an approach to generalize laws of large numbers for fuzzy random variables the formulation of strong and weak laws of large numbers for fuzzy random variables based on variance is given the concept of the variance of fuzzy random variables introduced by korner fuzzy sets and systems 92 1997 83 and by feng fuzzy sets and systems 120 2001 487 demonstrates its efficiency

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

lumpability and observability of linear systems large scale dynamic system models are plagued by problems of dimensionality and limitations of measurement it is common practice in applications to lump similar system state components together to get a reduced number of aggregate states an example of this approach is the pona scheme for modelling catalytic cracking of oil the thousands of chemical species involved in this process are lumped into four groups paraffins olefins naphthenes and aromatics pona for a survey of the use of lumping in the cracking problem the reader is referred to weekman 14971 attention has focused on conditions under which such lumping preserves structural properties of the system dynamics in particular when do the lumped states of a linear system satisfy linear dynamics if this occurs the system is said to be exactly lumpable for finite dimensional systems this issue has been resolved 2 1 51 but in practice the conditions specified are almost never satisfied in this paper we show that it is fruitful to consider the lumped variables as observations of the original system to our knowledge this viewpoint has not been taken previously we establish a relationship between lumpability and observability of the lumped observation system for both finite and infinite dimensional linear systems this relationship allows us to deduce the existence of a minimal size system which gives a true representation of the dynamics of the lumped variables definitions of observability and lumpability are given in sections 2 and 3 including extensions to infinite dimensional systems the relationship between these two concepts is discussed and we establish the main result concerning lumpability in section 4 in the case of finite dimensional systems in section 5 this discussion is carried over to infinite dimensional systems a summary and discussion follow in section 6

structure and stability of finite dimensional approximations for functional differential equations this paper deals with the structural and stability properties of the averaging approximation scheme for linear retarded functional differential equations both in the discrete and in the continuous time case the structure of the approximating systems is shown to be analogous to the structure of the underlying retarded equation moreover it is shown that the approximating systems are exponentially stable in a uniform sense if the original system is asymptotically stable

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

the solutions of linear fuzzy stochastic differential systems as a completion of previous work of feng fuzzy sets and systems 115 2000 351 on the general theory of fuzzy stochastic differential systems the solutions of linear fuzzy stochastic differential systems for general coefficient matrix are discussed

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

on the design of a stable adaptive filter for state estimation in high dimensional systems design of specific parametrized gain structures provides a new tool for ensuring stability of reduced order filter for high dimensional systems which is optimized adaptively in the minimum prediction error framework

lumpable hidden markov models model reduction and reduced complexity filtering this paper is concerned with filtering of hidden markov processes hmp which possess or approximately possess the property of lumpability this property is a generalization of the property of lumpability of a markov chain which has been previously addressed by others in essence the property of lumpability means that there is a partition of the atomic states of the markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned we prove necessary and sufficient conditions on the hmp for exact lumpability to hold for a particular class of hidden markov models hmm namely finite output alphabet models conditions for lumpability of all hmp representable by a specified hmm are given the corresponding optimal filter algorithms for the aggregated states are then derived the paper also describes an approach to efficient suboptimal filtering for hmp which are approximately lumpable by this we mean that the hmm generating the process may be approximated by a lumpable hmm this approach involves directly finding a lumped hmm which approximates the original hmm well in a matrix norm sense an alternative approach for model reduction based on approximating a given hmm by an exactly lumpable hmm is also derived this method is based on the alternating convex projections algorithm some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

on the convergence of fuzzy martingales the purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are lipschitz or continuous fuzzy numbers on rn this approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real valued random variables that appear as their support functions based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales the strong law of large numbers for fuzzy random variables is proved here the convergence is in the uniform metric but not the separable metric on fuzzy number space

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

discretisation of an infinite delay equation in this paper a banach phase space containing bc 0 and contained in c 0 is defined with which existence of a solution and convergence of a discrete scheme are proved for an infinite delay differential equation

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

markov property for a function of a markov chain a linear algebra approach abstract in this paper we address whether a probabilistic function of a finite homogeneous markov chain still enjoys a markov type property we propose a complete answer to this question using a linear algebra approach at the core of our approach is the concept of invariance of a set under a matrix in that sense the framework of this paper is related to the so called geometric approach in control theory for linear dynamical systems this allows us to derive a collection of new results under generic assumptions on the original markov chain in particular we obtain a new criterion for a function of a markov chain to be a homogeneous markov chain we provide a deterministic polynomial time algorithm for checking this criterion moreover a non standard notion of observability for a linear system will be used this allows one to show that the set of all stochastic matrices for which our criterion holds is nowhere dense in the set of stochastic matrices

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

note on sums of independent fuzzy random variables this note extends the results in feng fuzzy sets and systems 123 2001 11 about sums of independent fuzzy random variables by means of a convergence criterion in uniform hausdorff metric

coherency based decomposition and aggregation the relationship between coherency area decomposition and aggregation is investigated for a class of models of electromechanical networks including power system models it is established that coherency conditions and a physically meaningful area aggregate reproducing a prescribed set of system modes depend on external connections only coherency with respect to a set of slowest modes slow coherency and an earlier developed grouping algorithm are used to identify weakly coupled areas as demonstrated on a 48 machine model of the united states north east power system

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

semi order fuzzy supermartingales and submartingales with continuous time a new suitable semi order in the fuzzy number space en d the semi order fuzzy supermartingales and submartingales with continuous time and their characteristics are introduced the famous riesz decomposition and doob meyer decomposition theorems in standard martingale theory are generalized to semi order fuzzy supermartingales the proof of doob meyer decomposition is rather technical and so the theory of fuzzy valued measures generated by increasing fuzzy processes is developed

gaussian fuzzy random variables in this paper we define the concepts of a gaussian fuzzy random variable and a gaussian fuzzy random vector and discuss their properties the covariance operator the characteristic functional and the linear transformation of a gaussian fuzzy random variable are investigated

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

stability performance and sensitivity analysis of i i d jump linear systems abstractthis paper presents a symmetric kronecker product analysis of independent and identically distributed jump linear systems to develop new lower dimensional equations for the stability and performance analysis of this type of systems than what is currently available in addition new closed form expressions characterising multi parameter relative sensitivity functions for performance metrics are introduced the analysis technique is illustrated with a distributed fault tolerant flight control example where the communication links are allowed to fail randomly

aggregation of stochastic automata networks with replicas we present techniques for computing the solution of large markov chain models whose generators can be represented in the form of a generalized tensor algebra such as stochastic automata networks san many large systems include a number of replications of identical components this paper exploits replication by aggregating similar components this leads to a state space reduction based on lumpability we define san with replicas and we show how such san models can be strongly aggregated taking functional rates into account a tensor representation of the matrix of the aggregated markov chain is proposed allowing to store this chain in a compact manner and to handle larger models with replicas more efficiently examples and numerical results are presented to illustrate the reduction in state space and consequently the memory and processing time gains

justification of input output methods for systems with unbounded control and observation the authors present definitions of feedback stabilizability and detectability that are weaker than those previously appearing in the literature stabilizability and detectability in this sense are sufficient to ensure an equivalence between internal and external stability for well posed systems furthermore stabilizability and detectability are shown to be preserved under feedback and other connections these results are linked to present a justification of input output controller design methods for systems with unbounded control and observation

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

structure theory and duality for time varying retarded functional differential equations on etudie les equations differentielles fonctionnelles a retard variant dans le temps dans le cadre de l espace etat c des fonctions continues et de l espace produit r n l p dans l espace c on considere deux concepts d etat l etat donne par la fonction initiale et l etat donne par le terme de forcage la theorie de dualite est basee sur ces 2 concepts on utilise explicitement deux operateurs structuraux f t et g t

spline approximation for systems of linear neutral delay differential equations abstract we derive a new finite dimensional semidiscrete approximation scheme for systems of linear neutral delay differential equations and prove convergence results our construction extends to neutral delay equations results which were previously only available for retarded delay equations

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

statistical aspects of fuzzy monotone set valued stochastic processes application to birth and growth processes the paper considers a particular family of fuzzy monotone set valued stochastic processes the proposed setting allows us to investigate suitable a level sets of such processes modeling birth and growth processes a decomposition theorem is established to characterize the nucleation and the growth as a consequence different consistent set valued estimators are studied for growth process moreover the nucleation process is studied via the hitting function and a consistent estimator of the nucleation hitting function is derived

convergence theorems for fuzzy random variables and fuzzy martingales in this paper we discuss the properties of fuzzy random variables and fuzzy conditional expectation some extended results of dominated convergence theorems for fuzzy random variables are proved we define the concept of a right closed fuzzy martingale and give the necessary and sufficient conditions of right closability for fuzzy martingales the convergence theorem for fuzzy martingales is also extended

a twofold spline approximation for finite horizon lqg control of hereditary systems in this paper an approximation scheme is developed for the solution of the linear quadratic gaussian lqg control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay the solution here proposed is achieved by means of two approximating subspaces the first one to approximate the riccati equation for control and the second one to approximate the filtering equations since the approximating subspaces have finite dimension the resulting equations can be implemented the convergence of the approximated control law to the optimal one is proved simulation results are reported on a wind tunnel model showing the high performance of the method

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

model reduction identifying partitions for structured aggregates reduced order state vectors formed by partitioning the state and summing or averaging components in each partition group are considered evidence of reducibility in the form of invariant response is shown to propagate through ordered families of partitions making it possible to identify reducible partitions from very coarsely lumped system responses

a unified view of aggregation and coherency in networks and markov chains abstract this paper presents a unified treatment of aggregability lumpability coherency reversibility partial balance and similar properties appearing in different fields including power systems and queueing theory a coherency condition well known in power systems implies the existence of a finite state filter for markov chains while aggregability and coherency yield a new condition for decentralized computation of the invariant measure

identifiability of hidden markov information sources and their minimum degrees of freedom if only a function of the state in a finite state markov chain is observed then the stochastic process is no longer markovian in general this type of information source is found widely and the basic problem of its identifiability remains open that is the problem of showing when two different markov chains generate the same stochastic process the identifiability problem is completely solved by linear algebra where a block structure of a markov transition matrix plays a fundamental role and from which the minimum degree of freedom for a source is revealed

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

entropy of hidden markov processes via cycle expansion hidden markov processes hmp is one of the basic tools of the modern probabilistic modeling the characterization of their entropy remains however an open problem here the entropy of hmp is calculated via the cycle expansion of the zeta function a method adopted from the theory of dynamical systems for a class of hmp this method produces exact results both for the entropy and the moment generating function the latter allows to estimate via the chernoff bound the probabilities of large deviations for the hmp more generally the method offers a representation of the moment generating function and of the entropy via convergent series

observability and nonlinear filtering this paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability we consider a general class of hidden markov models in continuous time with compact signal state space and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process we demonstrate that observability implies stability of the filter i e the filtered estimates become insensitive to the initial measure at large times for the special case where the signal is a finite state markov process and the observations are of the white noise type a complete necessary and sufficient characterization of filter stability is obtained in terms of a slightly weaker detectability condition in addition to observability the role of controllability is explored finally the results are partially extended to non compact signal state spaces