State-based fault diagnosis of discrete-event systems with partially observable outputs

Abstract

This paper presents a state-based method for solving the problems of diagnosis and diagnosability of discrete-event systems (DES) with partially observable outputs, due to the lack or limitations of sensors. The diagnoser used for diagnosis consists of two parts: a state estimator and a failure decision-maker. The state estimator makes the state estimation of a system based on the observed output sequence and transfers the estimation to the failure decision-maker that determines whether a fault occurs or not. Moreover, the state or condition (failure status) of the system is not required to be known when launching the diagnoser; thus the system and the diagnoser do not have to be initialized simultaneously, i.e., the diagnoser may be initialized at any moment while the system is operational. Under the premise that the outputs of a system are partially observable, the notion of diagnosability is given and an efficient algorithm for verification of diagnosability is designed with the polynomial computational complexity with respect to the number of system states. Finally, the proposed algorithm is applied to a pump-valve-controller system.

Introduction

In the last three decades, modeling, control, and diagnosis of discrete-event systems (DES) [1], [2], [3], [4], [5], [6], [50], [51], [52] have been extensively studied by researchers and engineers from different domains. Typical applications include but are not restricted to aerospace systems, transportation systems manufacturing systems [43], [44], [45], [46], and real-time scheduling and reconfigurations [55], [56], [57], [58]. Informally, a DES is discrete in time and in state space, and event-driven rather than time-driven.

Fault diagnosis plays an vital role in maintaining the performance and enhancing the reliability of DES, especially the safety-critical systems. The problem of fault diagnosis has attracted considerable attention from industry and academia, and it has been widely investigated in the literature [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. Initially, the problem of fault diagnosis is studied in [7] and [9] where the concept of diagnosability is introduced and formalized in the DES setting. Later on, problems of modular [10], decentralized [11], [12], [14], state-based [13], distributed [15], hierarchical [16], [17], and robust approaches [18] to diagnosis have also been discussed. Moreover, the method to diagnosis is investigated for timed DES in [19] and [20]. In [21], the diagnosis technique that utilizes model checking is introduced. In [22], [23], [24], diagnosability is extended to stochastic, fuzzy, and bi-fuzzy DES settings, respectively. In [25], a broader spectrum of diagnoser-based approaches regarding the degree of reasoning performed offline is proposed. In [26], the diagnoser is constructed in the form of the symbolic observation graph (SOG), which combines symbolic and enumerative representations as a basis of efficient verification. In [27], the abstraction-based verification of diagnosability is presented for the purpose of reducing the computational cost in the case of complex DES. Most of the aforementioned diagnosis methods follow the event-based framework [7], i.e., the diagnosis is deployed based on the observation of event sequences. The system faults are characterized by pre-specified faulty events and thus the event set is partitioned into faulty and non-faulty parts. To get a general understanding on the literature, the reader can refer to the survey paper [28], where the state of the art of techniques and tools relevant to fault diagnosis of DES is well reviewed.

In the context of DES, automata and Petri nets (PN) are the most used modeling formalisms. The pioneer works [7], [9] on fault diagnosis are based on automata. In [7], a model called “diagnoser” is introduced to verify diagnosability by examining the existence of indeterminate cycles, and diagnosis is deployed online by a mapping relation between the online observations and the states of the diagnoser. However, the construction of a diagnoser suffers from the state explosion problem. To tackle this issue, an efficient algorithm for verification of diagnosability [29], [30] is proposed and the computational complexity is polynomial with respect to the number of system states and linear with respect to the number of failure types. Besides, a series of work [31], [32], [33], [34], [35], [36], [37], [38], [39] concerning diagnosis and diagnosability of DES modeled by PN ensues owing to graphical and mathematical representations of PN. Various diagnosis methods of DES using PN, such as P-invariants [31], the basic reachability graph [35], and the integer linear programming [36], [37], [38], are proposed. In [32], the issues of diagnosability and online diagnosis of DES using interpreted PN are addressed. In [33], online diagnosis of DES modeled by partially observed PN is investigated based on the capture and analysis of observation sequences. In [34], a general setting that markings and transitions are partially observable is considered and the diagnosis problem is studied by transforming partially observed PN into equivalent labeled PN. For more details, the reader can refer to the survey papers [4], [40] and the references therein.

In [13], a state-based method for fault diagnosis of a DES with fully observable outputs is introduced. The system to be diagnosed is modeled as a nondeterministic finite-state Moore automaton. It is assumed that the state set of the system can be partitioned according to the condition. The diagnosis problem is to decide that the state belongs to the normal or faulty partition when the last measurement is received via sensor readings. As the extension and reinforcement of [13], this work investigates the problems of diagnosis and diagnosability of a DES with partially observable outputs by considering more practical cases (partial observation) and the algorithm optimization (efficiency). The diagnoser in this research consists of two parts: a state estimator and a failure decision-maker for better expansibility and flexibility. In practice, a state estimator often needs to be renovated according to accessible information (e.g., output signals), while a failure decision-maker is fixed. The diagnosis is implemented online, i.e., the system is operational and the diagnosis decision is updated after a new output signal is observed. As the basis of online diagnosis, diagnosability analysis is performed. The notion of diagnosability is given and a polynomial-time algorithm is designed for verifying diagnosability without constructing a diagnoser. To demonstrate the proposed algorithm, a pump-valve-controller system [13] is provided.

The reminder of the paper is arranged as follows. Section 2 provides necessary concepts, terminologies, and particularly recalls the definition of the output projection function. In Section 3, an online diagnosis framework for detecting the occurrences of faults and localizing the cause of faults is introduced. Different from [13], the diagnoser in this research consists of two parts: a state estimator and a failure decision-maker. Section 4 gives the notion of diagnosability of a DES with partially observed outputs. In Section 5, an efficient algorithm for verification of diagnosability is designed and the computational complexity is polynomial with respect to the number of system states. Furthermore, a model reduction technique for detecting cycles that violate diagnosability condition efficiently is discussed. In Section 6, the developed algorithm is applied to a pump-valve-controller system. Further discussions on the comparison between state-based and event-based diagnosis approaches and the advantages of the proposed approach over the existing work are presented in Section 7. Finally, we draw the conclusion in Section 8. For the better readability, the related proofs have been moved to Appendix.