Detectability of Discrete Event Systems

In this note, we investigate the detectability problem in discrete event systems. We assume that we do not know initially which state the system is in. The problem is to determine the current and subsequent states of the system based on a sequence of observations. The observation includes partial event observation and/or partial state observation, which leads to four possible cases. We further define four types of detectabilities: strong detectability, (weak) detectability, strong periodic detectability, and (weak) periodic detectability. We derive necessary and sufficient conditions for these detectabilities. These conditions can be checked by constructing an observer, which models the estimation of states under different observations. The theory developed in this note can be used in feedback control and diagnosis. If the system is detectable, then the observer can be used as a diagnoser to diagnose the failure states of the system.     

Generalized Detectability for Discrete Event Systems

In our previous work, we investigated detectability of discrete event systems, which is defined as the ability to determine the current and subsequent states of a system based on observation. For different applications, we defined four types of detectabilities: (weak) detectability, strong detectability, (weak) periodic detectability, and strong periodic detectability. In this paper, we extend our results in three aspects. (1) We extend detectability from deterministic systems to nondeterministic systems. Such a generalization is necessary because there are many systems that need to be modeled as nondeterministic discrete event systems. (2) We develop polynomial algorithms to check strong detectability. The previous algorithms are based on an observer whose construction is of exponential complexity, while the new algorithms are based on a new automaton called a detector. (3) We extend detectability to D-detectability. While detectability requires determining the exact state of a system, D-detectability relaxes this requirement by asking only to distinguish certain pairs of states. With these extensions, the theory on detectability of discrete event systems becomes more applicable in solving many practical problems.     

离散事件系统的可测性

讨论基于自动机/形式语言模型的离散事件系统（DES）的可测性问题。可测性即为根据系统的可观事件和状态输出的信息估计系统的当前状态。定义了四种可测性：强可测性,弱可测性,强周期可测性,弱周期可测性。给出了这些可测性的充要条件,这些充要条件可通过构建观测器进行有效的判定。     

State estimation and detectability of probabilistic discrete event systems

A probabilistic discrete event system (PDES) is a nondeterministic discrete event system where the probabilities of nondeterministic transitions are specified. State estimation problems of PDES are more difficult than those of non-probabilistic discrete event systems. In our previous papers, we investigated state estimation problems for non-probabilistic discrete event systems. We defined four types of detectabilities and derived necessary and sufficient conditions for checking these detectabilities. In this paper, we extend our study to state estimation problems for PDES by considering the probabilities. The first step in our approach is to convert a given PDES into a nondeterministic discrete event system and find sufficient conditions for checking probabilistic detectabilities. Next, to find necessary and sufficient conditions for checking probabilistic detectabilities, we investigate the “convergence” of event sequences in PDES. An event sequence is convergent if along this sequence, it is more and more certain that the system is in a particular state. We derive conditions for convergence and hence for detectabilities. We focus on systems with complete event observation and no state observation. For better presentation, the theoretical development is illustrated by a simplified example of nephritis diagnosis.     

Detectability of networked discrete event systems

Detectability of discrete event systems, defined as the ability to determine the current and subsequent states, is very important in diagnosis, control, and many other applications. So far only detectability of non-networked discrete event systems has been defined and investigated. Non-networked discrete event systems assume that all the communications are reliable and instantaneous without any delays or losses. This assumption is often violated in networked systems. In this paper, we study detectability for networked discrete event systems. We investigate the impact of communication delays and losses on detectability. We define two classes of detectabilities: network detectability for determining the state of a networked discrete event systems and network D-detectability for distinguishing certain pairs of states of the systems. Necessary and sufficient conditions for network detectability and network D-detectability are derived. Methods to check network detectability and network D-detectability are also developed. Examples are given to illustrate the results.     

Control of nondeterministic discrete-event systems for bisimulation equivalence

Most prior work on supervisory control of discrete event systems is for achieving deterministic specifications, expressed as formal languages. In this paper we study supervisory control for achieving nondeterministic specifications. Such specifications are useful when designing a system at a higher level of abstraction so that lower level details of system and its specification are omitted to obtain higher level models that may be nondeterministic. Nondeterministic specifications are also meaningful when the system to be controlled has a nondeterministic model due to the lack of information (caused for example by partial observation or unmodeled dynamics). Language equivalence is not an adequate notion of behavioral equivalence for nondeterministic systems, and instead we use the finest known notion of equivalence, namely the bisimulation equivalence. Choice of bisimulation equivalence is also supported by the fact that bisimulation equivalence specification is equivalent to a specification in the temporal logic of /spl mu/-calculus that subsumes the complete branching-time logic CTL*. Given nondeterministic models of system and its specification, we study the design of a supervisor (possibly nondeterministic) such that the controlled system is bisimilar to the specification. We obtain a small model theorem showing that a supervisor exists if and only if it exists over a certain finite state space, namely the power set of Cartesian product of system and specification state spaces. Also, the notion of state-controllability is introduced as part of a necessary and sufficient condition for the existence of a supervisor. In the special case of deterministic systems, we provide an existence condition that can be verified polynomially in both system and specification states, when the existence condition holds.     

基于不确定观测下离散事件系统可诊断性的研究

在实际应用系统中,由于传感器故障、传感器限制和网络中的数据包丢失等原因,事件的可观测值变得不确定,使得观测系统行为变得尤为复杂。针对离散事件系统中,同个事件串可能有多个观测值以及不同状态下同个事件观测值也可能不同的问题,提出一种不确定观测下故障诊断验证的方法。首先对不确定观测的离散事件系统的可诊断性进行形式化,然后构建出用于上述故障诊断验证的验证器;基于验证器提出了系统基于不确定观测下可诊断的充要条件及验证算法;最后,实例说明不确定观测下故障诊断验证算法的应用。与现有研究相比,提出的方法对故障事件的观测值没有约束,可以为0个或多个观测值,使此方法应用的场景更为广泛。     

A Small Model Theorem for Bisimilarity Control Under Partial Observation

This paper extends our prior result on decidability of bisimulation equivalence control from the setting of complete observations to that of partial observations. Besides being control compatible, the supervisor must now also be observation compatible. We show that the "small model theorem" remains valid by showing that a control and observation compatible supervisor exists if and only if it exists over a certain finite state space, namely the power set of the Cartesian product of the system and the specification state spaces. Note to Practitioners-Non-determinism in discrete-event systems arises due to abstraction and/or unmodeled dynamics. This paper addresses the issue of control of non-deterministic systems subject to non-deterministic specifications, under a partial observation of events. Non-deterministic plant and specification are useful when designing a system at a higher level of abstraction so that lower level details of the system and its specification are omitted to obtain higher level models that are non-deterministic. The control goal is to ensure that the controlled system has an equivalent behavior as the specification system, where the notion of equivalence used is that of bisimilarity. Bisimilarity requires the existence of an equivalence relation between the states of the two systems so that transitions on common events beginning from a pair of equivalent states end up in a pair of equivalent successor states. Supervisors are also allowed to be nondeterministic, where the nondeterminism in control is implemented by selecting control actions nondeterministically from among a set of precomputed choices. The main contribution of this paper is to show that a supervisor exists if and only if one exists where the size of its state-space upper bounded and so it suffices to search over this state space. We illustrate our results through a manufacturing example     

Detectability in stochastic discrete event systems

A discrete event system possesses the property of detectability if it allows an observer to perfectly estimate the current state of the system after a finite number of observed symbols, i.e., detectability captures the ability of an observer to eventually perfectly estimate the system state. In this paper we analyze detectability in stochastic discrete event systems (SDES) that can be modeled as probabilistic finite automata. More specifically, we define the notion of A-detectability, which characterizes our ability to estimate the current state of a given SDES with increasing certainty as we observe more output symbols. The notion of A-detectability is differentiated from previous notions for detectability in SDES because it takes into account the probability of problematic observation sequences (that do not allow us to perfectly deduce the system state), whereas previous notions for detectability in SDES considered each observation sequence that can be generated by the underlying system. We discuss observer-based techniques that can be used to verify A-detectability, and provide associated necessary and sufficient conditions. We also prove that A-detectability is a PSPACE-hard problem.     

Opacity of discrete event systems and its applications

In this paper, we investigate opacity of discrete event systems. We define two types of opacities: strong opacity and weak opacity. Given a general observation mapping, a language is strongly opaque if all strings in the language are confused with some strings in another language and it is weakly opaque if some strings in the language are confused with some strings in another language. We show that security and privacy of computer systems and communication protocols can be investigated in terms of opacity. In particular, two important properties in security and privacy, namely anonymity and secrecy, can be studied as special cases of opacity. We also show that by properly specifying the languages and the observation mapping, three important properties of discrete event systems, namely observability, diagnosability, and detectability, can all be reformulated as opacity. Thus, opacity has a wide range of applications. Also in this paper we provide algorithms for checking strong opacity and weak opacity for systems described by regular languages and having a generalized projection as the observation mapping.     

Fault-tolerant supervisory control under C,D observability and its application

In this paper, a discrete-event dynamical system (DEDS) approach is employed to build a fault-tolerant system under partial observation with some diagnostic schemes. We address the issue of detectability for state identification and observability for achievable legal languages in DEDSs. Especially, we introduce C, D-observability conditions to determine the observability in view of supervisor synthesis. Based upon these, we study the diagnostic problems and the construction of fault-tolerant supervisory control systems with partial observations through the behavioural analysis upon a discrete-event model. A case study of a plasma etching system is described to illustrate the proposed approach.     

连续时间Markov决策过程互模拟等价及逻辑保持

模型检测中,Markov决策过程可以建模具有不确定性的系统,然而状态空间爆炸问题将会影响系统验证的成败与效率,互模拟等价可以用于系统状态的简约.在强互模拟关系的基础上,给出Markov决策过程模型弱互模拟等价关系的概念,导出了连续时间Markov决策过程及其内嵌离散时间Markov决策过程互模拟等价关系的内在联系;在强互模拟等价关系逻辑特征保持的基础上,给出弱互模拟等价关系下的逻辑保持性质,证明了弱互模拟等价的两个状态,同时满足除下一步算子外的连续随机逻辑公式,从而可以将原模型中的验证问题转换为简约后模型的验证问题,提高验证的效率.     

Strong planning under partial observability

Rarely planning domains are fully observable. For this reason, the ability to deal with partial observability is one of the most important challenges in planning. In this paper, we tackle the problem of strong planning under partial observability in nondeterministic domains: find a conditional plan that will result in a successful state, regardless of multiple initial states, nondeterministic action effects, and partial observability.We make the following contributions. First, we formally define the problem of strong planning within a general framework for modeling partially observable planning domains. Second, we propose an effective planning algorithm, based on and-or search in the space of beliefs. We prove that our algorithm always terminates, and is correct and complete. In order to achieve additional effectiveness, we leverage on a symbolic, bdd-based representation for the domain, and propose several search strategies. We provide a thorough experimental evaluation of our approach, based on a wide selection of benchmarks. We compare the performance of the proposed search strategies, and identify a uniform winner that combines heuristic distance measures with mechanisms that reduce runtime uncertainty. Then, we compare our planner mbp with other state-of-the art-systems. mbp is able to outperform its competitor systems, often by orders of magnitude.     

Bounds on the Number of Markings Consistent With Label Observations in Petri Nets

In this paper, we consider state estimation in discrete-event systems (DESs) modeled by labeled Petri nets and present upper bounds on the number of system states (or markings) that are consistent with an observed sequence of labels. Our analysis is applicable to Petri nets that may have nondeterministic transitions (i.e., transitions that share the same label) and/or unobservable transitions (i.e., transitions that are associated with the null label). More specifically, given knowledge of a labeled Petri net structure and its initial state, we show that the number of consistent markings in a Petri net with nondeterministic transitions is at most polynomial in the length of the observation sequence (i.e., polynomial in the number of labels observed). This polynomial dependency of the number of consistent markings on the length of the observation sequence also applies to Petri nets with unobservable transitions under the assumption that their unobservable subnets are structurally bounded. The bounds on the number of markings established in this paper imply that the state estimation problem in labeled Petri nets can be solved with complexity that is polynomial in the length of the observation sequence.     

判定非确定离散事件系统稳定性的多项式复杂度算法

本文讨论基于非确定自动机,形式语言模型的非确定离散事件系统稳定性的多项式算法.在引入拟距离的概念之后.根据拟距离形式化地定义了非确定离散事件系统稳定性.以往判定非确定离散事件系统稳定性的算法基于系统的观测器实现,该观测器在结构上具有指数复杂度,因此本文分析系统结构和观测器结构之间的关系,基于对系统状态对的讨论,提出了判定系统稳定性的有效多项式搜索算法.     

An algorithm for calculating indistinguishable states and clusters in finite-state automata with partially observable transitions

This paper presents a new algorithm for efficiently calculating pairs of indistinguishable states in finite-state automata with partially observable transitions. The need to obtain pairs of indistinguishable states occurs in several classes of problems related to control under partial observation, diagnosis, or distributed control with communication for discrete event systems. The algorithm obtains all indistinguishable state pairs in polynomial time in the number of states and events in the system. Another feature of the algorithm is the grouping of states into clusters and the identification of indistinguishable cluster pairs. Clusters can be employed to solve control problems for partially observed systems.     

不确定规划中非循环可达关系的求解方法

对一个不确定状态转移系统求多个规划问题,那么获得不确定状态转移系统的状态可达关系可以方便求解规划问题,减少冗余计算,建立系统的引导信息。提出一个关于矩阵求不确定领域的状态可达性关系的方法,主要思想是以矩阵乘法来模拟状态转移系统中状态转移,对不确定动作带来的扩散和确定关系带来的聚合进行了统计和处理,从而获得状态可达信息。证明了方法的正确性和有效性。在不确定规划中确定了状态之间的可达性关系,可以在求规划解时删除对规划没有用的状态节点和状态动作序偶;选择能到达目标节点的状态节点和状态动作序偶;进行启发式正向搜索;减少大量冗余计算;提高求解效率。     

The complexity of some reachability problems for a system on a finite group

We consider a system defined as the product of a finite set of periodic systems on cyclic groups. It is of interest to determine if certain subgroups and unions of subgroups of the state set are reachable from a specified initial state, and in particular to determine the computational complexity of verifying such reachability. These questions are motivated by certain problems that arise in the modelling and control of discrete event systems and certain forms of periodic scheduling. Our main result is that deciding whether or not the union of a certain set of subgroups is reachable or not is NP-complete.