On conditional decomposability
Introduction
In the Ramadge–Wonham supervisory control framework, discrete-event systems are represented by deterministic finite automata. Given a specification language (usually also represented by a deterministic finite automaton), the aim of supervisory control is to construct a supervisor so that the closed-loop system satisfies the specification [1]. The theory is widely developed for the case where the system (plant) is monolithic. However, large engineering systems are typically constructed compositionally as a collection of many small components (subsystems) that are interconnected by rules; for instance, using a synchronous product or a communication protocol. This is especially true for discrete-event systems, where different local components run in parallel. Moreover, examples of supervisory control of modular discrete-event systems show that a coordinator is often necessary for achieving the required properties because the purely decentralized control architecture may fail in achieving these goals.
The notion of separability of a specification language has been introduced in [2]; it says that a language over an alphabet, is separable if , where, for all , is a projection. A specification for a global system is separable if it can be represented (is fully determined) by local specifications for the component subsystems. It is very closely related to the notion of decomposability introduced in [3], [4] for decentralized discrete-event systems, which is also further studied in [5]. Decomposability is a slightly more general condition because it involves not only the specification, but also the plant language; that is, a language over an alphabet , is decomposable with respect to a plant language if : separability is then decomposability where is the set of all strings over the global alphabet. In this paper, we slightly abuse the terminology and call a separable language in the sense of [2] also decomposable. It has been shown in [2] that decomposability is important because it is computationally cheaper to compute locally synthesized supervisors that constitute a solution of the supervisory control problem for this decomposable specification. Recently, the notion of decomposability has also been extended to automata as an automaton decomposability in [6].
However, the assumption that a specification language is decomposable is too restrictive. Therefore, several authors have tried to find alternative techniques for general indecomposable specification languages; for instance, the approach of [7] is based on partial controllability, which requires that all shared events are controllable, or the shared events must have the same controllability status (but then an additional condition of so-called mutual controllability [8] is needed).
In this paper, we study a weaker version of decomposability, so-called conditional decomposability, which has recently been introduced in [9] and studied in [10], [11] in the context of coordination supervisory control of discrete-event systems. It is defined as decomposability with respect to local alphabets augmented by the coordinator alphabet. The word conditional means that, although a language is not decomposable with respect to the original local alphabets, it becomes decomposable with respect to the augmented ones, i.e., decomposability is only guaranteed (conditioned) by local event set extensions by coordinator events.
In the coordination control approach of modular discrete-event systems, the plant is formed as a parallel composition of two or more subsystems, while the specification language is represented over the global alphabet. Therefore, the property of conditional decomposability is required in this approach to distribute parts of the specification to the corresponding components to solve the problem locally. More specifically, we need to ensure that there exists a corresponding part of the specification for the coordinator and for each subsystem composed with the coordinator. Thus, if the specification is conditionally decomposable, we can take this decomposition as the corresponding parts for the subsystems composed with a coordinator and solve the problem locally.
Conditional decomposability depends on the alphabet of the coordinator, which can always be extended so that the specification is conditionally decomposable. In the worst (but unlikely) case, all events must be put into the coordinator alphabet to make a language conditionally decomposable. But in the case when the coordinator alphabet would be too large, it is better to divide the local subsystems into groups that are only loosely coupled, and introduce several coordinators on smaller alphabets. In this paper, a polynomial-time algorithm is provided for verification whether a language is conditionally decomposable. We make an important observation that the algorithm is linear in the number of local alphabets, while algorithms for checking similar properties (such as decomposability and co-observability) suffer from exponential-time complexity with respect to the number of local alphabets. This algorithm is then modified so that it extends the coordinator alphabet to make the specification language conditionally decomposable. Furthermore, we discuss a relationship of conditional decomposability to nonblockingness of a coordinated system, where a coordinated system is understood as a modular system composed of two or more subsystems and a coordinator.
Finally, since one of the central notions of this paper is the notion of a (natural) projection, the reader is referred to [12] for more information on the state complexity of projected regular languages.
The rest of this paper is organized as follows. In Section 2, basic definitions and concepts of automata theory and discrete-event systems are recalled. In Section 3, a polynomial-time algorithm for testing conditional decomposability for a general monolithic system is presented. In Section 4, this algorithm is modified to extend the coordinator alphabet so that the specification becomes conditionally decomposable. In Section 5, the relation of nonblockingness of a coordinated system with conditional decomposability is discussed. The conclusion with hints for future developments is presented in Section 6.
Section snippets
Preliminaries and definitions
In this paper, we assume that the reader is familiar with the basic concepts of supervisory control theory [13] and automata theory [14]. For an alphabet , defined as a finite nonempty set, denotes the free monoid generated by , where the unit of , the empty string, is denoted by . A language over is a subset of . A prefix closure of a language is the set of all prefixes of all words of , i.e., it is defined as the set . A language is said to be prefix
Polynomial test of conditional decomposability
In this section, we first construct a polynomial-time algorithm for the verification of conditional decomposability for alphabets , and , that is, for the case , and then show how this is used to verify conditional decomposability for a general . To this end, consider a language over , marked by a generator . To verify whether or not is conditionally decomposable with respect to , and , we construct a new structure as a parallel composition of two copies of ,
Extension of the coordinator alphabet
According to Theorem 8, we can again consider only the case . To compute an extension of so that the language becomes conditionally decomposable, we modify Algorithm 1 to Algorithm 2, which uses more structural properties of the structure . First, however, we explain the technique on an example.
Example 9 Consider the generator and of Example 3, Example 7. The main idea of this technique is to construct, step-by-step, the parallel composition of and , and to verify that all the steps
Relationship of nonblockingness of coordinated systems to conditional decomposability
In this section, we study the relation between conditional decomposability and nonblockingness of coordinated discrete-event systems. A coordinated modular discrete-event system is a system composed (by parallel composition) of two or more subsystems. In this section, we consider the case of one central coordinator. Let , and let , be generators over the respective alphabets . The coordinated system is defined as , where is the coordinator over an
Conclusion
The main contributions of this paper are polynomial-time algorithms for verification whether a language is conditionally decomposable and for an extension of the coordinator alphabet . Our approach to extend the alphabet is based on the successive addition of events to the alphabet . Another approach has recently been discussed in [29], where the problematic transitions are identified, and the events of these transitions are renamed. From the viewpoint of applications, however, our
Acknowledgments
The authors gratefully acknowledge very useful suggestions and comments of the anonymous referees. The research has been supported by the GAČR grants P103/11/0517 and P202/11/P028, and by RVO: 67985840.
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