Jan Komenda

The Czech Academy of Sciences | AVCR · Institute of Mathematics

P-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by max-plus linear-dual inequalities (LDIs), i.e., systems of linear dynamical inequalities in the primal and dual operations of the max-plus algebra. We define a new class of m...

Complex systems are often composed of many small communicating components called modules. We investigate the synthesis of supervisory controllers for modular systems under partial observation that, as the closed-loop system, realize the supremal normal sublanguage of the specification. We call such controllers maximally permissive normal supervisor...

Conditions preserving observability of specifications between the plant and its abstraction are essential for hierarchical supervisory control of discrete-event systems under partial observation. Observation consistency and local observation consistency were identified as such conditions. To preserve normality, only observation consistency is requi...

Complex systems are often composed of many small communicating components called modules. We investigate the synthesis of supervisory controllers for modular systems under partial observation that, as the closed-loop system, realize the supremal normal sublanguage of the specification. Such controllers are called maximally permissive normal supervi...

This article investigates diagnosability and $T$ -diagnosability for discrete-event systems modeled by unambiguous max-plus automata (UMPAs). More precisely, diagnosability requires that the occurrence of any fault can be detected within a finite number of events after the fault has occurred. $T$ -diagnosability requires that the occurrence of...

In this paper we will show that safe timed Petri nets (TPNs) with preselection semantics can be represented by deterministic two-level (max,+) automata (2l-MPA). In 2l-MPA, the outer dynamics, corresponding to the evolution of component clocks, is represented by deterministic Mealy automata with vector-valued outputs from (max,+) semiring. The oute...

In this contribution, we reconsider the notion of determinism for max-plus automata by including the quantitative aspect in its definition. A determinization procedure adapted to the so-called notion of weight-determinism is proposed. The motivation is to generalize the results based on the determinism of max-plus automata.

P-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by max-plus linear-dual inequalities (LDIs), i.e., systems of linear dynamical inequalities in the max-plus algebra and its dual, the min-plus algebra. We define a new class of...

These slides were used to present the paper "Switched Max-Plus Linear-Dual Inequalities: Application in Scheduling of Multi-Product Processing Networks" (DOI: 10.1016/j.ifacol.2022.10.343) at the 16th IFAC Workshop on Discrete Event Systems (WODES 2022).

We investigate modular supervisory control of discrete-event systems composed of several groups of components, where each group consists of similar modules. Because of the similar structures of the modules, such systems can be represented as a set of (group) templates. Supervisory control can then be performed on these templates, resulting in a set...

Let us consider a parametric weighted directed graph in which every arc (j,i) has weight of the form w((j,i))=max(Pij+λ,Iij−λ,Cij), where λ is a real parameter and P, I and C are arbitrary square matrices with elements in R∪{−∞}. In this paper, we design an algorithm that solves the Non-positive Circuit weight Problem (NCP) on this class of paramet...

P-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by systems of linear dynamical inequalities in the max-plus algebra and its dual, the min-plus algebra, referred to as max-plus linear-dual inequalities (LDIs). We define a new...

Conditions preserving observability of specifications between the plant and its abstraction are fundamental for hierarchical supervisory control of discrete-event systems under partial observation. Observation consistency and local observation consistency were identified as such conditions. To preserve normality, only observation consistency is req...

These slides were used to present the paper "Periodic Trajectories in P-Time Event Graphs and the Non-Positive Circuit Weight Problem" (DOI: 10.1109/LCSYS.2021.3085521) at the 60th IEEE Conference on Decision and Control (CDC 2021).

In this letter, we investigate multi-agent discrete-event systems with partial observation. The agents can be divided into several groups in each of which the agents have identical (isomorphic) state transition structures, and can thus be relabeled into the same template. Based on the template a scalable supervisor whose state size and computat...

P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers $d$ , the existence of consistent $d$ -periodic trajectories for a given P-TEG. In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weigh...

In this paper, we deal with state estimation of timed discrete event systems that are modeled by max-plus automata (MPAs), where only some events are observable. For a given MPA, a formal procedure is first proposed for constructing its observer by extending our previous concept of observer for unambiguous MPAs to polynomially ambiguous MPAs. As an...

In this paper we investigate multi-agent discrete-event systems with partial observation. The agents can be divided into several groups in each of which the agents have similar (isomorphic) state transition structures, and thus can be relabeled into the same template. Based on the template a scalable supervisor whose state size and computational co...

P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers $d$, the existence of consistent $d$-periodic trajectories for a given P-TEG. In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Prob...

Let us consider a parametric weighted directed graph in which every arc $(j,i)$ has weight of the form $w((j,i))=\max(P_{ij}+\lambda,I_{ij}-\lambda,C_{ij})$, where $\lambda$ is a real parameter and $P$, $I$ and $C$ are arbitrary square matrices with elements in $\mathbb{R}\cup\{-\infty\}$. In this paper, we design an algorithm that solves the Non-p...

In this paper, we investigate the behaviour of P-time event graphs, a class of time Petri nets with non-deterministic timing of places. Our approach is based on combined linear descriptions in both (max,+) and (min,+) semirings, where lower bounds on the state vector are (max,+)-linear and upper bounds are (min,+)-linear. We present necessary and s...

Weighted automata with weights (multiplicities) in the product semirings can be viewed as alternative models for timed systems, where concurrent time behaviors are modeled using nondeterminism, while in timed automata using multiple clocks. It is known that safe timed Petri nets can be modeled by means of (max,+) automata, i.e. weighted automata in...

The problem of checking the consistency property in a P-time event graph (P-TEG) is fundamental and has to be carried out before applying any control strategy. However, no generic algorithm that solves this problem is available yet. In this paper we propose a new, stronger, definition for consistency of P-TEGs that is both more useful and easier to...

These slides were used to present the paper "Bounded Consistency of P-time Event Graphs" (DOI: 10.1109/CDC42340.2020.9304309) at the 59th IEEE Conference on Decision and Control (CDC 2020).

In this work we investigate the behavior of P-time event graphs, a class of time Petri nets with nondeterministic timing of places. Our approach is based on combined linear descriptions in both (max,+) and (min,+) semirings, where lower bounds on the state vector are (max,+)-linear and upper bounds are (min,+)-linear. We present necessary and suffi...

It is a well known fact that not all max-plus automata can be determinized, i.e. transformed into deterministic max-plus automata with the same behavior. A classical sequentialization procedure, extended in the literature to max-plus automata, succeeds in computing an equivalent deterministic max-plus automaton for important subclasses of max-plus...

Modular supervisory control is motivated by the gain in complexity of control synthesis of supervisors. Sufficient conditions for maximal permissiveness of supervisors include mutual controllability and mutual normality. In this paper, we show how these conditions can be weakened. Namely, we can relax the requirement that the conditions hold for al...

The fundamental problem in hierarchical supervisory control under partial observation is to find conditions preserving observability between the original (low-level) and the abstracted (high-level) plants. Two conditions for observable specifications were identified in the literature - observation consistency (OC) and local observation consistency...

The fundamental problem in hierarchical supervisory control under partial observation is to find conditions preserving observability between the original (low-level) and the abstracted (high-level) plants. Two conditions for observable specifications were identified in the literature -- observation consistency (OC) and local observation consistency...

In this paper, a uniform approach to maximal permissiveness in modular control of discrete-event systems is proposed. It is based on three important concepts of modular closed-loops: monotonicity, distributivity, and exchangeability. Monotonicity of various closed-loops satisfying a given property considered in this paper holds whenever the underly...

Reports on recent CS society conference meetings and events.

This paper is a survey of the history of max-plus algebra and its role in the field of discrete event systems during the last three decades. It is based on the perspective of the authors but it covers a large variety of topics, where max-plus algebra plays a key role.

(Max,+)-automata are weighted automata over the (max,+) semiring. In this paper we investigate simulation like equivalences between (max,+)-automata. Since (max,+)-automata are nondetermin-istic (weighted) automata, there exist extensions of bisimilarity properties that are weaker than equality of their weighted languages (formal power series). The...

In decentralized supervisory control, several local supervisors cooperate to accomplish a common goal (specification). Controllability and coobservability are the key conditions to achieve a specification in the controlled system. We construct a controllable and coobservable sublanguage of the specification by using additional communications betwee...

The problem considered is to construct all solutions of an equation for a tuple of languages. The tuple in synchronous composition should equal a considered language. Of special interest are the maximal solutions with respect to a partial order relation on the set of solutions. The motivation of the problem is coordination control. The approach to...

We propose a new approach to the modeling and analysis of cycle times in interval P-time event graphs based on dual linear representations in the (max,+) and (min,+) semirings. We start with first order inequalities in the (max,+)-algebra for execution times of transitions based on the simplification at the event graph level. Then we show that the...

Control of a multilevel system is developed for a discrete-event system (DES) structured by an engineering model. In a multilevel system, each subsystem has a set of children at the next-lower level and a unique parent at the next-higher level. A coordinated multilevel DES is defined by the condition that a parent also is involved in the interactio...

In decentralized supervisory control, several local control agents (supervisors) cooperate to achieve a common goal expressed by a safety specification and/or nonblockingness. Coobservability is the key condition to achieve the specification as the resulting language of the controlled system. One of the most important problems is thus to compute a...

In this paper, we further develop the coordination control framework for discrete-event systems with both complete and partial observations. First, a weaker sufficient condition for the computation of the supremal conditionally controllable sublanguage and conditionally normal sublanguage is presented. Then we show that this condition can be impose...

Relative observability was introduced and studied in the framework of partially observed discrete-event systems as a condition stronger than observability and weaker than normality. Unlike observability, relative observability is closed under language unions, which makes it interesting for practical applications. In this paper, we investigate this...

Coordination control for multi-level discrete-event systems is generalized to supervisory control with partial observations. The multi-level system architecture is in this paper restricted to three levels, which include two levels of coordination on the two highest levels. At Level 1 a coordinator coordinates several coordinators of Level 2 and at...

In this paper we are interested in sequentialization of formal power series with coefficients in the semiring (Formula presented.) which represent the behavior of timed Petri nets. Several approaches make it possible to derive nondeterministic (max, + ) automata modeling safe timed Petri nets. Their nondeterminism is a serious drawback since determ...

In this paper, we discuss a supervisory control problem of modular discrete-event systems that allows for a distributed computation of supervisors. We provide a characterization and an algorithm to compute the supervisors. If the specification does not satisfy the properties, we make use of a relaxation of coordination control to compute a sublangu...

In this paper, supervisory control of (max,+) automata is studied. The synthesis of maximally permissive and just-in-time supervisor, as well as the synthesis of minimally permissive and just-after-time supervisor, are proposed. Results are also specialized to nondecreasing solutions, because only such supervisors can be realized in practice. The i...

Control of (max,+) automata is studied within a behavioral framework. The classical tensor product of their linear representations and its generalized version extends the parallel composition of logical automata to (max,+) automata. In terms of behaviors (formal power series) these correspond to Hadamard product and a generalized version of it is s...

(max,+) automata, an important class of both timed automata and weighted automata, are used as a model of a class of timed discrete event systems and their supervisory control is developped. The concept of supervised product is introduced to control (max,+) automata and its properties are studied. Decomposition of morphism matrix of the controlled...

Recently, we have proposed two complementary approaches, top-down and bottom-up, to multilevel supervisory control of discrete-event systems. In this paper, we compare and combine these approaches. The combined approach has strong features of both approaches, namely, a lower complexity of the top-down approach with the generality of the bottom-up a...

In this paper, we simplify the coordination control approach by removing the supervisor for the coordinator from the closed-loop system and relax the restrictions placed on a coordinator. This relaxation results in the simplification of the whole coordination control framework, including the notions of conditional controllability, conditional obser...

A multilevel coordination approach is proposed to lower the complexity of control synthesis of large-scale discrete-event systems. The bottom-up control synthesis method requires only conditional decomposability and conditional controllability of the system and of the specification unlike the top-down approach that requires the specification to be...

This paper presents a compositional modeling approach by means of (max, +) automata. The motivation is to be able to model a complex discrete event system by composing sub-models representing its elementary parts. A direct modeling of safe timed Petri nets using (max, +) automata is first introduced. Based on this result, two types of synchronous p...

Relative observability has been introduced and studied in the framework of partially observed discrete-event systems as a condition stronger than observability, but weaker than normality. However, unlike observability, relative observability is closed under language unions, which makes it interesting for practical applications. In this paper, we in...

In this paper we present a new approach to decentralized supervisory control of large automata with communicating supervisors. We first generalize the recently developed top-down architecture of multilevel coordination control with a hierarchical structure of groups of subsystems, their respective coordinators and supervisors. Namely, in the case w...

In this paper, we further develop the coordination control framework for discrete-event systems with both complete and partial observation. A new weaker sufficient condition for the computation of the supremal conditionally controllable sublanguage is presented. This result is then used for the computation of the supremal conditionally controllable...

A top-down approach to multilevel coordination control is presented along with the corresponding notions of conditional decomposability and conditional controllability. The multilevel structure makes the approach computationally more efficient in comparison with the approach of one central coordinator since fewer events need to be communicated amon...

In decentralized supervisory control, several local control agents (supervisors) cooperate to achieve a common goal, expressed by a safety specification and/or by nonblockingness. It is well-known that coobservability is the key condition to achieve the specification as the resulting language of the controlled system. One of the most important prob...

In this paper, we revise and further investigate the coordination control approach proposed for supervisory control of distributed discrete-event systems with synchronous communication based on the Ramadge-Wonham automata framework. The notions of conditional decomposability, conditional controllability, and conditional closedness ensuring the exis...

A coordinated distributed discrete-event system consists of a coordinator and two or more subsystems. The coordinator directly influences the dynamics of the other subsystems but the subsystems do not directly influence each other. Coordination control of a coordinated distributed discrete-event system is to synthesize supervisors for the coordinat...

A distributed discrete-event system consists of an interconnection of two or more subsystems. Control of a distributed system demands a set of controllers each receiving an observed event stream of a local subsystem and providing a subset of enabled events to that subsystem. Several control architectures are defined in this chapter: Distributed con...

In discrete-event systems it is often not realistic to assume that all events of such a system are observable. This is either because some events are typically not directly observable from their very nature (such as failure events, internal hidden events, etc.) or it is too costly to have sensors and observe every event that can occur in the system...

Note: Proposition 17 is not correct, see the full version of the paper "Coordination Control of Discrete-Event Systems Revisited" In this paper, we further develop the coordination control scheme for discrete-event systems based on the Ramadge-Wonham framework. The notions of conditional decomposability, conditional controllability, and conditiona...

Supervisory control of discrete-event systems with a global safety specification and with only local supervisors is a difficult problem. For global specifications the equivalent conditions for local control synthesis to equal global control synthesis may not be met. This paper formulates and solves a control synthesis problem for a generator with a...

The requirement of a language to be conditionally decomposable is imposed on a specification language in the coordination supervisory control framework of discrete-event systems. In this paper, we present a polynomial-time algorithm for the verification whether a language is conditionally decomposable with respect to given alphabets. Moreover, we a...

(Max,+)-automata are weighted automata over the (max,+) semiring. Deterministic (max,+)-automata are sequential models that need to be composed in order to model concurrent timed systems. In this paper, the synchronous product and the supervised product are defined by coinduction. Maximally permissive timed control then amounts to compute residuati...

In this paper, hierarchical control of both monolithic and modular discrete-event systems under partial observations is studied. Two new conditions, called observation consistency and local observation consistency, are proposed. These conditions are sufficient for the preservation of observability between the original and the abstracted plant. More...

Distributed systems consist of an interconnection of two or more subsystems. Control of such systems is structured by two or more controllers, each receiving an observation stream from a local subsystem and providing an input to the local subsystem. Coordinated distributed systems are defined for linear systems, for Gaussian systems, and for discre...

Synthesis of normal or controllable and normal sublanguages of global specification languages without computation of the global modular plant is a difficult problem. In this paper, these sublanguages are computed using a coordinator. We recall the notion of conditional controllability, introduce a notion of conditional normality, and prove necessar...

Decidability of existence, rationality of delay controllers and robust delay controllers are investigated for systems with time weights in the tropical and interval semirings. Depending on the (max,+) or (min,+)-rationality of the series specifying the controlled system and the control objective, cases are identified where the controller series def...

Linear description of interval P-time event graphs using a product idempotent semiring is proposed and applied to dead token detection. The dependence of dead token on initial condition is studied using residuation theory. Finally, the relationship with the spectral theory of matrices over product semirings is discussed.

In this paper, the supervisory control synthesis of modular or distributed discrete-event systems is discussed. The coordination-control architecture proposed by Komenda and Van Schuppen (2008) and studied for prefix-closed specification languages by Komenda et al. (2010a,b) is generalized to the case of non-prefix-closed global specification langu...

Modular control of discrete-event systems with a global Specification and with only local supervisors is a difficult problem. For global Specifications, however, the equivalent conditions may not be met. This paper formulates and solves a control synthesis problem for a generator with a global Specification and with a combination of a coordinator a...

In this paper, we study hierarchical supervisory control with partial observations. In particular, we are interested in preservation of supremal normal and supremal controllable and normal sublanguages from the abstracted system (high level) in the original (low level) system. Sufficient conditions are formulated under which the supremal normal or...

An important class of timed transition systems can be modeled by deterministic weighted automata, which are essentially partial Mealy automata, and their extensions using synchronous compositions defined over extended alphabets. From a coalgebraic viewpoint, behaviours of deterministic partial Mealy automata are causal and length preserving partial...

Interval weighted automata are introduced as automata with weights in a product dioid (idempotent semiring). They constitute an extension of (max,+) automata since they enable us to model temporal constraints (instead of exact durations) for transitions. Their synchronous composition, which coincides with the synchronous product of underlying (one...

In this paper 1-safe interval P-time Petri nets are described using linear description in an interval like idempotent semiring. This semiring, which is just the product of semirings Rmax and Rmin, enables linear letter driven description similar to the one known for timed Petri nets. Examples illustrating our approach show also an application of th...

A behavioral framework for control of (max,+) automata is proposed. It is based on behaviors (formal power series) and a generalized version of the Hadamard product, which is the behavior of a generalized tensor product of the plant and controller (max,+) automata in their linear representations. In the tensor product and the Hadamard product, the...

Synchronous products of (max,+) automata are introduced as multi-event (max,+) automata (which correspond to a class of timed automata with serveral clocks) and their behaviour is studied. A formula for the behavior of synchronous product is obtained.

Controllability of (max,+) automata and formal power series is studied within a behavioral framework. An extension of classical tensor product of their linear representations as a parallel compo-sition of controller with the plant (max,+) automaton is used. Controllability is studied using residuation theory of (multivariable) formal power series a...

KJB100190609, of the French-Czech bilateral project Barrande N. 14235XG and of the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503 are gratefully acknowledged.

The concept of a coordinator is proposed for control of modular discrete-event systems. The coordinator makes all subsystems conditionally independent generators as defined in the paper. The coordinator receives part of the partial observations of the subsystems and its task is to satisfy the global part of the specification and of the nonblockingn...

A new framework for control of (max,+) automata is introduced. The tensor product of their linear representations used in this paper is an extension of parallel composition from Boolean to (max,+) automata and can be nicely applied to both logical and timing aspects of supervisory control. Case of uncontrollable events that can neither be disabled...