Representation of Supervisory Controls using State Tree Structures, Binary Decision Diagrams, Automata, and Supervisor Reduction

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1 Representation of Supervisory Controls using State Tree Structures, Binary Decision Diagrams, Automata, and Supervisor Reduction Wujie Chao 1, Yongmei Gan 2, Zhaoan Wang 3, W. M. Wonham 4 1. School of Electrical Engineering, Xi an Jiaotong University, Xi an, China, chaowujie1013@163.com 2. School of Electrical Engineering, Xi an Jiaotong University, Xi an, China, ymgan@mail.xjtu.edu.cn 3. School of Electrical Engineering, Xi an Jiaotong University, Xi an, China, zawang@mail.xjtu.edu.cn 4. Department of Electrical and Computer Engineering, University of Toronto, ON, M5S 3G4, Canada. wonham@control.utoronto.ca Abstract: In the synthesis of an optimal nonblocking supervisor for a discrete-event system (DES), the problem of state explosion is a well-known computational obstacle. This problem can often be managed successfully by the use of state-tree structures (STS) and binary decision diagrams (BDD). Unfortunately BDD control functions may become quite large, and as such difficult to represent and interpret. In some cases it may, therefore, be convenient to convert an STS/BDD based controller to automaton form, and then apply a well known algorithm for supervisor reduction. In this paper we illustrate the advantage of this approach with a concrete example. Key Words: Discrete event systems, supervisory control, state tree structures, binary decision diagrams, automata, supervisor reduction 1 INTRODUCTION Supervisory control theory (SCT) was initiated by Ramadge and Wonham[1][2]. The plant and specification are modeled by finite deterministic automata, which is a convenient and transparent way to model and interpret the discrete-event system (DES). The principal function in SCT is the synthesis of an optimal (maximally permissive) nonblocking supervisor. In [3], Gohari and Wonham proved that this problem is in fact NP-hard. A complex DES is usually modeled as the product of a number of simpler components. So, the state size of the DES increases exponentially with the number of components. That is, the state space of a complex DES suffers exponential state explosion. In order to manage the problem of state space explosion, Chuan and Wonham develop state tree structures (STS) [4]. STS is a modeling method for DES. It is an extension of the automaton model in SCT which introduces natural hierarchical structure into the system model. The symbolic synthesis algorithm of STS is based on binary decision diagrams (BDD) [4]. The algorithm can often successfully manage exponential state explosion. It can design nonblocking optimal supervisors for systems with state space size, remarkably, up to [5]. Furthermore, the supervisor is both tractable to implement and often transparent to comprehend. But for some systems, such as the Flexible Manufacturing System of [6], the BDD of the control functions in the controller can contain up to five hundred This work is supported by China Scholarship Council under Grant [2010]3006. nodes. Such controls may be difficult to interpret and implement. So, in some cases it may be convenient to convert an STS/BDD based controller to automaton form. In this paper, we convert an STS/BDD controller to automaton form, as shown in Figure 1. That is, for a controlled system, we keep the plant as an STS, replace the tracker in the controller by an automaton and replace the BDD decision maker by a control data table which indicates which events must be disabled at each state. Usually, the automaton tracker in this controller represents the controlled behavior of the controlled system [7]. For a large scale system, the state size of the controlled behavior will be quite large. The controlled behavior of a plant, however, incorporates redundant information about transition constraints that the plant itself already enforces. If we project the plant structural information out of the controlled behavior, we often get a supervisor with many fewer states. This can be done using a well known algorithm for supervisor reduction [8] [9]. In this paper, we also get a reduced supervisor for the STS model. The remainder of this paper is organized as follows. Section 2 summarizes the preliminaries on supervisory control theory. Section 3 introduces the basics of STS. Section 4 presents the method to convert an STS/BDD based controller to automaton form. Section 5 explains how to compute the reduced supervisor based on automaton form. In section 6 we provide the concrete Flexible Manufacturing System (FMS) example to illustrate our method. Conclusions are presented in Section 7.

2 G(STS) Plant s enabled events Gtracker (STS) f f 1 2 b f n controller G(STS) Plant s enabled events Figure 1: Illustration of conversion 2 PRELIMINARIES ON SCT Decision maker controller In this section we give the preliminaries on SCT. Details may be found in the extended treatment [7]. In the SCT framework, a DES is represented by a deterministic automaton G = (Q, Σ, δ, q 0, Q m ), where Q is a finite set of states; Σ is finite set of events; δ is the (partial) transition function: δ : Q Σ Q; for a state q Q and event σ Σ, we write δ(q, σ)! to mean that δ(q, σ) is defined; q 0 Q is the initial state and Q m Q is the subset of marker states. Let Σ be the set of all finite strings of elements in Σ, together with the empty string ε. The transition function δ is extended to δ : Q Σ Q in the natural way. The closed behavior of G (the strings that are generated by G) is defined by L(G) := {s Σ δ(q 0, s)!} and the marked behavior (the strings that are generated by G and lead to a marker state) of G is defined by L m (G) := {s L(G) δ(q 0, s) Q m }. The control objective of SCT is, for a given plant G and the specification SP EC (also a DES in the SCT framework), to find a supervisor SUPER such that the closed loop language is, in the sense of set inclusion, the largest sublanguage of L m (G) L m (SP EC) which is controllable wrt to G and also nonblocking, namely L m (SUPER/G) = supc[l m (G), L m (SP EC)], where SUPER/G means G under the supervision of SUPER. In practice, SUPER is implemented by using a structure SUPER(SUPER, φ) consisting of an automaton SUPER = (X, Σ, ξ, x 0, X m ) together with a feedback map φ: X Γ [13]. Here Γ := {γ P wr(σ) γ Σ u }. Here, we consider SUPER as a tracker and φ as a decision maker. The tracker tracks the current state of the plant; the decision maker is a table called control data table that indicates which events must be disabled at this state. Usually, the tracker SUPER is exactly the closed loop behavior of G, that is, L m (SUPER) = L m (SUPER/G) = supc[l m (G), L m (SP EC)]. 3 BASICS OF STS In this section, we present the basics of STS theory. Details may be found in the extended treatment [10]. The state space of an STS is organized as a state tree, say ST. The nodes on a state tree are called states. A state tree has a unique root state. A state on a state tree is called an AND (resp. OR) superstate if it can be represented as the Cartesian product (resp. disjoint union) of its children. Each child state is an AND (resp. OR) component, except for the lowest level states, which must be OR components. If we remove some OR components and their children on a given state tree, we get a new state tree, called a sub-state-tree. We use ST (ST) to denote the set of all sub-state-trees of ST. If every OR superstate on a given sub-state-tree has exactly one child, then this sub-statetree is called a basic sub-state-tree; the set of all basic sub-state-trees b of ST is denoted by B(ST). Note that, an element in B(ST) is equivalent to a state in SCT framework [10]. Every OR superstate on a state tree is assigned a dynamic structure called a holon, namely an automaton with both boundary and local state transitions. On the set ST (ST), we define a forward state transition function : ST (ST) Σ ST (ST). If the event σ Σ occurs, the function maps a sub-state-tree S to another sub-state-tree S = (S, σ). The forward state transition function is extended to : ST (ST) Σ ST (ST) in the natural way. The formal definition of STS is now as follows. Definition 3.1 (State Tree Structure(STS))[10]: An STS is a 6-tuple G = (ST, H, Σ,, S 0, ST m ), where ST is a state tree. H is a set of holons assigned to the OR superstates in ST. Σ is the set of events. is the forward state transition function. S 0 is the initial sub-state-tree. ST m is the set of marker sub-state-trees. A predicate P defined on B(ST) is a function: B(ST) {0, 1}, where 0 and 1 stand for false and true respectively. Let b B(ST) be a basic sub-state-tree; if P (b) = 1 we say b satisfies P, written b = P. Write P red(st) for the set of all predicates defined on B(ST). Let P 1, P 2 P red(st). We say P 1 precedes P 2, or is stronger than P 2, if and only if ( b B(ST))b = P 1 b = P 2. If P 1 precedes P 2, we write P 1 P 2. We bring in the function Θ : ST (ST) P red(st) to convert a sub-statetree S to its predicate representation Θ(S) P red(st). This yields the symbolic forward state transition function : P red(st) Σ P red(st). Then, we can write the symbolic model of STS as G = (ST, H, Σ,, P 0, P m ), where is the symbolic forward state transition function, P 0 is the symbolic representation of S 0, and P m is the symbolic representation of ST m. Let f : B(ST) Π, be a state feedback control (SF- BC) for P where Π := {Σ Σ Σ u Σ }. The supervisor of the synthesis is implemented by a set of predicates {f σ σ Σ c }, called control functions of controllable events. For a given controllable event σ, the states where f σ = 1 (resp. 0) are exactly the states of the system at which event σ is enabled (resp. disabled). Write G f = (ST, H, Σ, f, P f 0, P m) for the controlled STS, where f is the closed loop transition function and P f 0 is the initial state of the closed loop system. Our STS synthesis problem is to find supc 2 P(P ), the largest subpredicate of P that is nonblocking and controllable.

3 4 AUTOMATON FORM OF SUPERVISOR In this section, we give a method to convert STS/BDD based supervisor to automaton form. The SUPER-enabled event set for x X is defined as:e : X 2 Σ : x E(x) = {σ Σ ξ(x, σ)!}. D(x) is the SUPER-disabled event set for x: D : X 2 Σ : x D(x) = {σ Σ ξ(x, σ)!&( s L(G))[sσ L(G)&ξ(x 0, s) = x]}. Let M : X {true, false} : x M(x) = true if x X m. Finally let T : X {true, false} : x T (x) = true if ( s Σ )ξ(x 0, s) = x& (S 0, s) = ST m. Let H(S) denote the set of events that is defined at substate-tree S. That is H : ST (ST) 2 Σ : S H(S) = {σ Σ (S, σ)!}. Let TT denote the transition table, which records the transitions of an automaton. For a given STS, having n basic sub-state-trees, the state feedback control is f and the controlled behavior is C. Then, the convert algorithm is as follows: Algorithm 1 (convert algorithm): (1) initialize X =, V =, TT = ; (2) x 0 = S 0 ; (3) X m = {b b = ST m }; (4) for i = 0 to n 1; (5) if b i = C; (6) X = X {b i }; (7) for all σ j Σ c ; (8) if f σj (b i ) = 0; (9) push (b i, σ j ) to control data table φ; (10) D(b i ) = D(b i ) {σ j }; (11) for all σ k H(b i ) D(b i ); (12) push (b i, σ k, (b i, σ k )) to transition table TT; (13) return x 0, X, X m, φ, TT. So, the automaton form supervisor SUPER(SUPER, φ) where SUPER = (X, Σ, ξ, x 0, X m ) is given as follows: x 0, X, X m, φ are generated by algorithm 1; Σ is given by the plant model; ξ is given by transition table TT. Then, in accordance with TCT software [12], we code the state in X using integers, with the initial state coded 0. 5 REDUCED VERSION OF SUPERVISOR Often there exists a supervisor, say SIM- SUP(SIMSUP,φ SIMSUP ), such that SIMSUP SUPER. Here means the state size of, and the following properties hold, L(SIMSUP) L(G) = L(SUPER) (1) L m (SIMSUP) L m (G) = L m (SUPER) (2) Any DES SIMSUP that satisfies (1, 2) is control equivalent to SUPER with respect to G. Let R X X be the binary relation such that for a pair x, x X, (x, x ) R iff 1) E(x) D(x ) = E(x ) D(x) = 2) T (x) = T (x ) M(x) = M(x ) Condition 1 says that for a pair of states x, x in R, the associated enable/disable control actions should be consistent, namely no event is enabled at x but disabled at x. Condition 2 means that states x, x in R are consistently marked either true of false in SUPER if they are both reachable by some strings s, s in L m (G), or else if neither is reachable by strings in L m (G). Recall that a cover of a set X is a family of nonempty subsets of X whose union is X. Definition 5.1 [8]: A cover C = {X i X i I} (I is a index set) is a control cover on SUPER if 1) ( i I)X i ( x, x X i )(x, x ) R 2) ( i I)( σ Σ)( j I)[( x X i )ξ(x, σ)! ξ(x, σ) X j ] The subset X i are the cells of C. A control cover is a control congruence if C is a partition on X, namely the X i are pairwise disjoint. Note that cells of a control cover may overlap: x X may belong to more than one cell X i. Given a control cover C = {X i X i I} on SUPER, we construct an induced supervisor J = (I, Σ, k, i 0, I m ) as follows. Let i 0 = some i I with x 0 X i, I m = {i I X i Xm } and k : I Σ I (pfn), with k(i, σ) = j I if ( x X i )ξ(x, σ) X j ( x X i )[ξ(x, σ)! ξ(x, σ) X j ]. Because the cells of a control cover are not necessarily disjoint, i 0 and k may not be uniquely determined; here we choose a fixed but arbitrary instance of J. If C is a control congruence then J will be unique. Proposition 5.1 [8]: J is control equivalent to SUPER with respect to G. Definition 5.2 [8]: A DES SIMSUP=(Z, Σ, ζ, z 0, Z m ) is normal with respect to SUPER if 1) ( z Z)( s L(SUPER))ζ(z 0, s) = z 2) ( z, z Z)( σ Σ)[ζ(z, σ) = z ( s Σ )[sσ L(SUPER)&ζ(z 0, s) = z]] 3) ( z Z m )( s L m (SUPER))ζ(z 0, s) = z Thus SIMSUP is normal if (1) its states are all reachable and (2) its transitions all taken, under strings in L(SUPER), namely no state or transition in SIMSUP is superfluous, (3) each marker state in SIMSUP is reachable by at least one string that is marked by SUPER. If SIMSUP is control e- quivalent to SUPER with respect to G but not normal, then simply deleting all superfluous states and transitions, and converting all unnecessary marked states into unmarked states in SIMSUP, will render SIMSUP normal, while preserving the control equivalence between SIMSUP and SUPER. Normalizing SIMSUP cannot increase its state size, so a minimal normal supervisor will be a minimal supervisor. Definition 5.3[8]: Let G A = (X A, Σ, ξ A, x A,0, X A,m ) and G B = (X B, Σ, ξ B, x B,0, X B,m ). G B is a DESepimorphic image of G A under DES-epimorphism θ : X A X B if 1) θ : X A X B is surjective.

4 2) θ(x A,0 ) = x B,0 and θ(x A,m ) = X B,m 3) ( x X A )( σ Σ)ξ A (x, σ)! [ξ B (θ(x), σ)!&ξ B (θ(x), σ) = θ(ξ A (x, σ))] 4) ( x X B )( σ Σ)ξ B (x, σ)! [( x X A )ξ A (x, σ)!&θ(x ) = x] In particular, G B is DES-epimorphic to G A if θ : X A X B is bijective. Theorem 5.1: Let SUPER be a supremal controllable sublanguage for G and let SIMSUP be any normal supervisor with respect to SUPER that is control equivalent to SUPER with respect to G. Then there exists a control cover C on SUPER for which some induced supervisor J is DES-isomorphic to SIMSUP. In [8], Su and Wonham have shown that the minimal supervision problem of computing a supervisor with minimal state size is NP-hard. Also in [8] Su and Wonham present a polynomial-time algorithm which can accomplish supervisor reduction for DES. In this part, we adapt this algorithm to STS and generate a control congruence, whose induced supervisor J is unique. Let SUPER = (X, Σ, ξ, x 0, X m ) as before and X = {x 1, x 2, x 3,, x n }, note that x i = b i as the result of algorithm 1. So, we can use the symbolic (predicate) representation of b i to denote x i, that is, b i = C x i X and use the symbolic (predicate) representation of [b i ] to denote [x i ]. Suppose C:={[b] C b = C b = [b]} be a control congruence on SUPER, initially set to be ( b = C)[b] := b. Let waitlist B(ST) B(ST) be a list of state pairs that are waiting to be merged, namely if all subsequent tests return true then each state pair will be placed in the same cell of the control congruence that is newly formed. The supervisor reduction algorithm is as follows: Algorithm 2: reduction algorithm: Main procedure: (1) for i = 1 to n 1 (2) { if (i > min{k I b k [b i ]}), (3) continue; (4) for j = i + 1 to n (5) { if j > min{k I b k [b j ]}, (6) continue; (7) waitlist := ; (8) flag = Check-Mergibility(b i, b j, waitlist, b i ); (9) if flag false; (10) { C := {[b] b :{(b,b ),(b,b)} waitlist [b ] [b], [b ] C}; (11) C:= C; (12) } (13) } (14) } Boolean Check-Mergibility ( b i, b j, waitlist, cnode ) (1) for each b p [b i ] b:{(b,b i),(b i,b)} waitlist [b] (2) { for each b q [b j ] b:{(b,b j ),(b j,b)} waitlist [b] (3) { if {(b p, b q ), (b q, b p )} waitlist, continue; (4) if (b p, b q ) R, (5) return false; (6) waitlist := waitlist {(b p, b q )}; (7) for each σ Σ with ξ(b p, σ)! and ξ(b q, σ)! (8) { if [ξ(b p, σ)] = [ξ(b q, σ)] or {(ξ(b p, σ), ξ(b q, σ)), (ξ(b q, σ), ξ(b p, σ))} waitlist, (9) continue; (10) if min{k b k [ξ(b p, σ)]} < cnode or min{k b k [ξ(b q, σ)]} < cnode, (11) return false; (12) flag = Check-Mergibility (ξ(b p, σ), ξ(b q, σ), waitlist, cnode) (13) if flag = false, (14) return false; (15) } (16) } (17) } (18) return true; Proposition 5.2[8]: algorithm 2 terminates and the result C is a control congruence. Note that, in the worst case the main procedure makes 1 2n(n 1) calls to subroutine Mergibility and the subroutine can make 1 2n(n 1) calls to itself. So the overall number of calls will be 1 2 n(n 1) 1 2 n(n 1) = 1 4 n2 (n 1) 2. Hence the complexity is O(n 4 ). 6 FMS EXAMPLE 6.1 System Description In the example of this paper (adapted from [6]), our flexible manufacturing system (FMS) is built up from ten components. They are: four machine tools M1-M4, three robots R1-R3 and three Input/Output buffers B1-B3. The logical layout of the FMS is shown in Figure 2. Production processes for the workpieces are shown in Figure 3. The event coding is displayed in Table 1. The total specification is divided into three groups of subspecifications as follows: 1. For the four machine tools M1-M4, each machine tool can hold two parts, possibly of different types, at the same time. 2. For the three buffers B1-B3, each buffer must not overflow or underflow. 3. For each of the three types of workpiece, the production sequence is as follows: P1: I1 M2 O1. P2: I2 R3 M4 M3 R1 O2. P3: Either I3 R1 M3 M4 R3 O3 or I3 R1 M1 M2 R3 O3. For further details of modeling, we refer to [11].

5 Input 1 Machine 1 Table 1: The event code event event code R1 U O2 11 R1 D M3 13 R1 D I3 15 R1 U M1 17 R1 U M3 19 D I1 21 U O1 23 D M2 25 U M2 P1 27 U M3 29 D M4 41 D M1 43 U M2 P3 45 D M3 47 U M4 49 R3 D I2 31 R3 U M4 33 R3 U O3 35 R3 D M2 37 R3 D M4 39 Robot 1 Output 2 Machine 2 Table 2: BDD size of control functions Controllable event BDD size of control function Table 3: control data of automaton based supervisor State Disabled events P1: Input 2 Robot 2 Output 2 Machine 3 Machine 4 Input 3 Robot 3 Output 3 I1 Figure 2: Logical layout of the FMS M2 P2: I2 R3 M4 M3 R1 O2 R1 M3 P3: I3 O3 R1 M1 Figure 3: Production processes for the workpieces O1 M4 M2 R3 R3 6.2 Supervisor Based on STS/BDD Form Using STSLib [5], we can get the supervisor for our FMS based on STS/BDD form. BDD sizes of the control functions are shown in Table Supervisor Based on Automaton Model Using the method we gave in section 4 we can get the supervisor based on automaton model. The tracker in this supervisor has states and transitions. The table of decision maker is as in Table Reduced Supervisor Based on Automaton Model Using the method we gave in section 5 we can get the reduced supervisor based on automaton model. Note that, because of the limitation of computer memory, TCT software cannot generate this reduced supervisor. The tracker in this supervisor has 2876 states and transitions. The table of decision maker is as in Table 4. In STS/BDD based supervisor, the most complicated control function is the control function of event 31, which has 530 BDD nodes. We cannot implement such a complicated control function. The number of states of the automaton based supervisor and reduced supervisor is and 2876 respectively. Compared with the STS/BDD based supervi-

6 Table 4: control data of reduced supervisor State Disabled events sor and automaton based supervisor, the reduced supervisor is much simpler and easier to be implemented. For example, in the reduced supervisor, if the tracker tells the decision maker that the current state is state 3, then, according to the control data table, Table 4, the decision maker will disable events 21, 27, 37, 39, 41, 43, 45 and 47. Implementation is much easier. 7 CONCLUSIONS In this paper, we propose to represent the STS/BDD based supervisor by using the automaton form. We give the method to convert an STS/BDD based supervisor to automaton form and then use the well known algorithm for supervisor reduction. The result is a simpler and more transparent supervisor for some very large DES, and meanwhile, maintains the efficiency of supervisor synthesis. REFERENCES [1] W. M. Wonham and P. J. Ramadge, On the supremal controllable sublanguage of a given language, SIAM J. Control and Optimization, Vol. 50, No. 4, , [2] P. J. Ramadge and W. M. Wonham, Supervisory control of a class of discrete event processes, SIAM J. Control and Optimization, Vol. 25, No. 1, , [3] Peyman Gohari and W. M. Wonham, On the Complexity of Supervisory Control Design in the RW Framework, IEEE Trans. Syst., Man, Cybern. B: Cybern., Vol. 30, No. 5, , [4] Chuan Ma and W. M. Wonham, Nonblocking Supervisory Control of State Tree Structures, IEEE Trans. on Automatic Control, Vol. 51, No. 5, , [5] Chuan Ma and W. M. Wonham, STSLib and Its Application to Two Benchmarks, Proceedings of the 9th International Workshop on Discrete Event Systems, Sweden, [6] ZhiWu Li, MengChu Zhou and NaiQi Wu, A Survey and Comparison of Petri Net-Based Deadlock Prevention Policies for Flexible Manufacturing Systems, IEEE Trans. Syst., Man, Cybern., C, Appl. Rev., Vol. 38, No. 2, , [7] W. M. Wonham, Supervisory Control of Discrete-Event Systems, Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, 2011, available at wonham/. [8] R. Su and W. M. Wonham, Supervisor Reduction for Discrete-Event Systems, Discrete Event Dynamic Systems: Theory and Application, No. 14, 31-53, [9] Ali Saadatpoor and W. M. Wonham, Supervisor State Size Reduction for Timed Discrete-Event Systems, Proceedings of the 2007 American Control Conference, New York City, [10] Chuan Ma and W. Murray Wonham, Nonblocking Supervisory Control of State Tree Structures, Springer-Verlag, Berlin, Germany, [11] Wujie Chao, Yongmei Gan, W. M. Wonham and Zhaoan Wang, Nonblocking Supervisory Control of Flexible Manufacturing Systems Based on State Tree Structures, Workshop on Discrete-Event Systems in Xidian University, Xi an, China, (to be published) [12] W. M. Wonham, Design Software: XPTCT, Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, 2011, available: wonham/. [13] W. M. Wonham and P. J. Ramadge, Modular Supervisory Control of Discrete-Event Systems, Mathematics of Control, Signals, and Systems, Vol. 1, No. 1, 13-30, 1988.