Yongzhi Cao

Observability and decentralized control of fuzzy discrete-event systems

Fuzzy discrete-event systems as a generalization of (crisp) discrete-event systems have been introduced in order that it is possible to effectively represent uncertainty, imprecision, and vagueness arising from the dynamic of systems. A fuzzy discrete-event system has been modeled by a fuzzy automaton; its behavior is described in terms of the fuzzy language generated by the automaton. In this paper, we are concerned with the supervisory control problem for fuzzy discrete-event systems with partial observation. Observability, normality, and co-observability of crisp languages are extended to fuzzy languages. It is shown that the observability, together with controllability, of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor. When a decentralized solution is desired, it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable. Moreover, the infimal controllable and observable fuzzy superlanguage, and the supremal controllable and normal fuzzy sublanguage are also discussed. Simple examples are provided to illustrate the theoretical development

Supervisory control of fuzzy discrete event systems

To cope with situations in which a plants dynamics are not precisely known, we consider the problem of supervisory control for a class of discrete event systems modeled by fuzzy automata. The behavior of such discrete event systems is described by fuzzy languages; the supervisors are event feedback and can only disable controllable events with any degree. In this new sense, we present a necessary and sufficient condition for a fuzzy language to be controllable. We also study the supremal controllable fuzzy sublanguage and the infimal controllable fuzzy superlanguage.

State-Based Control of Fuzzy Discrete-Event Systems

To effectively represent possibility arising from states and dynamics of a system, fuzzy discrete-event systems (DESs) as a generalization of conventional DESs have been introduced recently. Supervisory-control theory based on event feedback has been well established for such systems. Noting that the system state description, from the viewpoint of specification, seems more convenient, we investigate the state-based control of fuzzy DESs in this paper. An approach to finding all fuzzy states that are reachable by controlling the system is presented first. After introducing the notion of controllability for fuzzy states, a necessary and sufficient condition for a set of fuzzy states to be controllable is then provided. It was also found that event- and state-based controls are not equivalent, and the relationship between them was further discussed. Finally, we examine the possibility of driving a fuzzy DES under control from a given initial state to a prescribed set of fuzzy states and then keeping it there indefinitely

A Fuzzy Petri-Nets Model for Computing With Words

Motivated by Zadehs paradigm of computing with words (CWs) rather than numbers, several formal models of CWs have recently been proposed. These models are based on automata and, thus, are not well suited for concurrent computing. In this paper, we incorporate the well-known model of concurrent computing, which is called the Petri net, together with fuzzy-set theory and, thereby, establish a concurrency model of CWs-fuzzy Petri nets for CWs (FPNCWs). The new feature of such fuzzy Petri nets is that the labels of transitions are some special words modeled by fuzzy sets. By employing the methodology of fuzzy reasoning, we give a faithful extension of an FPNCW that makes computing with more words possible. The language expressiveness of the two formal models of CWs, i.e., fuzzy automata for CWs as well as FPNCWs, is compared. A few small examples are provided to illustrate the theoretical development.

Bisimulations for Fuzzy-Transition Systems

There has been a long history of using fuzzy-language equivalence to compare the behavior of fuzzy systems; however, the comparison at this level is too coarse. Recently, a finer behavioral measure, i.e., bisimulation, has been introduced to fuzzy-finite automata. However, the results obtained are applicable only to finite-state systems. In this paper, we consider bisimulation for general fuzzy systems, which may be infinite state or infinite event, by modeling them as fuzzy-transition systems (FTSs). To help understand and check bisimulation, we characterize it in three ways by enumerating whole transitions, comparing individual transitions, and using a monotonic function. In addition, we address composition operations, subsystems, quotients, and homomorphisms of FTSs and discuss their properties connected with bisimulation. The results presented here are useful to compare the behavior of general fuzzy systems. In particular, this makes it possible to relate an infinite fuzzy system to a finite one, which is easier to analyze, with the same behavior.

Retraction and Generalized Extension of Computing With Words

Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadehs paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.

A Behavioral Distance for Fuzzy-Transition Systems

In contrast with the existing approaches to exact bisimulation for fuzzy systems, we introduce a robust notion of behavioral distance to measure the behavioral similarity of nondeterministic fuzzy-transition systems which are a generalization of fuzzy automata. This behavioral distance provides a quantitative analogue of bisimilarity and is defined as the greatest fixed point of a suitable monotonic function. The behavioral distance has the important property that two systems are at zero distance if and only if they are bisimilar. Moreover, for any given threshold, we find that systems with behavioral distances bounded by the threshold are equivalent. In addition, we show that two system combinators-parallel composition and product-are nonexpansive with respect to our behavioral distance, which makes compositional verification possible. The theory developed here is applicable to the quantitative verification, approximate reduction, and reliability analysis of fuzzy-transition systems.

Nondeterministic fuzzy automata

To handle fuzzy uncertainty in system modeling, nondeterministic finite automata have been generalized into fuzzy automata. After a reexamination of the notions of fuzzy automata in the literature, we ascertain that the fundamental property-nondeterminism-in nondeterministic finite automata has not been well embodied in the generalization. In order to reflect nondeterminism in fuzzy automata, we introduce nondeterministic fuzzy automata with or without @e-moves and fuzzy languages recognized by them. Like nondeterministic finite automata, nondeterministic fuzzy automata provide a mathematical representation of nondeterministic dynamic fuzzy systems. Moreover, we show that (deterministic) fuzzy automata, nondeterministic fuzzy automata, and nondeterministic fuzzy automata with @e-moves are all equivalent in the sense that they recognize the same class of fuzzy languages, which is an extension of the well-known equivalence among finite automata, nondeterministic finite automata, and nondeterministic finite automata with @e-moves.

Simulation for lattice-valued doubly labeled transition systems

During the last decades, a large amount of multi-valued transition systems, whose transitions or states are labeled with specific weights, have been proposed to analyze quantitative behaviors of reactive systems. To set up a unified framework to model and analyze systems with quantitative information, in this paper, we present an extension of doubly labeled transition systems in the framework of residuated lattices, which we will refer to as lattice-valued doubly labeled transition systems (LDLTSs). Our model can be specialized to fuzzy automata over complete residuated lattices, fuzzy transition systems, and multi-valued Kripke structures. In contrast to the traditional yes/no approach to similarity, we then introduce lattice-valued similarity between LDLTSs to measure the degree of closeness of two systems, which is a value from a residuated lattice. Further, we explore the properties of robustness and compositionality of the lattice-valued similarity. Finally, we extend the Hennessy-Milner logic to the residuate lattice-valued setting and show that the obtained logic is adequate and expressive with lattice-valued similarity.

Lattice-valued simulations for quantitative transition systems

Quantitative (bi)simulations taking values from non-negative real numbers enjoy numerous applications in the analysis of labeled transition systems, whose transitions, states, or labels contain quantitative information. To investigate the simulation semantics of labeled transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled transition systems, called the quantitative transition systems (QTSs), whose labels are equipped with a residuated lattice-valued equality relation. We then establish a lattice-valued relation between states of a QTS, called approximate similarity, to quantify to what extent one state is simulated by another. One main contribution of this paper is to show that unlike the classic setting where similarity has both fixed point and logical characterizations, these results do not hold for approximate similarity on QTSs in general, but they hold for QTSs having truth values from finite Heyting algebras. Introduced an extension of labeled transition systems called quantitative transition systems.Presented an approximate similarity between states of quantitative transition systems.Discussed the fixed point and logical characterizations of approximate similarity.