Abstract-The paper gives a new characterization of solutions of the decentralized supervisory control problem and presents a special class.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 12, DECEMBER 1995
2094
Fully Decentralized Solutions of Supervisory Control Problems P. Kozhk and W. M. Wonham
Consider a decomposition of the event label set given as an ordered {E , c , z } L t ~where ) , I is a nonempty set triple D = ( I , { C o , z ) z t ~ ~ { C c , z } z t rare families of sets with index set I and { E 0 , 1 } 2 tand such that Eo, I C (z E I ) and C, = UzEl Ec,2 . 2 The decentralized supervisor is s d = (G, D , { Y ~ } ~ ~ Iwhere ), for each i E I , y t :P,(L) -+ and P, is the natural projection. The decentralized supervisor Sd consists of subsupervisors S,(z E I ) . Each subsupervisor S, observes only events having labels in and controls only events with labels in Cc,z. For all i E I and U E P,(L), r z ( u ) is the set of events disabled by the subsupervisor S, after observation of U . A centralized supervisor s d = (G, y) equivalent to the decentralized supervisor s d = (G, { Y z j z G l ) is defined using a feedback mapping ;Y:L + 2cc such that for all v E L,;U(w) = U I E Iy2[Pz(L)]. The closed-loop language generated by G and by D , { Y ~ } ~ E isI ) defined as the a decentralized supervisor s d = (G, closed-loop language generated by G and the equivalent centralized supervisor Sd . A new definition important for the existence of decentralized supervisors is decentralizable language w.r.t. G and D . A language K C L is decentralizable w.r.t. G and D if for all s E pref ( K ) and a E C , there exists i E I such that o E Cc, and for all s’ E pref (IC) 2 c c 5 t
Abstract-The paper gives a new characterization of solutions of the decentralized supervisory control problem and presents a special class of “fully decentralized” solutions, so called because the main idea is to avoid communication not only among subsupervisors during run-time but also among agents designing the subsupervisors. The solutions are constructed from optimal centralized ones using a worst-case philosophy. Their basic properties and comparison with the partially observable supervisory control problem are given. If it is known a priori that the fully decentralized solution exists, then the supervisor can be computed partly on-line. A sufficiency test of the existence of such a solution is provided.
I. INTRODUCTION It is well known [l] that the design procedure for a centralized supervisor has polynomial complexity but that design under partial observation (therefore also decentralized supervisory design) is NPcomplete [a]. Therefore specific solutions (useful typically for specific cases) become important for the decentralized supervisory control problem. A designer should be provided with a sufficiently rich set of such specific solutions, with rigorous description of their properties, some heuristic rules reflecting typical usage, and effective construction procedures. The proposed fully decentralized solutions are studied and compared with the infimal decentralized solutions in the paper. The design of fully decentralized solutions requires, in general, computing a number of language projections and so has exponential worst case complexity [3]. Therefore they may seem useless for automated synthesis. Fortunately, the computation can be broken into off-line and on-line parts with overall complexity proportional to the size of the centralized supervisor. Additional complexity reduction can be achieved by reduction of the centralized supervisor size using modular design [4] or on-line implementation in the sense of [5].
11. DECENTRALIZED CONTROL AND DECENTRALIZABILITY Consider plant given by the ordered triple G = (C. L, E,) with usual meaning.’ The prefix-closed language L models the uncontrolled behavior of G. A supervisor is defined as S = (G, -y), where y:L + 2cc gives disabled events. The closed-loop language generated by the feedback interconnection of G and S is defined and denoted as
C ( S / G )= { U
E LlVw’ E pref(v),
w’a E pref(w)
V a E C,
+a @ ~(v’)}.
v,
The definition means that for each event label a that must be disabled after some s E pref ( K )(to keep the closed-loop language in K ) there must exist some subsupervisor i E I which may disable a without removing from the closed-loop language any other word d o E li (where s‘a can be generated by the plant and s‘is not distinguishable by the subsupervisor i from s). The following theorem shows that decentralizability is a key part of the necessary and sufficient condition of solvability of the decentralized supervisory control problem, in the case of an arbitrary (finite) number of subsupervisors (cf., [9], [lo]). The concept of decentralizability and the proof of the following theorem (corresponding to Theorem 4.1 [9] are more concise and provide an alternative to the definition of co-observability [9], [lo]). Theorem I : Let K L. There exists a decentralized supervisor s d = (G, D , ( Y z j l E I ) realizing IC on G if and only if K is nonempty, prefix-closed (i.e., pref (I{) = K ) , controllable [i.e., pref(K)C, n L pref(K), where C , = C - CJ, and decentralizable w.r.t. D . Pro03 According to Proposition 5.1 [6],it remains to show that decentralizability is necessary and also sufficient. (If.) Let K 2 L be nonempty, prefix-closed, controllable, and decentralizable w.r.t. D . For each s E K and a E C such that s5
Manuscript received May 20, 1995. This work was supported in part by Bell Canada Research Contract 3-254-188-10 (University of Toronto). P. KozBk is with the Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodirenskou v6ii 4, 182 08 Prague 8, Czech Republic. This work was done when the author was on leave at the Systems Control Group, Department of Electrical Engineering, University of Toronto, Toronto, Canada. W. M. Wonham is with the Systems Control Group, Department of Electrical Engineering, University of Toronto, Toronto, Ontario, M5S 1A4 Canada. IEEE Log Number 9414678. See [6]-[9] for necessary background and details.
’
E L A s5
(1)
1%-
denote by ~ ( s a, ) the index i E I of a subsupervisor satisfying the condition of decentralizability (controllability implies that a E E,-), i.e., a E & t and for all S I E K [so E L A s a $?
Apt(.)
=Pz(s‘)A s‘a E L] s’o @ IC.
’The assumption does not restrict generality, because it is always possible to reduce the set of controllable events in the plant mode’l.
0018-9286/95$04.00 0 1995 E E E
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 12, DECEMBER 1995
Let each subsupervisor i E I of such that for all w E P,(L)
r t ( w ) ={a
sd
= (G, p,{ r t } z ~ber defined )
E CC,,13s E
Ii,
so E L A so $2 I
