-
Manufacturing Cell Supervisory Control A Timed Discrete Event System Approach
B. A. Brandin
W. M. Wonham
B. Benhabib
brandinOcontrol.utoronto.ca
[email protected]
benoQme.tmnto.edu
Computer Integrated Manufacturing Laboratory (CIMLab) University of Toronto Toronto, Canada M5S 1A4 Abstract
proposed in [2]; and an approach to workcell management which incorporates some of the features of [2] but which relies on the control theory for PES for supervisory control purposes [ll]is presented in [3]. The specific issue of designing supervisory control strategies may exploit techniques such as artificial intelligence [l,21, Petri nets [5], timed transition models (TTM)[8], real time temporal logic (RTTL) [9], and controlled automata [3, 7J. The advantage of considering a controlled-automaton based approach, over other approaches, lies mainly in the existence of design techniques which guarantee that the supervisory control strategies obtained are correct by construction, and that these are least restrictive subject to the speo ifications considered [ll].This paper proposes a new approach to the design of supervisory control strategies for manufacturing cells, based on the supervisory control of timed DES.The approach yields strategies which can be implemented on any management system based on closed-loop supervisory control. In particular, the strategies obtained are compatible with the workcell management system proposed in [3].
This article proposes a new approach to workcell supervisory control strategy design based on the control theory for discrete event systems (DES). A framework for the modelling and supervisory control of timed DES is presented in which the concept of forcing as a means of control is introduced, and in which hard temporal behaviouml specifications may be treated. Within this framework, supervisory control strategies are designed according to production specifications and subject to control enforcement constraints.
1 Introduction The flexible manufacturing concept advocates that, as one of the options, production operations be carried out within cells, each cell being responsiblefor the production of a specific part family. Supervisory control, communication and “housekeeping” constitute manufacturing cell management. Supervisory control consists of (i) the monitoring of the cell behaviour via sensory feedback, (ii) control synthesis in accord with the supervisory control strategies, and (iii) control enforcement via the downloading and execution of the appropriate device programs. Communication allows sensory feedback and control enforcement to be performed. “Housekeeping” is the set of tasks related to supervisory control and communicationwhich are necessary to their implementation, e.g. data-base management. Various approaches to workcell management in general are found in the literature: a knowledge-based on-line system which deals with execution monitoring, action scheduling and failure diagnosis and recovery is proposed in [l]; a workcell management system architecture based on the use of an expert system, a parallel processing system and specific hardware components, is advocated in [12]; a management system based on artificial intelligence techniques are
2
A Timed Discrete Event System Approach to Supervisory Control
DES are controlled as generators of a formal language; the adjunction of a control structure allows varying the language generated by the system within certain limits (specifications) by accordingly enabling and disabling events. The explicit introduction of time in the modelling of DES opens a new dimension in the supervisory control of DES. It allows temporal behavioural specifications to be considered; and it allows illegal plant behaviours to be either prevented by disablement or preempted by forcing [4]. The notion of forcing is investigated in [SI in a context which does not consider time explicitly. An a p proach to the formulation and control of real-time discrete event processes which defines semantics other 93 1
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In the present context: .All controllable events are disabled by default, i.e. are disabled unless enabled or forced. uncontrollable events are assumed always to be enabled and cannot be forced to occur: their upper and lower time bounds may not be altered. ~Prohibitibleevents may be disabled or enabled to occur by an external agent any time after their lower time bound is exceeded. The lower time bound may be increased 80 as to equal the upper time bound. Since the enablement of a prohibitible event does not guarantee its occurrence, it will happen sometime or possibly never; the upper time bound is set to 00 by default and cannot be modified. .Forcible events may be enabled to occur by an external agent any time after their lower time bound is exceeded. The lower time bound may be increased so as to equal the upper time bound, and the upper time bound may be decreased so as to equal the lower time bound. Since in the absence of specifications and control enforcement constraints, a forcible event will happen sometime or possibly never, its upper time bound is set to 00 in the open-loop system. The forcing of a forcible event guarantees its occurrence within the related window of opportunity, if it is not preempted by the occurrence of another event causing it to become physically impossible.
than the T T M semantics [8, 91 considered herein, is proposed in [3]. Production specifications are claasified into logiobased specificationeltemporalproduo tion specifications , and utility optimization specifications. Control enforcement constraints stem from the nature of the supervisory control and communication systems used: these are the effects of communication time delays between the cell devices and the manags ment system [lo].
2.1
The Modelling of Timed Discrete Event Systems
Assumptions: The following assumptions will be adopted when modelling physical DES with timing: supervisory tasks of finite duration are considered; time is sampled periodically by a clock; time is sampled persistently while the clock is operating; at most a finite number of activities or states may be visited by the supervised system in a finite time interval. Events: In order to model physical systems as DES we need to model on the one hand, events which are endogenous, i.e. their occurrence is caused by agents internal to the physical system, and on the other hand, events which are ezogenous, i.e. their occurrence is caused by agents external to the physical system. A machine breakdown is a typical example of an endogenous event. In the manufacturing context, external agents may consist of machine operators or supervisory control systems. From a control perspective, endogenous events can be thought of as uncontrollable, and exogenous events may be thought of as controllable. Controllable events are subdivided into two categories; events whose OG currence may be disabled or enabled, but not forced by an external agent (prohibitible), and events whose occurrence may be disabled or forced by an external agent (forcible). It is understood that the forcing of a forcible event implies its enablement. Let C, = CjUC, be the set of controllable events, where C j is the set of forcible events, and E, is the set of prohibitible events. Let C, be the set of uncontrollable events. Introduce the tick event which represents the clock tick. Let E = C,UE,U{tick}. Let N = {1,2,3 ,...}. Let N = NU{oo}. To every event U E C,OC, is associated a time interval (lo, U,,) E x N with 0 5 I,, 5 U,, 5 00, which represents the window of opportunity of the occurrence of U after the system has entered the exit state of U. The parameters Z,,, U, are respectively the lower, upper time bound of U. It is understood that U,, = 00 means that U will happen sometime or possibly never.
Timed Transition Models (TTM), Activity Transition Graphs (ATG) and Timed Transition Graphs (TTG): TTM are used to represent the processes of the plant and its controller [8]. Nevertheless, supervisory control synthesis in this framework lacks the development it has undergone in the RW framework [ll] ; mainly, no notion of a supremal controllable sublanguage of a given language is developed. On the other hand, time is not introduced explicitly for control purposes in the RW framework. Let a T T M of a sytem M be represented by a finite state activity transition graph (ATG). In the following, it will be shown how an ATG is translated into a timed transition graph (TTG). A TTG is a transition structure which can be represented by an automaton in which time appears explicitly, and which can be considered in the extended RW framework presented in Section 3. Let G = (E, Q,6, qo, Qm)be a generator which represents the ATG for a TTM M = (V,8,J), where V,8 and J have their usual meaning [SI. Let Q be the set of states q of G; these represent the activities of M. Let qo E Q represent the initial activity of M. Let Qm Q represent a subset of activities of M which mark the completion of a task. The states q E Qm are termed marker states [U]. Let C,UC, = {q, .. ,um} be the set of events from which
.
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Ri =
{
{(h,Ui)),
-
{(li, ui), (li - 1,ui), . (0, Ui)}, {(li, ili),(li- 1, - I), ...,(0, ~ie,
Then, (Vi E (1,. ..,m})l&l= ri
ri =
{
0 li ui
e
-
+ 1 where
Qr
(2)
Qcr
Define a choice point to be either the initial state of G or a state of G where a transition becomes defined or where it has just occurred and remains defined [8]. At a choice point for (Q,ui, 4’) E 7 set (si,ti) := ( l i s ui). Without lose of generality, let (Vu6 E C,)ui = 00. Replace the old states Q by Q = Q x @ E Q-is defined as k = (qo, (4, ui), ..,(lm, um)). Qp C Q is given by Qm = Qm X&. Then I Q I = I Q I nEl(ri 1). Define 7 to be the set of transitions of the new transition structure (TTG) with state set Q. I is defined as follows. Let q E Q. Let = (4, [ s l , t ~ ).,..,(sm,tm))E 0. A tick transition (Q, tick, q‘) E I is defined if and only if
.
(Vui E C)S(Ui, q)! jti (Vui E C)+(ai, a)!
(6)
n,“=,&.
nEl
> 0 01
if -6(Ui, a)! if si > 0 and 6(ui,q)! if Si = 0 and S(Ui, Q)!
= {e E QI(3sE E*)a(s,Q) E Qm}.
(7)
Let Qrm = Qr n Qm. Note: Since VU^ E &)Uj = 00, and because of the definition of 7 ,any state ij E Qr is the exit state either for some uncontrollakle event or for the tick event. Finally, let G = (E, Q r , 8, q,,, Qrm) represent the timed transition graph with transition set 7 and state set Qr. The generator1 6 can now be considered in the extended RW framework presented in Section 3. The Composition of Timed DES: The TTM convention for system composition [8, 91 is adopted [ll].Any two systems MIand Ma composed in parallel, generate “cooperatively” by agreeing to synchronize those transitions with label U which they possess in common. The time bounds associated with the latter transitions are assumed identical in both systems.
e
2.2
Controllability and Supervisory Control
In this case the timers are updated according to TTM semantica [SI as follows:
-1
= {q E Ql(3~ E P)~(s, qo) = Q}.
Let Qcr be the coreachable subset of Q, i.e.
if ui = li = 00 if li < 0O,ui = 00 otherwise.
+
. Fm,
Then 2 E Q is given by 3 = (q’, (si,t i ) ,. . , tA)). Let 6 : Q x + Q (pfn) be the transition funo tion corresponding to 7. Following [8] we say that (Cui,$) E I is pending if (q,ui,q/) E 7 and its current lower time bound is positive. We will say (q,ui,3) E is eligible if (4,ui,q’) E 7 and its current lower time bound is zero. In general, not all the states q E Q are reachable from qo. Let Qr be the reachable subset of Q,i.e. if ~i= lj = 00 if li < 00,ui = 00 (1) otherwise. li),(0, ui - li l),...,(0,O))
the transitions of M are derived. Let the transition function 6 : Q x C,SrC, + Q (pfn) be defined at each q E Q for some subset of the events U E C,UC,. The notation 6(u,q)! will mean that S(u,q) is defined. Let 7 = {(Q, ai, q’)l6(ui, q ) = q’, ai E CCUC,) be the set Of transitions of G. To every transition (q, ai, q’) E I are associated a lower and upper time bound l(q,#ilqt) = Ii9 u(9j~ip9’) = tli E A, with 0 5 li 5 Ui 5 00. Introduce a timer (si,ti) E & for ai E XcUCu where
In this Section, some of the results developed in the RW framework [ll]are extended in order to enhance supervisory control with timing and forcing. In the following, let G = (C,Q,6, QO, Qm),be a generator representing a TTG derived from an ATG according to the procedure illustrated above. Let G be a controlled generator [ll]. Let C = C,UC,U{tick}. A particular subset of events to be enabled or forced, can be 86 lected by specifying a subset of controllable events. It is convenient to adjoin with this all the uncontrollable events as these are automatically enabled. Each such subset of events is a control pattern; and we introduce the set of all control patterns
(4)
Then 2 E 8 is given by 2 7 (q,(si,ti),...,( s k , t k ) ) E 8. Now, let uj E C,UC,. A transition (q, ~j ,2) E I is defined if and only if (q,uj,q’) E 7 and sj = 0. The timers are updated according to TTM semantica as follows:
lThe occurrence of tick must not be indefinitely postponed: the timed transition graph (TTG)G of G must be restricted to rule out tick-free loops.
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For simplicity,it is assumed that the transfer of workpieces within the cell constitutes part of the processing of a workpiece by the machines. These are either available for production or unavailable. Machines MI and M2 become operational 1 time unit after being forced to start working; pl-parts take 3 time units to be processed on MI, and 1 time unit to be processed on Ma; pa-parts take 2 time units to be processed on MI, and 4 time units to be processed on M2. The closed-loop behaviour of the cell must satisfy the following specifications: (i) logic-based specifications: a part can only be proceased by one machine at a time; pl-parts must be processed by machines MI and Ma in that order; pa-parts must be processed by machines MZ and MI in that order; one pl-part and one pa-part must be produced in each production cycle; (ii) temporal specification: the production cycle must take at most 10 time units to complete; (iii) utility optimization specification: minimize the production cycle time. The control enforcement systems considered are composed by a centralized supervisory control system and a communication system, such that the communication time delays between the manufacturing cell and the supervisory system are negligible in comparison to the production times. The machines MI and M2 are modelled in Fig. 1,
r = (7 E 9 1 72 E,}. (8) A superuisory control for G is any map V :L(G) + I’. The pair (G,V )will be written V/G,to suggest “G under the supervision of V”. The closed behaviour of V/G is defined to be the language L(V/G) L(G) described as follows: (i) c E L(V/G), (ii) if s E L(V/G),U E V ( s ) ,and su E L(G) then su E L(V/G). (iii) no other strings belong to L(V/G). We always have {e} C L(V/G) 5 L(G). Clearly L(V/G) is nonempty and closed. The marked behaviour of V/G consists exactly of the strings of Lm(G) that “survive” under supervision by V . We always have 0 5 L(V/G) E L(G). We say that V is nonblocking for G if Lm(V/G)= L(V/G). An Extended Controllability Criterion: The proposed controllability notion supports two control mechanisms: the prevention of illegal strings by disablement and the preemption of illegal strings by forc ing. Let K C L(G) C C*. Let s E I?. Let AK(s) = { U E Clsu E I?}. Let AG(s) = { U E Clsu E L(G)}. Our main objective is to characterize those languages that qualify as the marked behaviour of some supervisory control V . To this end we define the language K to be controllable with respect to L(G) if
where a i j , i ,j = 1,2 E C j and p i j , i , j = 1 , 2 E C,. The timing information is given below: Mi: Wi(1, CQ), &1(3, 31, W2(1, CQ), &2(2,2); Ma: %1(1, CQ), P z I ( ~I), , W Z ( ~CQ,), ha(494). T h e aPpears explicitly in the machine timed transition graphs as illustrated’ in Fig. 2 for M I . The logic-based specifications are modelled as generators in Fig. 3, 4 and 5. The closed-loop cell behaviours which satisfy the specifications given in the problem description are obtained as follows: (i) M is the parallel composition of MI and M z , i.e. the cell open loop behaviour; it is a TTG with 81 states and 121 transitions. (ii) Noting that the specifications SP1 and Spa are enforced by specifications SP3 and SP4, the combined logic-based specifications is given by SP = SP3 n SP4 n SP5 (16 states, 121 transitions). (iii) The plant closed-loop behaviour which meets the logical specifications in the freest possible way is given3
The controllability of K means that the occurrence of an uncontrollable event never results in exit from K; and that if the occurrence of tick were to to result in exit from K then tick can be preempted by the (forced) occurrence of a forcible event. Existence of Non-Blocking Supervisors: Proposition: Let K 5 Lm(G),K # 0. There exists a nonblocking supervisory control V for G such that L,(V/G) = K if and only if: K is controllable with 0 respect to G and K is L,(G)-closed. Supremal Controllable Sublanguages: Let E 5 C*. Let C ( E ) = {K EIK is controllable with respect to G}. Proposition: C(E) is non-empty and closed under arbitrary unions. In particular C(E) contains a unique supremal element, which we denote by sup C(E).
3 Supervisory Control Design for a Two-Machine Workcell
‘In all the Fig. the tick event is labeled t for brevity. 3Here and below we write G3 = SUPCON(G1,Gz) to denote the operation that returns a TDES G3 whose marked behaviour is the supremal controllable sublanguage &upC(Lm(G1),Lm(G2));while its closed behaviour L(G3) = Lm (G3).
Problem Description: The manufacturing cell considered consists of two machines, an input conveyor (00source) and output conveyor (a sink). Two types of parts, p1 and p2, are processed by both machines.
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by SUP = S U P C O N ( M , S P ) (108 states, 104 transitions) , i.e. the supremal controllable language of the plant represented by M with respect to the combined logiobased specifications represented by S P . (iv) S U P is checked to determine whether it meets the constraints imposed by the control enforcement systems. If SUP does not satisfy these constraints, it is reduced until they are satisfied. An empty closedloop behaviour could be obtained since blocking is not admissible, in which case the plant or specifications need to be modified. Let RTSUP be the new closedloop behaviour obtained. Note that RTSUP C S U P . In the present example, according to the problem description, it can be verified that RTSUP = S U P . (v) Given R T S U P , we can now consider the tempe ral constraints given above. The cell behaviour which results in the production cycle being carried out in 10 time units or less is determined in this example as follows. Consider a timer W representing a 10time-unit sequence whose states are marked and selflooped with C,UC,. By intersecting the behaviour of W with the behaviour of M , the strings of M which correspond to a production cycle time of 10 time units or less are obtained since the end of the production cycle is marked. Of course, we must guarantee that the 10 time units deadline is actually met, if necessary by suitable forcing action. To determine whether such a guarantee is feasible, it suffices to check that the corresponding suprema1 controllable sublanguage is nonempty. Let the behaviour obtained be T S U P = S U P C O N ( M ,W ) (209 states, 263 transitions). Note that T S U P RTSUP. (vi) Again with reference to R T S U P , we next consider the utility optimization specification given above. Let OSUP be the cell behaviour which results in the production run being finished in minimum time. In this example, OSUP (19 states, 21 transitions) is obtained by iterating step (v): timers of durations 10,9,. .. time units are examined until the empty behaviour results. In this example, 7 time units is the minimum possible cycle time. Note: dynamic programming could have been used in step (vi) to determine the cell behaviour which results in the production run being finished in minimum time.
4
Conclusions
An approach to the design of supervisory control strategies, based on a new framework for the modelling and supervisory control of timed DES,was proposed. Strategies which can be implemented on any management system based on closed-loop supervisory control are obtained; in particular these are compat-
ible with the workcell management system proposed in [3]. The approach advocates the incorporation, within the timed DES framework, of logic-based specifications, control enforcement related constraints, and temporal and utility optimization behavioural specifications, in that order. It is based on the supervisory control of timed DES and yields supervisory control strategies which are least restrictive within given specifications, and which are correct by construction. It therefore offers potential advantages over other design approaches.
5
Acknowledghments
It is a pleasure to acknowledge helpful discussions with Siu O’Young, Kai Wong, Ben Schwartz, Mark Lawford and Chung-Yan Yuen. This work was partially supported by the Manufacturing Research Corporation of Ontario (MRCO).
References [I] R. Alami and H. Chochon, “NNS, A KnowledgeBased On-Line Systcm for an Assembly Workcell”, IEEE, Int. Conference on Robotics and Automation, 1986, pp. 603-609. [2] B. Benhabib, C.Y. Chen and W.R. Johnson, “An Integrated Manufacturing Workcell Management System”, ASME, Manufacturing Review, Vol. 2, No. 4, Dec. 1989, pp. 266-276. [3] B.A. Brandin, W.M. Wonham and B.Benhabib, “Discrete Event Supervisory Control Applied to The Management of Manufacturing Workcells”, 7th International Conference on Computer Aided Manufacturing Engineering, V.C. Venkatesh and J.A. McGeough Eds., Elsevier 1991, pp. 527-536. [4] Y. Brave and M. Heymann, “Formulation and Control of Real Time Discrete Event Processes”, Proc. 27th Conference on Decision and Control, Austin, Texas, December 1988, pp. 1131-1132. [5] P. Freedman, “Time Petri Nets, and Robotics”, IEEE, Transactions on Robotics and Automation, Vol. 7, NO. 4, August 1991, pp. 417-433. [6] C.H. Golaszewski and P. J. Ramadge, “Control of Discrete Event Processes with Forced Events”, Proc. 26th Conference on Decision and Control, Dec. 1987, pp. 247-251. [7] G. Hoffmann, C. Schaper and G. Franklin, “Discrete Event Controller for a Rapid Thermal Multipre cessor”, American Control Conference 1991, Boston, MA, June 26-28,1991, pp. 2936-2938. [8] J.S. Ostroff and W.M. Wonham, “A Framework for Real-Time Discrete Event Control”, IEEE, ’Ransactions on Automatic Control, Vol. 35, No. 4, April 1990, pp. 386-397.
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J.S. Ostroff, ‘Temporal Logic for Real-Time Syetems”, Research Studies Press Ltd, 1989.
Y. Li and W.M. Wonham, “On Supervisory Control of Real-Time DiscreteEvent Systems”, Information Sciences 46, pp. 159-183, 1988. P.J. h a d g e and W.M.Wonham, “The Control of Discrete Event Systems”, IEEE, Proc., vo1.77, No. 1, January 1989, pp. 81-98.
*{war&a, a z a , ha1
K. P. Valavanis and P. H. Yuau, “Hardware and Software for Intelligent Robotic Systems”, J. Intell. Robot. Syst., Theory Appl., Vol. 1, No. 4, 1989, pp. 343-373.
* * {all,&1, az1,Pa11
Fig. 3b) SP2
SPS
M i , i = 1,2
aij, i, j
= 1,2: Mi starts operating on a part of type j ;
Bij, i , j = 1,2: Mi finishes operating on a part of type j . Fig. 1 The machine activity transition graphs Ml &1
&a
Fig. 2 The TTG of MI
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