strong solvability of interval max plus systems and applications to optimal control abstract this paper considers the strong solvability of interval max plus systems the solvable entry and solvable interval are introduced to establish the necessary and sufficient condition for strong solvability of interval max plus systems it is proved that such a criterion is equivalent to the criterion proposed by cechlarova and cuninghame green the interval strong solution of interval max plus systems is described and the necessary and sufficient condition for uniquely strong solvability is given the method of testing strong solvability of interval max plus systems is constructive and leads to an effective algorithm to illustrate our results a workshop optimal control problem is addressed

energy representation for nonequilibrium brownian like systems steady states and fluctuation relations stochastic dynamics in the energy representation is used as a method to represent nonequilibrium brownian like systems it is shown that the equation of motion for the energy of such systems can be taken in the form of the langevin equation with multiplicative noise properties of the steady states are examined by solving the fokker planck equation for the energy distribution functions the generalized integral fluctuation theorem is deduced for the systems characterized by the shifted probability flux operator from this theorem a number of entropy and fluctuation relations such as the evans searles fluctuation theorem the hatano sasa identity and the jarzynski s equality are derived

max algebraic attraction cones of nonnegative irreducible matrices abstract it is known that the max algebraic powers ar of a nonnegative irreducible matrix are ultimately periodic this leads to the concept of attraction cone attr a t by which we mean the solution set of a two sided system t a ar x ar t x where r is any integer after the periodicity transient t a and a is the maximum cycle geometric mean of a a question which this paper answers is how to describe attr a t by a concise system of equations without knowing t a this study requires knowledge of certain structures and symmetries of periodic max algebraic powers which are also described we also consider extremals of attraction cones in a special case and address the complexity of computing the coefficients of the system which describes attraction cone

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

predicates and predicate transformers for supervisory control of discrete event dynamical systems discrete event systems are studied treating the state space as the fundamental modeling concept the control of discrete event systems using predicates and predicate transformers is treated predicates have the advantage that they can concisely characterize an infinite state space the notion of controllability of a predicate is defined and the supervisory predicate control problem is solved a closed form expression for the weakest controllable predicate is obtained the problem of controlling discrete event systems under incomplete state observation is also considered and observability of predicates is defined techniques for finding extremal solutions of boolean equations are used to derive minimally restrictive supervisors

from max plus algebra to nonexpansive mappings a nonlinear theory for discrete event systems discrete event systems provide a useful abstraction for modelling a wide variety of systems digital circuits communication networks manufacturing plants etc their dynamics stability equilibrium states cyclical behaviour asymptotic average delays are of vital importance to system designers however in marked contrast to continuous dynamical systems there has been little systematic mathematical theory that designers can draw upon in this paper we survey the development of such a theory based on the dynamics of maps which are nonexpansive in the l norm this has its origins in linear algebra over the max plus semiring but extends to a nonlinear theory that encompasses a variety of problems arising in other mathematical disciplines we concentrate on the mathematical aspects and set out several open problems

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

relative observability of discrete event systems and its supremal sublanguages we identify a new observability concept called relative observability in supervisory control of discrete event systems under partial observation a fixed ambient language is given relative to which observability is tested relative observability is stronger than observability but enjoys the important property that it is preserved under set union hence there exists the supremal relatively observable sublanguage of a given language relative observability is weaker than normality and thus yields when combined with controllability a generally larger controlled behavior in particular no constraint is imposed that only observable controllable events may be disabled we design new algorithms which compute the supremal relatively observable and controllable sublanguage of a given language which is generally larger than the normal counterpart we demonstrate the new observability concept and algorithms with a guideway and an agv example

similarity based supervisory control of discrete event systems due to the appearance of uncontrollable events in discrete event systems one may wish to replace the behavior leading to the uncontrollability of pre specified language by some quite similar one to capture this similarity we introduce metric to traditional supervisory control theory and generalize the concept of original controllability to spl lambda controllability where spl lambda indicates the similarity degree of two languages a necessary and sufficient condition for a language to be spl lambda controllable is provided we then examine some properties of spl lambda controllable languages and present an approach to optimizing a realization

mode coupling theory for heteropolymers we study the langevin dynamics of a heteropolymer by means of a mode coupling approximation scheme giving rise to a set of coupled integro differential equations relating the response and correlation functions the analysis shows that there is a regime at low temperature characterized by out of equilibrium dynamics with violation of time translational invariance and of the fluctuation dissipation theorem the onset of aging dynamics at low temperatures gives insight into the nature of the slow dynamics of a disordered polymer we also introduce a renormalization group treatment of our mode coupling equations which supports our analysis and might be applicable to other systems

supervisory control of nondeterministic systems with driven events via prioritized synchronization and trajectory models the supervisory control of nondeterministic discrete event dynamical systems dedss with driven events in the setting of prioritized synchronization and trajectory models introduced by heymann are studied prioritized synchronization captures the notions of controllable uncontrollable and driven events in a natural way and the authors use it for constructing supervisory controllers the trajectory model is used for characterizing the behavior of nondeterministic dedss since it is a sufficiently detailed model in contrast to the less detailed language or failures models and serves as a language congruence with respect to the operation of prioritized synchronization results concerning controllability and observability in this general setting are obtained

a characterization of realizable behavior in supervisory control of timed event graphs abstract we study supervisory control of timed event graphs in the max algebra framework desirable behavior is assumed to be specified as a set of firing time sequences of transitions we address a control problem which requires that behavior of the supervised system equal the specification set a necessary and sufficient condition for the existence of a control input set to solve our problem is presented we call the condition the strong controllability condition however the given specification is not necessarily strongly controllable we then obtain its supremal strongly controllable subset moreover we derive a necessary and sufficient condition for the supremal strongly controllable subset to be nonempty under the assumption that desirable behavior is specified by the upper time bound sequence

supervisory control of real time discrete event systems using lattice theory the behaviour of timed discrete event systems des s can be described by sequences of event occurrence times these sequences can be ordered to form a lattice since logical untimed des behaviours described by regular languages also form a lattice questions of controllability for timed des s may be treated in much the same manner as they are for untimed systems in this paper we establish conditions for the controllability of timed des performance specification which are expressed as inequations on the lattice of sequences these specifications may take the form of sets of acceptable event occurrence times maximum or minimum occurrence times or limits on the separation times between events optimal behaviours are found as extremal solutions to these inequations using fixed point results for lattices

spectral analysis for performance evaluation in a bus network this paper deals with the performance evaluation of a public transportation system in terms of waiting times at various connection points the behaviour of a bus network is studied in the framework of discrete event systems des two possible operating modes of buses can be observed at each connection stop periodic and non periodic mode two complementary tools petri nets and max algebra are used to describe the network by a non stationary linear state model this one can be solved after solving the structural conflicts associated to the graphical representation from the characteristic matrix of the mathematical model we determine eigenvalues and eigenvectors that we use to evaluate the connection times of passengers this work is finally illustrated with a numerical example

formulas for the extremal controllable sequences in timed event graphs this paper studies controllable sequences in the max algebra model of timed event graphs we present a formula for the supremal controllable sequence when a desirable behavior is specified as a single sequence as a result the supremal sequence can be computed without iterative computation a formula for the infimal controllable sequence is also obtained under some assumptions

discrete event systems with stochastic processing times discrete event dynamic systems are studied in which the underlying algebra is the max algebra and the coefficients in the system referring to processing times in practice are stochastic the processing times and or the transportation times within a network show stochastic fluctuations the restrictions are that the stochastic processing times of the nodes in the network are independent and identically distributed the asymptotic behavior of the system is investigated and the average duration of one cycle of the process is calculated a specific example of the theory is considered the state space is two dimensional and the probability distributions are exponential it is shown that the process approaches a stationary limit as time proceeds the case when the probability distributions are discrete is also treated several examples are given two dimensional systems and more generally finite dimensional systems are considered

paper a new approach to the analysis of discrete event dynamic systems we present a new time domain approach to the study of discrete event dynamical systems deds typified by queueing networks and production systems a general state space representation is developed and perturbation analysis is carried out observation of a single sample realization of such a system can be used to predict behavior over other sample realizations when some parameter is perturbed without having to make additional observations conditions under which this is always possible are investigated and explicit results for some special cases are included

supervisory control of modular systems with global specification languages the paper presents sufficient conditions for modular supervisory control synthesis to equal global control synthesis in modular control synthesis a supervisory control is synthesized for each module separately and the supervisory control consists of the parallel composition of the modular supervisory controls the general case of the specification that is indecomposable and not necessarily contained in the plant language which is often the case in practice is considered the usual assumption that all shared events are controllable is relaxed by introducing two new structural conditions relying on the global mutual controllability condition the novel concept used as a sufficient structural condition is strong global mutual controllability the main result uses a weaker condition called global mutual controllability together with local consistency of the specification an example illustrates the approach

language stability and stabilizability of discrete event dynamical systems this paper studies the stability and stabilizability of discrete event dynamical systems dedss modeled by state machines stability and stabilizability are defined in terms of the behavior of the dedss i e the language generated by the state machines sms this generalizes earlier work where they were defined in terms of legal and illegal states rather than strings the notion of reversal of languages is used to obtain algorithms for determining the stability and stabilizability of a given system the notion of stability is then generalized to define the stability of infinite or sequential behavior of a deds modeled by a buchi automaton the relationship between the stability of finite and stability of infinite behavior is obtained and a test for stability of infinite behavior is obtained in terms of the test for stability of finite behavior an algorithm of linear complexity for computing the regions of attraction is presented which is used for determining the stability and stabilizability of a giv

maximally permissive mutually and globally nonblocking supervision with application to switching control a supervisor is said to be mutually nonblocking with respect to a pair of specifications if upon completing a task in any of the specifications it can continue on to complete the task in the other specification i e the two specifications do not block each other the notion of mutually nonblocking supervisor was introduced in fabian and kumar 2000 automatica 36 12 1863 1869 in this paper we present an algorithm of polynomial complexity for computing a maximally permissive mutually and globally nonblocking supervisor in case such a supervisor does not exist we present a technique for relaxing the specifications for which a supervisor exists the algorithms are based on a notion of attractability and as a special case offer a new way of computing the maximally permissive nonblocking supervisors the results are then applied to design of maximally permissive switching supervisors so as to allow for switching between the specifications at any time while the system is executing

relative observability and coobservability of timed discrete event systems we study supervisory control of timed discrete event systems tdes under partial observation and propose new observability concepts effective for supervisor synthesis first we consider monolithic centralized supervisory control and introduce timed relative observability and timed relative weak observability the former concept extends our previous work to the timed case while the latter exploits choices of forcible events to preempt the clock event tick we prove that timed relative respectively weak observability is stronger than timed respectively weak observability weaker than normality and closed under set union hence there exists the supremal relatively respectively weakly observable sublanguage of a given language

formulas for a class of controllable and observable sublanguages larger than the supremal controllable and normal sublanguage abstract in this paper we give some new methods for synthesis of controllers of discrete event dynamical systems deds with partial event informations given a regular target language l we construct some effective computable algorithms for computing the controllable and observable sublanguages of l we show that any one of these controllable and observable sublanguages obtained by our algorithms is larger than the supremal controllable and normal sublanguage of l

an algebra for queueing networks with time varying service and its application to the analysis of integrated service networks we introduce a network model that allows us to capture the time varying service delivered to a traffic stream due to the presence of random perturbations e g cross traffic in a communication network we first present the model for a single network element and then describe how such elements may be interconnected using the operations offork andjoin such networks may be seen as a generalization of stochastic event graphs and of the class of fork join networks the departure processes in such a network satisfy a system of equations along with the exogenous arrival processes and the service processes of the various network elements this system can be seen as a general time varying linear system in the min plus semi field we obtain an explicit representation of the departure process in terms of the exogenous arrival processes and the service processes sufficient conditions are derived for this system to have a unique solution we also study liveness and absence of explosion in this class of networks under appropriate stationarity and ergodicity assumptions we establish stability theorems for such networks to this end we first obtain a rate result for the departure process in terms of the rates of the exogenous arrival process and the throughput of a saturated system we then use this result to show that in a network with a single exogenous arrival all queue lengths are finite with probability one if the arrival rate is less than the throughput of the saturated system and to give a representation of the queue length process this class of networks allows for a detailed description of controlled sessions in integrated service networks we also show that it contains several earlier discrete event models of the literature pertaining to stochastic petri nets service curves and fork join networks and show how the present model unifies them in a single algebraic structure this structure is that of a semi ring of functions of two real variables where the addition is the pointwise minimum and the multiplication a generalization of inf convolution

stabilization of max plus linear systems using model predictive control max plus linear mpl systems are a class of event driven nonlinear dynamic systems that can be described by models that are linear in the max plus algebra in this paper we derive a solution to a finite horizon model predictive control mpc problem for mpl systems where the cost is designed to provide a trade off between minimizing the due date error and a just in time production in general mpc can deal with complex input and states constraints however in this paper we assume that these are not present and it is only required that the input should be a nondecreasing sequence i e we consider the unconstrained case despite the fact that the controlled system is nonlinear by employing recent results in max plus theory we are able to provide sufficient conditions such that the mpc controller is determined analytically and moreover the stability in terms of lyapunov and in terms of boundedness of the closed loop system is guaranteed a priori

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

the ultimate rank of tropical matrices a tropical matrix is a matrix defined over the max plus semiring for such matrices there exist several non coinciding notions of rank the row rank the column rank the schein barvinok rank the kapranov rank or the tropical rank among others in the present paper we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix which does not depend on the underlying notion of rank furthermore we provide a simple formula for the ultimate rank of a matrix which can therefore be computed in polynomial time then we turn our attention to finitely generated semigroups of matrices for which our notion of ultimate rank is generalized naturally we provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank as a consequence we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal

on tropical supereigenvectors abstract the task of finding tropical eigenvectors and subeigenvectors that is non trivial solutions to a x x and a x x in the max plus algebra has been studied by many authors since the 1960s in contrast the task of finding supereigenvectors that is solutions to a x x has attracted attention only recently we present a number of properties of supereigenvectors focusing on a complete characterization of the values of associated with supereigenvectors and in particular finite supereigenvectors the proof of the main statement is constructive and enables us to find a non trivial subspace of finite supereigenvectors we also present an overview of key related results on eigenvectors and subeigenvectors

enforcing detectability in controlled discrete event systems in this technical note we investigate supervisory control that enforces strong detectability and strongly periodic detectability in a discrete event system the goal of control is to ensure that the current state of the closed loop system can be determined forever or periodically after some finite observations for all trajectories of the system we derive necessary and sufficient conditions for the existence of supervisory control and develop algorithms to synthesize such supervisors if they exist

stability of linear systems in maximum algebra in this paper we explore the concepts of block weak stability and block weak uniform stability for general discrete event dynamic systems deds by max algebra we have proved that the general deds is block weakly stable some sufficient conditions for block uniform stability are given

on computation of supremal controllable normal sublanguages in this paper we present an algorithm for the computation of the controllable normal sublanguage of a given language encountered in the solution of the supervisory control of discrete event systems under partial observation the algorithm produces the desired result under certain assumptions on the plant and the event projection map in particular the plant has to be nonblocking the advantage of the algorithm over the solution available in the literature is that it does not involve iterations on the supremal controllable sublanguage and supremal normal sublanguage operators

a note on the properties of the supremal controllable sublanguage in pushdown systems consider an event alphabet sigma the supervisory control theory of ramadge and wonham asks the question given a plant model g with language lm g sube sigma and another language k sube lm g is there a supervisor phi such that lm phi g k ramadge and wonham showed that a necessary condition for this to be true is the so called controllability of k with respect to lm g they showed that when g is a finite state automaton and k is a regular language also generated by a finite state automaton then there is a regular supremal controllable sublanguage supc k sube k that is effectively constructable from generators of k and g in this paper we show 1 there is an algorithm to compute the supremal controllable sublanguage of a prefix closed k accepted by a deterministic pushdown automaton dpda when the plant language is also prefix closed and accepted by a finite state automaton and 2 in this case we show that this supremal controllable sublanguage is also accepted by a dpda

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

synthesis of maximally permissive and robust supervisors for prefix closed language specifications j e r cury and b h krogh 1999 have formulated a robust supervisory control problem to synthesize a supervisor for the nominal plant which maximizes robustness they have solved the problem by unnecessarily restricting the upper bound of the legal behavior we show that the problem can be solved without restricting the upper bound of the legal behavior when the specification is described by prefix closed languages we synthesize a maximally permissive supervisor for the nominal plant which maximizes not only robustness but also permissiveness for the maximal set of admissible plant variations

on supervisory control of sequential behaviors the authors address the supervisory synthesis problem of controlling the sequential behaviors of discrete event dynamical systems dedss under complete and partial information through the use of synchronous composition of the plants and the supervisors the authors present the notion of complete languages discuss some of its algebraic properties and show its close relation to omega languages the authors prove that the supremal closed complete and controllable sublanguage of a given language exists and present an algorithm to compute it they present a closed form expression for the supremal omega controllable sublanguage of a given omega language in terms of the supremal closed complete and controllable sublanguage this closed form expression suggests that certain operations on a given omega language can alternatively be achieved by performing certain other similar operations on its prefix which is a finite language and then taking the limit to obtain the desired omega language a necessary and sufficient condition for the existence of a supervisor in case of partial observation is presented in terms of omega observability notion of omega normality is also introduced and a closed form expression for the supremal omega normal sublanguage in terms of the supremal closed complete and normal sublanguage is presented

on the supermal controllable sublanguage of a given language the concept of controllable language has been shown to play a basic role in the existence theory of supervisory controls for discrete event processes in this paper the supremal controllable sublanguage s of a given language l is characterized as the largest fixpoint of a monotone operator omega in the case where the languages involved are regular it is shown that the fixpoint s can be computed as the limit of the finite sequence k j given by k j 1 omega k j k 0 l an effective computational algorithm is developed and three examples are provided for illustration

large time nonequilibrium dynamics of a particle in a random potential we study the off equilibrium dynamics of a particle in a general n dimensional random potential when n to infty we demonstrate the existence of two asymptotic time regimes it i stationary dynamics it ii slow aging dynamics with violation of equilibrium theorems we derive the equations obeyed by the slowly varying part of the two times correlation and response functions and obtain an analytical solution of these equations for short range correlated potentials we find that it i the scaling function is non analytic at similar times and this behaviour crosses over to ultrametricity when the correlations become long range it ii aging dynamics persists in the limit of zero confining mass with universal features for widely separated times we compare with the numerical solution to the dynamical equations and generalize the dynamical equations to finite n by extending the variational method to the dynamics

synthesis of controllable and normal sublanguages for discrete event systems using a coordinator 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 necessary and sufficient conditions where such a computation is possible specifically we show that conditionally controllable and conditionally normal languages computed by our method are controllable and normal with respect to the global plant the optimality supremality of the resulting languages is also discussed

on controllability and normality of discrete event dynamical systems abstract this paper studies the supervisor synthesis problem of discrete event dynamical systems deds s through the use of synchronous composition of the plant and the supervisor and discusses some of the simplifications achieved by using the synchronous composition to model supervisory control an alternate definition of controllability is presented this definition of controllability is then used to derive an algorithm that is computationally more efficient than previously existing ones for the construction of the supremal controllable sublanguage the algorithm is also shown to be optimal the observability and normality issues arising due to the partial observation of the system dynamics under an arbitrary mask are then investigated closed form representations of the supremal normal and supremal closd and normal sublanguages are derived in the more general setting of arbitrary mask and nonclosed languages thus extending earlier results of the authors and others

the infimal prefix closed and observable superlanguage of given language abstract the supervisory control of discrete event systems under partial observation is examined in particular the condition of observability needed to solve the two main existing problems is investigated a fixed point characterization of observability is given and is shown to yield an effective test for observability where the system under consideration is specified by regular languages a formula for the infimal prefix closed observable superlanguage of a given language is presented where the system is specified by regular languages the formula for the infimal observable language is effectively computable

supersymmetry approach in the field theory of ergodicity breaking transitions abstract the supersymmetry susy self consistent approximation for the model of non equilibrium thermodynamic system with quenched disorder is derived from the dynamical action calculated by means of generalized second legendre transformation technique the equations for adiabatic and isothermal susceptibilities memory and field induced parameters are obtained on the basis of asymptotic analysis of dynamic dyson equations it is shown that the marginal stability condition that defines the critical point is governed by fluctuations violating fluctuation dissipation theorem fdt the temperature of ergodicity breaking transition is calculated as a function of quenched disorder intensities transformation of superfields related to the mapping between an instanton process and the corresponding causal solution is discussed

optimal supervisory control of discrete event dynamicalsystems the notion of optimal supervisory control of discrete event dynamical systems dedss is formalized in the framework of ramadge and wonham a deds is modeled as a state machine and is controlled by disabling some of its transitions two types of cost functions are defined a cost of control function corresponding to disabling transitions in the state machine and a penalty of control function corresponding to reaching some undesired states or not reaching some desired states in the controlled system the control objective is to design an optimal control mechanism if it exists so that the net cost is minimized since a deds is represented as a state machine a directed graph network flow techniques are naturally applied for designing optimal supervisors it is also shown that our techniques can be used to solve supervisory control problems under complete as well as partial observation in particular for the first time techniques for computing the supremal controllable and normal sublanguage and the infimal controllable and normal observable superlanguage without having to perform alternate computations of controllable and normal observable languages are obtained

synthesis of maximally permissive supervisors for partially observed discrete event systems we present new results on the synthesis of safe non blocking and maximally permissive supervisors for partially observed discrete event systems we consider the case where the legal language is a non prefix closed sublanguage of the system language i e non blockingness must be ensured in addition to safety to solve this problem we define a new bipartite transition system called the non blocking all inclusive controller nb aic that embeds all safe and non blocking supervisors we present an algorithm for the construction of the nb aic and discuss its properties we obtain the necessary and sufficient conditions for the solvability of the maximally permissive control problem we then provide a synthesis algorithm based on the nb aic that constructs a supervisor that is safe non blocking and maximally permissive if one exists this is the first algorithm with such properties

modeling and analysis of timed petri nets using heaps of pieces the authors show that safe timed petri nets can be represented by special automata over the max semiring which compute the height of heaps of pieces this extends to the timed case the classical representation a la mazurkiewicz of the behavior of safe petri nets by trace monoids and trace languages for a subclass including all safe free choice petri nets we obtain reduced heap realizations using structural properties of the net covering by safe state machine components the authors illustrate the heap based modeling by the typical case of safe jobshops for a periodic schedule the authors obtain a heap based throughput formula which is simpler to compute than its traditional timed event graph version particularly if one is interested in the successive evaluation of a large number of possible schedules

subadditivity and stability of a class of discrete event systems we investigate the stability of discrete event systems modeled as generalized semi markov processes with event epochs that satisfy max recursions we obtain three types of results under conditions we show that there exists for each event a cycle time which is the long run average time between event occurrences we characterize the rate of convergence to this limit bounding the error for finite horizons and we give conditions for delays i e differences between event epochs to converge to a stationary regime the main tools for the cycle time results are max matrix products and the subadditive ergodic theorem the convergence rate result which assumes bounded i i d inputs is based on a martingale inequality the stability of delays is derived from existing results on the stability of stochastic difference equations we discuss connections with these different fields with the general theory of random matrix products and with results for discrete event systems

polynomial synthesis of supervisor for partially observed discrete event systems by allowing nondeterminism in control we study the supervisory control of discrete event systems dess under partial observation using nondeterministic supervisors we formally define a nondeterministic control policy and also a control observation compatible nondeterministic state machine and prove their equivalence the control action of a nondeterministic supervisor is chosen online nondeterministically from among a set of choices determined offline also the control action can be changed online nondeterministically prior to any new observation in accordance with choices determined offline the online choices once made can be used to affect the set of control action choices in future we show that when control is exercised using a nondeterministic supervisor the specification language is required to satisfy a weaker notion of observability which we define in terms of recognizability and achievability achievability serves as necessary and sufficient condition for the existence of a nondeterministic supervisor and it is weaker than controllability and observability combined when all events are controllable achievability reduces to recognizability we show that both existence and synthesis of nondeterministic supervisors can be determined polynomially for deterministic supervisors only existence can be determined polynomially both achievability and recognizability are preserved under union and also under intersection when restricted over prefix closed languages using the intersection closure property we derive a necessary and sufficient condition for the range control problem for the prefix closed case unlike the deterministic supervisory setting where the complexity of existence is exponential our existence condition is polynomially verifiable and also a supervisor can be polynomially synthesized

optimistic optimization for model predictive control of max plus linear systems model predictive control for max plus linear discrete event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables which may be hard to solve efficiently an alternative approach is to transform the given problem into a mixed integer linear programming problem however the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon the focus of this paper is on making optimistic optimization suited to solve the given problem optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization a key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum we prove that optimistic optimization can be applied for the given problem by developing a dedicated semi metric and by proving it satisfies the necessary requirements for optimistic optimization moreover we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon hence using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large

the predictability of discrete event systems perturbation analysis and the automaton and language model are approaches developed recently for the study of discrete event systems dess the prediction of a trajectory of a new system is the essential idea of perturbation analysis the automaton theory models a trajectory of a des by a string in a particular language the author formulates the trajectory prediction as a projection of a string onto a language a sufficient condition is found for one language to be predictable from another language examples are given to show the application of this concept

robust nonblocking supervisory control of discrete event systems under partial observation in this paper we extend previous results on robust nonblocking supervisory control of discrete event systems with full observation to the case of control under partial observation where only a subset of events are observable and can be monitored the exact model of the plant is not known but is assumed to be among a finite set of possible models for each plant model a legal marked behavior is assumed given we manage to characterize the entire set of solutions of the robust control problem and obtain a set of necessary and sufficient conditions for the existence of a solution for the robust control problem

effective action for stochastic partial differential equations stochastic partial differential equations spdes are the basic tool for modeling systems where noise is important spdes are used for models of turbulence pattern formation and the structural development of the universe itself it is reasonably well known that certain spdes can be manipulated to be equivalent to nonquantum field theories that nevertheless exhibit deep and important relationships with quantum field theory in this paper we systematically extend these ideas we set up a functional integral formalism and demonstrate how to extract all the one loop physics for an arbitrary spde subject to arbitrary gaussian noise it is extremely important to realize that gaussian noise does not imply that the field variables undergo gaussian fluctuations and that these nonquantum field theories are fully interacting the limitation to one loop is not as serious as might be supposed experience with quantum field theories qfts has taught us that one loop physics is often quite adequate to give a good description of the salient issues the limitation to one loop does however offer marked technical advantages because at one loop almost any field theory can be rendered finite using zeta function technology we can sidestep the complications inherent in the martin siggia rose formalism the spde analog of the becchi rouet stora tyutin formalism used in qft and instead focus attention on a minimalist approach that uses only the physical fields this direct approach is the spde analog of canonical quantization using physical fields after setting up the general formalism for the characteristic functional partition function we show how to define the effective action to all loops and then focus on the one loop effective action and its specialization to constant fields the effective potential the physical interpretation of the effective action and effective potential for spdes is addressed and we show that key features carry over from qft to the case of spdes an important result is that the amplitude of the two point function governing the noise acts as the loop counting parameter and is the analog of planck s constant in this spde context we derive a general expression for the one loop effective potential of an arbitrary spde subject to translation invariant gaussian noise and compare this with the one loop potential for qft

supervisory control of timed discrete event systems the ramadge wonham framework for control of discrete event systems is augmented with timing features by use of ostroff s semantics for timed transition models it is shown that the rw concept of controllability and the existence of maximally permissive supervisory controls can be suitably generalized the enhanced setting admits subsystem composition and the concept of forcible event as an event that preempts the tick of a global clock an example of a simple manufacturing cell illustrates how the new framework can be used to solve synthesis problems which may include logic based temporal and quantitative optimality specifications

think globally act locally decentralized supervisory control decentralized supervisory control is investigated by considering problem formulations that model systems whose specifications are given as global constraints but whose solution is described by local controllers a necessary and sufficient condition is given for the existence of a solution to the problem of finding decentralized supervisors that ensure that the behavior of the closed loop system lies in a given range where the range of behavior can be described by regular languages it can be effectively tested whether the decentralized control problem is solvable in this case a procedure is given to compute the associated supervisors

transience bounds for long walks let g v e be a directed graph with rewards re for e e e we show that if there exists an s t walk in g of length k v 2 then there exists a maximum reward s t walk of length k with at least k v 2 arcs from a cycle with maximal mean reward if a is the reward incidence matrix of g then the maximum rewards of s t walks of length k can be determined by computing the k fold matrix product a k where the operations plus and times are replaced by max and plus denoted by and max algebra dynamic systems y k y k 1 a have been used to model certain transportation and automated manufacturing systems and can be interpreted as deterministic dynamic programming recursions if g is strongly connected then y k da y k da 1 for all k ka where is the maximum cycle mean in g and ka and da are the max algebra transient and period cyclicity of a for given y 0 the transient and period of the system may satisfy ka and da we derive a tight bound on ka and give polynomial algorithms for computing ka and and for solving deterministic dynamic programming problems we also give analogous results for walks of maximum discounted reward

permuted max algebraic eigenvector problem is np complete abstract let a b max a b and a b a b for a b r r and extend these operations to matrices and vectors as in conventional linear algebra the following eigenvector problem has been intensively studied in the past given a r n n find all x r n x t eigenvectors such that a x x for some r the present paper deals with the permuted eigenvector problem given a r n n and x r n is it possible to permute the components of x so that it becomes a max algebraic eigenvector of a using a polynomial transformation from bandwidth we prove that the integer version of this problem is np complete

dioids and semirings links to fuzzy sets and other applications besides the classical algebraic structures of groups rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling other algebraic structures such as dioids and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non classical problems e g in decision analysis fuzzy set theory operations research automatic control and mathematical physics the present paper aims at providing an overview of applications of dioid and semiring structures stressing links with fuzzy sets and emphasizing linear algebraic problems solving linear systems computing eigenvalues and eigenvectors non classical path finding problems using algebras of endomorphisms and connections between dioid structure and nonlinear analysis in view of solving problems in mathematical physics

extremal solutions of inequations over lattices with applications to supervisory control we study the existence and computation of extremal solutions of a system of inequations defined over lattices using the knaster tarski fixed point theorem we obtain sufficient conditions for the existence of supremal as well as infimal solution of a given system of inequations iterative techniques are presented for the computation of the extremal solutions whenever they exist and conditions under which the termination occurs in a single iteration are provided these results are then applied for obtaining extremal solutions of various inequations that arise in computation of maximally permissive supervisors in control of logical discrete event systems dess first studied by ramadge and wonham thus our work presents a unifying approach for computation of supervisors in a variety of situations

nonparametric estimation of a bivariate mean residual life function in many statistical studies involving failure data mean residual life function is of prime importance the bivariate mean residual life function has received relatively less attention in the literature in this article we use a simple nonparametric estimator for a bivariate mean residual life function the estimator is shown to be uniformly strongly consistent and on proper normalization converges weakly to a zero mean bivariate gaussian process numerical studies demonstrate that the estimator performs well even for moderate sample sizes results are applied to a real dataset related to cancer recurrence a few supporting results in connection with weak convergence proved in appendix c may be of independent interest

a note on deciding controllability in pushdown systems consider an event alphabet spl sigma the supervisory control theory of ramadge and wonham asks the question given a plant model g with language k l sub m g spl sube spl sigma sup and another language k spl sube l sub m g is there a supervisor spl psi such that l sub m spl psi g k ramadge and wonham showed that a necessary condition for this to be true is the so called controllability of k with respect to l sub m g they showed that when g is a finite state automaton and k is a regular language also generated by a finite state automaton then the controllability property was decidable for k the class of languages generated by pushdown automata properly includes the regular languages they are accepted by finite state machines coupled with pushdown stack memory this makes them interesting candidates as supervisory languages since the supervisor will have nonfinite memory in this note we show the following i if s is a specification given by a deterministic pushdown automaton and l is generated by a finite state machine then there is an algorithm to decide whether k s spl cap l is controllable with respect to l ii it is undecidable for an arbitrary specification s generated by a nondeterministic pushdown automaton and plant language l generated by a finite state machine whether k s spl cap l is controllable with respect to l

study of a np hard cyclic scheduling problem the recurrent job shop abstract this paper tackles a cyclic scheduling problem with disjunctive resources called the recurrent job shop it is aimed to show structural properties of feasible schedules on which efficient algorithms could be based the problem is proven to be np hard and is formalized as a mixed linear program we prove that any feasible solution is associated with a valuation called a conservative height on the graph of constraints computing the critical circuit provides the best solution associated with a given conservative height bounds on the feasible heights and on the throughput based on these properties are then stated finally we propose a branch and bound enumeration procedure and two heuristics solving the problem we report numerical experiments

opacity enforcing supervisory strategies via state estimator constructions state based notions of opacity such as initial state opacity and infinite step opacity emerge as key properties in numerous security applications of discrete event systems we consider systems that are modeled as partially observed nondeterministic finite automata and tackle the problem of constructing a minimally restrictive opacity enforcing supervisor moes which limits the system s behavior within some prespecified legal behavior while enforcing initial state opacity or infinite step opacity requirements we characterize the solution to moes under some mild assumptions in terms of the supremal element of certain controllable normal and opaque languages we also show that this supremal element always exists and that it can be implemented using state estimators the result is a supervisor that achieves conformance to the pre specified legal behavior while enforcing initial state opacity by disabling at any given time a subset of the controllable system events in a way that minimally restricts the range of allowable system behavior although infinite step opacity cannot be easily translated to language based opacity we show that by using a finite bank of supervisors the aforementioned approach can be extended to enforce infinite step opacity in a minimally restrictive way

an algebraic model for performance evaluation of timed event multigraphs this note demonstrated that a certain class of petri nets called timed event multigraphs can be represented as a set of linear equations by using a novel dioid with this new algebra some properties are presented in the text an example is discussed in the end and an interesting result is computed

the shapes and sizes of two dimensional pressurized self intersecting rings as models for two dimensional vesicles two dimensional vesicles are modelled as pressurized unrestricted closed random walks the model though suffering from undesirable physical characteristics has the decided advantage of yielding exact analytical statistical analysis of the size and shapes of these objects closed form expressions for the average of the square of the radius of gyration and the asphericity are derived in addition exact expressions for the full probability distribution are also obtained numerical computations of the probability function for several pressures are presented the authors arrive at a new exact formula for the distribution function p r12 r22 of the principal components of the square of the radius of gyration for two dimensional rings

comparison properties of stochastic decision free petri nets stochastic comparison problems arising in the analysis of stochastic decision free petri nets are treated the analysis is based on the evolution equations satisfied by firing times various structural properties are obtained including the association of the firing times and their stochastic and convex monotonicity with respect to the holding time sequences the initial marking and the topology the association and the stochastic monotonicity properties are extended to the counters using an inversion relation it is also proven that the counters and the throughput are stochastically concave in the initial marking provided that the holding times are independent identically distributed and belong to a subclass of the log concave distribution functions that are introduced various bounds for the asymptotic cycle time and the throughput are then derived from these stochastic ordering results comparative results are presented for the marking and response times

scaling relations for the slippery ballistic growth model abstract a slippery ballistic growth model is introduced in which deposited particles stick with probability ps to the sides of existing columns numerical simulations in 1 1 dimensions show that while this model belongs to the kpz universality class it also exhibits a short time scaling regime during which the width of the growth region obeys the power law w t 1 2 before crossing over to kpz behavior the crossover times between the three scaling regimes both scale with sticking probability as ps 1 the interface width at saturation ws is found to scale with ps in addition to its scaling with substrate size

brief on optimal control of a class of partially observed discrete event systems we are interested in a new class of optimal control problems for discrete event systems we adopt the formalism of supervisory control theory proc ieee 77 1 1989 81 and model the system as a finite state machine fsm our control problem is characterized by the presence of uncontrollable as well as unobservable events the notion of occurrence and control costs for events and a worst case objective function we first derive an observer for the partially unobservable fsm which allows us to construct an approximation of the unobservable trajectory costs we then define the performance measure on this observer rather than on the original fsm itself we then use the algorithm presented in sengupta and lafortune siam j control optim 36 2 1998 to synthesize an optimal submachine of the c observer this submachine leads to the desired supervisor for the system

supremum operators and computation of supremal elements in system theory constrained supremum and supremum operators are introduced to obtain a general procedure for computing supremal elements of upper semilattices examples of such elements include supremal a b invariant subspaces in linear system theory and supremal controllable sublanguages in discrete event system theory for some examples we show that the algorithms available in the literature are special cases of our procedure our iterative algorithms may also provide more insight into applications in the case of supremal controllable subpredicate the algorithm enables us to derive a lookahead policy for supervisory control of discrete event systems

linear matrix period in max plus algebra abstract linear periodicity of matrices in max plus algebra is studied it is proved that the linear period of a matrix can be computed in o n 3 time if the digraph of the matrix is strongly connected the general problem of deciding whether a given matrix is almost linear periodic is shown to be np complete

modular control of discrete event systems with coalgebra modular supervisory control of discrete event systems where the overall system is a synchronous parallel product of subsystems is considered the main results of this paper are formulations of sufficient conditions for the compatibility between the synchronous product and various operations stemming from supervisory control as supervised product and supremal controllable sublanguages these results are generalized to the case of modules with partial observations e g modular computation of supremal normal sublanguages is studied coalgebraic techniques based on the coinduction proof principle are used in our main results sufficient conditions are derived for modular to equal global control synthesis an algorithmic procedure for checking the new conditions is proposed and the computational benefit of the modular approach is discussed and illustrated by comparing the time complexity of modular and monolithic computation

characterization of tropical hemispaces by p r decompositions abstract we consider tropical hemispaces defined as tropically convex sets whose complements are also tropically convex and tropical semispaces defined as maximal tropically convex sets not containing a given point we introduce the concept of p r decomposition this yields to our knowledge a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays we characterize tropical hemispaces as tropically convex sets that admit a p r decomposition of certain kind in this characterization with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield satisfying an extended rank one condition our proof techniques are based on homogenization lifting a convex set to a cone and the relation between tropical hemispaces and semispaces

a note on an estimator of life expectancy with random censorship summary yang 1977 proposed an estimator of life expectancy based on the kaplan meier estimator under a random censorship model this note shows that this estimator considered as a function of age converges weakly to a gaussian process on the whole line a consistent estimator of the variance function of this limiting process is given and asymptotic confidence bands for the life expectancy are discussed

on the ultimate behavior of the sequence of consecutive powers of a matrix in the max plus algebra abstract we study the sequence of consecutive powers of a matrix in the max plus algebra which has maximum and addition as its basic operations if the matrix is irreducible then it is well known that the ultimate behavior of the sequence is cyclic for reducible matrices the ultimate behavior is more complex but it is also cyclic in nature we will give a detailed characterization of the rates and periods of the ultimate behavior for a general matrix

a simple nonparametric estimator of the bivariate survival function under univariate censoring summary in the presence of univariate censoring the bivariate survival function of paired failure times can be expressed as the ratio of the bivariate at risk probability to the survival function of the censoring time the use of this natural representation yields a very simple nonparametric estimator for the bivariate survival curve the estimator is strongly consistent and upon proper normalization converges weakly to a zero mean gaussian process with an easily estimated covariance function numerical studies demonstrate that both the survival curve estimator and its covariance function estimator perform markedly well for practical sample sizes applications to the correlation problem and to the interval estimation of the difference in median survival times are also studied

synthesis of maximally permissive supervisors for the range control problem we investigate the supervisor synthesis problem for centralized partially observed discrete event systems subject to safety specifications it is well known that this problem does not have a unique supremal solution in general instead there may be several incomparable locally maximal solutions one then needs a mechanism to select one locally maximal solution our approach in this paper is to consider a lower bound specification on the controlled behavior in addition to the upper bound for the safety specification this leads to a generalized supervisory control problem called the range control problem while the upper bound captures the prefix closed legal behavior the lower bound captures the prefix closed minimum required behavior we provide a synthesis algorithm that solves this problem by effectively constructing a maximally permissive safe supervisor that contains the required lower bound behavior this is the first algorithm with such properties as previous works solve either the maximally permissive safety problem with no lower bound or the lower bound containment problem without maximal permissiveness

brief paper input signal design for identification of max plus linear systems max plus linear systems efficiently describe the dynamics of event time sequences of a class of discrete event systems the present contribution addresses the problem of designing adequate input signals for state space identification of max plus linear systems it is shown that the input signal design problem can be rewritten as a set of upper bound constraints and therefore solved using an existing algorithm this input signal design method allows to incorporate additional objectives and constraints e g minimum or maximum input event separation time order constraints etc which are desirable or even required for the input signals and the resulting process behavior

parallel vlsi matrix pencil algorithm for high resolution direction finding algorithms to find the directions of arrival doas of multiple signals from measurements on an array of antenna doublets esprit method and their parallel vlsi implementation are discussed in particular algorithms that allow large scale pipelining and use only robust unitary transformations are identified this problem is solved by a matrix pencil approach in which the generalized eigenvalues of a pair of data matrices are determined a modified stewart jacobi approach is used for which convergence is improved and parameter computations are simplified the resulting architecture is a two layer jacobi array that can handle all the subproblems two qr factorizations two svds and a single generalized schur decomposition the mapping of the subproblems on a single parallel array of cordic processors is considered

on the boolean minimal realization problem in the max plus algebra one of the open problems in the max plus algebraic system theory for discrete event systems is the minimal realization problem in this paper we present some results in connection with the minimal realization problem in the max plus algebra first we characterize the minimal system order of a max linear discrete event system we also introduce a canonical representation of the impulse response of a max linear discrete event system next we consider a simplified version of the general minimal realization problem the boolean minimal realization problem i e we consider models in which the entries of the system matrices are either equal to the max plus algebraic zero element or to the max plus algebraic identity element we give a lower bound for the minimal system order of a max plus algebraic boolean discrete event system we show that the decision problem that corresponds to the boolean realization problem i e deciding whether or not a boolean realization of a given order exists is decidable and that the boolean minimal realization problem can be solved in a number of elementary operations that is bounded from above by an exponential of the square of any upper bound of the minimal system order we also point out some open problems the most important of which is whether or not the boolean minimal realization problem can be solved in polynomial time

the power algorithm in max algebra abstract it is proved that under certain conditions an algorithm resembling the power algorithm in conventional linear algebra can be used to find the eigenvector and eigenvalue of a max algebra system if the conditions are not satisfied the eigenvector can be found using an extension of the power algorithm

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

formulas for calculating supremal controllable and normal sublanguages abstract supremal controllable and normal sublanguages have been shown to play an important role in supervisor synthesis in this paper we discuss the computation of supremal controllable and normal sublanguages we derive formulas for both supremal controllable sublanguages and supremal normal sublanguages when the languages involved are closed as a result those languages can be computed without applying recursive algorithms we also discuss those aspects of these formulas