GM s.t. Lhi = L(Ghl ). L~ = Lm(Gltl) . High-level controllable events are L:~ :=
Definition 4.4. (Hierarchical Consistency (HC)). The hierarchical control system in Definition 4.2 is hierarchicallv consistent if the following applies: 8(L(SiO / G ))
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{s"i E L1711 (plti ) -l (i') r L~,
defined as "'i ;Ii := "'it r: "'ihi and L~' := LII n LhI • The high-level supervisor is denoted Shi : Lhi ~ r hi wi th the high-level closed- loop language L(Shi / Ghl ). A valid low-level supervisor 5 10 : U ~ r must fulfill 8(L(Slo / G')) ~ L(Slti / G/II ).
(5 )
There has been a variety of event-based contributions which tackle the complexity problem associated with the hierarchical abstraction and which provide appropriate definitions of the high-level marking as well as high-level controllable events in order to prove hierarchical consistenc y. Zhong and Wonham ( 1990) introduces the notion of strict-output-control-consistency to gU;lran tee a consistent low-level controller fulfilling a high-level specification. Blocking is not considered in this approach. da Cunha et al. (2002 ) utilizes a generalized model for controlled DES in the high-level to obtain hierarchical consistency without adding complexity by refining the hierarchy. Moor et al. (2003) uses Willems ' VO behaviours to establish a nonblocking hierarchical control architecture . Other state-based approaches (Hubbard and Caines. 2002) use state-aggregation for hierarchical abstrac ti on .
Lel1lma 5.1. (High Level Plant). Assume the control plzi( L::) . architecture of Definition 5.1 and let L7i Then the high level language is
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LI" = p hi( l i ~ ILil = 11;'= ,L;'i . Proof (Sketch): We use induction and the identity po(HIII H2 ) = po(Htl :po( H2 ) for languages Hand H2 with alphabets LN I and LH2 and the natural projection Po : (LH U LH, ~ (2:0 with "'io ~ (LHI U LH, ) and (LI-I I n Llic ) ~ LO as in Wonham C~OOl). C
r
r
The previous lemma enables the computation of the hierarchical abstraction by only using the hierarchically abstracted subsystems and it is not necessary to compute the full low-level plant automaton. Consequently the computational effort reduces from computi ng i!7=I and s ubsequent projection pIIlC ~, L; ) to first computing phl( Lj ) and then evaluating 1 ~~ , p hi ( L: '). While technically the computational complexity is of the same order. a significant computational benefit shows in applications where realizati ons of plz:( Lj') have less states than those of L;·.
5. HIERARCHICAL COl\TROL FOR DECEf'.<"TRALIZED SYSTEMS
L:
We discuss how a combination of the decentra lized approach in Section 3 w ith the hierarchical approach in Section 4 can improve the computatio nal effic iency of supervisor synthesis : see Figure 2 (b) for our proposed overall scheme. As our detailed plant model. we consider a structural decentralized DES G I : .. . G, . G Gi. subject to local feedback contro l by supen'isors 5 , .... . Sn. The abstraction G It i of the lowlevel closed loop is based on the observation of shared events L ItI \ia Inl ol". A high-level supervisor 5 111 is designed to contro l G iif . where effect on the low level is taken via Con / 'iio and a low-le vel supervisor 51" .
For our further discussi on. we assume that the lowlevel subsystems fulfill co nditions (i) and (ii ) in Theorem 3.1 and that the lov.i-Ie\·e/ supervisors have been synthesized to enforce a local specification Ei E FLi .", ; i.e. L: := USi/ GiJ KL,( E,) . Thus. as a result of Theo rem 3. 1. the lov..·-Ievel is nonblocking. i.e. L f Lin '
= ': 7=,
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The next theorem gives a possible choice of the lowleve l supervisor such that the proposed control architecture is hierarchically consistent and nonblocking.
Definirion 5.1. A Hierarchical and Decentralized Control System (HDCS ) consists of the following entities.
Theorem 5.1. (Main Result ): Let the hierarchical contro l architecture for structural decentralized DES be
• A detai led plant model is a decentral ized co ntro l system G := 1 ! ~~ , G i with subsystems G i O\'er respective a lphabets Li. i = 1: .... 1l and L :=
:; By cons~ruC', i o n ,~:;. is regul ar.
282
defined as in Definition 5.1 and define the low-level supervisor Slo for each 5 E L" as
SIO( s) :=
S ill (phI
(5))
l)
(~_ I hi).
Then s to is valid according to Definition 4.2, the HDCS is hierarchicallv consistent and the controller is maximally pem1i ssi~·e. In particular, Slo solves the hierarchical contro l problem in Definition 4.3
svnthesis: where N. N ili , Mil i denote number of states i~ Cc, Chi and the canonical recognizer of Ehi. respectively. Again, in the case of N hi < N we expect a computational benefit.
6. EXk\1PLE We consider the two cooperating machine cells C: and C~ given in Figure 3. Both machines have 6 states.
Proof: We use induction to show that Slo is valid, i.e. p'l!( L(SIO I Ce )) <;;; L(Shl I Chi). Obviously € E L(S'O I Ce) and phl(€ ) = € E L(Shi I C hi ). Pick any 5 and 0 E I with so E L(Slu I C t ) and phi (.I' ) E L(Shi I C hi ). Then either 0 E I 'l! or 0 E (I - I hl). In the first case, 0 E Slo(s) implies 0 E Shi( phl(S)) and, hence, ph/(SO ) = phi(S) O E L(ShijC hi ) . In the second case phi (so) phi(s) E L(Shil c hi ).
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We prove hierarchical consistency by induction. As val iditv has been established above, we are left to show ;/I!( L(S'O I C')) ;2 KL'Ij( £ II!) . It is c lear that € E phi( L(SIOICC )) and € E KLhj( £ /lI). Let shi E K/J,,( £ hi) and Shl E phi( L(Slo I Ce)) and ass ume for Ohl E Ihi. shia hf E KLhi(£ hi) . Then Oil! E Shl(SIl! ) and by definition of the low-level super.isor Slo , Vs E (p ll!) -I (Sill) r . L' we know that a hi E SIO(S). Further on, as slll a hl E Lhi it follows that 3.\' E L' S.t. .1" E (phi) - I (s"i) and s' Ohi E Le. But then s' a hl E L(S!() I C') and thus phi (SI a hi ) = pill (s' )a lII = Sill a hi E phi (L(s in I c e)). For proving non blocking behavior it must be shown L ( SIOI C')n L;~. As L(SloICC) ;2 that L(SIOI CC)
Fig. 3. local machines C I and C2 The event y represents the start of the cooperation of the two machines, 8 indicates that the cooperation terminated successfully and
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6.1 Hierarchical Method
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L(Slo I C') 'i L';" is obvious , we on ly prove L(sln I C') <;;; L ( Slol ce )n L~" Assume s E L(SIOI C' ) but s rt L ( S!l)l cr)~I L';". As L(SIOI C') <;;; L~I' .I' E L~I ho lds, and thus s rt L(Slo I er ) n L~, requires that Vu E I ' S.t. su EL';" . 3a hl E I hi and u' ,u" E I ' S.t. u = u'Ohiu" and a hi rt 510 (su') and VLI I ~ u', SLI I rt L:~ . .. :"l'o\'. let u be such that su E L';" and let a hi , u l , u" as above. Then it holds that Pi(SUI) E Lun for i such that a hi E I i because of (3) and Vi S.t. a hi rt I i (let il, .... i be
Following the lines of Lee and Wong (1002) (3) holds as all states before shared events are marked. The mutual controllability condition (4) for i,j 1.1 is also va lid. For examp le, from Figure 3 and Figure 4 5 it can readi ly be observed that ~ is controllable w.r.t. (pn -l(p12 (L2 )) andtheeventsetI 2"n II. 6
=
~~
m
the corresponding indices), 311 i E (I i - I hi ) . such that Pi(S)Ui E L: m as L;' V;:. Thus SUI Ui, II'2 ... u:." E L:n
=
and thus
S
E L(S I I Ct) r .L~" U
=
Thus G Cl ' !C2 constitutes a structural-deceno'ali zed co ntrol system and we identify C as the low-level plant of a HDCS. The low-level specifications £ I and E2 in Figure 5 are contro ll able w.r.t. their respective subsystems and are identical to the low-level control led subsystems C~ and C S.
For prO\'ing maximal perrnissiveness we assume that 3si () as in Definition 4.2 s. !. L(SloI C') C L(slo I C") <;;; (p ill) -I (L(Sh, I C lli )) with 510 from Theorem 5 .1 . Then 3s E L(Si() I C' ) with shi phi(s ) E L(Shi I Chi ) bur t .I' rt L(Slo I C ). Because of HC 3s' E L(Slo I C') with phi (SI ) s'" and .I' .1" u with 11 E (I - I hi)'. But we know that (VSI U' < SIll), (I - I hi) E SIO( su' ) and thus .I' s'u E L(Si" I C'}, which contradicts the assumption and 510 is maximal ly permissi\·e. C
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I" - ' ( 1 ' - \ Fig. 4. (P2) . PI- (L2.r11);
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?"ow, in addition to low-level supervision. the highlevel specification E i:; in F ig ure 6 is implemented by employing our hierarchical control method. £ hi states that after the occurrence of two successive failures in cooperation no more cooperation should take place.
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By the abO\e the orem. the complexity of synthesis for the high -level specification becomes Q( (/Vh·fll (Mh l) 2; compared to Q(.v2n(/\1 I!l) 2; for the local monolithi c
= {1I 1 . "2 ,a~.a", "5 .{/6 .(J; }. The con'c, ponding transition ;s labcled with a tic ~ as contains controllable c\·e nts . 6 T he re\'erse relation is ob\'ious because of symmetry. .<
'" T·::;
means a:: t!xtcn~ions
0:" S
lo a markcc string :11ust be disabled
and ai sabling. can on ly occur if a h igh-Ic\ ~: e\,ent;s disaoied 01' .~".
283
7 states for the case without high-level control. Thus, compared to our method the high-level specification must be decomposed, which leads to relatively large low-level supervisors.
7. CO:\,CLUSIONS Fig. 5. low-level specifications Eland E2
Fig. 6. high-level specification E h;
=
For the abstraction. I h' {y, 0 ,
G7'
Recent work in control of DES has been focused on exploiting structural properties for reducing the computational complexity of DES controller synthesis. Our approach embeds a decentralized approach Lee and Wong (2002) in a hierarchical control scheme and it was shown that our architecture guarantees nonblocking behavior of the controlled system. Furthennore we pointed out that hierarchical consistency need not be verified as it is directly implied by the proposed control architecture. The computational benefit of our method was illustrated by an example. Ongoing work aims for weaker conditions guaranteeing nonblocking behavior and the demonstration of our results by a laboratory case study of a manufacturing system.
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Fig. 7. high-level abstractions G'i' and G~i It can easily be verified that E hi is controllable W.r.t. L (G h') and the high-level controlled language L(S'''/ G hi ) equals E h, with as-state recognizer. Because of Theorem S. I the low-level supervisor S'O( s) Sh' (phi(S)) U (I - I '") realizes the desired high-level behavior and implements non-blocking and maximally pennissive behavior in the low level.
=
6.2 Comparisol/ WiTh other meThods For the standard RW control synthesis the overall plant and specification automata have 26 states and 14 states, respectively and the supervised plant has 77 states. For applying the method of Zhong and Wonham (1990): da Cunha et al. (2002) it is also necessary to compute the overall plant automaton. The reporter map for the high-level abstraction is defined by labeling the low-level strings and for Zhong and Wonham (1990) there is additional computational effort for verifying hierarchical consistency. whereas in da Cunha et al. (2002 ), a supervisory control problem must be soh'ed for each low-level subsystem. corresponding to a high-level string via the reporter map. For the technique of Lee and Wong (2002) the additional highlevel specification Eh; can be decomposed into two local specifications p ;( Eill), i = L 2 which leads to low-level specifications with 8 states. each , and supervised low'-Ievel plants with 17 states both. versus
REFERENCES c.G Cassandras and S. Lafortune . Introduction to discrete event systems. Klu vFer, 1999. A.E.C. da Cunha. J.E.R Cury, and B.H. Krogh. An assume guarantee reasoning for hierarchical coordination of discrete event systems. WODES.2002. P. Hubbard and P.E. Caines. Dynamical consistency in hierarchical supervisory control. IEEE TAc' 2002. S. Jiang. V Chandra, and R Kumar. Decentralized control of discrete event systems with multiple local specializations. Proc. ACe, 2001. S-H. Lee and K.c. Wong. Stuctural decentralised control of concurrent discrete-event systems. EJC, 2002. T. Moor. J. Raisch. and J.M. Davoren . Admissibility criteria for a hierarchical design of hybrid control systems. In Proc. ADHS'03, 2003. K. Rudie and WM. Wonham. Think globally, act locally: decentralized supervisory control. IEEE TAc'1992. K. Wong. On the complexity of projections of discrete-event systems. , 1997. lIRL c:' ::eseer r:ec . co::I.,·o:1g9 8c o IT,p~exl:: Y . :-r:r.2.. K.c. Wong and W.M. Wonham. Hierarchical control of discrete-event systems. Discrete El'enr Dynamic Systems. 1996. W.M Wonham. ~otes on control of di screte event systems. Departmem of EleCTrical & Computer Engineering, University of Toronto , 2001. T. Yoo and S. Lafortune. A generalized framework for decentralized supervisory control of discrete event systems. WODES.2000. H. Zhong and W.M. Won ham. On the consistency of hierarchical supervision in discrete-event systems. IEEE TAC, 1990.
.n: .
Copyright © IFAC Discrete Event Systems. Reims, France, 2004
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
HIERARCHICAL CO!'lTROL FOR STRVCTL'RAL DECE~TRALIZED DES Klaus Schmidt, Johann Reger, Thomas :\1oor
LelzrslUlzl fur Regelungstechllik. Unil'ersitiit £rlaflgefl-N unzberg CauersrrafJe 7. D-91 058 £rlallgell, Germany [email protected]
Abstract: In this contribution, we consider structural decentralized DES and supplement the existing control architecture with a two-level hierarchy. For the proposed overall system. we prove hierarchical consistency and that the closed- loop behavior is nonblocking. A comprehensive example demonstrates the computational benefit of our method. Copyright © 2004 fFAC Keyword s: supervisory control. structural decentralized control. hierarchical control.
1. I)JTRODUCTION
In hierarchical architecture s (Zhong and Wonham, 1990: da Cunha et a1.. 2002; Hubbard and Caines, 2002). controller synthesis is based on a plant abstraction (high-level model) , which is supposed to be less complex than the original plant model (lowlevel model ). Technicall y, abstractions can be defined as la nguage projection s. While from worst case scenarios projections are known to be of exponential computational complexity. application relevant cases with polynomial complexity are identified in Wong (1997 ). An important question is how to derive the plant abstraction. such that a high-level controller can be implemented by available low-level control actions (hierarchical consistency). A characterization of this property is given in Zhong and Wonham (1990).
In the past decade a great variety of ideas have been studied to reduce the complexity of synthesis algorithms fo r the supervisory control of discrete event systems (Rudie and Wonham. 1992: Jiang et aI., 2001: Yoo and Lafortune. 2000; Hubbard and Caines, 2002; da Cunha et a1.. 2002: Zhong and Wonham. 1990: Lee and Wong. 2002: Wo ng and Wonham. 1996). A key ingredient of promising approaches is to assume or to impose a particular contro l architecture. such that computationall y expensive product compositions of individual subsystems can be either avoided altogether or at least postponed to a more favorable stage in the design process. Our contribution builds on two recent results from this category. namely structural decentralized control and hie rarchical control.
In this paper, we consider the decentralized setting of Lee and Wong (2002), where the overall system is modelled by the synchronous product of the individual subsys tems. As an abstraction, we propose the natural language projection based on the shared events. We then show that this abstraction does comply with hierarchical consistency as defined in Zhong and Wonham (1990). Furthermore. th e high-level model can be computed by the synchronous product of the proJections of the individual subsystems (rather than the projection of the product of the individual subsystems ). Whenever the projections of the subsystems behave computati onally nicel y, this change of order
The decentralized control architecture proposed by Lee and Wong (2002) addresses plant models that are composed of a number of s ubsystems , which are coupled via shared events. Specifications are given for each subsystem individuall y. and the task is to synthesize indi vidual supervisors. As the subsystems are coupled, synthesis wi II in general need to refer to the sy nchronous product of all subsystems. Conditions under which such product can be avoided are given in Lee and Wong (2002).
279
promises a substantial computational benefit. This is demonstrated by an example. The outline of the paper is as follows. Basic notations and definitions of supervisory control theory are recalled in Section 2. Section 3 and Section 4 introduce structural decentralized and hierarchical control of discrete e vent systems, respectivel y. In Section 5, both methods are combined so as to form a decentralized and hierarchical control architecture. A comprehensive example in Section 6 illustrates our contribution.
We recall basic facts from supervisory control theory. (Wonham, 2001: Cassandras and Lafol1une. 1999). For a finite alphabet I. the set of all finite strings over I is denoted I'. We write SI S2 E I ' for the concatenation of two strings SI, S: E L/ . We write SI :::; S when Si is a prefix of s. i.e. if there exists a string S2 E I' with S = 51S2. The empty string is denoted I' E I'. i.e. sE ES S for all S E I'. A lallguage over I is a subset H ~ I-. The prefix closure of H is defined by H:= {SI E I - .:::s E H S. t. SI:::; s}. A language H is prefix closed if H = H.
= =
A language H is said to be controllable w.r.t. L(G) if there exists a supervisor S such that H L(S/ G). The set of all languages that are controllable w.r.t. L(G) is denoted C(L(G )) and can be characterized by C(L(G)) {H ;; L(G)! 3S st. H L(S/ G)}. Furthermore, the set C(L(G)) is closed under arbitrary union . Hence . for every specification language E there uniquely exists a supremal conrrollable sublan· guage of E w.r.t. L (G), which is formally defined as KL (C ) (E ) := L:{K E C(L(G)) : K;; E}. A supervisor S that leads to a closed-loop behavior K L( c ) ( E ) is said to be ma;rimal permissive. A maximal permissive supervisor can be realized on the basis of a generator of KL(C) (E ). The latter can be computed from G and a generator of E. The computational complexity is of order O(N 2 M2 ), where N and M are the number of states in G and the generator of E , respectivel y.
=
=
=
A language E is Lm-closed if En Lm E and the set of Lm(G) -closed languages is denoted FLo/(c) . For specifications E E FLo,( C ) ' the plant L(G) is non blocking under maximal permissive supervision.
=
The nalural projection p , : I- -1 If' i L 2, for the I I L Iz is defined (not necessari Iy disjoint) union I iterati vely: (1) let Pil E) := E: (2) for s E I ' , G E I, let Pi(SG) := pi(S) G if G E I i , or Pi(SG ) := plI) otherwise. The set-valued inverse of Pi is denoted p,- I : Ii' -122:' , pil(l ) := {s E I' lp ,(s ) = T}. The sVllchronous product HI IHl ~ I ' of two languages Hi ~ I f is Hl' H2 = PI I (H J) n p2,1 (H2 ) ~ I ' .
3. STRUCTlJRAL DECEr\TRALlZED DES Structural decentralized DES as proposed in Lee and Wong (2002) are composed of subsystems. realized by finite state automata G i , i 1, 2 , ... ,11 with respecti ve alphabets I i . The synchronization of each two subsystems G, and G j is organized via shared events I , n I j '
=
=
A filliTe aUTomaton is a tuple G (X,I , 8,xo ,Xm)' where X is the finite set of stales: I is the finite alphabet of evellls: 8 : X x I -1 X is the pal1ial transition jimCfion: Xo E X is the iniTial STate: and Xm ~ X is the set of marked STates. We write 8(x , G) ' if 8 is defined at (x, G). In order to extend 8 to a partial function on X x I ' , recursively let 8(x, E) := x and 8 (x , SG ) := 8(8(x,s ).G ), w henever both x' = 8(x,s ) and 8(x 1 O' )'. L(G) := {s El- : 8(xo , s)!} and LIn( G ) := {s E L(G) : 8(xo.s ) E Xm} are the closed and marked language generated by the finite automaton G. respectively. For a formal definition of the sy nchronous composition of two automata G , and Gl we refer to e.g. Cassandras and Lafortune (1999) and note that L (G I ll G2 )
Definition 3.1. A decenrralized col1trol system (DCS) consists of subsystems, modelled by finite state automata G" i 1, ... ,11 over the respective alphabets I i . The overall system is defined as G:= ! i~IG , over the alphabet I := U;'=I I i· The controllable and uncontrol lable events are Ii.c := I i r I e and L i.u := I i n I ll, respectively. where I e L.; I u I and I c r I u 0. For bre vity and convenience. let L:= L(G ), L o , := Lm(G;. Li := L(G,) . and L,," := Lm(G i ) .
=
=
=
=
For our applications Vie assume local specifications 1, .... 17 for each subsystem G , . E, E FL I.", ~ I j, i Relative to the overall alphabet I, the Ei become (Pi) -I ( Ei ) . where Pi : I' -1 I i' denotes the natural projection. Taking into account the language L of G , the global specification is E r:'=l (p,) -l (E,) rL.
L(G I JI' L(G2 ).
=
When L( Gl represents the plant behavior in a supervisory control context, we write I I , U I iJ • I , r Iu 0, to distinguish controllable (I ,) and ul7COlllrollable (L u) events. A control paTtern is a set '(. I u ;; Y ~ I, and the set of all control patterns is denoted r ~ 22:. A sllpervisor is a map S: L(G) -1 r , where S (s ) represents the set of enabled events after the occurrence of string s: i.e. a super;isor can disable controllable events only. The language L (S / G ) generated by G under supervision S is iteratively defined by ( J) I' E
=
=
=
2. PRELlM]]',ARlES
=
L(S/ G) and (2) SG E L(S/ G) iff sE L(S/ G),O' E S(s) and SG E L(G). Thus. L (S/G) represents the behavior of the closed· loop system. To take into account the marking of G. let Lm (S/ G) := L(S/G) n Lm( G). The closed-loop system is nonblocking if L In( S/ G) L(S/G) , i.e. if each string in L(S/ G) is the prefix ofa marked sning in Lm (S/ G).
=
=
There are two approaches for generating a supervisor imp le menting the given set of specifications: the synthesis of a monolithic supervisor S for the overall specification E leading to the closed lang uage L(S/ G) = KJ.( E ) of the supervised system. and the synthesis of
280
=
local supervisors Si for Gi and Ei, i 1, ... , 11 resulting in 1!;~lL(SiIGi ) n~ l (Pi)-I (KdE;)) n L: for the second approach see Figure I.
=
G
.--
I
S
GIl l ,
L"I
~.
.
J...-"'T , G2
L2 L'
S2 ,
~~~ ~
; Sn
r-4
two levels are interconnected via Com hilo and In/ohi. The former allows Shi to impose high-level control on Slo , the latter drives the abstract plant G hi in accordance to the detailed model. From the perspective of the high-level supervisor. the forward path sequence Con/'ilo, Con lo is usuall y designated "command and control", while the feedback path sequence In!loh,. In!hi is identified with "report and advise". Formally. the high-level abstraction is defined as follows.
=
(2)
Definition 4.1. (Hierarchical Abstraction). Let G (X ,L ,O,XO , Xm) be a finite automaton and L hi s; La set of high-level events. A reporter map 2 is a map 8: I' -+ (Ihi)' such that (1) 8 (£) £ and (2) either 8(s(j) = 8(s ) or 8(s(j) = 8(s)(jIIl, where (j E I. (jhi E Lhi. The high-level language is defined by Lhi := 8(L(G )) . The high-level marking is chosen S.t. L~ s; Lhi, where L~ is required to be regular. The canonical recognizer of L~~ is denoted G hi , and hence, L(Ghi ) = Lhi . Lm (Ghi ) L~~. Finally, high-level con-
Theorem 3. J. (Structural Decentralized DES Lee and
trollable and uncontrollable events are denoted L;" and I~i, respectively. where Lhi I~i U L~i, L;" n I~i 0.
Fig. I. Structural Decentralized Control Structure
=
In Lee and Wong (2002), conditions under which both approaches lead to the same overall result are 1, ... , n the following holds: developed. Thus, for i
=
11
n (Pi)-1 (KLi(Ei))
= KdE) ,
(1)
i=)
pi( KdE )) is nonblocking w.r.t. Li.m.
=
=
Wong (2002)). Using notation as in Definition 3.1, denote the natural projection (Ii L I j -t Li· Suppose that Ei E FLi.", and that for i, j 1, ... , n, i :p j
p;; :
(i)
=
For our control architecture, not only the high-level events and the reporter map, but also the choice of contro!lable and uncontrollable events L~i and L~i are essential. For the synchronous composition of subsystems. we will show in Section 5 that this choice can be based on shared events.
Li.m marks I ,"'" L j , i.e. Lj (L,n L j)r Li.m
(ii)
r
s;
Li,m(Li;! L J )
(3)
r-:::l
Li and L j are mutually controllable, i.e. L;(Lj.llr Li)r p; j((p;)-I( Lj))
s;
Li
=
In!hi
~~ HCoi?ii~
(4)
COl11 hilo
hi G
In! I 0 h'I
t
r-~-
Then (I) and (2) hold. )
Con lo
In order to discuss the computational complexity, let N and M denote bounds for the number of states in G i and generators of E,. respectively. Then Nil and M n are the respective bounds for the number of states in the overa!l system. The computational complexity of the monolithic synthesis procedure is O(N2nM2n). whereas it is O(nN 2M2 ) for the local synthesis approach. It is clear that this benefit does not come for free. The approach in Lee and Wong (2002) requires the computation of the language projection (with exponential worst case complexity) in condition (4). However, G 1:':1 Gi need not be computed.
/ nf'°
~~
(b)
(a)
Fig. 2. Hierarchical Control Schemes For our fw·ther discussion, we need to define the interconnection of low- and high-level supervisors with the plant.
DefilliTion 4.2. (Hierarchical Control System). Referring to the notation in Definition 4.1 , a hierarchical conflvl SYSTem (HCS) consists of G, G hi, Shi and 51". where the high-level supervisor Shi and the lowlevel supervisor Slo fulfill the following conditions: Shi: L hi -t rh, with the high-level control patterns rhi := {YII~i s; y s; I.hi } ; and 510 : L(G) -t r with
=
4. HIERARCHICAL CONTROL
s; L(Shi I Ghi ).
We consider the event-based hierarchical control scheme in Zhong and Wonham (1990) (Figure 2 (a)).
8(L(Slo I G))
The detailed plant model G and the supervisor 5 10 form a low-level closed-loop system, indicated by Con lo (co ntrol action) and Inf o (feedback information) . Similarly. the high-level closed loop consists of an abstract plant model G lli and the supervisor Sh'. The
Given a high-level specification Ehi EFL;;;' we can
J
=
KLhi(Ehi) with synthesize Shi such that L(SIIII G hi ) a nonblocking high-level closed-loop. At this stage. the remaining task is to implement high-level control actions for the low-level plant by means of 510. 2 In the secuel. we focus our allention on the reponer map where is the natural language projection into (E'i)".
I'
'iOtc also that in :his case E EFL,,'
281
e=
p",
Definition 4.3. (Hie rarchical Control Problem). Given G:Ghi:S hi . find a low-level supervisor 5 10 as in Definition 4.2 such that the low-level controlled language of the HCS, L(Slo / G). is nonblocking.
U7=ILi. Relevant languages are L:= L(G). Lm := Lm (G). L, := L(Gi ), and LI.In := Lm(G,), controllable and uncontrollable events are L i.f Li n Le and Li,u = Li n Lu as in Definition 3.1. • Local low-level controllers are denoted 5 i : Li ~
On the one hand, there may not exist a low-level supervisor s in such that 8(L(SI0 /G )) L(Shi/ GllI) , and we end up with a strict subset relation. This is interpreted as an overoptimistic high-level supervisor expecting low-level behavior that is not possible. On the other hand. if the above equation turns out true for some s in, it provides a powerful tool to show that the the low-level is nonblocking. This is one motivation for the following notion of hierarchical consistency.
r i, where r, are the respective control patterns. Low-level closed-loop languages are denoted
=
= L(SIIl/ G hi )
Lt:= L(Si/ Gi). Lj.m: = Lj' rLi,ln ' U:= T~ILj . = LCr: Lmf. Also let GCbe a genL(G ), L;~ Lm(Gf ) . erator s uch that U • For the abstraction the reporter map: 8 := plti is used. where phi : L' ~ (Llti) , denotes the natural projection and Lltl := L i,j,ii J( Li n L j ). The highlevel language is LIII := 8(U ) and 3 L~ := ! 1;~,L;:m
=
=
t 0), with canonical recognizer GM s.t. Lhi = L(Ghl ). L~ = Lm(Gltl) . High-level controllable events are L:~ :=
Definition 4.4. (Hierarchical Consistency (HC)). The hierarchical control system in Definition 4.2 is hierarchicallv consistent if the following applies: 8(L(SiO / G ))
=
{s"i E L1711 (plti ) -l (i') r L~,
defined as "'i ;Ii := "'it r: "'ihi and L~' := LII n LhI • The high-level supervisor is denoted Shi : Lhi ~ r hi wi th the high-level closed- loop language L(Shi / Ghl ). A valid low-level supervisor 5 10 : U ~ r must fulfill 8(L(Slo / G')) ~ L(Slti / G/II ).
(5 )
There has been a variety of event-based contributions which tackle the complexity problem associated with the hierarchical abstraction and which provide appropriate definitions of the high-level marking as well as high-level controllable events in order to prove hierarchical consistenc y. Zhong and Wonham ( 1990) introduces the notion of strict-output-control-consistency to gU;lran tee a consistent low-level controller fulfilling a high-level specification. Blocking is not considered in this approach. da Cunha et al. (2002 ) utilizes a generalized model for controlled DES in the high-level to obtain hierarchical consistency without adding complexity by refining the hierarchy. Moor et al. (2003) uses Willems ' VO behaviours to establish a nonblocking hierarchical control architecture . Other state-based approaches (Hubbard and Caines. 2002) use state-aggregation for hierarchical abstrac ti on .
Lel1lma 5.1. (High Level Plant). Assume the control plzi( L::) . architecture of Definition 5.1 and let L7i Then the high level language is
=
LI" = p hi( l i ~ ILil = 11;'= ,L;'i . Proof (Sketch): We use induction and the identity po(HIII H2 ) = po(Htl :po( H2 ) for languages Hand H2 with alphabets LN I and LH2 and the natural projection Po : (LH U LH, ~ (2:0 with "'io ~ (LHI U LH, ) and (LI-I I n Llic ) ~ LO as in Wonham C~OOl). C
r
r
The previous lemma enables the computation of the hierarchical abstraction by only using the hierarchically abstracted subsystems and it is not necessary to compute the full low-level plant automaton. Consequently the computational effort reduces from computi ng i!7=I and s ubsequent projection pIIlC ~, L; ) to first computing phl( Lj ) and then evaluating 1 ~~ , p hi ( L: '). While technically the computational complexity is of the same order. a significant computational benefit shows in applications where realizati ons of plz:( Lj') have less states than those of L;·.
5. HIERARCHICAL COl\TROL FOR DECEf'.<"TRALIZED SYSTEMS
L:
We discuss how a combination of the decentra lized approach in Section 3 w ith the hierarchical approach in Section 4 can improve the computatio nal effic iency of supervisor synthesis : see Figure 2 (b) for our proposed overall scheme. As our detailed plant model. we consider a structural decentralized DES G I : .. . G, . G Gi. subject to local feedback contro l by supen'isors 5 , .... . Sn. The abstraction G It i of the lowlevel closed loop is based on the observation of shared events L ItI \ia Inl ol". A high-level supervisor 5 111 is designed to contro l G iif . where effect on the low level is taken via Con / 'iio and a low-le vel supervisor 51" .
For our further discussi on. we assume that the lowlevel subsystems fulfill co nditions (i) and (ii ) in Theorem 3.1 and that the lov.i-Ie\·e/ supervisors have been synthesized to enforce a local specification Ei E FLi .", ; i.e. L: := USi/ GiJ KL,( E,) . Thus. as a result of Theo rem 3. 1. the lov..·-Ievel is nonblocking. i.e. L f Lin '
= ': 7=,
=
=
The next theorem gives a possible choice of the lowleve l supervisor such that the proposed control architecture is hierarchically consistent and nonblocking.
Definirion 5.1. A Hierarchical and Decentralized Control System (HDCS ) consists of the following entities.
Theorem 5.1. (Main Result ): Let the hierarchical contro l architecture for structural decentralized DES be
• A detai led plant model is a decentral ized co ntro l system G := 1 ! ~~ , G i with subsystems G i O\'er respective a lphabets Li. i = 1: .... 1l and L :=
:; By cons~ruC', i o n ,~:;. is regul ar.
282
defined as in Definition 5.1 and define the low-level supervisor Slo for each 5 E L" as
SIO( s) :=
S ill (phI
(5))
l)
(~_ I hi).
Then s to is valid according to Definition 4.2, the HDCS is hierarchicallv consistent and the controller is maximally pem1i ssi~·e. In particular, Slo solves the hierarchical contro l problem in Definition 4.3
svnthesis: where N. N ili , Mil i denote number of states i~ Cc, Chi and the canonical recognizer of Ehi. respectively. Again, in the case of N hi < N we expect a computational benefit.
6. EXk\1PLE We consider the two cooperating machine cells C: and C~ given in Figure 3. Both machines have 6 states.
Proof: We use induction to show that Slo is valid, i.e. p'l!( L(SIO I Ce )) <;;; L(Shl I Chi). Obviously € E L(S'O I Ce) and phl(€ ) = € E L(Shi I C hi ). Pick any 5 and 0 E I with so E L(Slu I C t ) and phi (.I' ) E L(Shi I C hi ). Then either 0 E I 'l! or 0 E (I - I hl). In the first case, 0 E Slo(s) implies 0 E Shi( phl(S)) and, hence, ph/(SO ) = phi(S) O E L(ShijC hi ) . In the second case phi (so) phi(s) E L(Shil c hi ).
=
We prove hierarchical consistency by induction. As val iditv has been established above, we are left to show ;/I!( L(S'O I C')) ;2 KL'Ij( £ II!) . It is c lear that € E phi( L(SIOICC )) and € E KLhj( £ /lI). Let shi E K/J,,( £ hi) and Shl E phi( L(Slo I Ce)) and ass ume for Ohl E Ihi. shia hf E KLhi(£ hi) . Then Oil! E Shl(SIl! ) and by definition of the low-level super.isor Slo , Vs E (p ll!) -I (Sill) r . L' we know that a hi E SIO(S). Further on, as slll a hl E Lhi it follows that 3.\' E L' S.t. .1" E (phi) - I (s"i) and s' Ohi E Le. But then s' a hl E L(S!() I C') and thus phi (SI a hi ) = pill (s' )a lII = Sill a hi E phi (L(s in I c e)). For proving non blocking behavior it must be shown L ( SIOI C')n L;~. As L(SloICC) ;2 that L(SIOI CC)
Fig. 3. local machines C I and C2 The event y represents the start of the cooperation of the two machines, 8 indicates that the cooperation terminated successfully and
=
6.1 Hierarchical Method
=
L(Slo I C') 'i L';" is obvious , we on ly prove L(sln I C') <;;; L ( Slol ce )n L~" Assume s E L(SIOI C' ) but s rt L ( S!l)l cr)~I L';". As L(SIOI C') <;;; L~I' .I' E L~I ho lds, and thus s rt L(Slo I er ) n L~, requires that Vu E I ' S.t. su EL';" . 3a hl E I hi and u' ,u" E I ' S.t. u = u'Ohiu" and a hi rt 510 (su') and VLI I ~ u', SLI I rt L:~ . .. :"l'o\'. let u be such that su E L';" and let a hi , u l , u" as above. Then it holds that Pi(SUI) E Lun for i such that a hi E I i because of (3) and Vi S.t. a hi rt I i (let il, .... i be
Following the lines of Lee and Wong (1002) (3) holds as all states before shared events are marked. The mutual controllability condition (4) for i,j 1.1 is also va lid. For examp le, from Figure 3 and Figure 4 5 it can readi ly be observed that ~ is controllable w.r.t. (pn -l(p12 (L2 )) andtheeventsetI 2"n II. 6
=
~~
m
the corresponding indices), 311 i E (I i - I hi ) . such that Pi(S)Ui E L: m as L;' V;:. Thus SUI Ui, II'2 ... u:." E L:n
=
and thus
S
E L(S I I Ct) r .L~" U
=
Thus G Cl ' !C2 constitutes a structural-deceno'ali zed co ntrol system and we identify C as the low-level plant of a HDCS. The low-level specifications £ I and E2 in Figure 5 are contro ll able w.r.t. their respective subsystems and are identical to the low-level control led subsystems C~ and C S.
For prO\'ing maximal perrnissiveness we assume that 3si () as in Definition 4.2 s. !. L(SloI C') C L(slo I C") <;;; (p ill) -I (L(Sh, I C lli )) with 510 from Theorem 5 .1 . Then 3s E L(Si() I C' ) with shi phi(s ) E L(Shi I Chi ) bur t .I' rt L(Slo I C ). Because of HC 3s' E L(Slo I C') with phi (SI ) s'" and .I' .1" u with 11 E (I - I hi)'. But we know that (VSI U' < SIll), (I - I hi) E SIO( su' ) and thus .I' s'u E L(Si" I C'}, which contradicts the assumption and 510 is maximal ly permissi\·e. C
=
=
I" - ' ( 1 ' - \ Fig. 4. (P2) . PI- (L2.r11);
=
?"ow, in addition to low-level supervision. the highlevel specification E i:; in F ig ure 6 is implemented by employing our hierarchical control method. £ hi states that after the occurrence of two successive failures in cooperation no more cooperation should take place.
=
By the abO\e the orem. the complexity of synthesis for the high -level specification becomes Q( (/Vh·fll (Mh l) 2; compared to Q(.v2n(/\1 I!l) 2; for the local monolithi c
= {1I 1 . "2 ,a~.a", "5 .{/6 .(J; }. The con'c, ponding transition ;s labcled with a tic ~ as
'" T·::;
means a:: t!xtcn~ions
0:" S
lo a markcc string :11ust be disabled
and ai sabling. can on ly occur if a h igh-Ic\ ~: e\,ent;s disaoied 01' .~".
283
7 states for the case without high-level control. Thus, compared to our method the high-level specification must be decomposed, which leads to relatively large low-level supervisors.
7. CO:\,CLUSIONS Fig. 5. low-level specifications Eland E2
Fig. 6. high-level specification E h;
=
For the abstraction. I h' {y, 0 ,
G7'
Recent work in control of DES has been focused on exploiting structural properties for reducing the computational complexity of DES controller synthesis. Our approach embeds a decentralized approach Lee and Wong (2002) in a hierarchical control scheme and it was shown that our architecture guarantees nonblocking behavior of the controlled system. Furthennore we pointed out that hierarchical consistency need not be verified as it is directly implied by the proposed control architecture. The computational benefit of our method was illustrated by an example. Ongoing work aims for weaker conditions guaranteeing nonblocking behavior and the demonstration of our results by a laboratory case study of a manufacturing system.
=
Fig. 7. high-level abstractions G'i' and G~i It can easily be verified that E hi is controllable W.r.t. L (G h') and the high-level controlled language L(S'''/ G hi ) equals E h, with as-state recognizer. Because of Theorem S. I the low-level supervisor S'O( s) Sh' (phi(S)) U (I - I '") realizes the desired high-level behavior and implements non-blocking and maximally pennissive behavior in the low level.
=
6.2 Comparisol/ WiTh other meThods For the standard RW control synthesis the overall plant and specification automata have 26 states and 14 states, respectively and the supervised plant has 77 states. For applying the method of Zhong and Wonham (1990): da Cunha et al. (2002) it is also necessary to compute the overall plant automaton. The reporter map for the high-level abstraction is defined by labeling the low-level strings and for Zhong and Wonham (1990) there is additional computational effort for verifying hierarchical consistency. whereas in da Cunha et al. (2002 ), a supervisory control problem must be soh'ed for each low-level subsystem. corresponding to a high-level string via the reporter map. For the technique of Lee and Wong (2002) the additional highlevel specification Eh; can be decomposed into two local specifications p ;( Eill), i = L 2 which leads to low-level specifications with 8 states. each , and supervised low'-Ievel plants with 17 states both. versus
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