A Uniform Approach for Computing Supremal Sublanguages Arising in Supervisory Control Theory∗ Tae-Sic Yoo, St´ephane Lafortune†, and Feng Lin‡

Abstract In this paper, the computation of supremal sublanguages arising in supervisory control theory is revisited and a new iterative algorithm computing the supremal controllable and/or normal sublanguages of a given non-prefix-closed language is presented. This algorithm provides a uniform tool for computing the above supremal sublanguages. The difficulties associated with the synthesis of a maximal controllable and observable sublanguage of a given language are also discussed.

1

Introduction

In this paper, we develop a new algorithm for computing certain supremal sublanguages of a given language; these supremal sublanguages arise in the synthesis of controllers for discreteevent systems (DES) in the framework of supervisory control theory (cf. Chapter 3 of [2]). Our algorithm applies to the supremal controllable, supremal normal, and supremal controllable and normal sublanguages of a given non-prefix-closed language with respect to a given DES and its associated sets of controllable and observable events. The available procedure for computing the supremal controllable and normal sublanguage of a given non-prefix-closed language presented in [3] involves iterating between the supremal normal and supremal controllable operations and does not take advantage of the formula for the prefix-closed supremal controllable and normal sublanguage of a given prefix-closed language that appeared in [1]. In contrast, our algorithm separates in some sense the computation of the supremal controllable and normal sublanguage (prefix-closed case) from the issue of having a desired language that is not prefix-closed. Our ∗

This research is supported in part by NASA under grant NAG 2-1279 (third author), and by the DDR&E

MURI on Low Energy Electronics Design for Mobile Platforms and managed by ARO under grant ARO DAAH0496-1-0377 and NSF under grant CCR-0082784 (first and second authors). † Department of Electrical Engineering and Computer Science, The University of Michigan, 1301 Beal Avenue, Ann Arbor, MI 48109–2122 USA, {tyoo, stephane}@eecs.umich.edu; www.eecs.umich.edu/umdes ‡ Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202 USA, [email protected]

1

algorithm also allows the computation of the supremal controllable and supremal normal sublanguages as a special case in the same “uniform” manner as for the supremal controllable and normal sublanguage. Moreover, it is possible to apply the formula in [1] for the computation of the prefix-closed supremal sublanguages by virtue of the above separation. These features of our algorithm are attractive from an implementation viewpoint as well as from a theoretical viewpoint. We also discuss the difficulties associated with the synthesis of maximal controllable and observable sublanguages of a given non-prefix-closed language. We show that our algorithm cannot in general be extended to the computation of such maximals using the VLP-PO algorithm in [4] (which works in the prefix-closed case).

2

Preliminaries

Consider a DES that is modeled by an automaton G = (Q, Σ, δ, q0 , Qm ) in the usual manner (see, e.g., [2], [8]1 ). If the set of states Q is finite, then the language generated by G, denoted by L(G), and the language marked by G, denoted by Lm (G), are said to be regular languages. The class of regular language over the event set Σ is denoted by R(Σ), that is R(Σ) := {L ⊆ Σ∗ : L is regular}. We denote the number of states of the canonical (minimal) recognizer of a language K by ||K||. To impose supervision on the plant, we identify some of its events as controllable and the rest as uncontrollable, thereby partitioning Σ into the disjoint sets Σc , the set of controllable events, and Σuc , the set of uncontrollable events. The event set Σ is also partitioned into disjoint sets Σo , the set of observable events, and Σuo , the set of unobservable events. A projection P is a function P : Σ∗ → Σo ∗ that erases from every trace the unobservable events. The inverse of P is denoted by ∗

∗

P −1 and is defined as P −1 : 2Σ → 2Σ with P −1 (L) := {t ∈ Σ∗ : (∃s ∈ L)[P (t) = s]}. Formally, a supervisor for a partially-observed DES is a function SP : Σo ∗ → Γ := {γ ∈ 2Σ : Σuc ⊆ γ}. That is, a supervisor assigns a set of enabled events to each observable trace s ∈ Σo ∗ , and uncontrollable events are always enabled. The language generated by the supervised DES, denoted by L(SP /G), is defined recursively in the usual manner (see e.g., [2]) and the supervised marked language is Lm (SP /G) := L(SP /G) ∩ Lm (G). The goal of supervisor synthesis is to design a supervisor SP for a given language K ⊆ Lm (G) such that L(SP /G) = K and Lm (SP /G) = K. It has been shown in [7] that such a supervisor exists if and only if K is controllable, observable, and Lm (G)-closed. The definitions of controllability, observability, and Lm (G)-closure can be found in Appendix. Let K ⊆ Σ∗ be an arbitrary language. It is known that controllability is preserved under union of languages. It implies that the supremal controllable sublanguage of K exists; it is denoted by (K)↑(C) . Unfortunately, observability is not preserved under union of languages. Therefore, 1

The reader is referred to these references for any undefined concepts or notations in this paper.

2

there is no supremal element for the class of observable sublanguages of K. For this reason, the property of normality, which is a stronger condition than observability, was introduced in [7]. The definition of normality can be found in Appendix. Normality is preserved under union. Therefore, the supremal normal sublanguage of K exists; it is denoted by (K)↑(N ) . We can also define the supremal controllable and normal sublanguage of K, which is denoted by (K)↑(CN ) . We define the subautomaton relation over two finite-state automata. Consider two finite-state automata over Σ, G1 = (QG1 , Σ, δ G1 , q0G1 ) and G2 = (QG2 , Σ, δ G2 , q0G2 ) (marked states are irrelevant for the purpose of this definition). We say that G1 is a subautomaton of G2 , denoted by G1 v G2 , if for all s ∈ L(G1 ), δ G1 (q0G1 , s) = δ G2 (q0G2 , s). Also we say that G1 is a strict subautomaton of G2 , denoted by G1 @ G2 , if for all s ∈ L(G2 ) \ L(G1 ), there exists s0 ≤ s such that δ Q2 (q0Q2 , s0 ) ∈ / QG1 in addition, where s0 ≤ s implies that s0 is a prefix of s. Moreover, we call two finite-state automata isomorphic, denoted by G1 ≈ G2 , if it is possible to rename the states of G1 or G2 and obtain G1 v G2 and G2 v G1 . Given two finite-state automata H and G such that L(H) ⊆ L(G), by following the technique presented in [3], we can construct H 0 and G0 such that L(H) = L(H 0 ), L(G) = L(G0 ) and H 0 @ G0 . The observer Gobs of an automaton G is a deterministic automaton generating the language P (L(G)). The construction algorithm for the observer can be found in [2]. We call an automaton G a state-partition automaton if for all q1 , q2 ∈ Qobs ⊆ 2Q where Qobs is the set of accessible observer states, q1 ∩ q2 = ∅ or q1 = q2 .

3

Problem Formulation

We start by reviewing known formulas and related convergence results in Sections 3.1 and 3.2.

3.1

Algorithm for (K)↑(C)

If K is prefix-closed (K = K), then it can be shown that (K)↑(C) is given by the formula presented in [1]: (K)↑(C) = K \ [(L(G) \ K)/Σ∗uc ]Σ∗ where the “/” symbol denotes the quotient operation on languages. That is, (L(G) \ K)/Σ∗uc := {s ∈ Σ∗ : (∃t ∈ Σ∗uc )[st ∈ L(G) \ K]}. For a general, i.e., not necessarily prefix-closed, language K, a lattice-theoretic characterization is exploited to find (K)↑(C) in [9]. Consider the following operator on languages, parameterized by languages K, L(G) and set Σuc : ∗

∗

Ω↑(C) : 2Σ → 2Σ and Ω↑(C) (L) = {t ∈ K : tΣuc ∩ L(G) ⊆ L}.

3

It turns out that the largest fixed point of this operator is K ↑(C) and an algorithm iterating the operator was proposed in [9] to compute (K)↑(C) : K0 = K; Ki+1 = Ω↑(C) (Ki ), for i ≥ 0.

(1)

If K and L(G) are regular languages, then the sequence {Ki , i ≥ 0} converges to K ↑(C) in finite steps.

3.2

Algorithm for (K)↑(CN )

If K is prefix-closed, then it can be shown that (K)↑(N ) can be expressed by the formula presented in [1]: (K)↑(N ) = K \ [P −1 {P (L(G) − K)}]Σ∗ . When the property of prefix-closure does not hold, there exists an iterative procedure for the computation of (K)↑(N ) . A lattice theoretic characterization, which is similar to that for (K)↑(C) , is used for the computation of (K)↑(N ) in [3]. Consider the following operator on languages, parameterized by language K, L(G) and projection P : ∗

∗

Ω↑(N ) : 2Σ → 2Σ and Ω↑(N ) (L) = {t ∈ K : P −1 P (t) ∩ L(G) ⊆ L}. It turns out that the largest fixed point of this operator is (K)↑(N ) and it can be computed in finite steps by iterating the operator as follows when K and L(G) are regular: K0 = K; Ki+1 = Ω↑(N ) (Ki ), for i ≥ 0.

(2)

In [5, 6], it is claimed that the language sequence generated by the following iteration converges to (K)↑(N ) . K0 = K; Ki+1 = Ki \ [{P −1 P (L(G) \ Ki )}Σ∗ ], i ≥ 0.

(3)

However, it turns out that the convergence to (K)↑(N ) is not true for the iteration scheme(3). We provide a counter example. Example 1 Consider L(G) = {αβ} and K = {αβ}. Let Σo = {α}. Iteration scheme (3) yields the empty set as the result. However, it is easily verified that (K)↑(N ) = K. The only known result on the computation of (K)↑(CN ) involves iteratively performing (·)↑(C) and (·)↑(N ) operations, which can be computed by (1) and (2), respectively. If K and L(G) are regular, then such an iteration stops in finite steps and converges to the language (K)↑(CN ) [3]. If K is prefix-closed, a compact formula for (K)↑(CN ) is available [1]: (K)↑(CN ) = L(G) ∩ P −1 [P ((K)↑(N ) )]↑(CP ) where for D ⊆ P (L(G)), D↑(CP ) is defined to be [ D↑(CP ) = {L ⊆ D : L is controllable w.r.t. P (L(G)) and P (Σuc )}. 4

(4)

3.3

Problem Statement

Instead of using the iteration scheme developed in [3], we consider the following approach for computing supremal controllable and/or normal sublanguages: K0 = K; Ki+1 = (Ki )↑(CN ) ∩ K, for i ≥ 0,

(5)

where K and Lm (G) are regular languages and K is Lm (G)-closed. Since a formula for (K)↑(CN ) is available and its computation consists of “usual” language operations, we may compute (K)↑(CN ) readily if the language sequence generated by (5) converges to (K)↑(CN ) in finite steps. The next subsection provides a general convergence result that will be used in Section 4 to prove the finite-step convergence of the language sequence generated by iteration (5).

3.4

Finite-Step Language Convergence Lemma

It is a well known mathematical fact that a monotone bounded sequence converges. Therefore, the convergence of a language sequence can be demonstrated by showing that the language sequence is bounded and monotone with respect to set inclusion. However, finite-step convergence is not guaranteed by the boundedness and monotonicity of the language sequence. Let Lm (Hi ) = {a}∗ \ {ai }, for i ≥ 0. Then the language sequence {Lm (Hi ) : i ≥ 0} converges to {} but not in finite steps. Although the language sequence is monotone and bounded, the number of the states of the canonical recognizer of Lm (Hi ) is not bounded. Therefore, the finite-step convergence is not achieved. However, if we assume that the state spaces of the canonical recognizers recognizing the languages generated by the sequence is bounded, we have the following finite convergence result that is proved in [9]. Lemma 1 [9] (Monotone Sequence Finite Convergence Lemma) Let K0 be a regular language and define the sequence Ki+1 := Ωi (Ki ), ∀i ≥ 0, where 1. Ωi : R(Σ) → R(Σ), ∀i ≥ 0; 2. ∃N < ∞ s.t. ||Ki || ≤ N, ∀i ≥ 0; 3. Ki+1 ⊆ Ki (or Ki+1 ⊇ Ki ), ∀i ≥ 0. Then, there exists m < ∞ such that for all m0 ≥ m, Km0 = Km .

4

Computing Supremal Sublanguages

In this section, we apply the result presented in Section 3.4 to prove that the language sequence generated by iteration (5) converges to (K)↑(CN ) in finite steps. G Consider two (trimmed) finite-state automata H = (QH , Σ, δ H , q0H , QH m ) and G = (Q , Σ,

δ G , q0G , QG m ) such that K := Lm (H) ⊆ Lm (G). We can assume that H @ G without loss of 5

generality, since it is always possible to transform the automata to satisfy the strict subautomaton P constraint. Let us denote GP := G k Gobs = (QGP , Σ, δ GP , q0GP , QG m ). It is shown in [3] (Lemma

3.4) that GP is a state-partition automaton. The following two facts are shown in [3] as well. Fact 1 If H @ G, then there exists an automaton HN such that HN @ GP and L(HN ) = Lm (HN ) = (K)↑(N ) . Fact 2 If H @ G and G is a state-partition automaton, then there exists an automaton HN such that HN @ G and L(HN ) = Lm (HN ) = (K)↑(N ) . In [9], we can find the following result: Fact 3 If H @ G, then there exists an automaton HC such that HC @ G and L(HC ) = Lm (HC ) = (K)↑(C) . In [3], an algorithm alternatively applying (·)↑(C) and (·)↑(N ) is devised to compute (K)↑(CN ) . With this algorithm and the facts stated above, it is straightforward to see that the following two propositions hold. Proposition 1 If H @ G, then there exists an automaton HCN such that HCN @ GP and L(HCN ) = Lm (HCN ) = (K)↑(CN ) . Proposition 2 If H @ G and G is a state-partition automaton, then there exists an automaton HCN such that HCN @ G and L(HCN ) = Lm (HCN ) = (K)↑(CN ) . Now, we prove the finite-step convergence of the language sequence produced by iteration (5). Lemma 2 For all Kn where n ≥ 1, there exists an automaton Hn such that Hn @ GP , L(Hn ) = Kn and Lm (Hn ) = Kn . Proof: We argue this by induction. (Base of induction) By Proposition 1, there exists HCN 1 such that HCN 1 @ GP and L(HCN 1 ) = Lm (HCN 1 ) = K0 ↑(CN ) . Then, K1 = (K0 )↑(CN ) ∩ K = (K0 )↑(CN ) ∩ K ∩ Lm (G) (∵ K is Lm (G)-closed) = (K0 )↑(CN ) ∩ Lm (G)

(∵ (K0 )↑(CN ) ⊆ K)

= L(HCN 1 ) ∩ Lm (G) = Lm (HCN 1 ) ∩ Lm (G) = Lm (HCN 1 × GP ). Let H1 = T rim(HCN 1 × GP ). Then, we get K1 = Lm (H1 ). 6

Since T rim(HCN 1 ×GP ) @ HCN 1 ×GP and HCN 1 ×GP ≈ HCN 1 , we get H1 @ (HCN 1 ×GP ) @ GP up to state renaming. This implies H1 @ GP up to state renaming. (Induction hypothesis) Assume that there exists Hn @ GP such that L(Hn ) = Kn and Lm (Hn ) = Kn . (Induction step) Since GP is a state-partition automaton and Hn @ GP , there exists HCN n+1 @ GP such that L(HCN n+1 ) = Lm (HCN n+1 ) = (Kn )↑(CN ) by Proposition 2. Therefore, Kn+1 = (Kn )↑(CN ) ∩ K = (Kn )↑(CN ) ∩ K ∩ Lm (G) (∵ K is Lm (G)-closed) = (Kn )↑(CN ) ∩ Lm (G)

(∵ (Kn )↑(CN ) ⊆ K)

= L(HCN n+1 ) ∩ Lm (G) = Lm (HCN n+1 ) ∩ Lm (G) = Lm (HCN n+1 × GP ). Let Hn+1 = T rim(HCN n+1 × GP ). Then, Lm (Hn+1 ) = Kn+1 and L(Hn+1 ) = Kn+1 . Moreover, Hn+1 @ (HCN n+1 × GP ) @ GP up to state renaming. This implies Hn+1 @ GN up to state renaming. Lemma 3 For all i ≥ 0, Ki is Lm (G)-closed. Proof: We prove this by induction. (Base of induction) Follows directly from the assumption that K is Lm (G)-closed. (Induction Hypothesis) Assume that Kn is Lm (G)-closed. (Induction Step) Since Kn+1 = (Kn )↑(CN ) ∩ K = (Kn )↑(CN ) ∩ Lm (G), Kn+1 ∩ K = Kn+1 ∩ Lm (G) = (Kn )↑(CN ) ∩ Lm (G) ∩ Lm (G) ⊆ (Kn )↑(CN ) ∩ Lm (G) = (Kn )↑(CN ) ∩ Lm (G) = (Kn )↑(CN ) ∩ K = Kn+1 . Moreover, since Kn+1 ⊆ Kn+1 and Kn+1 = (Kn )↑(CN ) ∩ Lm (G) ⊆ Lm (G), we get Kn+1 ⊆ Kn+1 ∩ Lm (G). Therefore, Kn+1 = Kn+1 ∩ Lm (G). Lemma 4 {Ki , i ≥ 0} is a monotone decreasing sequence w.r.t. set inclusion. Proof: Take any i ≥ 0, then Ki+1 = (Ki )↑(CN ) ∩ K = (Ki )↑(CN ) ∩ Lm (G) ⊆ Ki ∩ Lm (G) = Ki (by Lemma 3). 7

Theorem 1 If Ωi (Ki ) := (Ki )↑(CN ) ∩ K and K0 = K is Lm (G)-closed, then {Ki : i ≥ 0} converges in finite steps. Proof: Since we have Theorem 1, it is enough to show that ∃N < ∞ s.t. ||Ki || ≤ N, ∀i ≥ 0 and Ki+1 ⊆ Ki , ∀i ≥ 0. By Lemma 2, it is clear that, by setting N := |QGP | < ∞, we get ||Ki || ≤ |QHi | ≤ N, ∀i ≥ 1. Moreover, we have ||K0 || ≤ |QH0 | ≤ |QG | ≤ N. Therefore, ||Ki || ≤ N, ∀i ≥ 0. Monotonicity is proved in Lemma 4. To show that {Ki : i ≥ 0} converges to (K)↑(CN ) , we first prove the following two lemmas. Lemma 5 (K)↑(CN ) ⊇ K ↑(CN ) . Proof: By the monotonicity of the supremal controllable and normal operation, (K)↑(CN ) ⊇ K ↑(CN ) . Since prefix-closure is preserved under the supremal controllable and normal operation, we get (K)↑(CN ) ⊇ K ↑(CN ) . Lemma 6 If K is Lm (G)-closed, then K ↑(CN ) is Lm (G)-closed [3]. Now we argue that the language iteration (5) converges to (K)↑(CN ) . Theorem 2 The sequence {Ki : i ≥ 0} converges to (K)↑(CN ) in finite steps. Proof: First we claim that {Ki : i ≥ 0} is lower-bounded by K ↑(CN ) w.r.t. set inclusion. This can be shown by induction. (Base of induction) By the definition of K ↑(CN ) , it is obvious that K = K0 ⊇ K ↑(CN ) . (Inductive Hypothesis) Assume Kn ⊇ K ↑(CN ) . (Inductive Step) Similarly, Kn+1 = (Kn )↑(CN ) ∩ K = (Kn )↑(CN ) ∩ Lm (G) ⊇ Kn ↑(CN ) ∩ Lm (G)

(by Lemma 5)

⊇ K ↑(CN ) ∩ Lm (G)

(by inductive hypothesis)

= K ↑(CN )

(by Lemma 6).

Now we claim that Kn+1 = Kn implies that Kn = K ↑(CN ) . Let there exists n < ∞ such that Kn = Kn+1 . Then, we have Kn = (Kn )↑(CN ) ∩ K = (Kn )↑(CN ) ∩ Lm (G). This implies that Kn = (Kn )↑(CN ) ∩ Lm (G) ⊆ (Kn )↑(CN ) ∩ Lm (G) ⊆ (Kn )↑(CN ) . It is obvious that Kn ⊇ (Kn )↑(CN ) . So, we get Kn = (Kn )↑(CN ) . This implies that Kn is controllable and normal. Since Kn ⊆ K and controllable and normal, we get Kn ⊆ (K)↑(CN ) . Since {Ki : i ≥ 0} is lower-bounded by (K)↑(CN ) , we get Kn = K ↑(CN ) . 8

A notable feature of algorithm (5) is the separation of the prefix-closed supremal sublanguage computation (the computation of (Ki )↑(CN ) ) and the non-prefix-closed language generation (the intersection with K). By virtue of this separation, it is possible to apply the formula in [1] for computing (Ki )↑(CN ) . Another feature of algorithm (5) is that it applies to the computation of the supremal normal and the supremal controllable sublanguages by simply replacing (Ki )↑(CN ) with (Ki )↑(C) and (Ki )↑(N ) in (5). In this sense, algorithm (5) is “uniform” with all these operations (·)↑(CN ) , (·)↑(C) , and (·)↑(N ) for non-prefix-closed languages. These features may considerably simplify implementation. We provide an illustrating example of iteration (5). Example 2 Figure 1 depicts the system model G where Σuc = {α} and Σuo = {δ} as well as automaton H modeling the language of interest. We repsent marked states with doubly circled states. L(G) and Lm (G) represent the uncontrolled language and Lm (H) is the desired language. Obviously, H @ G, Lm (H) ⊆ Lm (G) and Lm (H) is Lm (G)-closed. Let us denote K = K0 = Lm (H). We apply formula (4) to compute (K0 )↑(CN ) . The result is shown in Fig. 2(a). We G

H

1 δ

1 β

2 β 4 γ 10

β 6

δ 3 γ

δ 5 β

β 2

β 8

4

γ

7

β 9

6

3 γ

δ 5 β 7

8 γ 9

α 11

Figure 1: System model and desired language model (Example 2) intersect (K0 )↑(CN ) with K to generate K1 which is shown in Fig. 2(b). This ends the first iteration. Now K1 is not normal since βδ ∈ P −1 P (β) ∩ L(G) and βδ ∈ / K1 . The language K2 generated by the second iteration is normal and controllable. The automaton generating K2 is shown in Fig. 2(c). This ends the iteration.

5

Synthesis of Maximal Sublanguages

Since iteration (5) works for the computation of (K)↑(CN ) , it is of interest to determine if it can be adapted for the computation of maximal controllable and observable sublanguages of a given language2 . That is, consider the iteration: K0 = K; Ki+1 = (Ki )M CO ∩ K, for i ≥ 0, 2

See [4] for definition and detailed discussion.

9

(6)

1

1 δ

δ

β 2

β 2

3

β

1 δ

3

2

δ

4

5

(a) (K0 )↑(CN )

(c) (K)↑(CN )

(b) K1

Figure 2: Iterations (Example 2)

where (Ki )M CO denotes a prefix-closed maximal controllable and observable sublanguage of Ki . An algorithm for computing (Ki )M CO (termed the VLP-PO algorithm) is presented in [4]. It turns out that in general algorithm (6) does not converge to a maximal controllable and observable sublanguage of K. Property 1 {Ki : i ≥ 0} generated by iteration (6) may not converge to a maximal controllable and observable sublanguage of K, in general. Proof: Consider the uncontrolled plant G in Fig. 3(a) and the specification H in Fig. 3(b) where Lm (H) = K. We set Σ = Σc and Σo = {γ}. Then, (Lm (H))M CO in Fig. 3(c) is a prefixclosed maximal controllable and observable sublanguage of Lm (H). Then, we get (Lm (H))M CO ∩ Lm (H) = ∅ by applying (6). However, ∅ is not a maximal controllable and observable sublanguage of K; such a maximal is shown in Fig. 3(d). 0 β

7

0

α

β

1

2

α

8

γ

α

β

1

2

0

γ 3 β

5

6

α

β

1

2

γ

3

α

0

α

α

β

5

6

3

β 9

10

(a) G

(c) (Lm (H))M CO

(b) H

(d) (Lm (H))M CO

Figure 3: Nonconvergence result

The proof of Property 1 also implies the following. Property 2 The existence of K M CO such that K M CO ⊆ (K)M CO is not guaranteed. 10

These undesirable properties suggest that marking can not be dealt with separately for the computation of (K)M CO . We believe that this “lack of separation” partly explains the difficulty of synthesizing maximal controllable and observable sublanguages of non-prefix-closed languages. We note that the beauty of the computation of supremal controllable and normal sublanguage is in the (forced) two-way separation of control and estimation (as is the standard LQG problem of conventional control theory, for instance). Roughly speaking, since the control action does not create a new estimation state (information state), i.e. does not result in further learning of the state of the system, we can continue iteration (5) without creating a new estimation state. Rigorous technical developments demonstrating these ideas in the computation of (K)↑(CN ) and (K)↑(CN ) can be found in [3] and [1]. We conclude this section with one last result. Even though iteration (6) may fail to synthesize (K)M CO , we are interested in its convergence properties. With the VLP-PO technique [4], we can compute a maximal controllable and observable sublanguage of the prefix-closure of a given language that is guaranteed to be a superlanguage of the supremal controllable and normal sublanguage of the prefix-closure of the given language. Suppose that we apply this procedure for the computation of (Ki )M CO in iteration (6). Let us assume that the language sequence generated by iteration (6) converges in finite steps3 to a certain language, denoted by K limit . Then we have the following result. Theorem 3 If algorithm (6) converges in finite steps to K limit , then K limit is controllable, observable, Lm (G)-closed, and (K)↑(CN ) ⊆ K limit . This result can be obtained in a straightforward manner and we omit its proof. The proof is contained in the addendum to assist reviewers.

6

Conclusion

In this paper, we presented a uniform approach for computing supremal sublanguages that arise in supervisory control of DES. The properties of this iterative procedure were discussed.

References [1] R. D. Brandt, B. K. Garg, R. Kumar, F. Lin, S. I. Marcus, and W. M. Wonham. Formulas for calculating supremal controllable and normal sublanguages. Systems and Control Letters, 15(8):111–117, 1990. [2] C. G. Cassandras and S. Lafortune. Introduction to Discrete Event Systems. Kluwer Academic Publishers, 1999. 3

The finite convergence of iteration (6) is an open issue.

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[3] H. Cho and S. I. Marcus. On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation. Math. Control Signals Systems, 2:47– 69, 1989. [4] N. B. Hadj-Alouane, S. Lafortune, and F. Lin. Centralized and distributed algorithm for online synthesis of maximal control policies under partial observation. Discrete Event Dynamic Systems: Theory and Applications, 6(41):379–427, 1996. [5] R. Kumar and V. K. Garg. Extremal solutions of inequations over lattices with applications to supervisory control. Theoretical Computer Science, 148:67–92, 1995. [6] R. Kumar and V. K. Garg. Modeling and Control of Logical Discrete Event Systems. Kluwer Academic Publishers, 1995. [7] F. Lin and W. M. Wonham. On observability of discrete-event systems. Information Sciences, 44(3):173–198, 1988. [8] P. J. Ramadge and W. M. Wonham. The control of discrete event systems. Proc. of the IEEE, 77(1):81–98, 1989. [9] W. M. Wonham and P. J. Ramadge. On the supremal controllable sublanguage of a given language. SIAM J. Control Optim., 25(3):637–659, 1987.

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Appendix: Definitions Definition 1 We call a language K is controllable (w.r.t. L(G) and Σc ), if KΣuc ∩ L(G) ⊆ K. Definition 2 A language K ⊆ L(G) is said to be observable w.r.t. L(G), Σo , and Σc , if for all s ∈ K and σ ∈ Σc , such that sσ ∈ L(G) \ K, P −1 P (s)σ ∩ K = ∅. Definition 3 A language K is said to be Lm (G)-closed, if K ∩ Lm (G) = K. Definition 4 A language K ⊆ L(G) is said to be normal w.r.t. L(G), Σo , P −1 P (K) ∩ L(G) = K.

Addendum: Proof of Theorem 3 (for the reviewers of this paper) We consider the following simple theorem first. Theorem 4 Consider Ωi : R(Σ) → R(Σ), i ∈ {1, 2}, where, for all K1 , K2 ∈ R(Σ), if K1 ⊆ K2 ⇒ Ω1 (K1 ) ⊆ Ω2 (K2 ). Define the sequences generated by Ω1 and Ω2 as follows: K1,n+1 := Ω1 (K1,n ) and K2,n+1 := Ω1 (K2,n ), where K1,0 = K2,0 = K. Then, if limn→∞ K1,n and limn→∞ K2,n exist, we get lim K1,n ⊆ lim K2,n .

n→∞

n→∞

Proof: It is enough to show that K1,n ⊆ K2,n for all n ≥ 0. This can be proved by a simple inductive argument as follows: (Base of induction) Follows from K1,0 = K2,0 = K. (Inductive hypothesis) Assume that for a given n ≥ 0, K1,n ⊆ K2,n . (Inductive Step) Since K1,n ⊆ K2,n by hypothesis, we get Ω1 (K1,n+1 ) ⊆ Ω2 (K2,n+1 ) by assumption. This directly implies that K1,n+1 ⊆ K2,n+1 . Suppose that we compute a maximal controllable and observable sublanguage of the prefixclosure of a given language, for iteration (6), that is (Ki )M CO , in a manner that is guaranteed 13

to be a superlanguage of the supremal controllable and normal sublanguage of the prefix-closure of the given language (as was mentioned earlier, this can be done using the VLP-PO algorithm in [4]). Let us denote the limits of the language sequences generated by the iterations (5) and (6) by lim K1,n and lim K2,n , respectively. Then, we have the following. Theorem 5 lim K2,n := K limit exists and (K)↑(CN ) ⊆ K limit . Proof: Using similar arguments as Lemma 3 and Lemma 4, we can show that {K2,n , n ≥ 0} is a monotone decreasing sequence w.r.t. set inclusion. Moreover, ∅ ⊆ K2,n ⊆ K for all n. Therefore, lim K2,n := K limit exists. Since lim K1,n := (K)↑(CN ) and given the result of Theorem 4, it is sufficient to show that, for all K1 , K2 ∈ Σ∗ , if K1 ⊆ K2 ⇒ (K1 )↑(CN ) ∩ K ⊆ (K2 )M CO ∩ K, to prove that (K)↑(CN ) ⊆ K limit .

By the monotonicity of the operator (·)↑(CN ) , we have

(K1 )↑(CN ) ⊆ (K2 )↑(CN ) . We apply the VLP-PO algorithm of [4] to synthesize a maximal controllable and observable sublanguage, denoted by (K2 )M CO , such that (K2 )↑(CN ) ⊆ (K2 )M CO for all K2 ⊆ Σ∗ . Combining the two preceding results, we have (K1 )↑(CN ) ⊆ (K2 )M CO . Therefore, (K1 )↑(CN ) ∩ K ⊆ (K2 )M CO ∩ K. Theorem 6 If {Ki : i ≥ 1} converges to K limit in finite steps, K limit is controllable, observable, and Lm (G)-closed. Proof: By arguing in a similar way as for Lemma 3, we can obtain that K limit is Lm (G)-closed. By hypothesis, there exists n < ∞ such that Kn+1 = Kn . That is, (Kn )M CO ∩ K = Kn . Then we have Kn = (Kn )M CO ∩ K ⊆ (Kn )M CO ⊆ Kn . Therefore, Kn = (Kn )M CO , which implies that Kn is controllable and observable. By combining Theorems 5 and 6, we obtain Theorem 3.

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