Infinite Server Semantics Research Articles

Structural controllability in timed continuous Petri nets

This work deals with the controllability analysis in timed continuous Petri nets (TCPNs) under infinite server semantics, a class of linear hybrid systems. In the literature, this problem has been addressed by analyzing each configuration (that defines a linear mode) of the system. However, the number of configurations may grow exponentially. Here, by using a global structural approach, we study the net rank-controllability (NRC), a structural property of the TCPN. Under the assumption of liveness, it is shown that NRC is a sufficient condition for controllability; nevertheless, if liveness is not fulfilled then controllability is not guaranteed by NRC The advantage of this approach is that NRC is characterized in terms of global structural objects of the net, thus, avoiding the analysis by configurations. In this sense, some new structural sufficient conditions for NRC are introduced for general TCPNs. Finally, polynomial-time algorithms for the verification of NRC are provided.

Problematic configurations and choice-join pairs on Mono-T-Semiflow nets: towards the characterization of behavior-structural properties

This work is concerned with the computation of problematic configurations and problematic choice-join pairs in timed continuous Petri nets under the infinite server semantics; those net structural objects explain possible bad/counter-intuitive behaviors of systems, such as non-monotonicities and discontinuities of the equilibrium throughput. The calculation of problematic configurations is a computationally complex task since their number grows exponentially with the input cardinality of join transitions. In order to alleviate this inconvenience, four type of reduction rules preserving the set of problematic configurations are addressed. Reduction rules are weighted versions of the macroplace, macrotransition, and fusion of transitions rules, the elimination of implicit places rule is also provided. Reduced nets are useful to better understand the net substructures leading to unexpected behaviors in the equilibrium throughput. They help to highlight that the structural objects named problematic choice-join pairs, defined in this work, are the actual responsible of these counter-intuitive throughput behaviors. The great advantage over the set of problematic configurations is that the set of problematic choice-join pairs grows polynomially in the size of the net.

On the Equilibrium Sets of Topologically Equal Conflict Timed Continuous Petri Nets

Abstract This work deals with the study of equilibrium markings (states) in a class of Timed Continuous Petri Nets (TCPNs) under infinite server semantics. First, by adopting a structural approach, a qualitative analysis of the equilibrium sets in Choice-Free (CF) TCPN systems is presented. It is based on the analysis of the slowest conservative subsystems of the system. This allows us to define a structural component named the Maximal Limiting Subnet (MLS). Then, some properties of the equilibrium sets in CF-TCPN systems are stated in terms of its MLS. Next, connectivity of the equilibrium sets, an important property for the global controllability of the system, is studied. It is shown that CF-TCPN systems always exhibit this property. Finally, the previous results are extended to Topologically Equal Conflict TCPN systems. However, it is also shown that connectivity is not necessarily fulfilled in general systems.

Structural Characterization of Controllability in Timed Continuous Petri Nets using Invariant Subspaces

Abstract This work deals with the controllability analysis in Timed Continuous Petri Nets (TCPNs) under infinite server semantics, a fluid relaxation that can model highly populated Discrete Event Systems. Here, the full rank-controllability property is defined, ensuring that the TCPN is controllable over the equilibrium markings in each of the regions of its reachability space. This allows forcing the TCPN systems to work at interesting operation points such as maximum production states, safety regions, to mention a few. Herein two structural conditions for full rank-controllability, one necessary and the other sufficient, are introduced, avoiding the enumeration of all the configurations required in other approaches. Finally, based on this, a polynomial algorithm to test the full rank-controllability is provided.

Performance Optimization for Timed Weighted Marked Graphs Under Infinite Server Semantics

This paper deals with the performance optimization of resource allocation systems with the aim of maximizing the system's throughput under a given budget for acquiring resources. Resources are assumed to be renewable, i.e., they are not consumed by the operations and become available again after they have been released. The systems under consideration are modeled by a subclass of timed Petri nets called deterministic timed weighted marked graphs. In addition, we take into account infinite server semantics, i.e., the degree of self-concurrency of each transition is infinite. We propose an approach that provides an optimal solution, but has a high computational cost. For this reason, we also present two different approaches that can find suboptimal solutions with a reduced computational cost. The performances of the proposed approaches are compared by means of numerical simulations.

Sufficient Conditions and Sensor Placement for Structural Fault Diagnosis in a class of Timed Continuous Petri Nets

This work addresses the characterization of the fault diagnosability property in Timed Continuous Petri Nets (TCPN) under infinite server semantics. In the approach herein presented, the potential faults are included in the model, however, their occurrence time, magnitude and location are unknown. Two novel results are presented. In the first one, sufficient conditions that guarantee the diagnosability in TCPN substructures are provided. These conditions are based on the net structure, avoiding the analysis of all linear systems describing the net dynamics. The second result is a methodology to place a reduced number of sensors in such a way that a class of TCPN becomes diagnosable. In order to show the applicability of the reported results, an illustrative example is presented.

On monotonicity and continuity: computing problematic configurations in timed continuous Petri nets

This work is concerned with the computation of problematic configurations in timed continuous Petri nets under the infinite server semantics. Those net structural objects are related to possible bad/paradoxical behaviors of systems, such as non-monotonicities and discontinuities of the equilibrium throughput. The calculation of problematic configurations is a computationally complex task because their number grows exponentially with the input cardinality of join transitions. This work introduces a branch and bound type algorithm to perform such computation and, as an improvement of the calculation, it proposes to reduce the net into a smaller one using three different types of reduction rules that preserve the set of problematic configurations. The transformed net system allows to better understand why undesired behaviors may appear.

Homothecy, bifurcations, continuity and monotonicity in timed continuous Petri nets under infinite server semantics

This work studies how equilibrium markings and throughputs change in Timed Continuous Petri Net (TCPN) systems as transition firing rates vary. In particular, it analyzes the bifurcations of the former, and the discontinuities and non-monotonicities of the latter; specifically, using structural objects of the net, such as P-semiflows, T-semiflows, and configurations, among others, the following properties can be obtained. For Join Free TCPN systems, a sufficient structural condition guaranteeing that the equilibrium markings do not bifurcate when firing rates vary, is derived. A dual result is obtained for Choice Free TCPN systems. For Mono-T-Semiflow TCPN systems, the equilibrium throughput is investigated; using a time-scale (a homothetic) property it is proven that a discontinuity of the equilibrium throughput implies its non-monotonicity, even if not evident at first glance. This is a connection of two timed behavioral properties of the equilibrium throughput. Moreover, a sufficient structural condition, parametrized by the equilibrium markings, ensures its continuity under firing rate variations. It is also proven that the monotonicity of the equilibrium throughput can be characterized by the previous structural condition. The convergence of the marking evolution of TCPN systems to its equilibrium markings is also discussed.

Fluid approximation of Petri net models with relatively small populations

Fluidization is an appealing relaxation technique based on the removal of integrality constraints in order to ease the analysis of discrete Petri nets. The result of fluidifying discrete Petri nets are the so called Fluid or Continuous Petri nets. As with any relaxation technique, discrepancies among the behaviours of the discrete and the relaxed model may appear. Moreover, such discrepancies may have a comparatively bigger effect when the population of the system, the marking in Petri net terms, is “relatively” small. This paper proposes two complementary approaches to obtain a better fluid approximation of discrete Petri nets. The first one focuses on untimed systems and is based on the addition of places that are implicit in the untimed discrete system but not in the continuous. The idea is to cut undesired spurious solutions whose existence worsens the fluidization. The second one focuses on a particular situation that can severely affect the quality of fluidization in timed systems. Namely, such a situation arises when the enabling degree of a transition is equal to 1. This last approach aims to alleviate such a state of affairs, which is termed the bound reaching problem, on systems under infinite servers semantics.

Individuals, populations and fluid approximations: A Petri net based perspective

Discrete Event Systems (DES) theory and engineering are mainly driven by needs that arise in many different human-made systems (manufacturing, communications, logistics, workflow management, traffic, etc.). With the accelerated increase in the complexity and size of new technological constructions, the state explosion problem in DES analysis and synthesis becomes more and more acute.Two traditional conceptual and complementary ways of dealing with the computational complexities in the Petri nets (PN) framework are structure theory (that investigates the relationship between the behavior of a net system and its structure) and fluid relaxations, here leading to particular classes of hybrid systems. In the second case, the expected computational gains for analysis and synthesis problems are usually achieved at the expense of the fidelity or accuracy of the relaxed model. This invited overview will mainly focus on the second strategy, nevertheless always interspersed with basic structural concepts and methods. Using an example-driven approach, starting with a DES “view of the system”, the legitimization and improvement of fluidization process, the aggregation of local states by symmetries and the decolorization of models will be briefly addressed, together with reflections about the analysis of the new models obtained.As the linearization of a continuous dynamical system, the fluidization of a DES is a relaxation that has to be used with care, depending on the problem at hand. This abstraction is here considered from two complementary perspectives: at logical and at performance levels, both for untimed and timed PNs. On the one hand, the expressive power of timed fluid PNs under infinite server semantics is such that the simulation of Turing machines is possible. From a complementary perspective, the expression of modeling capabilities such as non-monotonicities and bifurcations may also be revealed for steady-state behaviors. Symmetries (more generally, lumping) seek to group together “equivalent” behaviors and decolorization seeks to abstract identities, in order to create new collectivities of processes and resources. The synergy between symmetry-decolorization state-aggregation approaches and fluid relaxations is highlighted. In fact, the first approaches not only reduce the state space, but also “produce” populations, thus proceed upgrading the applicability of fluidization. Opening the window, related issues such as control, optimization, observation or diagnosis are briefly pointed out. For conciseness, this work is limited to fully fluid (or continuous) PN models and their relationships with the corresponding discrete systems.

Structural Fault Detection in Join Free Timed Continuous Petri Nets

Abstract: This work is concerned with fault detection in a class of timed continuous Petri nets (TCPN) under Infinite Server Semantics. Faults are deviations from the nominal system behavior due to undesired changes of the transition firing rates or place token losses. These fault types are represented by adding transitions, named fault transitions, to the TCPN model. This work presents a novel characterization of fault detectability in TCPN based on input-output observability results applied to faults. This paper also proposes the design of a diagnoser for detecting fault occurrences in TCPN which is based on a sliding mode observer.

Bifurcations in Timed Continuous Petri Nets

This paper is concerned with the steady-state throughput changes due to firing rate variations in Timed Continuous Petri Nets (TCPN) under infinite server semantics. Jumps in this value (equivalently in the steady state marking) are bifurcations. It presents a novel structural characterization of bifurcations based on the P-semiflows and T-semiflows appearing in the TCPN configurations. In order to give this characterization, first strongly connected and conservative JF-TCPN (join free TCPN) are studied. Bifurcations are not possible in this class of nets. Afterwards, strongly connected and consistent CF-TCPN (choice free TCPN) are addressed. It is shown that this class of nets does not exhibit bifurcations. This kind of result is extended to topologically Equal-conflict nets when we vary the firing rates of transitions in conflict in the same proportion. Based on these results, bifurcations in MTS-TCPN (Mono-T-Semiflow TCPN) including non-equal conflicts and joins are addressed. The main result in this class of nets states that bifurcations appear when the places restricting the transition's flow in a certain configuration do not belong to the support of any net P-semiflow. Finally, this work presents a relationship between bifurcations and non monotonic throughput in this class of systems.

Sliding Mode Diagnosers for Leakage Faults on Timed Continuous Petri Nets

Abstract: This paper considers the application of sliding mode diagnosers for leakage fault detection in systems modeled by Timed Continuous Petri Nets under infinite server semantics. In this case the Timed Continuous Petri Net describes a Switching Linear System. Two cases are addressed; in the first one all linear systems are distinguishable from each other (normal and faulty linear systems), while in the second one the linear systems representing the normal behavior are distinguishable from each other, and the linear systems representing the faulty behavior are observable. In the first case a single sliding mode diagnoser, based on a supertwisting switched linear system observer is proposed. For the second case, a combination between the previous diagnoser and models for linear systems representing the faulty behavior are used to detect and isolate the fault.

Control Design of Timed Petri Nets via Model Predictive Control with Continuous Petri Nets

This paper addresses the problem of control design for T-Timed Discrete Petri Net (T-TDPN) systems. The problem is studied using Model Predictive Control (MPC) and Timed Continuous Petri Net (TCPN) systems. Our goal is to find an approach which determines, in a systematic way, the firing sequence that drives a T-TDPN from an initial marking to a desired one. Since it had been proved that the MPC is an attractive feedback strategy in the case of TCPN under infinite server semantics and that this latter represents a good fluid approximation of the Discrete Event Systems, the proposed method deals with MPC applied on the TCPN equivalent to the discrete one and then, from the obtained results, deduces the firing sequence and computes the discrete markings.

Local control law for live and bounded timed continuous Petri nets

This work is concerned with the problem of driving a live and bounded Timed Continuous Petri Net (TCPN) from an initial state to a target ending state, and remaining on it, using only local information of the transitions. It is assumed that the TCPN is under the infinite server semantics and all the transitions are controllable. The proposed control law allows to start at any reachable initial marking (even at the boundaries of the reachability set), and ending at any reachable marking at the interior of the reachability set. The proposed control is a local one, thus the required information to allow the firing of an enabled transition depends only on its input and output places. In large TCPN this control law is suitable because it is not necessary to compute large amounts of information.

Sensor placement for distinguishability and single structure observer design in continuous timed Petri nets

This work is concerned with the sensor placement problem for distinguishability and the observer design in continuous timed Petri nets with infinite servers semantics (ContPN). The ContPN are represented by a family of linear systems (LS) switching among them. Depending on the ContPN marking, a LS evolves. Thus the distinguishability problem deals with the possibility of determining which LS is actually evolving, and the sensor placement for distinguishability problem deals with selecting which places must be measured to ensure distinguishability. On the other hand, the observer’s design problem is concerned with developing a mathematical entity devoted to compute the actual state of the ContPN by means of the knowledge of the marking in the places which have a sensor.This work presents three main contributions. The first one shows that there exist cases where the classical theorems in switched linear systems cannot be used in ContPN, since these theorems could establish that the ContPN is not distinguishable, even when, using the ContPN structure, the distinguishability is guaranteed. The second contribution is an algorithm devoted to place sensors (to build the output map) in such a way that the ContPN becomes distinguishable. The input to the algorithm is an output map S, that can be computed as in a previous work, which guarantees that every LS of the ContPN representation is observable. Then new sensors are included to gain distinguishability between every pair of LS. Thus the distinguishability and observability of the ContPN is guaranteed. Finally, the third main contribution is a single structure observer for the bounded and observable ContPN. This strategy is relevant to avoid the need of computing one observer per LS in the family that represents the ContPN, which in practice becomes unfeasible.

Controllability of timed continuous Petri nets with uncontrollable transitions

This paper is concerned with controllability of Timed Continuous Petri nets, under infinite server semantics, with uncontrollable transitions, which are a class of hybrid systems (piecewise-linear). This class of hybrid systems is suitable for representing biological systems, high traffic information networks, heavily loaded supply chains, etc. By adopting a Control Theory approach, the contribution of this paper is the characterisation of controllability over sets of equilibrium markings (potential equilibrium points), first inside a single marking region (or linear mode) and later extended to several regions.

About regulation and disturbance rejection for partially controlled contPNs

This article concerns control design for partially controlled timed continuous Petri nets under infinite server semantics and constant bounded input control actions. Domains of stationary markings and of their corresponding control actions are computed. Linear matrix inequalities and linear programming problems are used for that purpose. Then, disturbance rejection in the neighborhood of the stationary markings is also investigated.

Stochastic Continuous Petri Nets: An Approximation of Markovian Net Models

Fluidization constitutes a relaxation technique to study discrete event systems through a continuous approximated model, thus overcoming the state explosion problem. In this paper, the approximation of the average marking of Markovian Petri nets by the marking of the corresponding timed continuous Petri nets, under infinite-server semantics, is studied. This represents a sort of legitimization for the use of a continuous Petri net as a relaxation of a discrete Petri net. The main contribution is the addition of Gaussian noise in order to improve the approximation when the number of active servers (enabling degree) is large. The improvement is more evident when the system evolves “close” to the boundary of regions. In such a case, not only the expected value but also the probability distribution function of the marking may be approximated.

Discontinuity Induced Bifurcations in Timed Continuous Petri Nets

In this contribution the dynamics of timed continuous Petri nets with infinite server semantics under variation of firing rates are studied. The systems are described by continuous piecewise linear systems with polyhedral regions. It is well known that the throughput of such systems may change discontinuously by variation of a firing rate. We identify these jumps in the throughput with discontinuity induced bifurcations. They occur when the steady state marking, that depends on the firing rate, hits a boundary between regions of the piecewise linear system and loses hyperbolicity. Such bifurcations affect the robustness of the system with respect to a variation of firing rates and are thus critical for control problems.